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abstract: 'We study a three-dimensional dynamical system in two slow variables and one fast variable. We analyze the tangency of the unstable manifold of an equilibrium point with “the” repelling slow manifold, in the presence of a stable periodic orbit emerging from a Hopf bifurcation. This tangency heralds complicated and chaotic mixed-mode oscillations. We classify these solutions by studying returns to a two-dimensional cross section. We use the intersections of the slow manifolds as a basis for partitioning the section according to the number and type of turns made by trajectory segments. Transverse homoclinic orbits are among the invariant sets serving as a substrate of the dynamics on this cross-section. We then turn to a one-dimensional approximation of the global returns in the system, identifying saddle-node and period-doubling bifurcations. These are interpreted in the full system as bifurcations of mixed-mode oscillations. Finally, we contrast the dynamics of our one-dimensional approximation to classical results of the quadratic family of maps. We describe the transient trajectory of a critical point of the map over a range of parameter values.'
author:
- Ian Lizarraga
bibliography:
- 'shnfsaa4-preprint3.bib'
title: 'Tangency bifurcation of invariant manifolds in a slow-fast system'
---
> We study a | 0 | non_member_988 |
three-dimensional multiple timescale system in five parameters. A startling variety of behaviors can be identified as its five parameters are varied. Organizing this variety are the interactions between classical invariant manifolds (including fixed points, periodic orbits, and their (un)stable manifolds) and locally invariant slow manifolds. Here we focus on the interaction between the two-dimensional unstable manifold of a saddle-focus equilibrium point and a two-dimensional repelling slow manifold, in the presence of a stable periodic orbit of small amplitude.
>
> The images of global return maps, defined on carefully chosen two-dimensional cross-sections, are organized by the interactions of the attracting and repelling slow manifolds with these cross-sections. They are also influenced by the basin of attraction of the periodic orbit. We construct a symbolic map which partitions one such section according to the number and type of turning behaviors of the corresponding trajectories. We locate transverse homoclinic orbits to saddle points. On another cross-section, global returns are well-approximated by one-dimensional, nearly unimodal maps. We show that saddle-node bifurcations of periodic orbits and period-doubling cascades occur. Finally, we describe the dynamics of the critical point of the return map at carefully chosen parameters.
>
> Taking a broader view, our numerical | 0 | non_member_988 |
results continue to point to the fruitful connections that exist between multiple-timescale flows and low-dimensional maps.
\[sec:intro\] Introduction
==========================
We study slow-fast dynamical systems of the form
$$\begin{aligned}
{ \varepsilon}\dot{x} &=& f(x,y,{ \varepsilon})\\
\dot{y} &=& g(x,y,{ \varepsilon}),\end{aligned}$$
where $x \in R^m$ is the [*fast*]{} variable, $y \in R^n$ is the [*slow*]{} variable, ${ \varepsilon}$ is the [*singular perturbation parameter*]{} that characterizes the ratio of the timescales, and $f,g$ are sufficiently smooth. The [*critical manifold*]{} $C = \{f = 0\}$ is the manifold of equilibria of the fast subsystem defined by $\dot{x} = f(x,y,0)$. When ${ \varepsilon}> 0$ is sufficiently small, theorems of Fenichel[@fenichel1972] guarantee the existence of locally invariant [*slow manifolds*]{} that perturb from subsets of $C$ where the equilibria are hyperbolic. We may also project the vector field $\dot{y} = g(x,y,0)$ onto the tangent bundle $TC$. Away from folds of $C$, we may desingularize this projected vector field to define the [*slow flow*]{}. The desingularized slow flow is oriented to agree with the full vector field near stable equilibria of $C$. For sufficiently small values of ${ \varepsilon}$, trajectories of the full system can be decomposed into segments lying on the slow manifolds near $C$ together with fast jumps | 0 | non_member_988 |
across branches of $C$. Trajectory segments lying near the slow manifolds converge to solutions of the slow flow as ${ \varepsilon}$ tends to 0.
\(a) {width="45.00000%"} (b) {width="45.00000%"}
We now focus on the case of two slow variables and one fast variable ($m=1$, $n=2$). The critical manifold $C$ is two-dimensional and folds of $C$ form curves. Points on fold curves are called [*folded singularities*]{}. when the slow flow is two-dimensional we use the terms “folded node”, “folded focus”, and “folded saddle” to denote folded singularities of node-, focus-, and saddle-type, respectively. In analogy to classical bifurcation theory, folded saddle-nodes are folded singularities having a zero eigenvalue. When they exist, folded saddle-nodes are differentiated by whether they persist as equilibria in the full system of equations. We are interested here in folded-saddle nodes of type II (FSNII), which are true equilibria of the full system. It can be shown that [*singular Hopf bifurcations*]{} occur generically at distances $O({ \varepsilon})$ from the FSNII bifurcation in parameter space.[@guckenheimer2008siam] At this bifurcation, a pair of eigenvalues of the linearization of the flow crosses the imaginary axis, and a small-amplitude periodic orbit is born at the bifurcation point.
Normal forms are used to study the | 0 | non_member_988 |
local flow of full systems in neighborhoods of these folded singularities. Previous work by Guckenheimer[@guckenheimer2008chaos] analyzes the local flow maps and return maps of three-dimensional systems containing folded nodes and folded saddle-nodes. There, it is shown that the appearance of these folded singularities can give rise to complex and chaotic behavior. Characterizing the emergence of small-amplitude oscillations near a folded singularity has also been the subject of intense study. In the case of a folded node, Benoît[@benoit1990] and Wechselberger[@wechselberger2005] observed that the maximum number of small oscillations made by a trajectory passing through the folded node region is related to the ratio of eigenvalues of the folded node.
The present paper focuses on a dynamical system, defined in Sec. \[sec:shnf\], which contains folded singularities lying along a cubic critical manifold. The critical manifold serves as a global return mechanism. Parametric subfamilies of this dynamical system have served as important prototypical models of electrochemical oscillations, including the Koper model[@koper1992]. This system serves as a concrete, minimal example of a three-dimensional system having an $S$-shaped critical manifold as a global return mechanism. Trajectories leaving a neighborhood of the folded singularities do so by jumping between branches of the critical manifold, before ultimately | 0 | non_member_988 |
being reinjected into the regions containing the folded singularities. This interplay between local and global mechanisms gives rise to [*mixed-mode oscillations*]{} (MMOs), which are periodic solutions of the dynamical system containing large and small amplitudes and a distinct separation between the two. These solutions may be characterized by their signatures, which are symbolic sequences of the form $L_1^{s_1}L_2^{s_2} \cdots$. This notation is used to indicate that a particular solution undergoes $L_1$ large oscillations, followed by $s_1$ small oscillations, followed by $L_2$ large oscillations, and so on. The distinction between ‘large’ and ‘small’ oscillations is dependent on the model. Nontrivial aperiodic solutions are referred to as [*chaotic MMOs*]{}, and may be characterized as limits of families of MMOs as the lengths of the signatures grow very large.
The classification of routes to MMOs with complicated signatures as well as chaotic MMOs continues to garner interest. Global bifurcations have been identified as natural starting points in this direction. Even so, the connection between these bifurcations and interactions of slow manifolds—which organize the global dynamics for small values of ${ \varepsilon}$—remains poorly understood. Period-doubling cascades, torus bifurcations,[@guckenheimer2008siam] and most recently, Shilnikov homoclinic bifurcations,[@guckenheimer2015] have been shown to produce MMOs with complex signatures. In | 0 | non_member_988 |
the last case, one-dimensional approximations of return maps were used to analyze a Shilnikov bifurcation in a system which exhibits singular Hopf bifurcation.
In this paper, we use a similar technique to analyze a tangency of invariant manifolds. Our starting point is a study by Guckenheimer and Meerkamp[@guckenheimer2012siam], which comprehensively classifies local and global unfoldings of singular Hopf bifurcation. We describe the changes in the phase space as the unstable manifold of the saddle-focus equilibrium point crosses the repelling slow manifold of the system. Our approach takes for granted the complicated crossings of these two-dimensional manifolds, instead focusing directly on the influence of these crossings on the global dynamics. The main tool in our analysis is the approximation of the two-dimensional return map by a map on an interval, which parametrizes trajectories beginning on the attracting slow manifold. We show that in the presence of a small-amplitude stable periodic orbit, the one-dimensional return map has a rich topology. The domain of the map is disconnected, with components separated by finite-length gaps. Intervals where the return map is undefined correspond to bands of initial conditions in the full system whose forward trajectories asymptotically approach the small-amplitude stable periodic orbit without making | 0 | non_member_988 |
a large-amplitude passage. The first and second derivatives of the map grow very large outside of large subintervals where the map is unimodal.
We also interpret classical bifurcations of the one-dimensional map as routes to chaotic behavior in the full system. We show that a period-doubling cascade occurs in this map, which gives rise to chaotic MMOs. This cascade is reminiscent of the classical cascade in the family of quadratic maps, even though on small subsets, our return map is far from unimodal. Saddle-nodes of mixed-mode cycles, defined as fixed points of the return map with unit derivative, are also shown to occur. Finally, we identify a parameter set for which the full dynamics is close to the dynamics of a unimodal map with a critical point having dense forward orbit.
\[sec:shnf\] Three-Dimensional System of Equations
==================================================
We study the following three-dimensional flow: $$\begin{aligned}
{ \varepsilon}\dot{x} &=& y - x^2 - x^3 \nonumber\\
\dot{y} &=& z - x \label{eq:shnf}\\
\dot{z} &=& -\nu - ax -by - cz,\nonumber\end{aligned}$$
where $x$ is the fast variable, $y,z$ are the slow variables, and ${ \varepsilon},\nu,a,b,c$ are the system parameters. This system exhibits a singular Hopf bifurcation.[@braaksma1998; @guckenheimer2008siam; @guckenheimer2012dcds] The critical manifold is the S-shaped | 0 | non_member_988 |
cubic surface $C = \{y = x^2 + x^3\}$ having two fold lines $L_0 := S \cap \{x = 0\}$ and $L_{-2/3} := S \cap \{ x = -2/3\}$. When ${ \varepsilon}> 0 $ is sufficiently small, nonsingular portions of $C$ perturb to families of slow manifolds: near the branches $S\cap \{x > 0\}$ (resp. $S \cap \{x < -2/3\}$), we obtain the [*attracting slow manifolds*]{} $S^{a+}_{{ \varepsilon}}$ (resp. $S^{a-}_{{ \varepsilon}}$) and near the branch $S\cap\{-2/3<x<0\}$ we obtain the [*repelling slow manifolds*]{} $S^r_{{ \varepsilon}}$. Nearby trajectories are exponentially attracted toward $S^{a\pm}_{{ \varepsilon}}$ and exponentially repelled from $S^r_{{ \varepsilon}}$. One derivation of these estimates uses the Fenichel normal form.[@jones1994] Within each family, these sheets are $O(-\exp(c/ { \varepsilon}))$ close[@jones1994; @jones1995], so we refer to any member of a particular family as ‘the’ slow manifold. This convention should not cause confusion.
We focus on parameters where forward trajectories beginning on $S^{a+}_{{ \varepsilon}}$ interact with a ‘twist region’ near $L_0$, a saddle-focus equilibrium point $p_{eq}$, or both. A folded singularity $n =(0,0,0) \in L_0$ is the governing center of this twist region. The saddle-focus $p_{eq}$ has a two-dimensional unstable manifold $W^u$ and a one-dimensional stable manifold $W^s$. This notation disguises the dependence of | 0 | non_member_988 |
these manifolds on the parameters of the system.
Tangency bifurcation of invariant manifolds
===========================================
Guckenheimer and Meerkamp[@guckenheimer2012siam] drew bifurcation diagrams of the system in a two-dimensional slice of the parameter space defined by ${ \varepsilon}= 0.01$, $b = -1$, and $c = 1$. Codimension-one tangencies of $S^r_{{ \varepsilon}}$ and $W^u$ are represented in Figure 5.1 of their paper by smooth curves (labeled T) in $(\nu,a)$ space. For fixed $a$ and increasing $\nu$, this tangency occurs after $p_{eq}$ undergoes a supercritical Hopf bifurcation. A parametric family of stable limit cycles emerges from this bifurcation. Henceforth we refer to ‘the’ small-amplitude stable periodic orbit $\Gamma$ to refer to the corresponding member of this family at a particular parameter set. The two-dimensional stable manifolds of $\Gamma$ interact with the other invariant manifolds of the system. Guckenheimer and Meerkamp identify a branch of period-doubling bifurcations as $\nu$ continues to increase after the first slow-manifold tangency. We show that the basin of attraction of the periodic orbit has a significant influence on the global returns of the system.
Fixing $a = -0.03$, the tangency occurs within the range $\nu \in \left[ 0.00647, 0.00648\right]$. The location of the tangency may be approximated by studying the asymptotics | 0 | non_member_988 |
of orbits beginning high up on $S^{a+}_{{ \varepsilon}}$. Fix a section $\Sigma = S^{a+}_{{ \varepsilon}}\cap \{x = 0.27\}$. Before the tangency occurs, trajectories lying on and sufficiently near $W^u$ must either escape to infinity or asymptotically approach $\Gamma$; these trajectories cannot jump to the attracting branches of the slow manifold, as they must first intersect $S^r_{{ \varepsilon}}$ before doing so. Trajectories beginning in $\Sigma$ first flow very close to $p_{eq}$. As shown in Figure 1, these trajectories then leave the region close to $W^u$. We observe that before the tangency, $W^u$ forms a boundary of the basin of attraction of $\Gamma$. Therefore, all trajectories sufficiently high up on $S^{a+}_{{ \varepsilon}}$ must lie inside the basin of attraction (Figure \[fig:tangbif\]a).
After the tangency has occurred, isolated trajectories lying in $W^u$ will also lie in $S^r_{{ \varepsilon}}$. These trajectories will bound sectors of trajectories which can now make large-amplitude passages. Trajectories within these sectors jump ‘to the left’ toward $S^{a-}_{{ \varepsilon}}$ or ‘to the right’ toward $S^{a+}_{{ \varepsilon}}$. Trajectories initialized in $\Sigma$ that leave neighborhoods of $p_{eq}$ near these sectors contain [*canard*]{} segments, which are solution segments lying along $S^r_{{ \varepsilon}}$. Examples of such trajectories are highlighted in green in Figure 1. | 0 | non_member_988 |
We can now establish a dichotomy between those trajectories in $\Sigma$ that immediately flow to $\Gamma$ and never leave a small neighborhood of the periodic orbit, versus those that make a global return. In Figure \[fig:tangbif\]b, only two of the thirty sample trajectories are able to make a global return. Near the boundaries of these subsets, trajectories can come arbitrarily close to $\Gamma$ before escaping and making one large return. Note however that such trajectories might still lie inside the basin of attraction of $\Gamma$, depending on where they return on $\Sigma$. Such trajectories escape via large-amplitude excursions at most finitely many times before tending asymptotically to $\Gamma$. We now focus on the parameter regime where the tangency has already occurred. In Figure 5.1 of the paper of Guckenheimer and Meerkamp, this corresponds to the region to the right of the $T$ (manifold tangency) curve.
![\[fig:retmap\] The return map $R: \Sigma_+ \to \Sigma_+$ of the system with $\Sigma_+ = \{x = 0.3\}$. Points in the two-dimensional section are parametrized by their $z$-coordinates. The dashed black line is the line of fixed points $\{(z,z)\}$. Parameter set: $\nu = 0.00802$, $a = -0.3$, $b = -1$, $c = 1$.](2png.png){width="50.00000%"}
\[sec:maps\] Singular and | 0 | non_member_988 |
Regular Returns
=========================================
\(a) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3apng.png "fig:"){width="45.00000%"} (b) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3bpng.png "fig:"){width="45.00000%"}\
(c) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is | 0 | non_member_988 |
the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3cpng.png "fig:"){width="45.00000%"} (d) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3dpng.png "fig:"){width="45.00000%"}
Approximating points on $\Sigma$ by their $z$-coordinates, the return map $R: \Sigma \to \Sigma$ is well-approximated by a one-dimensional map on an interval, also denoted $R$. In the presence of the small-amplitude stable periodic orbit $\Gamma$, we now compare our one-dimensional approximation to return maps in the case of folded nodes[@wechselberger2005] and folded | 0 | non_member_988 |
saddle-nodes[@guckenheimer2008chaos; @krupa2010]. Where the return map is defined, trajectories beginning in different components of the domain of $R$ make different numbers of small turns before escaping the local region. These subsets are somewhat analogous to the rotation sectors arising from twists due to a folded node.[@wechselberger2005] However, in the present case there is a folded singularity as well as a saddle-focus as well as a small-amplitude periodic orbit. Each of these local objects plays a role in the twisting of trajectories that enter neighborhoods of the fold curve $L_0$.
When the small-amplitude stable periodic orbit exists, the domain of the return map is now disconnected, with components separated by finite-length gaps (Figure \[fig:retmap\]). The gaps where $R$ is undefined correspond to those trajectories beginning on $S^{a+}_{{ \varepsilon}}$ that asymptotically approach $\Gamma$ without making a large-amplitude oscillation. The second difference concerns the extreme nonlinearity near the boundaries of the disconnected intervals where $R$ is defined (Figure \[fig:retmap2\]a). Portions of the image lie below the local minima in these local concave segments, resulting in tiny regions near the boundaries where the derivative changes rapidly. These points arise from canard segments of trajectories resulting in a jump from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$ | 0 | non_member_988 |
and hence to $\Sigma$. Fixing the parameters and iteratively refining successively smaller intervals of initial conditions, this pattern of disconnected regions where the derivative changes rapidly seems to repeat up to machine accuracy. One consequence of this structure is the existence of large numbers of unstable periodic orbits, defined by fixed points of $R$ at which $|R'(z)| > 1$. This topological structure also appears to be robust to variations of the parameter $\nu$.
This complicated structure arises from the interaction between the basin of attraction of $\Gamma$, the twist region near the folded singularity and $W^{u,s}$. As an illustration of this complexity, consider an unstable fixed point $z \approx 0.05939079$ of the return map as defined in Figure \[fig:retmap2\](b), interpreted as an unstable periodic orbit in the full system of equations (Figure \[fig:retmap2\](c)-(d)). The orbit is approximately decomposed according to its interactions with the (un)stable manifolds of $p_{eq}$ and the slow manifolds. One possible forward-time decomposition of this orbit proceeds as follows:
- A segment (red) that begins on $S^{a+}_{{ \varepsilon}}$ and flows very close to $p_{eq}$ by remaining near $W^s$,
- a segment (gray) that leaves the region near $p_{eq}$ along $W^u$, then jumping right from $S^r$ to $S^{a+}_{{ | 0 | non_member_988 |
\varepsilon}}$,
- a segment (green) that flows from $S^{a+}_{{ \varepsilon}}$ to $S^{r}_{{ \varepsilon}}$, making small-amplitude oscillations while remaining a bounded distance away from $p_{eq}$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$,
- a segment (blue) that flows back down into the region near $p_{eq}$, making small oscillations around $W^s$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$,
- a segment (magenta) with similar dynamics to the green segment, making small-amplitude oscillations while remaining a bounded distance away from $p_{eq}$, then jumping right from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$, and
- a segment (black) making a large-amplitude excursion by jumping left to $S^{a-}_{{ \varepsilon}}$, flowing to the fold $L_{-2/3}$, and then jumping to $S^{a+}_{{ \varepsilon}}$.
A linearized flow map can be constructed[@glendinning1984; @silnikov1965] in small neighborhoods of the saddle-focus $p_{eq}$, which can be used to count the number of small-amplitude oscillations contributed by orbit segments approaching the equilibrium point. However, the small-amplitude periodic orbit and the twist region produce additional twists, as observed in the green and magenta segments of the example above.
