# THE DIFFEOMORPHISM FIELD

Delalcan Kilic

(2018)

Physics Professor Vincent G. J. Rodgers Vincent G. J. Rodgers Yannick Meurice
Wayne N. Polyzou Hao Fang Yasar Onel May 2018 dedication thesisAck
thesisAbstract publicAbstract

##  Chapter 1 introduction

The diffeomorphism field is introduced to the physics literature in
[rairodgers90]. The authors obtained geometric actions by integrating the
Kirillov form on the coadjoint orbits of Kac-Moody (KM) and Virasoro algebras,
two infinite-dimensional and centrally extended Lie algebras111A similar
analysis was also done in [alekseev].. These algebras are reviewed in Sections
2.3 and 2.4.

The parts of the geometric actions coming from the centers of KM and Virasoro
algebras are, respectively, Wess-Zumino-Witten (WZW) action [WZNW] and
Polyakov’s two dimensional quantum gravity (P2DG) action in lightcone gauge
(LCG) [polyakov2Dgravity], describing bosonization of the gauge and
gravitational coupling of the chiral fermions in 2D. WZW and P2DG theories are
reviewed in Sections 3.1 and 3.2, and the construction of the geometric
actions on the coadjoint orbits of KM and Virasoro algebras in Sections 3.3
and 3.4.

The remaining terms in the geometric actions suggest the following. The non-
central part of the KM coadjoint element can be identified as a background
Yang-Mills (YM) field coupled to the WZW field [divecchia], [redlich87].
Similarly, the non-central part of the Virasoro coadjoint element can be
identified as a background rank-two field coupled to the Polyakov field. This
rank-two field and its higher dimensional extensions are called the
diffeomorphism field, or the diff field in short. P2DG action in LCG is
considered [knizhnik] as the gravitational analog of WZW action. Diff field
is, in the same sense, the gravitational analog of YM field, which is central
to the Standard Model.

In 2D, Einstein tensor identically vanishes so Einstein’s theory of gravity
does not provide dynamics for the spacetime metric. Einstein-Hilbert action
yields Euler-characteristic, providing only topological information about the
spacetime. Therefore, dynamics for gravity can arise only from quantum
anomalies. P2DG action is originally introduced as the effective action
encoding the conformal anomaly [polyakov81bosonic] and carries dynamical
information. Since the background diff field couples to the Polyakov metric,
it provides a source for cosmological constant and its dynamics would affect
the spacetime. In particular, it may solve the dark energy puzzle.

If the 2D result, that the diff field is the gravitational analog of the YM
field, holds in higher dimensions then the graviton may not be described by
the spacetime metric or derivable from it, as has been thought. Alternatively,
the diff field theory may be constructed in a way to encompass Einstein’s
theory to fix the quantization problem. These are currently speculative
statements, yet suggesting the motivation of pursuing research on this
subject. See, for instance, [hendersonrajeev] for a quantum gravity theory on
a circle suggested along a similar idea.

There are two distinct approaches for constructing a dynamical theory for the
diff field, leaving aside the most recent approach to be discussed at the end.
In the first approach, Virasoro coadjoint action is considered as the Lie
derivative of a rank-two object. This rank-two object is not a tensor due to
the central term in its Lie derivative. Therefore, covariantization222By
covariantization we mean lifting the space and time indices of tensors to
spacetime indices, and lifting the partial spatial and temporal derivatives to
covariant spacetime derivatives. can not yield a scalar under general
coordinate transformations (GCTs) formed only from the diff field and its
derivatives. One needs to introduce other objects, which also do transform
inhomogeneously, into the theory in order to get a GCT-scalar Lagrangian.

Affine connection (metric or not) also transforms inhomogeneously. In fact, in
1D, it is easy to obtain a particular functional of connection coefficients
that transform in the same way as a Virasoro coadjoint element (Section
2.4.4). However, extension of this relation to higher dimensions is highly
nontrivial (Section 2.4.5), so this approach has been mostly evaded. Two
places, where this approach is held, are [lanophd] and [rodgers19942d]. The
former claims to obtain a covariant action for the diff field. The latter is
the theory of a $(1,3)$-field. We restrict our attention to the examination of
rank-two proposals for the diff field in this thesis.

In the second approach, one considers a rank-two tensor whose field theory
yields a constraint equation such that this constraint equation reduces in 1D
to the isotropy equation on Virasoro coadjoint orbits. This constraint is
called the diff-Gauss law since the analogous constraint of YM theory is the
Gauss law, which reduces in 1D to the isotropy equation on the KM coadjoint
orbit. Since the main field of the theory is proposed to be a tensor,
covariantization yields a GCT-scalar. This is the approach that has been
followed the most.

There are two subcases to consider in the second approach. One can introduce
the diff-Gauss law as an implicit constraint i.e. one imposed on the phase
space via an equivalence relation (invariance under the field lift of the
Virasoro coadjoint transformation). In [LR95] authors followed this approach
using [rajeev88] as a guide. They introduced the Virasoro analog of the Wilson
loop, and obtained a finite reduction (i.e. a theory with a finite-dimensional
phase space) of the diff field theory. We review [rajeev88] in Section 5.2,
and [LR95] in Section 5.3.

In the other subcase ([LR96], [BLR97], [BLR00], [rodgers2007general],
[takeshithesis] ) the diff-Gauss law is made explicit i.e. introduced into the
action. This method is called the transverse action method. By this method one
recovers the gauge-invariant YM action from the gauge-fixed contents of it. We
do this construction in Section 4.2. An important part of this thesis,
Chapters 4, 5 and 6, is devoted to the examination of the tranverse formalism.
Let us outline the transverse formalism procedure to obtain the YM Lagrangian
from the KM algebra, and the diff field Lagrangian from the Virasoro algebra.

It is known that the coadjoint action, $\text{ad}^{*}_{\Lambda}A$ of KM
algebra is equivalent to the gauge transformation, $\delta_{\Lambda}A_{1}$ of
a YM field in 1D. In 2D, it can be identified as the residual gauge
transformation of the spatial component of a YM field $A_{\mu}$, in the
temporal gauge $A_{0}=0$. To build the transverse Lagrangian associated with
the KM algebra one uses the latter identification. Introducing a conjugate
momentum $\pi^{1}$ to $A_{1}$ one can obtain the gauge transformation of
$A_{1}$ through the Poisson bracket (PB) relation $\delta
A_{1}=\\{A_{1},\int\Lambda G\\}$. The generator $G$ of the transformation is
the well-known Gauss law.

Next, one introduces a Lagrangian formed from the symplectic term
$\partial_{0}A_{1}\pi^{1}$, the Hamiltonian $\pi^{1}\pi_{1}$ and the Gauss law
times a Lagrange multiplier $\lambda$. Introducing333In the references listed
above YM form of the momentum was directly assumed. Since in the diff field
case we do not know the final result to be reached, we proposed an ansatz that
could deduce the YM form. an ansatz of the form
$\pi^{1}=\partial^{0}A^{1}+...$ one recomputes the momentum from the
constructed Lagrangian. This yields the YM momentum for $\lambda=A_{0}$, and
the YM Lagrangian in 2D, $\sim\pi^{1}\pi_{1}=F^{01}F_{01}$. One can
straightforwardly lift this to higher dimensions, and covariantize. Note that
upon covariantization gauge structure of the YM theory is preserved. That is,
$A_{0}$ is still nondynamical, and the Gauss law is still a first-class
constraint generating time-independent gauge transformations of $A_{i}$.

With the lead from KM-YM pair, one lifts the transformation of a Virasoro
coadjoint element $D$ to the Lie derivative of a rank-two object $D_{\mu\nu}$,
the diff field. This transformation has a third-order inhomogeneous term
$\partial_{\mu}\partial_{\nu}\partial_{\lambda}\xi^{\lambda}$. Hence,
$D_{\mu\nu}$ does not transform as a tensor at this point. In 2D, one can
recover the transformation of $D$ as the Lie derivative of $D_{11}$ under
spatial, time-independent coordinate transformations. Introducing a conjugate
momentum $X^{11}$ to $D_{11}$, one obtains the operator $G_{1}$ generating
$\delta D_{11}$, namely, the diff-Gauss law. This operator is called the diff-
Gauss law because it is introduced precisely in the same way as the ordinary
Gauss law. Therefore, it is expected, in the end, to be a first-class
constraint generating the Lie derivative $\delta D_{ij}$ as a local symmetry
of the theory just as the first-class constraint Gauss law generates the gauge
symmetry $\delta A_{i}$.