(a)![\[fig:z0\] (a) Geometry in the section $\Sigma_0 = \{(x,y,z): x \in \left[-0.07,0.11\right], y \in \left[-0.005,0.01\right], z=0\}$. Gray points sample the subset of $\Sigma_0$ whose corresponding | 0 | non_member_988 |
forward trajectories tend asymptotically close to the stable periodic orbit without returning to $\Sigma_0$. Green points denote the first forward return of the remaining points in $\Sigma_0$ with the orientation $\dot{z} < 0$. (b) Color plot of maximal height ($y$-coordinate) obtained by trajectories that return to $\Sigma_0$ as defined in (a). Cross-sections of $S^{a+}_{{ \varepsilon}}$ (red) and $S^r_{{ \varepsilon}}$ (black) at $\Sigma_0$ are shown, and the tangency of the vector field with $\Sigma_0$ (i.e. the set $\{ax + by = -\nu\}$) is given by the magenta dashed line. Parameter set: $\nu \approx 0.00870134$, $a = 0.01$, $b = -1$, $c = 1$.](4apng.png "fig:"){width="48.00000%"}\
(b)![\[fig:z0\] (a) Geometry in the section $\Sigma_0 = \{(x,y,z): x \in \left[-0.07,0.11\right], y \in \left[-0.005,0.01\right], z=0\}$. Gray points sample the subset of $\Sigma_0$ whose corresponding forward trajectories tend asymptotically close to the stable periodic orbit without returning to $\Sigma_0$. Green points denote the first forward return of the remaining points in $\Sigma_0$ with the orientation $\dot{z} < 0$. (b) Color plot of maximal height ($y$-coordinate) obtained by trajectories that return to $\Sigma_0$ as defined in (a). Cross-sections of $S^{a+}_{{ \varepsilon}}$ (red) and $S^r_{{ \varepsilon}}$ (black) at $\Sigma_0$ are shown, and the tangency of the vector field with | 0 | non_member_988 |
$\Sigma_0$ (i.e. the set $\{ax + by = -\nu\}$) is given by the magenta dashed line. Parameter set: $\nu \approx 0.00870134$, $a = 0.01$, $b = -1$, $c = 1$.](4bpng.png "fig:"){width="48.00000%"}
We will return to one-dimensional approximations of the return map in Sec. \[sec:ret\], but now we focus on two-dimensional maps, and show that we can illuminate key features of their small-amplitude oscillations. Let us fix a cross-section and define the geometric objects whose interactions organize the return dynamics. Define $\Sigma_0$ to be a compact subset of $\{ z = 0\}$ containing the first intersection (with orientation $\dot{z} >0$) of $W^s$ . Let $B_0$ denote the [*immediate basin of attraction*]{} of the stable periodic orbit $\Gamma$, which we define as the set of points in $\Sigma_0$ whose forward trajectories under the flow of Eq. asymptotically approach $\Gamma$ without returning to $\Sigma_0$, and let $\partial B_0$ denote its boundary. The periodic orbit itself does not intersect our choice of cross-section.
Since we wish to study trajectory segments that return to the cross-section, $B_0$ functions as an escape subset. Rigorously, the forward return map $R: \Sigma_0 \to \Sigma_0$ is undefined on the subset $B_0$, and points landing in $B_0$ under forward iterates | 0 | non_member_988 |
of $R$ ‘escape’. Obviously the trajectories with initial conditions inside $\cup_{i=0}^{\infty}R^{-i}(B_0)$ are contained within the basin of attraction of $\Gamma$, and furthermore the $j$-th iterate of the return map $R^j$ is defined only on the subset $\Sigma_0 - \cup_{i=0}^j R^{-j}(B_0)$. Finally, we abuse notation slightly and denote by $S^{a+}_{{ \varepsilon}}$ (resp. $S^r_{{ \varepsilon}}$) the intersections of the corresponding slow manifolds with $\Sigma_0$. We also refer to the intersection of $S^{a+}_{{ \varepsilon}}$ (resp. $S^r_{{ \varepsilon}}$) with $\Sigma_0$ as the [*attracting*]{} (resp. [*repelling*]{}) [*spiral*]{} due to its distinctive shape (see Figure \[fig:z0\]). The immediate basin of attraction $B_0$ is depicted by gray points in Figure \[fig:z0\](a). This result implies that the basin of attraction of the periodic orbit contains at least a thickened spiral which $S^{a+}_{{ \varepsilon}}$ intersects transversely in interval segments, accounting for the disconnected images of the one-dimensional return maps.
The slow manifolds also intersect transversely. Segments of the attracting spiral can straddle both $B_0$ and the repelling spiral. In Fig. \[fig:z0\](b), we color initial conditions based on the maximum $y$-coordinate achieved by the corresponding trajectory before its return to $\Sigma_0$. Due to the Exchange Lemma, only thin bands of trajectories are able to remain close enough to $S^r_{{ \varepsilon}}$ | 0 | non_member_988 |
to jump at an intermediate height. We choose the maximum value of the $y$-coordinate to approximately parametrize the length of the canards. This parametrization heavily favors trajectories jumping left (from $S^r_{{ \varepsilon}}$ to $S^{a-}_{{ \varepsilon}}$) rather than right (from $S^r_{{ \varepsilon}}$ to $S^{a+}_{{ \varepsilon}}$), since trajectories jumping left can only return to $\Sigma_0$ by first following $S^{a-}_{{ \varepsilon}}$ to a maximal height, and then jumping from $L_{-2/3}$ to $S^{a+}_{{ \varepsilon}}$. This asymmetry is useful: in Figure \[fig:z0\], $S^r_{{ \varepsilon}}$ serves as a boundary between the (apparently discontinuous) blue and yellow regions, clearly demarcating those trajectories which turn right rather than left before returning to $\Sigma_0$. Summarizing, $\partial B_0$ and $S^r_{{ \varepsilon}}$ partition this section according to the behavior of orbits containing canards.
\(a) ![\[fig:sao\] (a) Two phase space trajectories beginning and ending on the section $\{z = 0\}$ with stopping condition $\dot{z} < 0$ and (b) the normalized time series of their normalized $y$-coordinates of each trajectory. Initial conditions: blue, $(x,y,z) = (0.000553, 0.000201, 0)$; red, $(x,y,z) = (0.000553, 0.003065, 0)$. Parameter set: $\nu \approx 0.00870134$, $a = 0.01$, $b = -1$, $c = 1$.](5apng.png "fig:"){width="45.00000%"}\
(b) ![\[fig:sao\] (a) Two phase space trajectories beginning and ending on the section $\{z | 0 | non_member_988 |
= 0\}$ with stopping condition $\dot{z} < 0$ and (b) the normalized time series of their normalized $y$-coordinates of each trajectory. Initial conditions: blue, $(x,y,z) = (0.000553, 0.000201, 0)$; red, $(x,y,z) = (0.000553, 0.003065, 0)$. Parameter set: $\nu \approx 0.00870134$, $a = 0.01$, $b = -1$, $c = 1$.](5bpng.png "fig:"){width="45.00000%"}
Trajectories beginning in $\Sigma_0$ either follow $W^s$ closely and spiral out along $W^u$ or remain a bounded distance away from both the equilibrium point and $W^s$, instead making small-amplitude oscillations consistent with a folded node. Differences between these two types of small-amplitude oscillations have been observed in earlier work. The transition from the first kind of small-amplitude oscillation to the second is a function of the distance from the initial condition to the intersection of $W^s$ with the cross-section. Two initial conditions are chosen on a vertical line embedded in the section $\{z=0\}$, having the property that the resulting trajectory jumps right from $S^r_{{ \varepsilon}}$ at an intermediate height before returning to the section with orientation $\dot{z} < 0$. These initial conditions are found by selecting points in Figure \[fig:z0\](b) in the blue regions lying on a ray that extends outward from the center of the repelling spiral. The corresponding | 0 | non_member_988 |
return trajectories are plotted in Figure \[fig:sao\]. The production of small-amplitude oscillations is dominated by the saddle-focus mechanism: in the example shown, the red orbit exhibits four oscillations before the (relatively) large-amplitude return, whereas the blue orbit exhibits seven oscillations. We can select trajectories with increasing numbers of small-amplitude oscillations by picking points closer to $W^s \cap \{z=0\}$. A complication in this analysis is that jumps at intermediate heights, which are clearly shown to occur in these examples, blur the distinction between ‘large’ and ‘small’ oscillations in a mixed-mode cycle.
\(a) ![\[fig:2dmap\] (a) Partition of a compact subset of the cross-section $\Sigma_0 = \{z = 0\}$. Black dashed line is the tangency of the vector field $\{\dot{z} = 0\}$, separating the subset $\{\dot{z} > 0\}$ (black points) from the other subsets. Yellow (resp. green): points above (resp. below) the line $\{y = 0\}$ with winding number less than three. Red (resp. blue): points whose forward trajectories reach a maximal height greater than (resp. less than) 0.18 and have winding number three or greater. (b) Overlay of red and blue subsets of domain (points) with images of yellow, green, and black subsets (crosses). (c) Overlay of red and blue subsets of | 0 | non_member_988 |
domain (points) with the image of the blue subset (crosses). (d) Overlay of attracting spiral (magenta), repelling spiral (dark green), and image of red subset (crosses). Note the change in scale of the final figure. Generated from a $500 \times 500$ grid of initial conditions beginning on $\Sigma_0$. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](6apng.png "fig:"){width="40.00000%"} (b) ![\[fig:2dmap\] (a) Partition of a compact subset of the cross-section $\Sigma_0 = \{z = 0\}$. Black dashed line is the tangency of the vector field $\{\dot{z} = 0\}$, separating the subset $\{\dot{z} > 0\}$ (black points) from the other subsets. Yellow (resp. green): points above (resp. below) the line $\{y = 0\}$ with winding number less than three. Red (resp. blue): points whose forward trajectories reach a maximal height greater than (resp. less than) 0.18 and have winding number three or greater. (b) Overlay of red and blue subsets of domain (points) with images of yellow, green, and black subsets (crosses). (c) Overlay of red and blue subsets of domain (points) with the image of the blue subset (crosses). (d) Overlay of attracting spiral (magenta), repelling spiral (dark green), and image of red subset (crosses). Note | 0 | non_member_988 |
the change in scale of the final figure. Generated from a $500 \times 500$ grid of initial conditions beginning on $\Sigma_0$. Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](6bpng.png "fig:"){width="40.00000%"}\
(c) ![\[fig:2dmap\] (a) Partition of a compact subset of the cross-section $\Sigma_0 = \{z = 0\}$. Black dashed line is the tangency of the vector field $\{\dot{z} = 0\}$, separating the subset $\{\dot{z} > 0\}$ (black points) from the other subsets. Yellow (resp. green): points above (resp. below) the line $\{y = 0\}$ with winding number less than three. Red (resp. blue): points whose forward trajectories reach a maximal height greater than (resp. less than) 0.18 and have winding number three or greater. (b) Overlay of red and blue subsets of domain (points) with images of yellow, green, and black subsets (crosses). (c) Overlay of red and blue subsets of domain (points) with the image of the blue subset (crosses). (d) Overlay of attracting spiral (magenta), repelling spiral (dark green), and image of red subset (crosses). Note the change in scale of the final figure. Generated from a $500 \times 500$ grid of initial conditions beginning on $\Sigma_0$. Parameter set: $\nu \approx 0.00870134$, $a | 0 | non_member_988 |
---
abstract: 'We propose a technique to improve the search efficiency of the bag-of-words method for image retrieval. We introduce a notion of difficulty for the image matching problems and propose methods that reduce the amount of computations required for the feature vector-quantization task in BoW by exploiting the fact that easier queries need less computational resources. Measuring the difficulty of a query and stopping the search accordingly is formulated as a stopping problem. We introduce stopping rules that terminate the image search depending on the difficulty of each query, thereby significantly reducing the computational cost. Our experimental results show the effectiveness of our approach when it is applied to appearance-based localization problem.'
author:
- 'Kiana Hajebi and Hong Zhang[^1]'
title: '**Stopping Rules for Bag-of-Words Image Search and Its Application in Appearance-Based Localization** '
---
INTRODUCTION
============
Bag-of-Words (BoW) was originally proposed for document retrieval. In recent years, the method has been successfully applied to image retrieval tasks in computer vision community [@sivic; @nister]. The method is attractive because of its efficient image representation and retrieval. BoW represents an image as a sparse vector of visual words, and thus images can be searched efficiently using an inverted index file system. | 0 | non_member_989 |
Other major application areas of the BoW method are appearance-based mobile robot localization and SLAM[^2] problems.
The fundamental issue involved with the appearance-based approach to both visual SLAM and global localization is the place recognition. Robot should be able to recognize the places it has visited before to localise itself or refine the map of the environment. This task is performed by matching the current view of the robot to the existing map that contains the images of the previously visited locations. In this paper we consider the problem of appearance-based localization in which the map is known *a priori*.
In large-scale environments maps contain too many images to match. The image search in such a large map is still a challenging and open problem. Matching images by comparing the local features of each image directly to the local features of all other images in the map is not practical. Bag-of-words proposes a more efficient approach; first, rather than matching with a pool of million visual features extracted from many thousands of images, the local features are mapped to a smaller number of vocabulary words that are built in an offline phase. This process is called vector-quantization. Once the visual | 0 | non_member_989 |
words are identified in the query image, they are used as indices into the image database, to directly retrieve the images that share the same words.
When the vocabulary is large, the vector-quantization process can be a computationally expensive task in real-time localization. Considerable research has been done to speed up the search; some papers [@fabmap] do approximate nearest-neighbor search using structures like vocabulary trees [@nister]; some methods reduce the number of local image features by selecting only a fraction of features that are highly discriminative [@Achar; @BoRF]; another group makes use of more compact feature descriptors like [@bi-BoW; @fabmap]. In this paper, we first show that some image retrieval tasks are [*easier*]{} for BoW method. The hardness criteria, defined later, concerns how distinctive the image query is among all images in the dataset. Given this criteria, we show that the BoW search can be terminated earlier for easier queries. This means, in such queries, mapping only a portion of features can be sufficient to yield a relatively good result. The stopping rule saves considerable amount of computational resources.
The intuition behind this is that when there are many similar images to the query image, in terms of the number | 0 | non_member_989 |
of common words they share, the search becomes more difficult as more processing is required to find the closest match candidate. Whereas when the query image has only a few match candidates, i.e., it shares its visual words with only a few images, the search becomes easy as vector-quantizing only a small number of features is sufficient to find the closest match to the query. By exploiting this fact, we can stop the vector-quantization when the search is easy and the nearest neighbor to the query is easy to find. Our method acts as an approximate image search algorithm. Our experimental results show that the accuracy decreases only slightly while the computational cost decreases dramatically.
Our approach can be best compared with the approach of Cummins and Newman [@accel_fabmap] who use concentration inequalities (Bennett’s inequality in their case) for early bail-out in multi-hypothesis testing that excludes unlikely location hypotheses from further evaluation. However, we use a different bail-out strategy for the process of vector-quantization. In the next section, we briefly review the image representation and the inverted-index search algorithm used in BoW framework. This is followed by a review of the localization algorithms that employ BoW for the image search. | 0 | non_member_989 |
Our proposed method to improve the efficiency of BoW is described in Section \[sec:method\]. Section \[sec:results\] presents the experimental results and the evaluation criteria, and the result of our comparisons. Finally, we conclude the paper in Section \[sec:conclud\].
BACKGROUND {#sec:bakgnd}
==========
Bag-of-Words for image retrieval {#sec:bow}
--------------------------------
Bag-of-words is a popular model that has been used in image classification, objection recognition, and appearance-based navigation. Because of its simplicity and search efficiency it has been used as a successful method in Web search engines for large-scale image and document retrieval [@sivic; @nister; @Sivic2]. Bag-of-words model represents an image by a sparse vector of visual words. Image features, e.g., SIFTs [@lowe], are sampled and clustered (e.g., using k-means) in order to quantize the space into a discrete set of visual words. The centroids of clusters are then considered as visual words which form the visual vocabulary. Once a new image arrives, its local features are extracted and vector-quantized into the visual words. Each word might be weighed by some score which is either the word frequency in the image (i.e., *tf*) or the “term frequency-inverse document frequency” or *tf-idf* [@sivic]. A histogram of weighted visual words, which is typically sparse, is then | 0 | non_member_989 |
built and used to represent the image.
An inverted index file, used in the BoW framework, is an efficient image search tool in which the visual words are mapped to the database images. Each visual word serves as a table index and points to the indices of the database images in which the word occurs. Since not every image contains every word and also each word does not occur in every image, the retrieval through inverted-index file is fast.
Bag-of-Words for Image-based Localization {#sec:relwork}
-----------------------------------------
Bag-of-words model has been extensively used as the basis of the image search in appearance-based localization or SLAM algorithms [@fabmap; @fabmap2; @Achar; @Angeli; @bi-BoW]. Cummins and Newman [@fabmap; @fabmap2] propose a probabilistic framework over the bag-of-words representation of locations, for the appearance-based place recognition. Along with the visual vocabulary they also learn the Chow Liu tree to capture the co-occurrences of visual words. Similarly, Angeli *et al.* [@Angeli] develop a probabilistic approach for place recognition in SLAM. They build two visual vocabularies incrementally and use two BoW representations as an input of a Bayesian filtering framework to estimate the likelihood of loop closures.
Assuming each image has hundreds of SIFT features, mapping the features to | 0 | non_member_989 |
the visual words, using a linear search method, is computationally expensive and not practical for real-time localization. Researchers have tackled this problem with different approaches that speeds up the search but at the expense of accuracy. A number of papers have employed compact feature descriptors that speeds up the search. Gálvez-López and Tardós in [@bi-BoW] propose to use FAST [@FAST] and BREIF [@BRIEF] binary features and introduce a BoW model that descritizes a binary space. Similarly, [@fabmap; @fabmap2] use SURF [@Surf] to have a more compact feature descriptor. Another approach is to use approximate nearest neighbor search algorithms, like hierarchical k-means [@nister], KD-trees [@fabmap] or graph-based search methods [@GNNS] and [@SGNNS], to speed up the quantization process.
Achar *et al.* [@Kosecka] and Zhang [@BoRF] propose reducing the number of features in each image, thereby reducing (removing) the vector-quantization process. They keep track of the features that are repeatable over time. However, these approaches are more specific to the navigation problems where the data is sequential and there exists considerable overlaps between consecutive images. Our approach is more similar to this group as we also reduce the amount of feature mapping. However rather than only selecting a small set of features, | 0 | non_member_989 |
we use all features but we stop the mapping process when necessary. Our approach is also more general as it does not depend on the sequential property of the data.
PROPOSED METHOD {#sec:method}
===============
Vector-quantization (VQ) is an expensive process when the BoW-based image retrieval is performed in large-scale environments, in which hundreds of features extracted from an image need to be matched against hundreds of thousands of visual words. The question is if we really need to vector-quantize all features? Depending on the difficulty level of the search, the number of features to be converted to visual words may vary. We call a search difficult when there are many similar images to the query and thus finding the nearest neighbor among all those candidates requires more computations[^3]. Whereas in an easy search, the query image is similar to only a few images in the database and it can find its true match after processing only a small percentage of features. Figure \[diff\_search\] and Figure \[easy\_search\] show examples of easy and difficult searches. The histograms show the similarity of each database image to the query based on the *tf-idf* score. The difficult search needs to process at least $89\%$ of | 0 | non_member_989 |
features to find the closest match to the query (indicated by the peak of the histogram), however the easy search can stop the search after processing only $12\%$ of features as the peak does not change until the end of the search. Comparing the distance between the peak and average of the histograms, it can be seen that in difficult search this distance is smaller than that in easy search, which is expected. Initially each image starts with no vote. In original BoW, image features are converted to visual words one by one and the histogram (of images’ scores) is built incrementally. Each bin of the histogram corresponds to one of the database images and indicates the score of that image. Each visual word will cast a distance-weighted vote for multiple images, i.e., histogram bins. This process continues until all words cast their votes and then the peak of the histogram (the bin with the highest score) determines the nearest neighbor to the query.[^4]
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![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does | 0 | non_member_989 |
not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/846.jpg "fig:"){width=".21\textwidth"} ![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/633.jpg "fig:"){width=".21\textwidth"}
(a) (b)
![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/diff_query11percent_newcol_odd_846_peak633_firsthist.eps "fig:"){width=".22\textwidth"} ![Easy search. The Query image (a) has been matched to | 0 | non_member_989 |
(b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/diff_query11percent_newcol_odd_846_peak633_lasthist.eps "fig:"){width=".223\textwidth"}
(c) (d)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
-- --
-- --
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![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/996.jpg "fig:"){width=".21\textwidth"} ![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last | 0 | non_member_989 |
88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/430.jpg "fig:"){width=".21\textwidth"}
(e) (f)
![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/easy_query88percent_newcol_odd_996_peak430_firsthist.eps "fig:"){width=".22\textwidth"} ![Easy search. The Query image (a) has been matched to (b). (c) shows the histogram of images’ scores after vector-quantizing 12% of features. The peak of the histogram (shown by the red arrow) does not change until the last histogram (d) when all features have been processed. This means processing the last 88% of features (or the votes of the last 88% of words) are not necessary to select the highest scored image.[]{data-label="easy_search"}](./images/easy_query88percent_newcol_odd_996_peak430_lasthist.eps "fig:"){width=".223\textwidth"}
(g) (h)
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[c]{}\
\
Our method builds upon the voting scheme of BoW; See Table \[alg:mapping\]. The features are selected randomly to be vector-quantized and are | 0 | non_member_989 |
used to score the images (performed by `extract-feature`, `random-permute`, and `vector-quantize` functions in Table \[alg:mapping\]). In an easy search, as the query has many common words with its nearest neighbor and not with other images, after processing only some small percentage of the features the score of the true matching image becomes significantly larger than the mean of the histogram. However, when the query shares its words with many images, i.e., a difficult search, a higher percentage of the words need to cast their vote to find the true match. We stop the feature mapping once the distance between the peak of the histogram to the average of the other bins is greater than some threshold (`stop-time` function). Other stopping rules might be defined, which are described in Section \[sec:rules\].