Using the corresponding ingredients, i.e. the symplectic term
$\partial_{0}D_{ij}X^{ij}$, the Hamiltonian $X^{ij}X_{ij}$, and the diff-Gauss
law $G_{i}$ times its Lagrange multiplier $D_{0}^{\ i}$, one obtains a
Lagrangian. Then one introduces the analogous ansatz444Note that in the
references stated above, the momentum was taken as $X=\dot{D}$. We observed
that with this choice one does not recover back the same momentum from the
constructed Lagrangian. In fact the same kind of choice in the gauge theory
case does not yield YM Lagrangian. The new ansatz leads to the momentum
squared form of the diff Lagrangian, and one obtains three and four-point
self-interaction terms of the diff field, again in analogy with the YM theory.
To distinguish the modified and old theories in the analysis, we call the
latter, the BLRY theory. Whenever we would like to exemplify a computational
technique we use BLRY theory rather than the full theory for simplicity.
$X^{11}=\partial^{0}D^{11}+\cdots$ for the momentum, inserts it in the
Lagrangian, and recomputes the momentum from the constructed Lagrangian. With
this momentum the Lagrangian attains the same form as in YM theory $\sim
X^{ij}X_{ij}$.

The following step is covariantization just as in the YM case. However, at the
starting point we had a nontensor rank-two field. Covariant derivative is not
defined on such an object and even if we blindly applied the covariant
derivative formula of a rank-two tensor to it, such a derivative would
preserve non-covariance of the object. Similarly, contraction of spacetime
indices of such an object and its derivatives will not yield a GCT-scalar
Lagrangian. Hence, at this point diff field is regarded as a tensor. Moreover,
upon covariantization the Lagrange multiplier $D_{0}^{\ i}$ becomes dynamical,
and the diff-Gauss law is no more obtained as a constraint. Hence, contrary to
the YM case, at this step we lose connection to the origins of the theory.
This is expected because in the diff field case the local symmetry itself is
coordinate invariance.

We construct the transverse action for the diff field in Section 4.3. Its
supersymmetric extension is obtained in Section 4.5 using [GR01] as a guide.
We analyze the transverse diff theory in 2D Minkowski spacetime before
covariantization in Section 6.2, and the covariantized theory in Sections 5.4
and 5.5.

Interactions of the diff field is obtained by a prescription that emerges from
examining the structure of the self-interaction of the diff field [BLR00] in
the transverse action. When this prescription is applied to the point particle
and spinor interactions, the resulting expression suggests that the diff field
is a perturbation to the spacetime metric. We review interactions of the diff
field in Section 4.4. Motivated by the coadjoint action of the semidirect
product of Virasoro and KM algebras, we treat the diff field as transforming
nontrivially under gauge transformations555In [lanophd] also the diff field is
treated this way.. We examine application of the interaction prescription to
the spin-one coupling with this treatment.

Note that even if we turn off covariantization, and treat diff field as a
nontensor, the diff-Gauss law turns out to be inconsistent for the chosen
standard kinetic term (Section 6.3). In the Dirac-Hamiltonian analysis, new
constraints arise, and these are all derivable from the kinetic term. The
diff-Gauss law turns out to be second-class unless the kinetic term itself is
a constraint. Hence, we turn the kinetic term into a constraint. Then the
diff-Gauss law becomes a first-class constraint, and no new constraints arise.
In this case, however, dynamics is lost (Section 6.4). We investigate an
alternative kinetic term in Section 6.5.

Note that although covariant transverse diff theory is inconsistent with its
own philosophy, it has mathematically consistent subcases i.e. gauge-fixed
reductions without constraint inconsistencies. Namely, it is not a theory with
local Virasoro symmetry, and the motivation coming from geometric actions is
lost (i.e. diff field being the gravitational analog of the gauge field), but
it still provides subcases with dynamical content (momenta, field equations
and so on) that is related to the Virasoro algebra in some way (e.g.
appearance of the KdV equation and its variants). One such case is in a gauge
we call the chiral gauge. The diff-Gauss law does not arise, as the field
equation of $D_{0}^{\ i}$ component reduces to $0=0$. In 2D, in this gauge,
covariant transverse theory reduces to a theory with two decoupled fields (one
a function of time only and the other a function of space only) which seems to
be related to the geometric action associated with the direct product of two
Virasoro algebras (Section 3.6). We investigate this in Section 5.4.

In Section 6.6, we review the tranverse method, outline all its problems and
discuss how they are related. We decide that the most important issue in the
theory is covariantization. We abandon covariantization and go back to the
approach of treating the diff field as a nontensor. We look for alternative
ways to recover covariance. We investigate complementing666As we mentioned
above, in [lanophd], a covariant theory of the diff field is proposed along
similar lines i.e. by introducing connection coefficients into the action. We
could not verify their result, but it would be interesting to investigate how
this theory may be related to the transverse theory, or whether it actually
fulfills our goal by providing a gauge theory of the diff field. For this one
needs to check whether the theory provides the diff-Gauss law as a first-class
constraint generating a coordinate transformation that reduces in 1D to the
Virasoro coadjoint action.  the diff field with connection coefficients (using
the results of Section 2.4.5) and propose modifications which recover full
covariance for the interactions of the diff field while keeping spatial
covariance of the diff Lagrangian.

Problems of the transverse theory lead us to investigate alternative methods
to obtain a theory of the diff field. One such method is the Euler-Poincare
formalism, an alternative Lagrangian formalism suited for Lie groups (Section
7.1). Application of this formalism to diff field, however, leads to a
dynamical theory within a coadjoint orbit, rather than producing dynamics with
gauge degrees of freedom lying on the orbits i.e. with the diff-Gauss law
being a first-class constraint generating the Virasoro coadjoint
transformation as a local symmetry.

Next, we extensively examine the analog of the Wilson loop for the diff field
in Section 7.2. For this we follow the references [scherer88],
[hendersonrajeev]. The latter claims to obtain the Virasoro analog of the
theory in [rajeev88], just as [LR95], but we believe the Wilson loop to be
used for such a theory should be associated with the operator $\nabla^{(3)}$
(introduced in Section 7.2.2 ) rather than the Hill operator $\nabla^{(2)}$.
Indeed, in [LR95] a Wilson loop associated with $\nabla^{(3)}$ was introduced,
but we believe its implementation was incomplete (Section 7.2.6). We produce
results associated with $\nabla^{(3)}$ that may be needed for future research
on this project.

As the final part of the thesis we review an entirely different approach
proposed recently [brensinger] to obtain a dynamical diff theory. Diff field
is identified as part of a TW projective connection. The authors introduce a
curvature-squared type action for the diff field based on this identification.
We are going to provide a quick summary of this work in Section 7.3. Our focus
in this thesis is on the clarification of the relationship between TW
connections and diff field. This is investigated in Section 7.4.

Let us also briefly discuss the notation and the conventions used in the
thesis. Throughout the thesis, summation convention is used both for algebraic
and tensorial sums unless there is potential confusion. Derivative of a
quantity with respect to a variable is frequently denoted by a subscript,
e.g., $\partial/\partial\rho=:\partial_{\rho}$. In 1D, derivative with respect
to a single coordinate is denoted by a prime unless it is a temporal
parameter, in which case it is denoted by a dot. In 2D, derivative with
respect to time and space is also be denoted by a dot and a prime,
respectively.

The sign convention for the metric is $(+t,-x,-x,\cdots)$. To avoid
culmination of negative signs, in any dimensions we write $\sqrt{g}$ for the
metric determinant even when $g$ is negative; what is implied is $\sqrt{|g|}$.
Certain sections require additional notational and conventional changes, they
are noted beforehand.

Analogs of objects of the YM theory are named with ”diff-…” in the case of the
diff field e.g. diff-Gauss law, diff-Wilson loop etc.