Our method is different from the naive approach of stopping after quantizing a fixed percentage of the features. Based on our stopping criterion when the search is easy, the VQ stops sooner, otherwise it stops after processing more features. With the naive approach, a search is forced to stop even if more processing is required to find the nearest neighbor.
0 The problem that we study in this paper can | 0 | non_member_989 |
be formulated as a stopping problem [@Shiryaev]. The stopping problem is a decision making problem where at each round, the decision maker observes an input and a (possibly noisy) reward and decides whether he wants to see the next input or not. The objective is to maximize the expected reward when he decides to [*stop*]{}. The stopping problem is studied in various settings in statistics, decision theory, and economics.
More specifically, we can formulate our problem as finding the best option from a pool on options based on some measurements. Assume a number of distributions are given, from which we want to find the one with the highest mean. At each round, we observe a new sample from these distributions. As sampling can be expensive, the objective is to stop sampling and choose the distribution with the highest mean as quick as possible . The common approach is to employ certain concentration inequalities (such as Hoeffding’s inequality [@Hoeffding]) to approximate the mean of each distribution after a number of observations. When the approximations are accurate enough and the decision maker has enough confidence, he stops and chooses the distribution with the highest empirical mean [@Hoeffding_races].
In our problem, given a | 0 | non_member_989 |
query image, we want to find an image that is closest in terms of an approximate cosine distance. The BoW procedure processes the features one by one, increasing the score of each candidate image by some number. We can view the candidate images as different distributions and the new scores as the new samples. At each round, we have a new estimate for the similarity of each candidate image to the query image. The problem is to stop sampling and return the true match as quick as possible.
The problem that we study in this paper can be formulated as a stopping problem [@Shiryaev]. The stopping problem is a decision making problem where at each round, the decision maker observes an input and a (possibly noisy) reward and decides whether he wants to see the next input or not. The objective is to maximize the expected reward when he decides to [*stop*]{}. The stopping problem is studied in various settings in statistics, decision theory, and economics.
More specifically, we can formulate our problem as finding the best option from a pool of options based on some measurements. Assume a number of distributions are given, from which we want to find | 0 | non_member_989 |
the one with the highest mean. At each round, we observe a new sample from each distribution. As sampling can be expensive, the objective is to stop sampling and find the distribution with the highest mean as quick as possible. The common approach is to employ certain concentration inequalities (such as Hoeffding’s Inequality [@Hoeffding]) to construct confidence bands around the empirical mean of each distribution after a number of observations. When the approximations are accurate enough and the decision maker has enough confidence, he stops and chooses the distribution with the highest empirical mean [@Hoeffding_races].
To illustrate the ideas, assume we are given two distributions, $p_1$ and $p_2$, and are asked to find the one with the higher mean. Let the expected values of these distributions be $\mu_1$ and $\mu_2$, respectively. Assume that $\mu_1>\mu_2$. Let $x_s\in[0,1], 1\le s\le t$ be samples from $p_1$ and $y_s\in[0,1], 1\le s\le t$ be samples from $p_2$. Define the empirical means by $\overline{X}_t = \frac{1}{t}\sum_{s=1}^t x_s$ and $\overline{Y}_t = \frac{1}{t}\sum_{s=1}^t y_s$ and the empirical gap by $g_t = \overline{X}_t - \overline{Y}_t$. Without loss of generality, assume that $g_t > 0$. From Hoeffding’s Inequality, we get that
$${{\mathbb P}\left(\overline{X}_t - \mu_1 \le g_t/2\right)} \ge 1 - | 0 | non_member_989 |
e^{-g_t^2 t/2}\,,$$ and $${{\mathbb P}\left(\overline{Y}_t - \mu_2 \ge -g_t/2\right)} \ge 1 - e^{-g_t^2 t/2}\;.$$ Thus, with probability at least $1-2e^{-g_t^2 t/2}$, $\overline{X}_t - \mu_1 \le g_t/2$ and $\overline{Y}_t - \mu_2 \ge -g_t/2$, which implies that $$\mu_1 > \mu_2\;.$$ Thus, we have correctly identified the distribution with the highest mean (here $p_1$) with probability at least $1-2e^{-g_t^2 t/2}$. If we demand that this probability be at least $1-\delta$ for some $\delta\in (0,1)$, then we need to have $$\label{eq:num_samples}
g_t\ge \sqrt{\frac{2\log(2/\delta)}{t}}\;.$$ Equation can be used as a stopping condition to make the right decisions with high probability. We might ask how many samples are needed before Condition is satisfied? By applying Hoeffding’s Inequality, it is not difficult to see that if the true gap ($g=\mu_1-\mu_2$) is small, then the number of samples $t$ in Condition needs to be larger. This implies that the identification is more difficult when there is a small gap between the two distributions.
0 $$\label{eq:num_samples}
t\ge \frac{2\log(2/\delta)}{g^2},$$ with probability at least $1-\delta/2$, it holds that $$\label{eq:error1}
\overline{X}_t - \mu_1 \ge -g/2\;.$$ Similarly, it can be shown that if holds, then with probability at least $1-\delta/2$, it holds that $$\label{eq:error2}
\overline{Y}_t - \mu_2 \le g/2\;.$$ Then, by and , | 0 | non_member_989 |
we get that if holds, with probability at least $1-\delta$, it holds that $$\overline{X}_t \ge \overline{Y}_t\,,$$ which implies that we have identified the distribution with the highest mean correctly. Equation determines the number of samples that is required to make the right decision with high probability. Notice that this number scales like $1/g^2$, which implies that the identification is more difficult (more samples are required) when there is a small gap between the two distributions.
In our image matching problem, given a query image, we want to find an image that is closest in terms of an approximate cosine distance. The BoW procedure processes the features one by one, increasing the score of each candidate image by some number. We can view the candidate images as different distributions and the new scores as the new samples. At each round, we have a new estimate for the similarity of each candidate image to the query image. The problem is to stop sampling and return the true match as quick as possible.
It might seem natural to use the stopping condition for this problem. However, we found out that these theoretical results are very conservative in practice and often require the processing | 0 | non_member_989 |
of a large number of features before stopping, so we do not test Hoeffding’s inequality experimentally. In the next section, we propose a number of rules that show better performance.
EXPERIMENTAL RESULTS {#sec:results}
====================
In this section, we compare the performance of the original BoW method with the BoW method that uses our stopping rules, when applied to the appearance-based localization problem. We will describe the datasets we used for performance evaluation of both methods, followed by discussion of our experimental results. We also describe the different stopping rules that we used in our experiments.
Datasets
--------
We performed our experiments on four datasets. Two datasets have been selected from the Oxford City Center dataset and two from the New College dataset [@fabmap] (see Figure \[fig:datasets\]). Both have been used as the benchmark for localization and SLAM evaluations. The ground truth data is also available for each of them[^5]. Each dataset contains two sets of image sequences. One sequence is taken from the right camera and the other from the left camera mounted on the robot. Each sequence of the City Center dataset contains $1237$ images and each of the New College contains $1073$. The resolution of images is $640\times480$. | 0 | non_member_989 |
For each dataset we used the first half of the images as training data on which we performed the *k-means* clustering and generated a vocabulary of $5000$ visual words. We used $128$-dimensional SIFT feature descriptors as the input to the clustering. Each image has $\sim 400$ SIFT descriptors on average.
The other half of the sequences have been used as the test data, i.e., query images. Each query is matched to the earlier images in the sequence. For each query there are multiple matches in ground truth. If the match that we find for each query is among those correct(ground truth) matches, we call the match correct otherwise incorrect.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/college_0007.jpg "fig:"){width=".15\textwidth"} ![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/college_0442.jpg "fig:"){width=".15\textwidth"} ![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/college_1732.jpg "fig:"){width=".15\textwidth"}
![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/city_0301.jpg "fig:"){width=".15\textwidth"} ![Images from the New College (top row) and City Center (bottom row) | 0 | non_member_989 |
sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/city_1840.jpg "fig:"){width=".15\textwidth"} ![Images from the New College (top row) and City Center (bottom row) sequences used in the reported experiment.[]{data-label="fig:datasets"}](./images/city_0048.jpg "fig:"){width=".15\textwidth"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Performance Evaluations
-----------------------
On each dataset we ran the BoW with the original voting scheme (inverted-index) and BoW with our voting scheme that employs stopping rules. Each visual word in the query image has been weighed with the *tf-idf* score and the top-$n$ images with highest scores have been returned as match candidates. We set $n$ to 3, 5 and 10 in our experiments. We select the candidates whose similarity to the query is above some threshold. Our experimental results have been summarized in Table \[exp1:city1\] to Table \[exp1:newcol2\]. The reported recall values are for precisions above 90%. Top 3, top 5 and top 10 in Tables \[exp1:city1\] to \[exp1:newcol2\] indicate the top-3, -5 and -10 nearest neighbors to the query image that we retrieved. We used different stopping thresholds to generate different recalls. The percentage of the features that we processed and the accuracy of the localization have been computed and compared with the original BoW. All the results have been produced by averaging over $10$ Monte Carlo runs. The | 0 | non_member_989 |
experiments show that we have improved the computational cost significantly at the expense of slight reduction in accuracy.
4.5pt
-- --------- ------------- -------- -------- -------- --------
BoW
(Inv. Indx)
top 3: 0.7897 0.7861 0.7540 0.7273 0.6934
top 5: 0.8378 0.8342 0.8111 0.7932 0.7647
top 10: 0.9055 0.9037 0.8806 0.8610 0.8503
1 0.8655 0.6879 0.5911 0.4965
- 0.25 0.2 0.18 0.16
-- --------- ------------- -------- -------- -------- --------
: Comparison of our approach to the original BoW on the New College dataset, left-side sequence, 1395 words.[]{data-label="exp1:newcol2"}
[cc]{} &\
4.5pt
-- --------- ------------- -------- -------- -------- --------
BoW
(Inv. Indx)
top 3: 0.8324 0.8253 0.8164 0.7879 0.7219
top 5: 0.8556 0.8538 0.8485 0.8235 0.7772
top 10: 0.8895 0.8877 0.8895 0.8717 0.8307
1 0.8395 0.6606 0.5001 0.3240
- 0.28 0.20 0.15 0.10
-- --------- ------------- -------- -------- -------- --------
: Comparison of our approach to the original BoW on the New College dataset, left-side sequence, 1395 words.[]{data-label="exp1:newcol2"}
[cc]{} &\
4.5pt
-- --------- ------------- -------- -------- -------- --------
BoW
(Inv. Indx)
top 3: 0.9104 0.8920 0.8668 0.8455 0.7743
top 5: 0.9395 0.9322 0.9211 0.8988 0.8416
top 10: 0.9709 0.9642 0.9613 0.9535 0.9138
1 0.7821 0.6670 0.4994 0.3030
- 0.12 0.10 0.08 0.06
-- --------- | 0 | non_member_989 |
------------- -------- -------- -------- --------
: Comparison of our approach to the original BoW on the New College dataset, left-side sequence, 1395 words.[]{data-label="exp1:newcol2"}
[cc]{} &\
4.5pt
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BoW
(Inv. Indx)
top 3: 0.9487 0.9397 0.9299 0.8778 0.8150
top 5: 0.9780 0.9707 0.9544 0.9356 0.8818
top 10: 0.9902 0.9894 0.9804 0.9658 0.9487
1 0.8567 0.7374 0.5517 0.3211
- 0.25 0.20 0.15 0.10
-- --------- ------------- -------- -------- -------- --------
: Comparison of our approach to the original BoW on the New College dataset, left-side sequence, 1395 words.[]{data-label="exp1:newcol2"}
Different Stopping Rules {#sec:rules}
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![Accuracy (recall) vs. % of features used, City Center dataset[]{data-label="fig:rules2"}](./images/feat_Acc_3rules_top5_im.eps "fig:"){width=".40\textwidth"}
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![Accuracy (recall) vs. % of features used, City Center dataset[]{data-label="fig:rules2"}](./images/feat_Acc_city_odd.eps "fig:"){width=".40\textwidth"}
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We used different stopping rules in our experiments and compared the efficiency of each of them when applied to the localization problem. Three rules have been used that are explained below:
- Rule 1: computes the distance between the peak and the average of the similarity histogram (i.e., the histogram that shows the similarity of database images to the query). Once the distance is above some threshold, it stops the search: Stop if $|\max(hist)-\mbox{mean}(hist)| > | 0 | non_member_989 |
T$ and return the image corresponding to the peak.\
- Rule 2: computes the relative distance between the peak and the average of the histogram. Once the distance is above some threshold it stops the search: Stop if $|\max(h)-\mbox{mean}(h)|/\mbox{mean}(h) > T$ and return the image corresponding to the peak.\
- Rule 3: keeps track of the peak of the histogram. If the peak does not change after processing $T$ features, the image corresponding to the peak is returned as the nearest neighbor to query.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- --
![Our experiment on City Center dataset with our stopping method that shows only $60\%$ of images need to process $90\%$$100\%$ of their features to get the accuracy of BoW.[]{data-label="fig:feat_im"}](./images/feat_im3.eps "fig:"){width=".35\textwidth"}
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The experimental results shown in Tables \[exp1:city1\] to \[exp1:newcol2\] have been performed by Rule 1. Figures \[fig:rules\] and \[fig:rules2\] show the result of applying different rules to the BoW search on the New College (right-side sequence) and City Center (right-side sequence) datasets, respectively. The graphs have been generated by varying the stopping thresholds. As can be seen, stopping rules \#1 and \#3 generate better results. By reducing the number of processed features to half the accuracy only decreases slightly.
Figure | 0 | non_member_989 |
\[fig:feat\_im\] shows another experiment that validates our proposed stopping method. Figure shows the relation between the number of images and the percentage of features they need to process to obtain the same result as that of the original BoW. The experiment has been done on City Center dataset (left-side sequence) with the stopping rule \#1 used. We set the threshold to $0.29$ to get exactly the same accuracy of original BoW. As can be seen, only $338$ images need to process $90\%$$100\%$ of their features, another $223$ only need to vector-quantize smaller percentages of features.
CONCLUSION {#sec:conclud}
==========
We observed that the computational requirements of the BoW method can be significantly reduced by allocating less computations to easier search queries. Deciding when to terminate the search is treated as a stopping problem. We proposed several stopping rules and showed their effectiveness on an appearance-based localization problem.
[99]{}
S. Achar, C. V. Jawahar and K. Madhava Krishna, “Large scale visual localization in urban environments”, in [*Proc. of the IEEE International Conference on Robotics and Automation (ICRA)*]{}, 2011.
A. Angeli, D. Filliat, S. Doncieux, and J.-A. Meyer, “A fast and incremental method for loop-closure detection using bags of visual words”, [*IEEE Transactions | 0 | non_member_989 |
---
abstract: |
We consider multi-variable sigma function of a genus $g$ hyperelliptic curve as a function of two group of variables - jacobian variables and parameters of the curve. In the theta-functional representation of sigma-function, the second group arises as periods of first and second kind differentials of the curve. We develop representation of periods in terms of theta-constants. For the first kind periods, generalizations of Rosenhain type formulae are obtained, whilst for the second kind periods theta-constant expressions are presented which are explicitly related to the fixed co-homology basis.\
We describe a method of constructing differentiation operators for hyperelliptic analogues of $\zeta$- and $\wp$-functions on the parameters of the hyperelliptic curve. To demonstrate this method, we gave the detailed construction of these operators in the cases of genus 1 and 2.
address:
- 'Steklov Mathematical Institute, Moscow'
- 'National University of Kyiv-Mohyla Academy'
- Institute of Magnetism NASU
author:
- 'V.M.Buchstaber'
- 'V.Z. Enolski'
- 'D.V.Leykin'
title: 'Multi-variable sigma-functions: old and new results'
---
..
Introduction
============
Our note belongs to an area in which Emma Previato took active part in the development. Since the time of first publication of the present authors [@bel97] she has inspired them, | 0 | non_member_990 |
and given them a lot of suggestions and advices.
The area under consideration is the construction of Abelian functions in terms of multi-variable $\sigma$-functions. Similarly to the Weierstrass elliptic function, the multi-variable sigma keeps the same main property - it remains form-invariant at the action of the symplectic group. Abelian functions appear as logarithmic derivatives like the $\zeta, \wp$-functions of the Weierstrass theory and similarly to the standard theta-functional approach which lead to the Krichever formula for KP solutions, [@kr1977]. But a fundamental difference between sigma and theta-functional theories is the following. They are both constructed by the curve and given as series in Jacobian variables, but in the first case the expansion is purely algebraic with respect to the model of the curve since its coefficients are polynomials in parameters of the defining curve equation, whilst in the second case coefficients are transcendental, being built in terms of Riemann matrix periods which are complete Abelian integrals.
In many publications, in particular see [@oni05; @eemop07; @bel12; @ny12; @ehkklp12; @an13; @bef13; @mp14; @nak16; @nak18] and references therein, it was demonstrated that multi-variable $\wp$-functions represent a language which is very suitable to speak about completely integrable systems of KP type. In particular, | 0 | non_member_990 |
very recently, using the fact that sigma is an entire function in the parameters of the curve (in contrast with theta-function), families of degenerate solutions for solitonic equations were obtained [@bl18; @ben18]. In recent papers [@bm17] and [@bm18], an algebraic construction of a wide class of polynomial Hamiltonian integrable systems was given, and those of them whose solutions are given by hyperelliptic $\wp$-functions were indicated.
Revival of interest in multi-variable $\sigma$-functions is in many respects guided by H.Baker’s exposition of the theory of Abelian functions, which go back to K.Weierstrass and F.Klein and is well documented and developed in his remarkable monographs [@bak998; @bak903]. The heart of his exposition is the representation of fundamental bi-differential of the hyperelliptic curve in algebraic form, in contrast with the representation as a double differential of Riemann theta-function developed by Fay in the monograph, [@fay973]. Recent investigations have demonstrated that Baker’s approach can be extended beyond hyperelliptic curves to wider classes of algebraic curves.
The multi-variable $\sigma$-function of algebraic curve $\mathcal{C}$ is known to be represented in terms of $\theta$-function of the curve as a function of two groups of variables - the Jacobian of the curve, $\mathrm{Jac}(\mathcal{C})$ and the Riemann matrix $\tau$. In | 0 | non_member_990 |
the vast amount of recent publications, properties of the $\sigma$-function as a function of the first group of variables were discussed, whilst modular part of variables and relevant objects like $\theta$-constant representations of complete Abelian integrals are considered separately. In this paper we deals with $\sigma$ as a function over both group of variables.
Due to the pure modular part of the $\sigma$-variables, we consider problem of expression of complete integrals of first and second kind in terms of theta-constants. A revival of interest in this classically known material accords to many recent publications reconsidering such problems such as the Schottky problem [@fgs17], the Thomae [@eg06; @ef08; @ekz18] and Weber [@nr17] formulae, and the theory of invariants and its applications [@ksv05; @ef16].
The paper is organized as follows. In the Section 2 we consider the hyperelliptic genus $g$ curve and the complete co-homology basis of $2g$ meromorphic differentials, with $g$ of them chosen as holomorphic ones. We discuss expressions for periods of these differentials in terms of $\theta$-constants with half integer characteristics. Theta-constants representations of periods of holomorphic integrals is known from the Rosenhain memoir [@ros851], where the case of genus two was elaborated. We discus this case and generalize | 0 | non_member_990 |
the Rosenhain expressions to higher genera hyperelliptic curves. Theta constant representations of second kind periods is known after F.Klein [@klein888], who presented closed formula in terms of derivated even theta-constants for the non-hyperelliptic genus three curve. We re-derive this formula for higher genera hyperelliptic curves.
Section 3 is devoted to a classically known problem, which was resolved in the case of elliptic curves by Frobenius and Stickelberger [@fs882]. The general method of the solution of this problem for a wide class, the so-called $(n, s)$ - curves, has been developed in [@bl08] and represents extensions of the Weierstrass’s method for the derivation of system of differential equations defining the sigma-function. All stages of derivation are given in details and the main result is that the sigma-function is completely defined as the solution of a system of heat conductivity equations in a nonholonomic frame. We also consider there another widely known problem - the description of the dependence of the solutions on initial data. This problem is formulated as a description of the dependence of the integrals of motion, which levels are given as half of the curve parameters, from the remaining half of the parameters. The differential formulae obtained permit | 0 | non_member_990 |
us to present effective solutions of this problem for Abelian functions of the hyperelliptic curve. It is noteworthy that because integrals of motion can be expressed in terms of second kind periods in this place, the results of Section 2 are required. The results obtained in Section 3 are exemplified in details by curves of genera one and two. All consideration is based on explicit uniformization of the space of universal bundles of the hyperelliptic Jacobian.