##  Chapter 2 Preliminaries

###  2.1 Coadjoint Orbits and Kirillov Form

####  2.1.1 Adjoint Action

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Consider the
conjugation map by a fixed element $g\in G$

$\displaystyle C_{g}:G\rightarrow G:h\mapsto ghg^{-1}$ (2.1)

and its pushforward

$\displaystyle\text{Ad}_{g}\equiv(C_{g})_{*}:\mathfrak{g}\rightarrow\mathfrak{g}$
(2.2)

For $X\in\mathfrak{g}$ we can explicitly write

$\displaystyle\text{Ad}_{g}X=\frac{d}{dt}(R_{g^{-1}}\circ
L_{g}\circ\exp(tX))\bigg{|}_{t=0}$ (2.3)

For matrix groups this simplifies to

$\displaystyle\text{Ad}_{g}X=gXg^{-1}$ (2.4)

The map

$\displaystyle\text{Ad}:G\times\mathfrak{g}\rightarrow\mathfrak{g}:(g,X)\mapsto\text{Ad}_{g}X$
(2.5)

defines an action of $G$ on its Lie algebra $\mathfrak{g}$ called the adjoint
action. Any group action on a vector space defines a representation of the
group; $\mathfrak{g}$ is a vector space. The representation for the adjoint
action is defined by

$\displaystyle\text{Ad}:G\mapsto\mathfrak{gl}(\mathfrak{g}):g\mapsto\text{Ad}_{g}$
(2.6)

and is called the adjoint representation. Using the properties of the
pushforward and the group one can show that Ad indeed satisfies the properties
of a representation

$\displaystyle\text{Ad}_{g}\circ\text{Ad}_{h}=\text{Ad}_{gh}\hskip
21.68121pt\text{and}\hskip 21.68121pt(\text{Ad}_{g})^{-1}=\text{Ad}_{g^{-1}}$
(2.7)

Adjoint representation of $G$ induces a representation of its Lie algebra
$\mathfrak{g}$ defined by

$\displaystyle\text{ad}\equiv\text{Ad}_{*}:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g}):X\mapsto\text{ad}_{X}$
(2.8)

We shall call this the infinitesimal adjoint action and it can be explicitly
written as

$\displaystyle\text{ad}_{X}=\frac{d}{dt}\bigg{|}_{t=0}\text{Ad}_{\exp tX}$
(2.9)

One can show using the flow of $X\in\mathfrak{g}$ that its action on
$Y\in\mathfrak{g}$ yields

$\displaystyle\text{ad}_{X}Y=\mathcal{L}_{X}Y=[X,Y]$ (2.10)

Using the Jacobi identity on $\mathfrak{g}$ one can show that ad is indeed a
representation of $\mathfrak{g}$

$\displaystyle\text{ad}_{[X,Y]}=[\text{ad}_{X},\text{ad}_{Y}]$ (2.11)

####  2.1.2 Coadjoint Action

Let $V$ and $W$ be vector spaces, and let $A:V\rightarrow W$ be a linear map.
The dual map $A^{*}:W^{*}\rightarrow V^{*}$ is defined by

$\displaystyle(A^{*}b)(v)\equiv b(Av)$ (2.12)

where $b\in W^{*}$, $v$ $\in$ $V$. $\mathfrak{g}$ is a vector space and the
adjoint action $\text{Ad}_{g}:\mathfrak{g}\rightarrow\mathfrak{g}$ is a linear
map. Hence we can define its dual,
$(\text{Ad}_{g})^{*}:\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}$ , as

$\displaystyle[(\text{Ad}_{g})^{*}b]\ (u)\equiv b(\text{Ad}_{g}u)$ (2.13)

where $b$ $\in$ $\mathfrak{g}^{*}$ and $u$ $\in$ $\mathfrak{g}$.

Vectors in $\mathfrak{g}$ are often called adjoint vectors and ones in
$\mathfrak{g}^{*}$ are called coadjoint vectors. Thus, a coadjoint vector is a
linear functional on $\mathfrak{g}$. This is often written in the form of a
’pairing’, a linear map $\left<\ |\
\right>:\mathfrak{g}^{*}\times\mathfrak{g}\rightarrow\mathbb{R}$

$\displaystyle\left<b|u\right>\equiv b(u)$ (2.14)

The coadjoint action of $g$ $\in$ $G$ on $\mathfrak{g}^{*}$, denoted
$Ad^{*}_{g}$, is defined by

$\displaystyle\text{Ad}^{*}_{g}\equiv(\text{Ad}_{g^{-1}})^{*}$ (2.15)

The reason for $g^{-1}$ on the right hand side is to make the pairing
invariant under the action of the group. That is, if $b\in\mathfrak{g}^{*}$
and $u\in\mathfrak{g}$ then we have

$\displaystyle\left<\text{Ad}^{*}_{g}b|\text{Ad}_{g}u\right>=\left<b|\text{Ad}_{g^{-1}}\text{Ad}_{g}u\right>=\left<b|u\right>$
(2.16)

Practically one first introduces a pairing, then obtain the coadjoint action
by declaring invariance of the pairing.

The set $\mathfrak{g}^{*}$ is a vector space just as $\mathfrak{g}$.
Therefore, similar to the adjoint case, we can define the coadjoint
representation of the group from the coadjoint action.

The induced infinitesimal coadjoint action
$\text{ad}^{*}_{v}:\mathfrak{g}^{*}\rightarrow
T\mathfrak{g}^{*}\cong\mathfrak{g}^{*}$ is defined by

$\displaystyle(\text{ad}^{*}_{v}b)u\equiv-b(\text{ad}_{v}u)=-b([v,u])$ (2.17)

One can obtain this from the invariance condition (2.16) by considering the
one-parameter subgroup generated by $v\in\mathfrak{g}$ i.e. $g=\exp(tv)$,
differentiating with respect to $t$, and evaluating at $t=0$. Then the
infinitesimal form of invariance follows:

$\displaystyle v*\left<b|u\right>=$
$\displaystyle\left<v*b|u\right>+\left<b|v*u\right>$ $\displaystyle=$
$\displaystyle\left<\text{ad}^{*}_{v}b|u\right>+\left<b|\text{ad}_{v}u\right>$
$\displaystyle=$ $\displaystyle-b([v,u])+b([v,u])=0$ (2.18)

The isotropy group $G_{b}$ of $b\in\mathfrak{g}^{*}$ under the coadjoint
action is defined by

$\displaystyle G_{b}\equiv\\{g\in G\ |\ \text{Ad}^{*}_{g}b=b\\}$ (2.19)

and is a subgroup of $G$. The isotropy algebra $\mathfrak{g}_{b}$ of $b$ is
the Lie subalgebra of $\mathfrak{g}$ that generates the isotropy group
$G_{b}$. It is given by

$\displaystyle\mathfrak{g}_{b}=\\{u\in\mathfrak{g}\ \ |\ \
\text{ad}^{*}_{u}b=0\\}$ (2.20)

The equation $\text{ad}^{*}_{u}b=0$ is called the isotropy equation for the
coadjoint element $b$. We will construct transverse actions in Chapter 4 by
lifting isotropy equations of algebras to constraint equations of the
corresponding field theory.

####  2.1.3 Coadjoint Orbits and Kirillov Form

The coadjoint orbit of $b_{0}\in\mathfrak{g}^{*}$ is defined by

$\displaystyle\text{Orb}(b_{0})\equiv\\{b\in\mathfrak{g}^{*}\ |\ \exists g\in
G\ \ \text{st}\ \ b=\text{Ad}^{*}_{g}b_{0}\\}$ (2.21)

and is a subspace of $\mathfrak{g}^{*}$. Kirillov [kirillovorbit] showed that
every coadjoint orbit of a Lie group $G$ is naturally equipped with a
symplectic structure $\Omega$ (called the Kirillov form) that is invariant
under the action of $G$. A symplectic structure is a two-form that is non-
degenerate and closed.