Modular representation of periods of hyperelliptic co-homologies
================================================================
Modular invariance of the Weierstrass elliptic $\sigma$-function, $\sigma=\sigma(u;g_2,g_3)$ follows from its defining in in [@wei885] in terms of recursive series in terms of variables $(u,g_2,g_3)$. Alternatively $\sigma$-function can be represented in terms of Jacobi $\theta$-function and its modular invariance follows from transformation properties of $\theta$-functions. The last representation involves complete elliptic integrals of first and second kind and their representations in terms of $\theta$-constants are classically known. In this section we are studying generalizations of these representations to hyperelliptic curves of higher genera realized in the form $$\begin{aligned}
y^2 = P_{2g+1}(x)= (x-e_1)
\cdots (x-e_{2g+1}) \label{HCurve} \end{aligned}$$ Here $P_{2g+1}(x)$ - monic polynomial of degree $2g+1$, $e_i\in \mathbb{C}$ - branch points and the curve supposed to be non-degenerate, | 0 | non_member_990 |
i.e. $e_i\neq e_j$.
Representations of complete elliptic integrals of first and second kind in terms of Jacobi $\theta$-constants are classically known. In particular, if elliptic curve is given in Legendre form[^1] $$y^2=(1-x^2)(1- k^2x^2)$$ where $k$ is Jacobian modulus, then complete elliptic integrals of the first kind $K=K(k)$ is represented as $$\begin{aligned}
K= \int_{0}^1 \frac{\mathrm{d} x}{ \sqrt{ (1-x^2)(1-k^2 x^2) }}= \frac{\pi}{2} \vartheta_3^2(0;\tau), \label{FirstKind}
\end{aligned}$$ and $\vartheta_3=\vartheta_3(0|\tau)$ and $\tau=\imath \frac{K'}{K}$, $K'=K(k'), k^2+{k'}^2=1$.
Further, for elliptic curve realized as Weierstrass cubic $$y^2= 4 x^3-g_2 x - g_3= 4 (x-e_1)(x-e_2)(x-e_3) \label{WCubic}$$ recall standard notations for periods of first and second kind elliptic integrals $$\begin{aligned}
\begin{split}
2\omega &= \oint_{\mathfrak{a}} \frac{\mathrm{d}x}{y}, \quad 2\eta = -\oint_{\mathfrak{a}} \frac{x\mathrm{d}x}{y}\\
2\omega' &= \oint_{\mathfrak{b}} \frac{\mathrm{d}x}{y}, \quad 2\eta' = -\oint_{\mathfrak{b}} \frac{x\mathrm{d}x}{y}
\end{split} \hskip1.5cm \tau = \frac{\omega'}{\omega}\end{aligned}$$ and Legendre relation for them $$\quad\omega\eta'-\eta\omega' = - \frac{\imath \pi}{2} \label{LegendreRel}$$ Then the following Weierstrass relation is valid $$\begin{aligned}
\eta=- \frac{1}{12\omega} \left(\frac{\vartheta_2''(0)}{\vartheta_2(0)}
+ \frac{\vartheta_3''(0)}{\vartheta_3(0)}+ \frac{\vartheta_4''(0)}{\vartheta_4(0)}\right)
\label{SecondKind}\end{aligned}$$ In this section we are discussing generalization of these relations to higher genera hyperelliptic curves realized as in (\[HCurve\]).
In this subsection we reproduce H.Baker [@bak907] notations. Let $\mathcal{C}$ be genus $g$ non-degenerate hyperelliptic curve realised as double cover of Riemann sphere, $$\begin{aligned}
y^2= 4 \prod_{j=1}^{2g+1}(x-e_j)\equiv 4x^{2g+1}+ \sum_{i=0}\lambda_i x^i,\quad e_i\neq | 0 | non_member_990 |
e_j,\; \lambda_i
\in \mathbb{C}
\label{hyperelliptic}\end{aligned}$$ Let $({\mathfrak{a}};{\mathfrak{b}})
=( \mathfrak{a}_1,\ldots,\mathfrak{a}_g; \mathfrak{b}_1,\ldots,\mathfrak{b}_g ) $ be canonic homology basis. Introduce co-homology basis ([*Baker co-homology basis*]{}) $$\begin{aligned}
\begin{split} \mathrm{d}{u} (x,y)=( \mathrm{d}u_1 (x,y), \ldots,
\mathrm{d}u_g (x,y))^T, \mathrm{d}{r} (x,y)=( \mathrm{d}r_1 (x,y), \ldots,
\mathrm{d}r_g (x,y))^T\\
\mathrm{d}u_i(x,y)=\frac{x^{i-1}}{y}\mathrm{d}x,\quad \mathrm{d}r_j(x,y)=\sum_{k=j}^{2g+1-j}(k+1-j)
\frac{x^k}{4y}\mathrm{d}x,\quad i,j=1,\ldots,g\end{split}\label{bakerbasis}\end{aligned}$$ satisfying to the generalized Legendre relation, $$\begin{aligned}
\mathfrak{M}^TJ\mathfrak{M}=-\frac{\imath \pi}{2} J,\qquad \mathfrak{M}=\left(\begin{array}{cc} \omega&\omega'\\
\eta& \eta' \end{array}\right), \quad J = \left( \begin{array}{cc} 0_g& 1_g \\ -1_g&0_g
\end{array} \right)\end{aligned}$$ where $g\times g$ period matrices $\omega,\omega', \eta, \eta'$ are defined as $$\begin{aligned}
2\omega=\left( \oint_{\mathfrak{a}_j}\mathrm{d}u_i \right),\quad
2\omega'=\left( \oint_{\mathfrak{b}_j}\mathrm{d}u_i \right), \quad
2\eta=-\left( \oint_{\mathfrak{a}_j}\mathrm{d}r_i \right),\quad
2\eta'=-\left( \oint_{\mathfrak{b}_j}\mathrm{d}r_i \right)\end{aligned}$$ We also denote $\mathrm{d}v=(\mathrm{d}v_1,\ldots,\mathrm{d}v_g )^T = (2\omega)^{-1} \mathrm{d}u$ vector of normalized holomorphic differentials.
Define Riemann matrix $\tau= \omega^{-1}\omega'$ belonging to Siegel half-space $\mathcal{S}_g =\{ \tau^T=\tau,\; \mathrm{Im} \tau>0 \} $. Define Jacobi variety of the curve $\mathrm{Jac}(\mathcal{C})=\mathbb{C}^g/ 1_g\oplus \tau $. Canonic Riemann $\theta$-function is defined on $\mathrm{Jac}(\mathcal{C})\times \mathcal{S}_g$ by Fourier series $$\theta({z};\tau) = \sum_{\mathbb{n} \in \mathbb{Z}^n} \mathrm{e}^{ \imath \pi {n}^T\tau {n}
+ 2\imath \pi {z}^T {n} }\label{thetacan}$$ We will also use $\theta$-functions with half-integer characteristics $[\varepsilon] =
\left[\begin{array}{c} {\varepsilon'}^T\\ {\varepsilon''} \end{array} \right]$, $\varepsilon_i', \varepsilon_j'' = 0$ or $1$ defined as $$\theta[\varepsilon]({z};\tau) = \sum_{\mathbb{n} \in \mathbb{Z}^n} \mathrm{e}^{ \imath \pi
( {n+\varepsilon'}/2)^T
\tau( {n+\varepsilon'}/2)
+ 2\imath \pi ({z+\varepsilon''}/2)^T( n+\varepsilon'/2) }\label{thetachar}$$ | 0 | non_member_990 |
Characteristic is even or odd whenever $ {\varepsilon'}^T\varepsilon'' = 0 $ (mod 2) or 1 (mod 2) and $\theta[\varepsilon](z;\tau)$ as function of $z$ inherits parity of the characteristic.
Derivatives of $\theta$-functions by arguments $z_i$ will be denoted as $$\theta_i[\varepsilon](z;\tau) = \frac{\partial}{\partial z_i} \theta[\varepsilon](z;\tau),
\quad \theta_{i,j}[\varepsilon](z;\tau) = \frac{\partial^2}{\partial z_i\partial z_j} \theta[\varepsilon](z;\tau) , \quad\text{etc.}$$
Fundamental bi-differential $\Omega(P,Q)$ is uniquely definite on the product $(P,Q)\in \mathcal{C}\times \mathcal{C}$ by following conditions:
[**i**]{} $\Omega$ is symmetric, $\Omega(P,Q)=\Omega(Q,P)$
[**ii**]{} $\Omega$ is normalized by the condition $$\begin{aligned}
\oint_{\mathfrak{a}_i} \Omega(P,Q)=0, \quad i=1,\ldots,g\end{aligned}$$
[**iii**]{} Let $P=(x,y)$ and $Q=(z,w)$ have local coordinates $\xi_1=\xi(P)$, $\xi_2=\xi(Q)$ in the vicinity of point $R$, $\xi(R)=0$, then $\Omega(P,Q)$ expands to power series as $$\Omega(P,Q)= \frac{\mathrm{d}\xi_1 \mathrm{d}\xi_2}{( \xi_1-\xi_2)^2 } + \; \text{homorphic 2-form}$$
Fundamental bi-differential can be expressed in terms of $\theta$-function [@fay973] $$\Omega(P,Q)= \mathrm{d}_x\mathrm{d}_z\theta\left( \int_{Q}^P\mathrm{d}{v} + {e}\right), \quad P=(x,y), Q=(z,w)$$ where $\mathrm{d}v$ is normalized holomorphic differential and ${e}$ any non-singular point of the $\theta$-divisor $(\theta)$, i.e. $\theta({e})=0$, but not all $\theta$-derivatives, $ \partial_{z_i}\theta({z})\vert_{{z}={e}} $, $i=1,\ldots,g$ vanish.
In the case of hyperelliptic curve $\Omega(P,Q)$ can be alternatively constructed as $$\Omega(P,Q) =\frac12 \frac{\partial}{\partial z} \frac{ y+w }{ y(x-z)} \mathrm{d}x\mathrm{d}z
+ \mathrm{d}{r}(P)^T \mathrm{d}{u} (Q) + 2\mathrm{d}{u}^T (P)\varkappa
\mathrm{d}{u} (Q)\label{omega1}$$ where first two terms are given as rational functions | 0 | non_member_990 |
of coordinates $P,Q$ and necessarily symmetric matrix $\varkappa^T=\varkappa$, $\varkappa=\eta(2\omega)^{-1}$ is introduced to satisfy the normalization condition ${\bf ii}$. In shorter form (\[omega1\]) cab be rewritten as $$\Omega(P,Q) =\frac{2 yw +F(x,z)}{ 4 (x-z)^2 yw } \mathrm{d}x\mathrm{d}z + 2\mathrm{d}{u}^T (P)\varkappa
\mathrm{d}{u} (Q)\label{omega1}$$ where $F(x,z)$ is so-called Kleinian 2-polar, given as $$F(x,z)=\sum_{k=0}^g x^kz^k \left( 2\lambda_{2k}+\lambda_{2k+1}(x+z) \right) \label{polar}$$ Recently algebraic representation for $\Omega(P,Q)$ similar to (\[omega1\]) found in [@suz17], [@eyn18] for wide class on algebraic curves, included $(n,s)$-curves [@bel999].
Main relation lying in the base of the theory is Riemann formula represented third kind Abelian integral as $\theta$-quotient written in terms of described above realization of the fundamental differential $\Omega(P,Q)$.
(Riemann) Let $P'=(x',y')$ and $P''=(x'',y'')$ are two arbitrary distinct points on $\mathcal{C}$ and let $\mathcal{D}'=\{ P_1'+\ldots+P_{g}'\}$ and $\mathcal{D}''=\{ P_1''+\ldots+P_{g}''\}$ are two non-special divisors of degree $g$. Then the following relation is valid $$\begin{aligned}
\begin{split}
&\int_{P''}^{P'} \sum_{j=1}^g\int_{P_j'}^{P_j''} \left\{ \frac{2yy_i+F(x,x_i)}{4(x-x_i)^2}\frac{\mathrm{d}x}{y}\frac{\mathrm{d}x_i}{y_i} + 2 \mathrm{d}{u}(x,y)\varkappa \mathrm{d} {u}(x_i,y_i) \right\} \\
&=\mathrm{ln}
\left(\frac{ \theta(\mathcal{A}(P')-\mathcal{A}(\mathcal{D}') +{K}_{\infty} )}
{ \theta(\mathcal{A}(P')-\mathcal{A}(\mathcal{D}'') +{K}_{\infty} ) } \right)
-\mathrm{ln}
\left(\frac{\theta(\mathcal{A}(P'')-\mathcal{A}(\mathcal{D}') +{K}_{\infty} )}
{\theta(\mathcal{A}(P'')-\mathcal{A}(\mathcal{D}'') +{K}_{\infty} ) } \right)
\end{split} \label{Riemann1}\end{aligned}$$ where $\mathcal{A}(P)= \int_{\infty}^{P} \mathrm{d}{v}$ Abel map with base point $\infty$, ${K}_{\infty}$ - vector of Riemann constants with bases point $\infty$ which is a half-period.
Introduce multi-variable fundamental | 0 | non_member_990 |
$\sigma$-function, $$\sigma({u}) = C \theta[{K}_{\infty}]( (2\omega)^{-1}{u})
\mathrm{e}^{ {u}^T\varkappa{u} },$$ where $[{K}_{\infty}]$ is characteristic of the vector of Riemann constants, ${u}= \int_{\infty}^{P_1} \mathrm{d}{u} + \ldots + \int_{\infty}^{P_g} \mathrm{d}{u} $ with non-special divisor $P_1+\ldots+P_g$. The constant $C$ is chosen so that expansion $\sigma(\boldsymbol{u})$ near ${u}\sim 0$ starts with a Schur-Weierstrass polynomial [@bel999]. The whole expression is proved to be invariant under the action of symplectic group $\mathrm{Sp}(2g,\mathbb{Z})$. Klein-Weierstrass multi-variable $\wp$-functions are introduced as logarithmic derivatives, $$\begin{aligned}
\wp_{i,j}({u})=-\frac{\partial^2}{\partial u_i\partial u_j}, \quad \wp_{i,j,k}
({u})=-\frac{\partial^3}{\partial u_i\partial u_j\partial u_k},\quad \text{etc.} \quad
i,j,k = 1,\ldots g\end{aligned}$$
For $r\neq s \in \{1,\ldots, g\}$ the following formula is valid $$\sum_{i,j=1}^g\wp_{i,j} \left( \sum_{k=1}^g \int_{\infty}^{(x_k,y_k)} \mathrm{d}{u} \right)x_s^{i-1} x_r^{j-1}
= \frac{F(x_s,x_r)-2y_sy_r}{ 4(x_s-x_r)^2}$$
Jacobi problem of inversion of the Abel map $\mathcal{D} \rightarrow \mathcal{A}( \mathcal{D})$ with $\mathcal{D}= (x_1,y_1)+ \ldots+ (x_g,y_g)$ is resolved as $$\begin{aligned}
\begin{split}
&x^g-\wp_{g,g}({u})x^{g-1} - \ldots -\wp_{g,1}({u})=0\\
&y_k= \wp_{g,g,g}({u})x_k^{g-1}+ \ldots + \wp_{g,g,1}({u})
, \quad k=1,\ldots,g\end{split}\end{aligned}$$
In this section we present generalization of Weierstrass formulae $$\wp(\omega)=e_1,\; \wp(\omega+\omega')=e_2,\; \wp(\omega')=e_3$$ to the case of genus $g$ hyperelliptic curve (\[hyperelliptic\]). To do that introduce partitions $$\begin{aligned}
\begin{split} \{1,\ldots, 2g+1\} = \mathcal{I}_0\cup \mathcal{J}_0, \quad \mathcal{I}_0\cap \mathcal{J}_0=\emptyset\\
\mathcal{I}_0=\{ i_1,\ldots, i_g \}, \quad \mathcal{J}_0=\{ j_1,\ldots, j_{g+1} \} \end{split}
\end{aligned}$$ Then any non-singular even half-period $\Omega_{\mathcal{I}}$ is given as | 0 | non_member_990 |
$$\begin{aligned}
\Omega_{\mathcal{I}_0} = \int_{\infty}^{( e_{i_1},0 )}\mathrm{d} {u}
+\ldots+ \int_{\infty}^{( e_{i_g},0 )}\mathrm{d} {u},\quad \mathcal{I}_0= \{i_1,\ldots, i_g\}\subset \{1,\ldots,2g+1\}
\end{aligned}$$
Denote elementary symmetric functions $s_n(\mathcal{I}_0)$, $S_{n}(\mathcal{J}_0)$ of order $n$ built on branch points $\{ e_{i_k} \}$, $i_k\in \mathcal{I}_0$, $\{ e_{j_k} \}$, $j_k\in \mathcal{J}_0$ correspondingly. In particular, $$\begin{aligned}
\begin{split}
s_1(\mathcal{I}_0)&=e_{i_1}+\ldots+ e_{i_g}, \hskip1.65cm S_1(\mathcal{J}_0)=e_{j_1}+\ldots+ e_{j_{g+1}}\\
s_2(\mathcal{I}_0)&=e_{i_1}e_{i_2}+\ldots+e_{i_{g-1}} e_{i_g},
\hskip0.5cm S_2(\mathcal{J}_0)=e_{j_1}e_{j_2}+\ldots+e_{j_{g}} e_{i_{g+1}}\\
&\vdots \hskip5.4cm\vdots\\
s_g(\mathcal{I}_0)&=e_{i_1}\cdots e_{i_g} \hskip 2.62cm
S_{g+1}(\mathcal{J}_0)=e_{j_1}\cdots e_{j_{g+1}}
\end{split}\end{aligned}$$ Because of symmetry, $\wp_{p,q}({\Omega}_{\mathcal{I}_0})=\wp_{q,p}({\Omega}_{\mathcal{I}_0})$ enough to find these quantities for $ p\leq q \in \{1,\ldots,g\} $. The following is valid
(Conjectural Proposition ) Let even non-singular half-period ${\Omega}_{\mathcal{I}_0}$ associate to a partition $\mathcal{I}_0\cup \mathcal{J}_0=\{1,\ldots,2g+1\}$. Then for all $k,j\geq k, k,j = 1\ldots,g$ the following formula is valid
$$\begin{aligned}
\begin{split}
&\wp_{k,j}({\Omega}_{\mathcal{I}_0})\\&=(-1)^{k+j}\sum_{n=1}^k n \left( s_{g-k+n}(\mathcal{I}_0)S_{g-j-n+1}(\mathcal{J}_0)
+s_{g-j-n}(\mathcal{I}_0)S_{g+n-k+1}(\mathcal{J}_0) \right),
\end{split} \label{formula}\end{aligned}$$
Klein formula written for even non-singular half period ${\Omega}_{\mathcal{I}_0}$ leads to linear system of equations with respect to Kleinian two-index symbols $ \wp_{i,j}({\Omega}_{\mathcal{I}_0}) $ $$\begin{aligned}
\sum_{i=1}^g\sum_{j=1}^g \wp_{i,j}({\Omega}_{\mathcal{I}_0}) e_{i_r}^{i-1}e_{i_s}^{j-1} =\frac{F(e_{i_r}, e_{i_s})}
{4(e_{i_r} - e_{i_s})^2}\quad \; i_r, i_s \in \mathcal{I}_0\label{Kleineq}\end{aligned}$$
To solve these equation we note that $$\wp_{k,g}({\Omega}_{\mathcal{I}_0})= (-1)^{k+1}s_k(\mathcal{I}_0), \quad k=1,\ldots,g$$ Also note that $F(e_{i_r}, e_{i_s})$ is divisible by $(e_{i_r}- e_{i_s})^2$ and $$\frac{F(e_{i_r}, e_{i_s})}{4(e_{i_r} - e_{i_s})^2}= e_{i_r}^{g-1} e_{i_s}^{g-1}\mathfrak{S}_1+
e_{i_r}^{g-2} e_{i_s}^{g-2}\mathfrak{S}_2+\ldots + \mathfrak{S}_{2g-1}\label{JIP1}$$ where $\mathfrak{G}_k$ are order $k$ elementary symmetric functions of | 0 | non_member_990 |
elements $e_i$ $ i\in \{1, \ldots, 2g+1 \} - \{ i_r,i_s \} $
Let us analyse equations (\[Kleineq\]) for small genera, $g\leq 5$. One can see that plugging to the equation (\[JIP1\]) to (\[Kleineq\]) we get non-homogeneous linear equations solvable by Kramer rule and the solutions can be presented in the form (\[formula\]).
Now suppose that (\[formula\]) is valid for higher $g>5$ were computer power is insufficient to check that by means of computer algebra. But that’s possible to check (\[formula\]) for arbitrary big genera numerically giving to branch points $e_i, i=1,\ldots, i=2g+1$ certain numeric values. Many checking confirmed (\[formula\]) .