$\Omega$ is defined as follows. The (coadjoint) action of two adjoint vectors
$u,u^{\prime}\in\mathfrak{g}$ on $b\in\mathfrak{g}^{*}$ yield two coadjoint
vectors $a,a^{\prime}$ that are tangent to the orbit at $b$ :

$\displaystyle a\equiv\text{ad}^{*}_{u}b\hskip 21.68121pt\text{and}\hskip
21.68121pta^{\prime}\equiv\text{ad}^{*}_{u^{\prime}}b$ (2.22)

Then the Kirillov form $\Omega$ is defined as

$\displaystyle\Omega(a,a^{\prime})\equiv\left<b|[u,u^{\prime}]\right>$ (2.23)

$\Omega$ is antisymmetric because of the commutator on the right. The pairing
is $G$-invariant by definition, so $\Omega$ is $G$-invariant :

$\displaystyle\Omega(a,a^{\prime})=\left<b|[u,u^{\prime}]\right>=\left<b_{g}|[u,u^{\prime}]_{g}\right>=\left<b_{g}|[u_{g},u^{\prime}_{g}]\right>=\Omega(a_{g},a^{\prime}_{g})$
(2.24)

where $b_{g}\equiv\text{Ad}^{*}_{g}b$ , $u_{g}\equiv\text{Ad}_{g}u$ ,
$a_{g}\equiv\text{ad}^{*}_{u_{g}}b_{g}$ and $g,g^{\prime}\in G$. If
$a=\text{ad}^{*}_{u}b$ is a nonzero coadjoint vector then $b$ is nonzero by
linearity of $\text{ad}^{*}$. Then there must be an adjoint vector $v$ (that
does not commute with $u$) such that
$(\text{ad}^{*}_{u}b)v=-\left<b|[u,v]\right>\neq 0$ so that $\Omega$ is
nondegenerate.

In order to prove111The proof here is from [witten88]. For a rigorous proof
see e.g. [marsdenratiu] Chapter 14. closure of $\Omega$ we use the invariant
formula for exterior derivatives. For a two-form $\lambda$ and vector fields
$u,v,w$ on a manifold $M$ it reads

$\displaystyle d\lambda(u,v,w)$
$\displaystyle=u\cdot\nabla(\lambda(v,w))+v\cdot\nabla(\lambda(w,u))+w\cdot\nabla(\lambda(u,v))$
$\displaystyle\ \ \ -\lambda([u,v],w)-\lambda([w,u],v)-\lambda([v,w],u)$
(2.25)

The adjoint vectors $u,v,w\in\mathfrak{g}$ define the coadjoint tangent
vectors $b_{u}$, $b_{v}$, $b_{w}$ on the orbit, where
$b_{u}\equiv\text{ad}^{*}_{u}b\in T_{b}\mathfrak{g}^{*}\cong\mathfrak{g}^{*}$.
The Kirillov form $\Omega\in\Lambda^{2}(\text{Orb}(b))$ acts on a pair of
(tangent) coadjoint vectors on the orbit, and $d\Omega$ on three of them. The
invariant formula in this case reads

$\displaystyle d\Omega(b_{u},b_{v},b_{w})$
$\displaystyle=b_{u}\cdot\nabla(\Omega(b_{v},b_{w}))+b_{v}\cdot\nabla(\Omega(b_{w},b_{u}))+b_{w}\cdot\nabla(\Omega(b_{u},b_{v}))$
$\displaystyle\ \
-\Omega([b_{u},b_{v}],b_{w})-\Omega([b_{w},b_{u}],b_{v})-\Omega([b_{v},b_{w}],b_{u})$
(2.26)

Consider the first term on the right, $b_{u}\cdot\nabla(\Omega(b_{v},b_{w}))$.
It represents the change of $\Omega(b_{v},b_{w})=\left<b|[v,w]\right>$ in
$b_{u}$ direction, so is equal to the action of the adjoint element $u$ on the
pairing, which is zero by invariance of the pairing. Thus the first line on
the right in (2.1.3) vanishes.

Now, consider the first term in the second line. Since $\text{ad}^{*}$ is a
representation of $\mathfrak{g}$ we have

$\displaystyle\Omega([b_{u},b_{v}],b_{w})$
$\displaystyle=\Omega([\text{ad}^{*}_{u}b,\text{ad}^{*}_{v}b],\text{ad}^{*}_{w}b)$
$\displaystyle=\Omega(\text{ad}^{*}_{[u,v]}b,\text{ad}^{*}_{w}b)=\Omega(b_{[u,v]},b_{w})=b([[u,v],w])$
(2.27)

Thus the terms in the second line add up to zero by Jacobi identity on
$\mathfrak{g}$ and linearity of $b$. This completes the proof of $d\Omega=0$.