Quantities $\wp_{i,j}({\Omega}_{\mathcal{I}_0})$ are expressed in terms of even $\theta$-constants as follows $$\begin{aligned}
\wp_{i,j}({\Omega}_{\mathcal{I}_0})=-2\varkappa_{i,j}
- \frac{1}{\theta[\varepsilon_{\mathcal{I}_0}]({0})}
\partial_{{U}_i,{U_j} }^2 \theta[\varepsilon_{\mathcal{I}_0}]({0}), \quad \forall \mathcal{I}_0,\; i,j=1,\ldots, g.\end{aligned}$$ Here $[\varepsilon_{\mathcal{I}_0}]$ is characteristic of the vector $\left[{\Omega}_{\mathcal{I}_0}+{K}_{\infty} \right]$, where ${K}_{\infty}$ is vector of Riemann constants with base point $\infty$ and $\partial_{{U}}$ is directional derivative along vector $\boldsymbol{U}_i$, that is $i$th column vector of inverse matrix of $\mathfrak{a}$-periods, $\mathcal{A}^{-1} = ( {U}_1,\ldots, {U}_g )$. The same formula is valid for all possible partitions $\mathcal{I}_0\cup \mathcal{J}_0$, there are $N_g$ of that, that is number of non-singular even characteristics, $$N_g= \left( \begin{array}{c} 2g+1\\ g \end{array} \right)$$ Therefore one | 0 | non_member_990 |
can write
$$\begin{aligned}
\begin{split}
\varkappa_{i,j} &=\frac{1}{8N_g} \Lambda_{i,j}- \frac{1}{2 N_g}
\sum_{ \text{All even non-singular} \;\; [\varepsilon] } \frac{\partial^2_{{U}_i,{U_j}}
\theta[\varepsilon_{\mathcal{I}_0}]({0})}{\theta[\varepsilon_{\mathcal{I}_0}]({0})}\\
\end{split}\end{aligned}$$
where $$\begin{aligned}
\Lambda_{i,j}=-4\sum_{ \text{All partitions} \; \mathcal{I}_0 } \wp_{i,j}({\Omega}_{\mathcal{I}_0})\end{aligned}$$ Denote by $\Lambda_g$ symmetric matrix $$\Lambda_g = ( \Lambda_{i,j} )_{i,j=1,\ldots,g} \label{Lambda1}$$
Entries $\Lambda_{k,j}$ at $k\leq j$ to the symmetric matrix $\Lambda$ are given by the formula $$\begin{aligned}
\begin{split}
\Lambda_{k,j}&= \lambda_{k+j} \frac{ \left( \begin{array}{c} 2g+1\\g \end{array} \right) }
{ \left( \begin{array}{c} 2g+1\\2g+1-k-j \end{array} \right) }
\sum_{n=1}^k n \left[ \left( \begin{array}{c} g\\ g-k+n \end{array} \right) \left( \begin{array}{c} g+1\\ g-j-n+1 \end{array} \right) \right. \\&\hskip 5.5cm + \left.\left( \begin{array}{c} g \\g-j-n \end{array} \right)
\left( \begin{array}{c} g+1\\ g-k+n+1 \end{array} \right) \right]\end{split}\label{Lambda2}\end{aligned}$$
Execute summation in (\[formula\]) and find that each $\Lambda_{k,j}$ proportional to $\lambda_{k+j}$ with integer coefficient.
Matrix $\Lambda_g$ exhibits interesting properties of sum of anti-diagonal elements implemented at derivations in [@eil16], $$\begin{aligned}
\sum_{i,j, \;i+j=k}\Lambda_{g;i,j}=\lambda_k \frac{N_g}{4g+2}
\left[ \frac12 k(2g+2-k)+\frac14(2g+1)( (-1)^k-1 ) \right]\end{aligned}$$
Lower genera examples of matrix $\Lambda$ were given in [@ehkklp12], [@eil16], but method implemented there unable to get expressions for $\Lambda$ at big genera.
At $g=6$ we get matrix $$\begin{aligned}
\Lambda_6= \left( \begin{array}{cccccc} 792\lambda_2&330\lambda_3&120\lambda_4&36\lambda_5&8\lambda_6&\lambda_7\\
330\lambda_3&1080\lambda_4&492\lambda_5&184\lambda_6&51\lambda_7&8\lambda_8\\
120\lambda_4&492\lambda_5&1200\lambda_6&542\lambda_7&184\lambda_8&36\lambda_9\\
36\lambda_5&184\lambda_6&542\lambda_7&1200\lambda_8&492\lambda_9&120\lambda_{10}\\
8\lambda_6&51\lambda_7&184\lambda_8&492\lambda_9&1080\lambda_{10}&330\lambda_{11}\\
\lambda_7&8\lambda_8&36\lambda_9&120\lambda_{10}&330\lambda_{11}&792\lambda_{12}
\end{array} \right)\end{aligned}$$
Collecting all these together we get the following
$\varkappa$-matrix defining multi-variate $\sigma$-function admits the following | 0 | non_member_990 |
modular form representation
$$\begin{aligned}
\varkappa=\frac{1}{8 N_g}\Lambda_g-\frac{1}{2N_g} {(2\omega)^{-1}}^T . \left[
\sum_{N_g\;\text{even}\;[\varepsilon]} \frac{1}{\theta[\varepsilon]}
\left(\begin{array}{ccc}
\theta_{1,1}[\varepsilon]&\cdots&\theta_{1,g}[\varepsilon]\\
\vdots&\cdots&\vdots\\
\theta_{1,g}[\varepsilon]&\cdots&\theta_{g,g}[\varepsilon]\end{array}\right)\right].(2\omega)^{-1}
\label{formula2}\end{aligned}$$
[*where $2\omega$ is matrix of $\mathfrak{a}$-periods of holomorhphic differentials and $ \theta_{i,j}[\varepsilon]= \partial^2_{z_i,z_j}\theta[\varepsilon]({z})_{{z}=0} $.*]{}
Note that modular form representation of period matrices $\eta, \eta'$ follows from the above formula,
$$\eta=2\varkappa\omega, \qquad \eta'=2\varkappa \omega'-\imath\pi{(2\omega)^T}^{-1}\label{etamodular}$$
At $g=2$ for the curve $y^2=4x^5+\lambda_4x^4+\ldots+\lambda_0$ the following representation of $\varkappa$-matrix is valid $$\begin{aligned}
\varkappa= \frac{1}{80}\left( \begin{array}{cc} 4\lambda_2&\lambda_3\\
\lambda_3&4\lambda_4 \end{array} \right)-\frac{1}{20}\sum_{10\;\text{even}\;[\varepsilon]}\frac{1}{\theta[\varepsilon]}
\left(\begin{array}{cc} \partial^2_{{U}_1^2}\theta[\varepsilon]& \partial^2_{{U}_1,{U}_2}
\theta[\varepsilon]\\
\partial^2_{{U}_1,{U}_2}
\theta[\varepsilon]& \partial^2_{{U}_2^2}\theta[\varepsilon]\end{array}\right)\end{aligned}$$ with $\varkappa=\eta(2\omega)^{-1}$, $\mathcal{A}^{-1}=(2\omega)^{-1}=({U}_1,{U}_2 ) $ and directional derivatives $\partial_{{U}_i}$, $i=1,2$.
Representation of $\varkappa$ matrix of genus 2 and 3 hyperelliptic curves in terms of directional derivatives of non-singular odd constant was found in [@eee13]
Co-homologies of Baker and Klein
--------------------------------
Calculations of $\varkappa$-matrix for the hyperelliptic curve (\[hyperelliptic\]) were done in co-homology basis introduced by H.Baker (\[bakerbasis\]). When holomorphic differentials, $ \mathrm{d}{u} (x,y) $ are chosen meromorphic differentials, $ \mathrm{d}{r} (x,y)$ can be find from the symmetry condition $\mathbf{I}$. One can check that symmetry condition also fulfilled if meromorphic differentials will be changed as $$\mathrm{d}{r}(x,y) \rightarrow \mathrm{d}{r}(x,y)+M\mathrm{d}{u}(x,y),$$ where $M$ is arbitrary constant symmetric matrix $M^T=M$. One can then choose $$M=-\frac{1}{8 N_g}\Lambda_g$$ Then $\varkappa$ will change to $$\begin{aligned}
\varkappa=
-\frac1{2}\frac1{N_g}\sum_{N_g\;\text{even}\;\left[\varepsilon_{\mathcal{I}_0}\right]} \frac{1}{\theta[\varepsilon_{\mathcal{I}_0}]({0})} \left(\partial_{{U}_i}\partial_{{U_j}} \theta[\varepsilon_{\mathcal{I}_0}]({0})\right)_{i,j=1,\ldots,g} .
| 0 | non_member_990 |
\label{Kleinian}\end{aligned}$$
Following [@eil16] introduce co-homology basis of Klein $$\begin{aligned}
\mathrm{d}{u} (x,y), \quad \mathrm{d}{r} (x,y)
-\frac{1}{8N_g}\Lambda_g\mathrm{d}{u} (x,y)\label{Kleinbasis}\end{aligned}$$ with constant matrix, $\Lambda_g=\Lambda_g({\lambda})$ given by (\[Lambda1\],\[Lambda2\]). Therefore we proved
$\varkappa$-matrix is represented in the modular form (\[Kleinian\]) in the co-homology basis (\[Kleinbasis\]).
Formula (\[Kleinian\]) first appears in F.Klein ([@klein886], [@klein888]), it was recently revisited in a more general context by Korotkin and Shramchenko ([@ksh12]) who extended representation for $\varkappa$ to non-hyperelliptic curves. Correspondence of this representation to the co-homology basis to the best knowledge of the authors was not earlier discussed.
Rewrite formula (\[Kleinian\]) in equivalent form, $$\begin{aligned}
\omega^T\eta=-\frac{1}{4N_g}
\sum_{N_g\;\text{even}\;[\varepsilon]} \frac{1}{\theta[\varepsilon]}
\left(\begin{array}{ccc}
\theta_{1,1}[\varepsilon]&\cdots&\theta_{1,g}[\varepsilon]\\
\vdots&\cdots&\vdots\\
\theta_{1,g}[\varepsilon]&\cdots&\theta_{g,g}[\varepsilon]\end{array}\right)\label{omegaeta}\end{aligned}$$ where $\omega,\eta$ are half-periods of holomorphic and meromorphic differentials in Kleinian basis.
For the Weierstrass cubic $y^2=4x^3-g_2x-g_3$ (\[omegaeta\]) represents Weierstrass relation (\[SecondKind\]).
At $g=2$ (\[omegaeta\]) can be written in the form $$\begin{aligned}
\omega^T\eta = -\frac{\imath \pi}{10} \left(\begin{array}{cc}
\partial_{\tau_{1,1}} & \partial_{\tau_{1,2}}\\
\partial_{\tau_{1,2}} & \partial_{\tau_{2,2}}
\end{array} \right)\; \mathrm{ln}\; \chi_{5}\end{aligned}$$ where $\chi_5$ is relative invariant of weight 5, $$\begin{aligned}
\chi_{5} = \prod_{ 10\;\text{even}\; [\varepsilon] } \theta[\varepsilon]\end{aligned}$$
Worth to mention how equations of KdV flows looks in both bases. For example at $g=2$ and curve $y^2=4x^5+\lambda_4x^4+\ldots+\lambda_0$ in Baker basis we got [@bel997] $$\begin{aligned}
\begin{split}
\wp_{2222}&=6\wp_{2,2}^2+4\wp_{1,2}+\lambda_4\wp_{2,2}+\frac12\lambda_3 \\
\wp_{1222}&=6\wp_{2,2}\wp_{1,2}-2\wp_{1,1}+\lambda_4\wp_{1,2} \end{split}\end{aligned}$$ In Kleinian basis the | 0 | non_member_990 |
same equations change only in linear in $\wp_{i,j}$-terms $$\begin{aligned}
\begin{split}
\wp_{2222}&=6\wp_{2,2}^2+4\wp_{1,2}-47\lambda_4\wp_{2,2}+92\lambda_4^2-\frac72\lambda_3\\
\wp_{1222}&=6\wp_{2,2}\wp_{1,2}-2\wp_{1,1}-23\lambda_4\wp_{1,2}-6\lambda_3\wp_{2,2}+23\lambda_3\lambda_4+8\lambda_2 \end{split}\end{aligned}$$
Rosenhain modular form representaion of first kind periods
----------------------------------------------------------
Rosenhain [@ros851] was the first who introduced $\theta$-functions with characteristics at $g=2$. There are 10 even and 6 odd characteristics in that case. Let us denote each from these characteristics as $$\varepsilon_j = \left[\begin{array}{c} {\varepsilon_j'}^T\\ {\varepsilon''_j}^T \end{array} \right],
\quad j=1,\ldots 10$$ where $\varepsilon'_j$ and $\varepsilon''_j$ are column 2-vectors with entries equal to 0 or 1.
Rosenhain fixed the hyperelliptic genus two curve in the form $$y^2=x(x-1)(x-a_1)(x-a_2)(x-a_3)$$ and presented without proof expression $$\begin{aligned}
\mathcal{A}^{-1} =\frac{1}{2\pi^2 Q^2 } \left( \begin{array}{rr} -P\theta_2[\delta_2]&Q\theta_2[\delta_1]\\
P\theta_1[\delta_2]&-Q\theta_1[\delta_1] \end{array} \right) \label{RosenhainFormula}\end{aligned}$$ with $$\begin{aligned}
P&=\theta[\alpha_1]\theta[\alpha_2]\theta[\alpha_3],\quad
Q=\theta[\beta_1]\theta[\beta_2]\theta[\beta_3]\end{aligned}$$ and 6 even characteristics $[\alpha_{1,2,3}], [\beta_{1,2,3}]$ and two odd $[\delta_{1,2}]$ which looks chaotic. One of first proofs can be found in H.Weber [@web859]; these formulae are implemented in Bolza dissertation [@bol885] and [@bol886]. Our derivation of these formulae are based on the [*Second Thomae relation*]{} [@tho870], see [@er08] and [@eil18]. To proceed we give the following definitions.
A triplet of characteristics $[\varepsilon_1]$, $[\varepsilon_2,]$, $[\varepsilon_3]$ is called azygetic if
$$\mathrm{exp}\; \imath \pi \left\{
\displaystyle{\sum_{j=1}^3{\varepsilon_j'}^T \varepsilon''_j + \sum_{i=1}^3{\varepsilon_i'}^T
\sum_{i=1}^3\varepsilon''_i } \right\} =- 1$$
A sequence of $2g + 2$ characteristics $[\varepsilon_1],\ldots,[\varepsilon_{2g+2} ] $ | 0 | non_member_990 |
is called a [*special fundamental system*]{} if the first $g$ characteristics are odd, the remaining are even and any triple of characteristics in it is azygetic.
(Conjectural Riemann-Jacobi derivative formula) Let $g$ odd $[\varepsilon_1], \ldots, [\varepsilon_g]$ and $g+2$ even $ [\varepsilon_{g+1}], \ldots, [\varepsilon_{2g+2}]$ characteristics create a special fundamental system. Then the following equality is valid $$\mathrm{Det}\left. \frac{ \partial( \theta[\varepsilon_1]({v}),\ldots,
\theta[\varepsilon_g]({v})) }
{\partial (v_1,\ldots, v_g)}\right|_{{v}=0}= \pm \prod_{k=1\ldots g+2}\theta[\varepsilon_{g+k}]({0})
\label{RiemannJacobi}$$
(\[RiemannJacobi\]) proved up to $g=5$ [@fro885], [@igu980], [@fay979]
Jacobi derivative formula for elliptic curve $$\begin{aligned}
\vartheta'_1(0)=\pi \vartheta_2(0)\vartheta_3(0)\vartheta_4(0)\end{aligned}$$
Rosenhain derivative formula for genus two curve is given without proof in the memoir [@ros851], namely, let $[\delta_1]$ and $[\delta_2]$ are any two odd characteristics from all 6 odd, then $$\begin{aligned}
\theta_1[\delta_1] \theta_2[\delta_2]-\theta_2[\delta_1] \theta_1[\delta_2]
= \pi^2\theta[\gamma_1]\theta[\gamma_2]\theta[\gamma_3]\theta[\gamma_4]
\label{RosenhainDerivative}\end{aligned}$$ where 4 even characteristics $[\gamma_1], \ldots, [\gamma_4]$ are given as $[\gamma_i]=[\delta_1]+[\delta_2]+[\delta_i], \; 3\leq i \leq 6$. There are 15 Rosenhain derivative formulae.
The following geometric interpretation of the special fundamental system can be given in the case of hyperelliptic curve. Consider genus two curve, $$\mathcal{C}:\quad y^2= (x-e_1) \cdots ( x-e_{6})$$ Denote associated homology basis as $( \mathfrak{a}_1, \mathfrak{a}_2;\mathfrak{b}_1,\mathfrak{b}_2 )$. Denote characteristics of Abelian images of branch points with base point $e_6$ as $ \mathfrak{A}_k$, $k=1,\ldots,6$. These are half-periods | 0 | non_member_990 |
given by their characteristics, $[\mathfrak{A}_k]$ with $$\begin{aligned}
\mathfrak{A}_k=
\int_{(e_6,0)}^{(e_k,0)} {u}= \frac12 \tau {\varepsilon'}_k
+ \frac12 {\varepsilon''}_k, \quad k=1,\ldots,6\end{aligned}$$
For the homology basis drawn on the Figure we have
0.7mm
(150.00,80.00) (9.,33.)[(1,0)[12.]{}]{} (9.,33.) (21.,33.) (10.,29.)[(0,0)\[cc\][$e_1$]{}]{} (21.,29.)[(0,0)\[cc\][$e_2$]{}]{} (15.,33.)[(20,30.)]{} (8.,17.)[(0,0)\[cc\][$\mathfrak{ a}_1$]{}]{} (15.,48.)[(1,0)[1.0]{}]{} (32.,33.)[(1,0)[9.]{}]{} (32.,33.) (41.,33.) (33.,29.)[(0,0)\[cc\][$e_3$]{}]{} (42.,29.)[(0,0)\[cc\][$e_4$]{}]{} (37.,33.)[(18.,26.)]{} (30.,19.)[(0,0)\[cc\][$\mathfrak{a}_2$]{}]{} (36.,46.)[(1,0)[1.0]{}]{} (100.,33.00) [(1,0)[33.]{}]{} (100.,33.) (133.,33.) (101.,29.)[(0,0)\[cc\][$e_{5}$]{}]{} (132.,29.)[(0,0)\[cc\][$e_{6}$]{}]{} (59.,58.)[(0,0)\[cc\][$\mathfrak{b}_1$]{}]{} (63.,62.)[(1,0)[1.0]{}]{} (15.,33.00)(15.,62.)(65.,62.) (65.00,62.)(119.00,62.00)(119.00,33.00) (15.,33.00)(15.,5.)(65.,5.) (65.00,5.)(119.00,5.00)(119.00,33.00) (70.,44.)[(0,0)\[cc\][$\mathfrak{b}_2$]{}]{} (74.00,48.)[(1,0)[1.0]{}]{} (37.,33.00)(37.,48.)(76.00,48.) (76.00,48.)(111.00,48.00)(111.00,33.00) (37.,33.00)(37.,19.)(76.00,19.) (76.00,19.)(111.00,19.00)(111.00,33.00)
$$\begin{aligned}
[{\mathfrak A}_1]= \frac{1}{2}
\left[\begin{array}{cc}1&0\\
0&0\end{array}\right],\quad
[{\mathfrak A}_2]= \frac{1}{2}
\left[\begin{array}{cc}1&0\\
1&0\end{array}\right],\quad
[{\mathfrak A}_3]= \frac{1}{2}
\left[\begin{array}{cc}0&1\\
1&0\end{array}\right],\end{aligned}$$
$$\begin{aligned}
[{\mathfrak A}_4]= \frac{1}{2}
\left[\begin{array}{cc}0&1\\
1&1\end{array}\right],\quad
[{\mathfrak A}_5]= \frac{1}{2}
\left[\begin{array}{cc}0&0\\
1&1\end{array}\right],\quad
[{\mathfrak A}_6]= \frac{1}{2}
\left[\begin{array}{cc}0&0\\
0&0\end{array}\right]\end{aligned}$$
$${\mathfrak{A}}_k= \int_{\infty}^{e_k}{u}= \frac12 \tau.
{\varepsilon}_k
+ \frac12 {\varepsilon'}_k , \quad [ {\mathfrak{A}}_k ] =
\left[ \begin{array}{c}
{\varepsilon}_k^T\\ {\varepsilon'}_k^T \end{array} \right]$$ One can see that the set of characteristics $[\varepsilon_k] = [ \mathfrak{A}_k]$ of Abelian images of branch point contains two odd $[\varepsilon_2]$ and $[\varepsilon_4]$ and remaining four characteristics are even. One can check that the whole set of these $6=2g+2$ characteristics is azygetic and therefore the set constitutes special fundamental system. Hence one can write Rosenhain derivative formula $$\theta_1[\varepsilon_2]\theta_2[\varepsilon_4]-\theta_2[\varepsilon_2]\theta_1[\varepsilon_4]=\pm \pi^2
\theta[\varepsilon_1]\theta[\varepsilon_3]\theta[\varepsilon_5]\theta[\varepsilon_6]$$ In this way we gave geometric interpretation of Rosenhain derivative formula and associated set of characteristics in the case when one from | 0 | non_member_990 |
even characteristic is zero. The same structure is observed for higher genera hyperelliptic curves even for $g>5$.
The characteristics entering to the Rosenhain formula are described as follows. Take any of 15 Rosenhain derivative formula, say, $$\theta_1[p] \theta_2[q]-\theta_2[p] \theta_1[q]
= \pi^2\theta[\gamma_1]\theta[\gamma_2]\theta[\gamma_3]\theta[\gamma_4]$$ Then 10 even characteristics can be grouped as $$\underbrace{ [\gamma_1],\ldots,[\gamma_4]}_4, \; \underbrace{ [\alpha_1],[\alpha_2],[\alpha_3] }_{[\alpha_1]+ [\alpha_2]+ [\alpha_3]=[p]},\; \;
\underbrace{ [\beta_1],[\beta_2],[\beta_3] }_{ [\beta_1]+ [\beta_2]+ [\beta_3]=[q]},$$ Then matrix of $\mathfrak{a}$-periods $$\begin{aligned}
\mathcal{A} &= \frac{2Q}{PR} \left( \begin{array}{rr}
Q\theta_1[q]&Q \theta_2[q]\\
P\theta_1[p]& P\theta_2[p]
\end{array} \right)\end{aligned}$$ with $$\begin{aligned}
P=\prod_{j=1}^3\theta[\alpha_j], \;\;Q=\prod_{j=1}^3\theta[\beta_j],\; \;R=\prod_{j=1}^4\theta[\gamma_j]\label{PQR}\end{aligned}$$
Note, that the 15 curves are given as $$\begin{aligned}
\mathcal{C}_{p,q}: \quad y^2=x(x-1)(x-a_1)(x-a_2)(x-a_3) \label{Bolza} \end{aligned}$$ where branch points are computed by [*Bolza formulae*]{} [@bol886], $$\begin{aligned}
e[\delta_j]= - \frac{\partial_{{U}_1} \theta[\delta_j] }{\partial_{\boldsymbol{U}_2} \theta[\delta_j]} , \quad
j=1,\ldots,6
\label{bolza1}
\end{aligned}$$ where $\partial_{{U}_i}$ directional derivative along vector ${U}_i$, $i=1,2$, $\mathcal{A}^{-1}=({U}_1,{U}_2)$ and $$e[p]=0,\; e[q]=\infty,$$ All 15 curves $\mathcal{C}_{p,q}$ are Möbius equivalent.
-
Generalization of the Rosenhain formula to higher genera hyperelliptic curve was found in [@er08] and developed further in [@eil18].
$$\begin{aligned}
\begin{split} y^2&=\phi(x) \psi(x)\\ \phi(x)&= \prod_{k=1}^g(x-e_{2k}), \;
\psi(x)= \prod_{k=1}^{g+1}(x-e_{2k-1}) \label{curve-g} \end{split} \end{aligned}$$
Denote $ R=\prod_{k=1}^{g+2}\theta[\gamma_k]$ monomial in left had side of Riemann-Jacobi formula (\[RiemannJacobi\])
Let genus $g$ hyperelliptic curve is given in (\[curve-g\]). Then winding vectors $(U_1,\ldots,U_g)=\mathcal{A}^{-1}$ are given by the | 0 | non_member_990 |
formula $$\begin{aligned}
{U}_m=\frac{\epsilon}{ 2\pi^g R} \mathrm{Cofactor}\left( \left. \frac{ \partial( \theta[\varepsilon_1]({v}),\ldots,
\theta[\varepsilon_g]({v})) }
{\partial (v_1,\ldots, v_g)}\right|_{{v}=0}
\right)
\left( \begin{array}{c} s_{m-1}^{2} \sqrt[4]{\chi_1}\\ \vdots
\\s_{m-1}^{2g} \sqrt[4]{ \chi_g}\end{array} \right)\end{aligned}$$ Here $s_k^i$ -order $k$ symmetric function of elements $\{e_{2}, \ldots e_{2g} \} / \{ e_{2i} \}$ and $$\chi_{i} = \frac{\psi(e_{2i})}{\phi'(e_{2i})} , \quad i=1,\ldots, g$$ $s_k^i, \chi_i$ are expressible in $\theta$-constants via Thomae formulae [@tho870].
Typical answer of multi-gap integration includes $\theta$-function $ \theta({U}x +{V}t+{W};\tau )$ where winding vectors ${U}, {V}$ expressed in terms of complete holomorphic integrals and constant ${W}$ defined by initial data. Rosenhain formulae and their generalization, express $U,V$ in terms of $\theta$-constants and parameters the equation defining $\mathcal{C}$. In this way the problem of [*effectivization of finite gap solutions*]{} [@dub81] can be solved in that way at least for hyperelliptic curves.
Other application of the Rosenhain formula (\[RosenhainFormula\]) presented in [@be994] where two-gap Lamé and Treibich-Verdier potentials were obtained by the reduction to elliptic functions of general Its-Matveev representation [@im975] of finite-gap potential to the Schrödinger equation in terms of multi-variable $\theta$-functions.
Another application relevant to a computer algebra problem. In the case when, say in Maple, periods of holomorphic differentials are computed them periods of second kind differentials can be obtained | 0 | non_member_990 |
by Rosenhain formula (\[RosenhainFormula\]) and its generalization.
Sigma-functions and the problem of differentiatiion of Abelian functions.
=========================================================================
Consider the curve $$\label{F-1}
V_\lambda = \left\{(x,y)\in\mathbb{C}^2\, : y^2 = \mathcal{C}(x;\lambda) = x^{2g+1}+\sum_{k=2}^{2g+1}
\lambda_{2 k} x^{2g - k + 1} \right\}$$ where $g\geqslant 1$ and $\lambda=(\lambda_4,\ldots,\lambda_{4g+2})\in \mathbb{C}^{2g}$ are the parameters. Set $\mathcal{D} = \{ \lambda\in \mathbb{C}^{2g}\,: \mathcal{C}(x;\lambda)\; \text{has multiple roots}\}$ and $\mathcal{B} = \mathbb{C}^{2g}\setminus\mathcal{D}$. For any $\lambda\in \mathcal{B}$ we obtain the affine part of a smooth projective hyperelliptic curve $\overline{V}_\lambda$ of genus $g$ and the Jacobian variety $Jac(\overline{V}_\lambda) = \mathbb{C}^{g}/\Gamma_g$, where $\Gamma_g \subset \mathbb{C}^{g}$ is a lattice of rank $2g$ generated by the periods of the holomorphic differential on cycles of the curve $V_\lambda$.
In the general case, an *Abelian function* is a meromorphic function on a complex Abelian torus $T^g=\mathbb{C}^g\!/\Gamma$, where $\Gamma\subset\mathbb{C}^g$ is a lattice of rank $2g$. In other words, a meromorphic function $f$ on $\mathbb{C}^g$ is Abelian iff $f(u)=f(u+\omega)$ for all $u=(u_1,\ldots,u_g)\in\mathbb{C}^{g}$ and $\omega\in\Gamma$. Abelian functions on $T^g$ form a field $\mathcal{F} = \mathcal{F}_g$ such that:
\(1) let $f\in\mathcal{F}$, then $\partial_{u_i} f\in \mathcal{F}$, $i=1,\dots,g$;
\(2) let $f_1,\dots,f_{g+1}$ be any nonconstant functions from $\mathcal{F}$, then there exists a polynomial $P$ such that $P(f_1,\dots, f_{g+1})(u)=0$ for all $u\in T^g$;
\(3) let $f\in\mathcal{F}$ | 0 | non_member_990 |
be a nonconstant function, then any $h\in\mathcal{F}$ can be expressed rationally in terms of $(f,\partial_{u_1}f,\dots,\partial_{u_g}f)$;
\(4) there exists an entire function $\vartheta\colon\mathbb{C}^g\to\mathbb{C}$ such that $\partial_{u_i,u_j}\log\vartheta \in \mathcal{F}$, $i,j=1,\dots,g$.
For example, any elliptic function $f\in \mathcal{F}_1$ is a rational function in the Weierstrass functions $\wp(u; g_2, g_3)$ and $\partial_u\wp(u; g_2, g_3)$, where $g_2$ and $g_3$ are parameters of elliptic curve $$V = \{ (x,y) \in \mathbb{C}^2\;|\; y^2 = 4 x^3 - g_2 x - g_3 \}.$$ It is easy to see that the function $\frac{\partial}{\partial g_2}\wp(u; g_2, g_3)$ will no longer be elliptic. This is due to the fact that the period lattice $\Gamma$ is a function of the parameters $g_2$ and $g_3$. In [@fs882] Frobenius and Stickelberger described all the differential operators $L$ in the variables $u,\;g_2$ and $g_3$, such that $Lf\in \mathcal{F}_1$ for any function $f\in \mathcal{F}_1$ (see below Section 7.3).
In [@bl07; @bl08] the classical problem of differentiation of Abelian functions over parameters for families of $(n,s)$-curves was solved. In the case of hyperelliptic curves this problem was solved more explicitly.
All genus $2$ curves are hyperelliptic. We denote by $\pi\colon \mathcal{U}_g \to \mathcal{B}_g$ the universal bundle of Jacobian varieties $Jac(\overline{V}_\lambda)$ of hyperelliptic curves. Let us consider the | 0 | non_member_990 |
mapping $\varphi\colon \mathcal{B}_g\times \mathbb{C}^g \to \mathcal{U}_g$, which defines the projection $\lambda\times \mathbb{C}^g \to \mathbb{C}^g/\Gamma_g(\lambda)$ for any $\lambda\in \mathcal{B}_g$. Let us fix the coordinates $(\lambda;u)$ in $\mathcal{B}_g\times \mathbb{C}^g\subset \mathbb{C}^{2g}\times \mathbb{C}^g$ where $u = (u_1,\ldots,u_{2g-1})$. Thus, using the mapping $\varphi$, we fixed in $\mathcal{U}_g$ the structure of the space of the bundle whose fibers $J_\lambda$ are principally polarized Abelian varieties.
We denote by $F = F_g$ the field of functions on $\mathcal{U}_g$ such that for any $f\in F$ the function $\varphi^*(f)$ is meromorphic, and its restriction to the fiber $J_\lambda$ is an Abelian function for any point $\lambda \in \mathcal{B}_g$.
Below, we will identify the field $F$ with its image in the field of meromorphic functions on $\mathcal{B}\times \mathbb{C}^g$.
The following [**Problem I**]{}:
[*Describe the Lie algebra of differentiations of the field of meromorphic functions on $\mathcal{B}_g\times \mathbb{C}^g$, generated by the operators $L$, such that $Lf\in F$ for any function $f\in F$*]{}
was solved in [@bl07; @bl08].
From the differential geometric point of view, Problem I is closely related to [**Problem II**]{}:
[*Describe the connection of the bundle $\pi\colon \mathcal{U}_g \to \mathcal{B}_g$.*]{}
The solution of Problem II leads to an important class of solutions of well-known equations of mathematical physics. In the | 0 | non_member_990 |
case $g=1$, the solution is called the Frobenius-Stikelberger connection (see [@dubr96]) and leads to solutions of Chazy equation.
The space $\mathcal{U}_g$ is a rational variety, more precisely, there is a birational isomorphism $\varphi\colon \mathbb{C}^{3g} \to \mathcal{U}_g$. This fact was discovered by B. A. Dubrovin and S. P. Novikov in [@dn974]. In [@dn974], a fiber of the universal bundle is considered as a level surface of the integrals of motion of $g$th stationary flow of KdV system, that is, it is defined in $\mathbb{C}^{3g}$ by a system of $2g$ algebraic equations. The degree of the system grows with the growth of genus. In [@bel97], [@bel997] the coordinates in $\mathbb{C}^{3g}$ were introduced such that a fiber is defined by $2g$ equations of degree not greater than $3$.
The Dubrovin-Novikov coordinates and the coordinates from [@bel97], [@bel997] are the same for the universal space of genus $1$ curves. But already in the case of genus 2, these coordinates differ (see [@bl08]).
The integrals of motion of KdV systems are exactly the coefficients $\lambda_{2g+4},\ldots,\lambda_{4g+2}$ of hyperelliptic curve $V_\lambda$, in which the coefficients $\lambda_{4},\ldots,\lambda_{2g+2}$ are free parameters (see ). Choosing a point $z\in \mathbb{C}^{3g}$ such that the point $\varphi(z)\in \mathcal{B}_g$ is defined, one can calculate | 0 | non_member_990 |
---
author:
- Alain Dresse
- |
Marc Henneaux[Also at Centro de Estudios Científicos de Santiago, Casilla 16443, Santiago 9, Chile]{}\
Faculté des Sciences, Université Libre de Bruxelles,\
Campus Plaine C.P. 231, B-1050 Bruxelles (Belgium)
title: BRST Structure of Polynomial Poisson Algebras
---
\#1\#2[[\#1\#2]{}]{} =0 ${\global\advance\parenthesis by1\left(}
\def$[by-1)]{} $${\relax} \def$$ \#1 \#1[0=\#1sp0 by 30]{}
Introduction
============
Polynomial algebras with a Lie bracket fulfilling the derivation property $$[f g, h] = f [g,h] + [f,h]g$$ are called polynomial Poisson algebras and play an increasingly important role in various areas of theoretical physics [@Nak:; @Pri:; @Skl:; @Zam:; @FatZam:; @Oh:; @TarTakFad:; @BakMat:; @BhaRam:; @GraZhe:]. In terms of a set of independent generators $G_a$, $a = 1, \ldots, n$, the brackets are given by $$\label{basic_bracket}
[G_a, G_b] = C_{ab}(G)$$ where $C_{ab} = - C_{ba}$ are polynomials in the $G$’s[^1]. If the polynomials $C_{ab}(G)$ vanish when the $G$’s are set equal to zero, i.e. if they have no constant part, the polynomial algebra is said to be first class, in analogy with the terminology for constrained Hamiltonian systems (see e.g. [@HenTei:QuaGauSys]). An important class of first class Poisson algebras are symmetric algebras over a finite dimensional Lie algebra. In that case, the bracket (\[basic\_bracket\]) belongs | 0 | non_member_991 |
to the linear span of the $G_a$’s, i.e. the $C_{ab}(G)$ are homogeneous of degree one in the $G$’s, $[G_a, G_b] = C_{ab}{}^c G_c$. We shall call this situation the “Lie algebra case”, and refer to the non Lie algebra case as the “open algebra case” using again terminology from the theory of first class constrained systems [@HenTei:QuaGauSys][^2].
The purpose of this paper is to investigate the BRST structure of first class Poisson algebras. The BRST formalism has turned out recently to be the arena of a fruitful interplay between physics and mathematics (see e.g. [@HenTei:QuaGauSys] and references therein). A crucial ingredient of BRST theory is the recursive pattern of homological perturbation theory [@Sta:] which allows one to construct the BRST generator step by step. In most applications, however, this recursive construction collapses almost immediately, and, to our knowledge, no example has been given so far for which the full BRST machinery is required (apart from the field-theoretical membrane models [@Hen:PhyLet; @FujKub:]). We show in this paper that Poisson algebras—actually, already quadratic Poisson algebras—offer splendid examples illustrating the complexity of the BRST construction. While Lie algebras yield a BRST generator of rank 1 (see e.g. [@HenTei:QuaGauSys]), the BRST charge for quadratic | 0 | non_member_991 |
Poisson algebras can be of arbitrarily high rank. We also point out that BRST concepts provide intrinsic characterizations of Poisson algebras.
In the next section, we briefly review the BRST construction. We then discuss how it applies to Poisson algebras, even when the generators $G_a$ are not realized as phase space functions of some dynamical system. We analyze the BRST cohomology and introduce the concepts of covariant and minimal ranks, for which an elementary theorem is proven. Quadratic algebras are then shown to provide models with arbitrarily high rank. These contain “self-reproducing” algebras for which the bracket of $G_a$ with $G_b$ is proportional to the product $G_a G_b$. The first few terms in the BRST generator are also computed for more general algebras by means of a program written in REDUCE. The paper ends with some concluding remarks on the quantum case.
A Brief Survey of the BRST Formalism
====================================
We follow the presentation of [@HenTei:QuaGauSys], to which we refer for details and proofs. Given a set of independent functions $G_a(q,p)$ defined in some phase space $P$ with local coordinates $(q^i, p_i)$ and fulfilling the first class property $[G_a, G_b] \approx 0$, where $\approx$ denotes equality on the surface $G_a(q,p) | 0 | non_member_991 |
= 0$, one can introduce an odd generator $\Omega$ (“the BRST generator”) in an extended phase space containing further fermionic conjugate pairs $(\eta^a, {{\cal P}}_a)$ (the “ghost pairs”) which has the following properties : $$\begin{aligned}
[\Omega, \Omega] &=& 0 \label{nilpotency} \\
\Omega &=& G_a \eta^a + \mbox{``more''}.\end{aligned}$$ Here, “more” stands for terms containing at least one ghost momentum ${{\cal P}}_a$. We take the ghosts $\eta^a$ to be real and their momenta imaginary, with graded Poisson bracket $$[{{\cal P}}_a, \eta^b] = - \delta_a{}^b$$
The BRST derivation $s$ in the extended phase space is generated by $\Omega$, $$s \bullet = [ \bullet, \Omega]$$ and is a differential ($s^2 = 0$) because of (\[nilpotency\]). One also introduces a grading, the “ghost number” by setting $${\mbox{gh}}\eta^a = - {\mbox{gh}}{{\cal P}}_a = 1, \quad {\mbox{gh}}q^i = {\mbox{gh}}p_i = 0.$$ The ghost number of the BRST generator is equal to 1.
The BRST generator $\Omega$ is constructed recursively as follows. One sets $$\Omega = {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} + {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}} + \cdots$$ where ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ contains $k$ ghost momenta. One has ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} = G_a \eta^a$. The | 0 | non_member_991 |
nilpotency condition becomes, in terms of ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$, $$\label{delta-om=d}
\delta {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}} + {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 0$$ where ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}$ involves only the lower order ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s)} \\ \Omega \end{array}}}$ with $s \leq p$ and is defined by $$\label{d-p}
{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 1 / 2 \left[
\sum^p_{k=0} [{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} , {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p-k)} \\ \Omega \end{array}}}]_{\mbox{orig}} +
\sum^{p-1}_{k=0} [ {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ \Omega \end{array}}} ,
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p-k)} \\ \Omega \end{array}}}]_{{{\cal P}}, \eta}
\right].$$
Here, the bracket $[\; , \;]_{\mbox{orig}}$ refers to the Poisson bracket in the original phase space, which only acts on the $q^i$ and $p_i$, and not on the ghosts, whereas $[\; , \;]_{{{\cal P}}, \eta}$ refers to the Poisson bracket acting only on the ghost and ghost momenta arguments and not on the original phase space variables. The “Koszul” differential $\delta$ in (\[delta-om=d\]) is defined by $$\label{koszul}
\delta q^i = \delta p_i = 0, \quad \delta \eta^a = 0, \quad \delta
{{\cal P}}_a = - G_a$$ and is extended to | 0 | non_member_991 |
arbitrary functions on the extended phase space as a derivation. One easily verifies that $\delta^2 = 0$.
Given ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s)} \\ \Omega \end{array}}}$ with $s \leq p$, one solves (\[delta-om=d\]) for ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}}$. This can always be done because $\delta
{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = 0$, and because $\delta$ is acyclic in positive degree. One then goes on to ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+2)} \\ \Omega \end{array}}}$ etc... until one reaches the complete expression for $\Omega$. The last function ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ that can be non zero is ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (n-1)} \\ \Omega \end{array}}}$ where $n$ is the number of constraints. Indeed, the product $\eta^{a_1} \cdots \eta^{a_n} \eta^{a_{n+1}}$ of $n+1$ anticommuting ghost variables in ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (n)} \\ \Omega \end{array}}$ is zero. The function ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}$ is determined by (\[delta-om=d\]) up to a $\delta$-exact term. This amounts to making a canonical transformation in the extended phase space.
First Class Polynomial Poisson algebras
=======================================
The standard BRST construction recalled in the previous section assumes that the $G_a$’s are realized as functions on some phase space, and allows the $C^c{}_{ab}$ in $$[G_a, | 0 | non_member_991 |
G_b] = C^c{}_{ab} G_c$$ to be functions of $q^i$ and $p_i$. However, when the $C^c{}_{ab}$’s depend on the $q$’s and $p$’s only through the $G_a$’s themselves, as is the case when the $G_a$’s form a first class polynomial Poisson algebra, one can define the BRST generator directly in the algebra $\Bbb{C}\,({{\cal P}}_a) \otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta^a)$ of polynomials in the $G$’s, the $\eta$’s and the ${{\cal P}}$’s without any reference to the explicit realization of the $G$’s as phase space functions[^3]. That is, the BRST generator can be associated with the Poisson algebra itself.
The reason for which this can be done is that both the Koszul differential $\delta$ defined by (\[koszul\]) [*and*]{} the ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}$ in (\[d-p\]) involve only $G_a$ and not $q^i$ or $p_i$ individually. Thus, ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}$ can be taken to depend only on $G_a$. The BRST generator is defined accordingly in the algebra $\Bbb{C}\,({{\cal P}}_a)
\otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$.
One can give an explicit solution of (\[delta-om=d\]) in terms of the homotopy $\sigma$ defined on the generators by $$\sigma G_a = - {{\cal P}}_a, \quad \sigma {{\cal P}}_a = \sigma G_a = 0$$ and extended to | 0 | non_member_991 |
the algebra $\Bbb{C}\,({{\cal P}}_a) \otimes \Bbb{C}\,(G_a) \otimes
\Bbb{C}\,(\eta_a)$ as a derivation, $$\label{sigma}
\sigma = - {{\cal P}}_a \frac{\partial}{\partial G_a}.$$ One has $$\sigma \delta + \delta \sigma = N$$ where $N$ counts the degree in the $G$’s and the ${{\cal P}}$’s. Hence, if ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D_m \end{array}}$ is the term of degree $m$ in $(G, {{\cal P}})$ of ${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}$, a solution of (\[delta-om=d\]) is given by $$\label{om-p}
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}} = - \sum_m 1/m \left( \sigma {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}_m
\right)$$ since $\delta {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}}=0$ [@HenTei:QuaGauSys] and $m >
0$ (one has $m \geq p$ and for $p=0$, $m \geq 1$ because $[{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}},
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}]$ contains $G_a$ by the first class property).
It should be stressed that the partial derivations $\partial/\partial
G_a$ in(\[sigma\]) are well defined because the functions on which they act depend only on on $G_a$. For an arbitrary function of $q^i,
p_i$, $\partial F / \partial G_a$ would not be well defined even if the constraints $G_a$ are independent (i.e. irreducible) as here. One must specify | 0 | non_member_991 |
what is kept fixed. For example, if there is one constraint $p_1 = 0$ on the four-dimensional phase space $(q^1, p_1),
(q^2, p_2)$, then $\partial p_2 / \partial p_1 = 0$ if one keeps $q^1,
q^2$ and $p_2$ fixed, but $\partial p_2 / \partial p_1 = 1$ if one keeps $q^1, q^2$ and $p_2 - p_1$ fixed. Note that the subsequent developments require only that the $C^a{}_{b c}$ be functions of the $G_a$, but not that these functions be polynomials. We consider here the polynomial case for the sole sake of simplicity.
As mentioned earlier, the solution (\[om-p\]) of the equation (\[delta-om=d\]) is not unique. We call it the “covariant solution” because the homotopy $\sigma$ defined by (\[sigma\]) is invariant under linear redefinitions of the generators.
[**Example:**]{} for a Lie algebra $$[G_a, G_b] = C^c{}_{ab} G_c$$ the covariant BRST generator is given by $$\label{L-A-omega}
\Omega = G_a \eta^a - 1/2 {{\cal P}}_a C^a{}_{bc} \eta^c \eta^b.$$ Its nilpotency expresses the Jacobi identity for the structure constants $C^a{}_{bc}$. One has ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ \Omega \end{array}}} = 0$ for $p \geq
2$.
In general, the BRST generator $\Omega$ for a generic Poisson algebra contains higher order terms whose calculation may be | 0 | non_member_991 |
quite cumbersome. However, because the procedure is purely algorithmic, it can be performed by means of an algebraic program like REDUCE.
The cohomology of the Poisson algebra may be defined to be the cohomology of the BRST differential $s$ in the algebra $\Bbb{C}\,({{\cal P}}_a)
\otimes \Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$. Because $s$ contains $\delta$ as its piece of lowest antighost number (with ${\mbox{antigh}}({{\cal P}}_a) = 1, {\mbox{antigh}}(\mbox{anything else}) = 0$), and because $\delta$ provides a resolution of the zero-dimensional point $G_a =
0$, standard arguments show that the cohomology of $s$ is isomorphic to the cohomology of the differential $s'$ in $\Bbb{C}\,(\eta^a)$, $$\label{eq:sPrime}
s' \eta^a = 1/2 C^a{}_{bc} \eta^b \eta^c$$ where $C^a{}_{bc}$ is defined by $$C^a{}_{bc} = \left.\frac{\partial C_{bc}}{\partial G_a}\right|_{G = 0}$$
The $C^a{}_{bc}$ fulfill the Jacobi identity so that $s'^2 = 0$. Hence, they are the structure constants of a Lie algebra, which is called the Lie algebra underlying the given Poisson algebra.
Because of (\[eq:sPrime\]), the BRST cohomology of a Poisson algebra is isomorphic to the cohomology of the underlying Lie algebra. For a different and more thorough treatment of Poisson cohomology, see [@Hue:].
Rank
====
Again in analogy with the terminology used in the theory of constrained systems, we | 0 | non_member_991 |
shall call [*“covariant rank”*]{} of a first class polynomial Poisson algebra the degree in ${{\cal P}}_a$ of the covariant BRST generator. This concept is invariant under linear redefinitions of the generators because the covariant BRST generator is itself invariant if one transforms the ghosts and their momenta as $$\begin{aligned}
G_a &\rightarrow& \bar{G}_a = A_a{}^b G_b \\
{{\cal P}}_a &\rightarrow& \bar{{{\cal P}}}_a = A_a{}^b {{\cal P}}_b \\
\eta^a &\rightarrow& \bar{\eta}^a = (A^{-1})_b{}^a \eta^b\end{aligned}$$
We shall call [*“minimal rank”*]{} the degree in ${{\cal P}}_a$ of the solution of $[ \Omega, \Omega] = 0$ of lowest degree in ${{\cal P}}$ (i.e., one chooses at each stage ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}}$ in such a way that $\Omega$ has lowest possible degree in ${{\cal P}}$). It is easy to see that for a Lie algebra, the concepts of covariant and minimal ranks coincide. As we shall see on an explicit example below, they do not in the general case.
Now, for a Lie algebra, the rank is not particularly interesting in the sense that it does not tell much about the structure of the algebra : the rank of a Lie algebra is equal to zero if and only if the algebra | 0 | non_member_991 |
is abelian. It is equal to one otherwise. For non linear Poisson algebras, the rank is more useful. All values of the rank compatible with the trivial inequality $$rank \leq n-1$$ may occur. Thus, the rank of the BRST generator provides a non trivial characterization of Poisson algebras. Conversely, non linear Poisson algebras yield an interesting illustration of the full BRST machinery where higher order terms besides ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}$ are required in $\Omega$ to achieve nilpotency.
Upper bound on the rank
=======================
One can understand the fact that the rank of a Lie algebra is at most equal to one by introducing a degree in $\Bbb{C}\,({{\cal P}}_a) \otimes
\Bbb{C}\,(G_a) \otimes \Bbb{C}\,(\eta_a)$ different from the ghost degree as follows.
Assume that one can assign a “degree” $n_a \geq 1$ to the generators $G_a$ in such a way that the bracket decreases the degree by at least one, $$\label{deg-g=n}
\deg G_a = n_a, \; \deg([G_a, G_b]) \leq n_a + n_b - 1.$$ Then, one can bound the covariant and minimal ranks of the algebra by $\sum_a (n_a - 1) + 1$, $$r \leq \sum_a (n_a - 1) + 1$$
In the case of a Lie algebra, one | 0 | non_member_991 |
can take $n_a = 1$ for all the generators since $\deg([G_a, G_b])$ is then equal to one and fulfills (\[deg-g=n\]). The theorem then states that the rank is bounded by one, in agreement with (\[L-A-omega\]).
Assign the following degrees to $\eta^a$ and ${{\cal P}}_a$, $$\deg \eta^a = - n_a + 1, \; \deg {{\cal P}}_a = n_a - 1$$ If $\delta A = B$ and $\deg B = b$, then $\deg A = b - 1$ since $\delta$ increases the degree by one. Now ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} = G_a
\eta^a$ is of degree one. It follows that $[{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}},
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}] = [{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}},{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}]_{\mbox{orig}}$ is of degree $\leq 1$ and hence, by (\[delta-om=d\]) and (\[d-p\]), $\deg {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}
\leq 0$. More generally, one has $\deg {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \leq -k + 1$. Indeed, if this relation is true up to order $k-1$, then it is also true at order $k$ because in $$\delta {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \sim [{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (r)} \\ | 0 | non_member_991 |
\Omega \end{array}}},{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s)} \\ \Omega \end{array}}}]_{\mbox{orig}} + [{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (r')} \\ \Omega \end{array}}},
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (s')} \\ \Omega \end{array}}}]_{{{\cal P}}, \eta}$$ ($r+s = k-1, \; r' + s' = k$), the right hand side is of degree $\leq
-k+2$. Thus $\deg {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \leq -k + 2 - 1 = -k + 1$.
But the element with most negative degree in the algebra is given by the product of all the $\eta$’s, which has degree $-\sum_a(n_a - 1)$. Accordingly, ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ is zero whenever $-k+1 >= - \sum_a(n_a - 1)$, which implies $r \leq \sum_a(n_a-1)+1$ as stated in the theorem.
[**Remarks:**]{}
1. One can improve greatly the bound by observing that the $\eta$’s do not come alone in ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$. There are also $k$ momenta ${{\cal P}}_a$ which carry positive degree. This remark will, however, not be pursued further here.
2. One can actually assign degrees smaller than one to the generators $G_a$. For instance, in the case of an Abelian Lie algebra, one may take $deg G_a = 1/2, \; \deg \eta^a = 1/2, \deg {{\cal P}}_a = -
1/2$. | 0 | non_member_991 |
Because the degree of a ghost number one object is necessarily greater than or equal to $1/2$, the condition $\deg {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \leq -k+1$ (if ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} \neq 0$) implies ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} = 0$ for $k > 0$.
Self-reproducing algebras
=========================
While Lie algebras are characterized by the existence of a degreee that is decreased by the bracket, one may easily construct examples of Poisson algebras for which such a degree does not exist. The simplest ones are quadratic algebras for which $[G_a, G_b]$ is proportional to $G_a, G_b$ $$[G_a, G_b] = M_{ab} G_a G_b \quad\quad\mbox{no summation on $a,b$}$$ with $M_{ab} = -M_{ba}$. The Jacobi identity is fulfilled for arbitrary $M$’s. Since $\deg(G_a G_b) = n_a + n_b$, the inequality (\[deg-g=n\]) is violated for any choice of $n_a$. Because $[G_a, G_b]$ is proportional to $G_a G_b$, we shall call these algebras “self-reproducing algebras”.
The most general self-reproducing algebra with three generators is given by $$\begin{aligned}
[G_1, G_2] &=& \alpha\, G_1 G_2 \\{}
[G_2, G_3] &=& \beta \, G_2 G_3 \\{}
[G_3, G_1] &=& \gamma \, G_1 G_3.\end{aligned}$$ This Poisson algebra can be realized on a | 0 | non_member_991 |
six-dimensional phase space by setting $$G_1 = \exp(p_2 + \alpha q_3), G_2 = \exp(p_3 + \beta q_1), G_3 =
\exp(p_1 + \gamma q_2).$$ The covariant BRST charge for this model is equal to $$\begin{aligned}
\Omega &=&\eta^1 \, G_1 + \eta^2 \, G_2 + \eta^3 \, G_3 + \\
\nonumber
& & 1/2 \,
(\alpha \,\eta^{2}\,\eta^{1}\,{{\cal P}}_{2}\,G_{1}
-\alpha \,\eta^{2}\,\eta^{1}\,{{\cal P}}_{1}\,G_{2}
-\beta \,\eta^{3}\,\eta^{2}\,{{\cal P}}_{3}\,G_{2} \\ \nonumber
& &\mbox{~~~~~~}
-\beta \,\eta^{3}\,\eta^{2}\,{{\cal P}}_{2}\,G_{3}
+\gamma \,\eta^{3}\,\eta^{1}\,{{\cal P}}_{3}\,G_{1}
+\gamma \,\eta^{3}\,\eta^{1}\,{{\cal P}}_{1}\,G_{3}) + \\ \nonumber
& & 1/12 \, (
( - \alpha \,\beta +2\,\alpha \,\gamma -\beta \,\gamma )
\,\eta^{3}\,\eta^{2}\,\eta^{1}\,{{\cal P}}_{3}\,{{\cal P}}_{2}\,G_{1} + \\
\nonumber & &\mbox{~~~~~~~~~}
( -2\,\alpha \,\beta +\alpha \,\gamma +\beta \,\gamma )
\,\eta^{3}\,\eta^{2}\,\eta^{1}\,{{\cal P}}_{3}\,{{\cal P}}_{1}\,G_{2} +\\
\nonumber & &\mbox{~~~~~~~~~}
( -\alpha \,\beta -\alpha \,\gamma +2\,\beta \,\gamma )
\,\eta^{3}\,\eta^{2}\,\eta^{1}\,{{\cal P}}_{2}\,{{\cal P}}_{1}\,G_{3}
)\end{aligned}$$ and is of rank 2 (the maximum possible rank) unless $\alpha = \beta =
\gamma$, or $\alpha = \beta = 0$, $\gamma \neq 0$, in which case it is of rank 1.
Examples
========
We now give the BRST charge (or the first terms of the BRST charge) for some particular Poisson algebras. The examples have been treated using REDUCE, using the treatment of summation over dummy indices developed in [@Dre:CanExp; @Dre:Imacs]. Details | 0 | non_member_991 |
of the implementation of the BRST algorithm can be found in [@BurCapDre:]. All dummy variables are noted as $d_i$ where $i$ is an integer. Unless stated otherwise, there is an implicit summation on all dummy variables. For the examples in which the Jacobi identity is not trivially satisfied, the expressions have been normalized so that no combinations of terms in a polynomial belongs to the polynomial ideal generated by the left hand side of the Jacobi identity. In particular, polynomials in this ideal are represented by identically null expressions.
Self-Reproducing Algebras
-------------------------
As we have just defined, the basic Poisson brackets for the generators $G_d$ of the [*self-reproducing algebra*]{} are given by $$[G_{d_1}, G_{d_2}] = M_{d_1 d_2} G_{d_1} G_{d_2}$$ without summation over the dummy variables $d_1$ and $d_2$. The matrix $M$ is antisymmetric, but otherwise arbitrary.
The seven first orders of the covariant BRST charge are given by $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}=
\[G_{d_{1}}\,\eta^{d_{1}}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}=
\[\frac{G_{d_{1}}\,M_{d_{1}d_{2}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,{{\cal P}}_{d_{2}}}{
2}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (2)} \\ \Omega \end{array}}}=
\[\frac{-
\(G_{d_{1}}\,M_{d_{1}d_{2}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,\eta^{d_{3}}\,{{\cal P}}_{d_{2}}\,
{{\cal P}}_{d_{3}}\,
\(M_{d_{1}d_{3}}
+M_{d_{2}d_{3}}
\)
\)
}{
12}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (3)} | 0 | non_member_991 |
\\ \Omega \end{array}}}=
\[\frac{-
\(G_{d_{1}}\,M_{d_{1}d_{2}}\,M_{d_{1}d_{4}}\,
M_{d_{2}d_{3}}\,\eta^{d_{1}}\,\eta^{d_{2}}\,
\eta^{d_{3}}\,\eta^{d_{4}}\,{{\cal P}}_{d_{2}}\,
{{\cal P}}_{d_{3}}\,{{\cal P}}_{d_{4}}
\)
}{
24}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (4)} \\ \Omega \end{array}}}=
\[\(G_{d_{1}}\,\eta^{d_{1}}\,\eta^{d_{2}}\,
\eta^{d_{3}}\,\eta^{d_{4}}\,\eta^{d_{5}}\,
{{\cal P}}_{d_{2}}\,{{\cal P}}_{d_{3}}\,{{\cal P}}_{d_{4}}\,
{{\cal P}}_{d_{5}}\,\nl
\off{3499956}
\(-
\(M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{1}d_{4}}\,
M_{d_{1}d_{5}}
\)
+4\,M_{d_{1}d_{2}}\,M_{d_{1}d_{4}}\,M_{d_{1}d_{5}}\,
M_{d_{2}d_{3}}\nl
\off{3827636}
+2\,M_{d_{1}d_{2}}\,M_{d_{1}d_{4}}\,M_{d_{2}d_{3}}\,
M_{d_{4}d_{5}}
+M_{d_{1}d_{2}}\,M_{d_{1}d_{5}}\,M_{d_{2}d_{3}}\nl
\off{3827636}
\,M_{d_{2}d_{4}}
-M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{2}d_{5}}
-M_{d_{1}d_{4}}\,M_{d_{1}d_{5}}\,\nl
\off{3827636}
M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}
-2\,M_{d_{1}d_{4}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{4}d_{5}}
+M_{d_{1}d_{5}}\,\nl
\off{3827636}
M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,M_{d_{2}d_{5}}
-M_{d_{1}d_{5}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{4}d_{5}}
\)
\)
/720
\]
\Nl}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (5)} \\ \Omega \end{array}}}=
\[\(G_{d_{2}}\,M_{d_{1}d_{2}}\,M_{d_{3}d_{4}}\,
\eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
\eta^{d_{4}}\,\eta^{d_{5}}\,\eta^{d_{6}}\,
{{\cal P}}_{d_{1}}\,{{\cal P}}_{d_{3}}\,{{\cal P}}_{d_{4}}\,
{{\cal P}}_{d_{5}}\,{{\cal P}}_{d_{6}}\nl
\off{3499956}
\,
\(-
\(M_{d_{2}d_{3}}\,M_{d_{2}d_{5}}\,M_{d_{2}d_{6}}
\)
+2\,M_{d_{2}d_{3}}\,M_{d_{2}d_{5}}\,M_{d_{5}d_{6}}
+M_{d_{2}d_{3}}\nl
\off{4100703}
\,M_{d_{2}d_{6}}\,M_{d_{3}d_{5}}
-M_{d_{2}d_{3}}\,M_{d_{3}d_{5}}\,M_{d_{3}d_{6}}
-M_{d_{2}d_{5}}\nl
\off{4100703}
\,M_{d_{2}d_{6}}\,M_{d_{3}d_{5}}
-2\,M_{d_{2}d_{5}}\,M_{d_{3}d_{5}}\,M_{d_{5}d_{6}}\nl
\off{4100703}
+M_{d_{2}d_{6}}\,M_{d_{3}d_{5}}\,M_{d_{3}d_{6}}
-M_{d_{2}d_{6}}\,M_{d_{3}d_{5}}\,M_{d_{5}d_{6}}
\)
\)
/1440
\]
\Nl}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (6)} \\ \Omega \end{array}}}=
\[\(G_{d_{3}}\,\eta^{d_{1}}\,\eta^{d_{2}}\,
\eta^{d_{3}}\,\eta^{d_{4}}\,\eta^{d_{5}}\,
\eta^{d_{6}}\,\eta^{d_{7}}\,{{\cal P}}_{d_{1}}\,
{{\cal P}}_{d_{2}}\,{{\cal P}}_{d_{4}}\,{{\cal P}}_{d_{5}}\,
{{\cal P}}_{d_{6}}\,{{\cal P}}_{d_{7}}\,\nl
\off{3499956}
\(-
\(M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{1}d_{6}}\,
M_{d_{2}d_{4}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
\)
-M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\nl
\off{3827636}
\,M_{d_{2}d_{4}}\,M_{d_{2}d_{6}}\,M_{d_{4}d_{5}}\,
M_{d_{6}d_{7}}
+2\,M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{2}d_{4}}\,
\nl
\off{3827636}
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
+M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{2}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{2}}\,M_{d_{1}d_{3}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-2\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{2}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-13\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,
M_{d_{2}d_{4}}\,M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,
M_{d_{6}d_{7}}\nl
\off{3827636}
+4\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-4\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+4\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-2\,M_{d_{1}d_{2}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
+M_{d_{1}d_{3}}\,M_{d_{1}d_{4}}\,M_{d_{1}d_{6}}\,
M_{d_{2}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{3}}\nl
\off{3827636}
\,M_{d_{1}d_{4}}\,M_{d_{2}d_{4}}\,M_{d_{4}d_{5}}\,
M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
+5\,M_{d_{1}d_{3}}\,M_{d_{1}d_{4}}\,\nl
\off{3827636}
M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,
M_{d_{4}d_{6}}
+M_{d_{1}d_{3}}\,M_{d_{1}d_{6}}\,M_{d_{2}d_{3}}\,
M_{d_{2}d_{4}}\nl
\off{3827636}
\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{3}}\,M_{d_{1}d_{6}}\,M_{d_{2}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
\,M_{d_{6}d_{7}}
-5\,M_{d_{1}d_{3}}\,M_{d_{1}d_{6}}\,M_{d_{2}d_{7}}\,
M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
+M_{d_{1}d_{3}}\,M_{d_{1}d_{7}}\,M_{d_{2}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
-M_{d_{1}d_{3}}\,M_{d_{1}d_{7}}\nl
\off{3827636}
\,M_{d_{2}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,
M_{d_{6}d_{7}}
-M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{2}d_{6}}\,\nl
\off{3827636}
M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
-8\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
| 0 | non_member_991 |
+2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{4}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{2}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{3}}\nl
\off{3827636}
\,M_{d_{2}d_{3}}\,M_{d_{2}d_{7}}\,M_{d_{4}d_{5}}\,
M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
+2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\nl
\off{3827636}
\,M_{d_{2}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,
M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
\nl
\off{3827636}
M_{d_{3}d_{5}}\,M_{d_{3}d_{6}}\,M_{d_{3}d_{7}}
+12\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,
M_{d_{3}d_{4}}\,M_{d_{3}d_{6}}\nl
\off{3827636}
\,M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}
-6\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\nl
\off{3827636}
\,M_{d_{6}d_{7}}
-5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
+5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{4}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{6}}\,
M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
+13\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,
M_{d_{6}d_{7}}\nl
\off{3827636}
-5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+5\,M_{d_{1}d_{3}}\,M_{d_{2}d_{3}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{4}}\,M_{d_{3}d_{4}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
-8\,M_{d_{1}d_{3}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
+2\,M_{d_{1}d_{3}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}\nl
\off{3827636}
-M_{d_{1}d_{3}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}\nl
\off{3827636}
+M_{d_{1}d_{3}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{4}}\,\nl
\off{3827636}
M_{d_{2}d_{4}}\,M_{d_{3}d_{4}}\,M_{d_{4}d_{5}}\,
M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
+M_{d_{1}d_{4}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{4}}\nl
\off{3827636}
\,M_{d_{3}d_{6}}\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
+5\,M_{d_{1}d_{6}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{4}}\,
M_{d_{3}d_{6}}\nl
\off{3827636}
\,M_{d_{4}d_{5}}\,M_{d_{6}d_{7}}
-8\,M_{d_{1}d_{6}}\,M_{d_{2}d_{6}}\,M_{d_{3}d_{6}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
\,M_{d_{6}d_{7}}
-2\,M_{d_{1}d_{7}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{6}}\,
M_{d_{3}d_{7}}\,M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\nl
\off{3827636}
+6\,M_{d_{1}d_{7}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{4}d_{7}}
-6\,\nl
\off{3827636}
M_{d_{1}d_{7}}\,M_{d_{2}d_{7}}\,M_{d_{3}d_{7}}\,
M_{d_{4}d_{5}}\,M_{d_{4}d_{6}}\,M_{d_{6}d_{7}}
\)
\)
/60480
\]
\Nl}$$
These expressions are not particularly illuminating but are of interest because they generically do not vanish and hence, define higher order BRST charges. This can be seen by means of the following example, in which only the brackets of the first generator with the other ones are non vanishing, and taken equal to $$\begin{aligned}
\label{eq:maxquad}
[G_1, G_{\alpha}] &=& G_1 G_{\alpha} = - [ G_{\alpha}, G_1]
\quad (\alpha = 2,3,\ldots,n),\\{}
[G_{\alpha}, G_{\beta}] &=& 0.\end{aligned}$$ For this particular self-reproducing algebra, all orders of the covariant BRST charge can be explicitly computed. One finds $$\begin{aligned}
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} &=& \eta^a G_a \\
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}} &=& \alpha_k ({\renewcommand{\arraystretch}{.7} | 0 | non_member_991 |
\begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}} + (-)^{k+1} {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_2 \end{array}})\end{aligned}$$ where $$\begin{aligned}
{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}} &=& G_1 \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^1
{{\cal P}}_{\alpha_1} \cdots {{\cal P}}_{\alpha_k} \\
{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_2 \end{array}} &=& G_{\alpha_k} \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^1
{{\cal P}}_1 {{\cal P}}_{\alpha_1} \cdots {{\cal P}}_{\alpha_{k-1}}\end{aligned}$$ and $$\begin{aligned}
\alpha_1 &=& -1/2, \; \alpha_2 = -1/12, \; \alpha_3 = 0 \\
\alpha_k &=& -\frac{1}{k+1} \sum_{l=1}^{k-3} \alpha_{l+1} \alpha_{k-l-1}
\quad \mbox{for $k > 3$}\label{eq:AlpRec}.\end{aligned}$$
This can be seen as the only non zero brackets involved in the construction of the BRST charge are $$\begin{aligned}
[{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}}, {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}]_{\mbox{orig}} &=& (-)^{k+1}{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} \\{}
[{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ T_1 \end{array}}, {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (l)} \\ T_2 \end{array}}]_{\eta{{\cal P}}} &=&
(-)^{l(k+1)}{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+l-1)} \\ S \end{array}},\end{aligned}$$ where $${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} = G_1 G_\alpha \eta^{\alpha_1} \cdots \eta^{\alpha_k} \eta^\alpha
\eta^1
{{\cal P}}_{\alpha_1} \cdots {{\cal P}}{\alpha_k}.$$ We further have $$\label{eq:sigmaS}
\sigma {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ S \end{array}} = (-)^{k+1} {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ T_2 \end{array}} - {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k+1)} \\ T_1 \end{array}}$$ Given these relations, | 0 | non_member_991 |
it is straightforward to verify (39). First, one easily checks that (39) is correct for $k=1$ with $\alpha_1$ equal to 1/2. Let us then assume that (39) is true for $k = 0, 1 ...$ up to $p$. One then obtains $${\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ D \end{array}} = \beta_p {\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p)} \\ S \end{array}}$$ with $\beta_p$ given by $$\beta_p = (-)^{p+1} \alpha_p - \sum_{k=0}^{p-1} (-)^{p(k+1)} \alpha_{k+1}
\alpha_{p-k} = - \sum_{k=1}^{p-2} (-)^{p(k+1)} \alpha_{k+1} \alpha_{p-k}$$ from which one gets, using (\[eq:sigmaS\]), that ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (p+1)} \\ \Omega \end{array}}}$ is indeed given by (39) with $\alpha_{p+1}$ equal to $$\alpha_{p+1} = \frac{\beta_p}{p+2}$$ Observe now that $\alpha_k = 0$ for $k$ odd, $k \neq 1$. This can again be shown by recurrence. First note that $\alpha_3$ = 0. Now let $p$ be even, $p > 3$. Suppose $\alpha_k = 0$ for $k$ odd, $1 < k < p$. All terms in the relation defining $\beta_{p}$ are proportional to an $\alpha_m$ with $m$ odd, $1 < m < p$, since $k+1$ and $p-k$ have opposite parities. Therefore, $\beta_p = 0 = \alpha_{p+1}$ and thus $\alpha_k
= 0$ for $k$ odd, $k > 1$. Accordingly only $\alpha_k $ with $k$ even can be | 0 | non_member_991 |
different from zero. The expression for $\alpha_k$ reduces then to (43) since $k+1$ must be even in (49).
Although $\alpha_k = 0$ for $k$ odd, $k > 1$, one easily sees that $\alpha_k < 0$ for $k$ even. This is true for $k =
2$ as $\alpha_2 = -1/12$. Let $p$ be even, and suppose $\alpha_m < 0$ for $1 < m < p$, $m$ even. Then, all terms in the sum in the recurrence relation (\[eq:AlpRec\]) are strictly positive, so that $\alpha_p < 0$. Since $\alpha_p \neq 0$, the quadratic algebra (\[eq:maxquad\]) provides examples of systems with arbitrarily high covariant rank.
Note also that the minimal rank is equal to one: indeed, the non covariant BRST charge given by $$\tilde{\Omega}={{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}} + {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}} + ({{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_1 - {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_2)/2 =
\eta^a G_a - G_a \eta^a \eta^1 {{\cal P}}_1$$ is nilpotent and $$\delta({{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_1 - {{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ T \end{array}}}_2) = 0$$ so $\tilde{\Omega}$ is indeed a valid BRST charge. This shows that the minimal and covariant ranks are in general different.
| 0 | non_member_991 |
Finally, it is easy to modify slightly the basic brackets so as to induce non zero covariant ${{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (k)} \\ \Omega \end{array}}}$ with $k$ odd. One simply replaces (\[eq:maxquad\]) by $$\begin{aligned}
[G_{n-1}, G_n] &=& G_{n-1} G_n = - [G_n, G_{n-1}] \\{}
[G_1, G_n] &=& - G_1 G_n, \\{}
[G_1, G_{\alpha}] &=& G_1 G_{\alpha} \quad (\alpha \neq n).\end{aligned}$$
Purely Quadratic Algebras
-------------------------
A generalization of the above is the pure quadratic algebra. The basic Poisson brackets are then given by $$[G_{d_1}, G_{d_2}] = D_{d_1 d_2}^{d_3 d_4} G_{d_3} G_{d_4}$$ where $D_{d_1 d_2}^{d_3 d_4}$ is antisymmetric in $d_1, d_2$ and symmetric in $d_3, d_4$. The Jacobi identity implies that $$D_{d_4 d_1}^{d_5 d_6} D_{d_2 d_3}^{d_4 d_7} + \mbox{symm}(d_5, d_6,
d_7) + \mbox{cyclic}(d_1,d_2,d_3) = 0$$
The first orders of the covariant BRST charge are given by $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (0)} \\ \Omega \end{array}}}=
\[G_{d_{1}}\,\eta^{d_{1}}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (1)} \\ \Omega \end{array}}}=
\[\frac{D_{d_{1}d_{2}}^{
d_{4}d_{3}}\,G_{d_{4}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,{{\cal P}}_{d_{3}}}{
2}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (2)} \\ \Omega \end{array}}}=
\[\frac{D_{d_{2}d_{3}}^{
d_{6}d_{5}}\,D_{d_{6}d_{1}}^{
d_{7}d_{4}}\,G_{d_{7}}\,
\eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
{{\cal P}}_{d_{4}}\,{{\cal P}}_{d_{5}}}{
6}
\]
\cr}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (3)} \\ \Omega \end{array}}}=
\[\frac{-
\(D_{d_{2}d_{3}}^{
d_{8}d_{7}}\,D_{d_{4}d_{1}}^{
d_{9}d_{5}}\,D_{d_{8}d_{9}}^{
d_{10}d_{6}}\,G_{d_{10}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,\eta^{d_{3}}\,\eta^{d_{4}}\,
{{\cal P}}_{d_{5}}\,{{\cal P}}_{d_{6}}\,{{\cal P}}_{d_{7}}
\)
}{
24}
\]
\cr}$$ | 0 | non_member_991 |
$$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (4)} \\ \Omega \end{array}}}=
\[\(\eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
\eta^{d_{4}}\,\eta^{d_{5}}\,{{\cal P}}_{d_{6}}\,
{{\cal P}}_{d_{7}}\,{{\cal P}}_{d_{8}}\,{{\cal P}}_{d_{9}}\,
\nl
\off{3499956}
\(3\,D_{d_{1}d_{2}}^{
d_{13}d_{6}}\,D_{d_{4}d_{5}}^{
d_{12}d_{9}}\,
D_{d_{10}d_{3}}^{
d_{11}d_{8}}\,D_{d_{12}d_{13}}^{
d_{10}d_{7}}\,G_{d_{11}}
+4\,D_{d_{3}d_{4}}^{
d_{10}d_{9}}\,D_{d_{5}d_{2}}^{
d_{13}d_{6}}\,
D_{d_{10}d_{11}}^{
d_{12}d_{8}}\nl
\off{3827636}
\,D_{d_{13}d_{1}}^{
d_{11}d_{7}}\,G_{d_{12}}
-4\,D_{d_{4}d_{5}}^{
d_{12}d_{9}}\,D_{d_{10}d_{3}}^{
d_{11}d_{8}}\,
D_{d_{12}d_{1}}^{
d_{13}d_{7}}\,D_{d_{13}d_{2}}^{
d_{10}d_{6}}\,G_{d_{11}}
\)
\)
/360
\]
\Nl}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (5)} \\ \Omega \end{array}}}=
\[\(D_{d_{12}d_{13}}^{
d_{14}d_{10}}\,G_{d_{14}}\,\eta^{d_{1}}\,
\eta^{d_{2}}\,\eta^{d_{3}}\,\eta^{d_{4}}\,
\eta^{d_{5}}\,\eta^{d_{6}}\,{{\cal P}}_{d_{7}}\,
{{\cal P}}_{d_{8}}\,{{\cal P}}_{d_{9}}\,{{\cal P}}_{d_{10}}\,
{{\cal P}}_{d_{11}}\nl
\off{3499956}
\,
\(-
\(2\,D_{d_{1}d_{2}}^{
d_{16}d_{7}}\,D_{d_{5}d_{6}}^{
d_{15}d_{11}}\,
D_{d_{15}d_{3}}^{
d_{12}d_{8}}\,D_{d_{16}d_{4}}^{
d_{13}d_{9}}
\)
+3\,D_{d_{1}d_{3}}^{
d_{16}d_{8}}\,D_{d_{4}d_{5}}^{
d_{12}d_{11}}\,
\nl
\off{4100703}
D_{d_{6}d_{2}}^{
d_{15}d_{7}}\,D_{d_{15}d_{16}}^{
d_{13}d_{9}}
-4\,D_{d_{4}d_{5}}^{
d_{12}d_{11}}\,D_{d_{6}d_{1}}^{
d_{15}d_{7}}\,
D_{d_{15}d_{3}}^{
d_{16}d_{9}}\,D_{d_{16}d_{2}}^{
d_{13}d_{8}}
\)
\)
/720
\]
\Nl}$$ $$\displaylines{\qdd
{{\renewcommand{\arraystretch}{.7} \begin{array}[b]{@{}c@{}} {\scriptscriptstyle (6)} \\ \Omega \end{array}}}=
\[\(\eta^{d_{1}}\,\eta^{d_{2}}\,\eta^{d_{3}}\,
\eta^{d_{4}}\,\eta^{d_{5}}\,\eta^{d_{6}}\,
\eta^{d_{7}}\,{{\cal P}}_{d_{8}}\,{{\cal P}}_{d_{9}}\,
{{\cal P}}_{d_{10}}\,{{\cal P}}_{d_{11}}\,{{\cal P}}_{d_{12}}\,
{{\cal P}}_{d_{13}}\,\nl
\off{3499956}
\(6\,D_{d_{1}d_{2}}^{
d_{17}d_{8}}\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,
D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{17}}^{
d_{18}d_{11}}\,D_{d_{18}d_{4}}^{
d_{19}d_{10}}\,D_{d_{19}d_{3}}^{
d_{14}d_{9}}\,G_{d_{15}}\nl
\off{3827636}
-9\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{3}d_{4}}^{
d_{19}d_{9}}\,
D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{17}}^{
d_{14}d_{10}}\,D_{d_{18}d_{19}}^{
d_{17}d_{11}}\,G_{d_{15}}\nl
\off{3827636}
+12\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{3}d_{4}}^{
d_{19}d_{9}}\,
D_{d_{6}d_{7}}^{
d_{17}d_{13}}\,D_{d_{14}d_{15}}^{
d_{16}d_{12}}\,D_{d_{17}d_{18}}^{
d_{15}d_{11}}\,D_{d_{19}d_{5}}^{
d_{14}d_{10}}\,G_{d_{16}}\nl
\off{3827636}
-10\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,
D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{3}}^{
d_{17}d_{9}}\,D_{d_{17}d_{19}}^{
d_{14}d_{11}}\,D_{d_{18}d_{4}}^{
d_{19}d_{10}}\,G_{d_{15}}\nl
\off{3827636}
+24\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,
D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{17}}^{
d_{14}d_{11}}\,D_{d_{18}d_{4}}^{
d_{19}d_{10}}\,D_{d_{19}d_{3}}^{
d_{17}d_{9}}\,G_{d_{15}}\nl
\off{3827636}
-4\,D_{d_{1}d_{2}}^{
d_{18}d_{8}}\,D_{d_{6}d_{7}}^{
d_{17}d_{13}}\,
D_{d_{14}d_{15}}^{
d_{16}d_{12}}\,D_{d_{17}d_{4}}^{
d_{15}d_{9}}\,D_{d_{18}d_{5}}^{
d_{19}d_{11}}\,D_{d_{19}d_{3}}^{
d_{14}d_{10}}\,G_{d_{16}}\nl
\off{3827636}
+4\,D_{d_{1}d_{2}}^{
d_{19}d_{8}}\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,
D_{d_{14}d_{5}}^{
d_{15}d_{12}}\,D_{d_{16}d_{17}}^{
d_{18}d_{11}}\,D_{d_{18}d_{3}}^{
d_{14}d_{9}}\,D_{d_{19}d_{4}}^{
d_{17}d_{10}}\nl
\off{3827636}
\,G_{d_{15}}
-6\,D_{d_{1}d_{3}}^{
d_{18}d_{9}}\,D_{d_{5}d_{6}}^{
d_{14}d_{13}}\,
D_{d_{7}d_{2}}^{
d_{17}d_{8}}\,D_{d_{14}d_{15}}^{
d_{16}d_{12}}\,D_{d_{17}d_{18}}^{
d_{19}d_{11}}\nl
\off{3827636}
\,D_{d_{19}d_{4}}^{
d_{15}d_{10}}\,G_{d_{16}}
-4\,D_{d_{1}d_{3}}^{
d_{19}d_{9}}\,D_{d_{5}d_{6}}^{
d_{14}d_{13}}\,
D_{d_{7}d_{2}}^{
d_{17}d_{8}}\,D_{d_{14}d_{15}}^{
d_{16}d_{12}}\,\nl
\off{3827636}
D_{d_{17}d_{18}}^{
d_{15}d_{11}}\,D_{d_{19}d_{4}}^{
d_{18}d_{10}}\,G_{d_{16}}
-16\,D_{d_{5}d_{6}}^{
d_{14}d_{13}}\,D_{d_{7}d_{1}}^{
d_{17}d_{8}}\,
D_{d_{14}d_{15}}^{
d_{16}d_{12}}\nl
\off{3827636}
| 0 | non_member_991 |
\,D_{d_{17}d_{4}}^{
d_{18}d_{11}}\,D_{d_{18}d_{3}}^{
d_{19}d_{10}}\,
D_{d_{19}d_{2}}^{
d_{15}d_{9}}\,G_{d_{16}}
+16\,D_{d_{6}d_{7}}^{
d_{16}d_{13}}\,D_{d_{14}d_{5}}^{
d_{15}d_{12}}\nl
\off{3827636}
\,D_{d_{16}d_{4}}^{
d_{17}d_{11}}\,D_{d_{17}d_{3}}^{
d_{19}d_{10}}\,
D_{d_{18}d_{1}}^{
d_{14}d_{8}}\,D_{d_{19}d_{2}}^{
d_{18}d_{9}}\,G_{d_{15}}
\)
\)
/15120
\]
\Nl}$$
Again, these expressions are not particularly illuminating. The point emphasized here is that the calculation of the BRST charge is purely algorithmic and follows a general, well-established pattern.
Since homogeneous quadratic algebras contain the self-reproducing algebras as special case, they are generically of maximal covariant rank. More on this in [@ADresse3].
L-T algebras
------------
We now consider adding a linear term to the quadratic algebra above. The basic Poisson brackets for the generators $G_d$ are given by $$[G_{d_1}, G_{d_2}] = C_{d_1 d_2}^{d_3} \, G_{d_3} +
D_{d_1 d_2}^{d_3 d_4} G_{d_3} G_{d_4}$$ where $ C_{d_1 d_2}^{d_3} $ and $D_{d_1 d_2}^{d_3 d_4}$ are antisymmetric in $d_1, d_2$, and $D_{d_1 d_2}^{d_3 d_4}$ is symmetric in $d_3, d_4$. A particular instance of such an algebra is given by Zamolodchikov algebras [@Zam:]. We will start with a specific example, and consider general quadratically nonlinear Poisson algebras next.
The generators in the example are assumed to split into $L_{a}$ and $T_b$, $a = 1, \ldots n_1$, $b = n_1 + 1, \ldots, n$, with the brackets $$\begin{aligned}
[L_{a_1}, L_{a_2}] &=& \tilde{C}_{a_1 a_2}^{a_3} L_{a_3} \nonumber \\{}
[L_{a_1}, T_{b_1}] | 0 | non_member_991 |
Subsets and Splits