###  2.2 Construction of Geometric Actions on Coadjoint Orbits

####  2.2.1 Mechanics on Space of Paths in Phase Space

Symplectic structure $\Omega$ is the main ingredient of Hamiltonian mechanics.
Hamilton’s equations describing the dynamics of a physical system can be
written as

$\displaystyle\iota_{X_{H}}\Omega=dH$ (2.28)

where $H$ is the Hamiltonian, $X_{H}$ is the Hamiltonian vector field whose
flow describes the evolution. Let $\Gamma=\\{\xi^{i}\\}$ be the phase space.
Then we can explicitly write

$\displaystyle\Omega_{ij}\dot{\xi}^{j}=\partial_{\xi^{i}}H$ (2.29)

In most cases, the symplectic structure is not only closed but also (globally)
exact so that we can write it as the exterior derivative of the so called
”canonical one-form”, denoted by $\theta$ , i.e. $\Omega=d\theta$. Then the
action can be written as

$\displaystyle S=\int\theta-Hdt$ (2.30)

Consider a simple example, that of a two-dimensional phase space
$\Gamma=\\{(p,q)\\}$ with $\omega=dp\wedge dq$ and $\theta=pdq$. The action
then reads

$\displaystyle S=\int(pdq-Hdt)=\int dt\ (p\dot{q}-H)=\int dt\ L$ (2.31)

For coadjoint orbits, however, we do not, in general, enjoy this
simplification. Balachandran et al [zaccaria] [balagauge] discusses an
extension of symplectic mechanics when symplectic structure is not exact.
Below is the outline.

Instead of the phase space $\Gamma$ we consider the space of paths on
$\Gamma$, denoted $P\Gamma$. The points on $P\Gamma$ can be defined by fixing
a point $P_{0}$ in $\Gamma$ . Then an element of $P\Gamma$ is a path from
$P_{0}$ to some other point $\xi$ in $\Gamma$. We may parametrize these paths
as

$\displaystyle\gamma\in\\{\gamma(\lambda)\ |\ 0\leq\lambda\leq
1,\gamma(0)=P_{0},\gamma(1)=\xi\\}$ (2.32)

Introducing also the time coordinate $\tau$ we get time-dependent paths,
$\gamma(\lambda,\tau)$ where $\gamma(\lambda=0,\tau)=P_{0}$ and
$\gamma(\lambda=1,\tau)=\xi(\tau)$. Here, $\\{\xi(\tau)\\}$ is a possible
trajectory to be followed by the system. That is, we would like to obtain an
action functional whose extremization yields equations only on
$\\{\xi(\tau)\\}$. As $\lambda$ and $\tau$ vary, the paths
$\gamma(\lambda,\tau)$ sweep out a two-surface $m$ in $\Gamma$ (See Figure
2.1). Its boundary is given by

$\displaystyle\partial m=\partial m_{1}\cup\partial m_{2}\cup\partial m_{3}$
(2.33)

where

$\displaystyle\partial m_{1}$ $\displaystyle=\\{\xi(\tau)\ |\
\tau_{i}\leq\tau\leq\tau_{f}\\}=AB$ $\displaystyle\partial m_{2}$
$\displaystyle=\\{\gamma(\lambda,\tau_{i})\ |\ 0\leq\lambda\leq 1\\}=P_{0}A$
$\displaystyle\partial m_{3}$ $\displaystyle=\\{\gamma(\lambda,\tau_{f})\ |\
0\leq\lambda\leq 1\\}=P_{0}B$ (2.34)

Figure 2.1: The Two-surface $m$ Traced by the Paths

The Hamiltonian $H$ is lifted to a functional $\tilde{H}$ on paths, as

$\displaystyle\int_{0}^{1}d\lambda\
\tilde{H}[\gamma(\lambda,\tau)]=H[\gamma(1,\tau)]=H[\xi(\tau)]$ (2.35)

Then the action functional can be defined as

$\displaystyle S=\int_{m}(\Omega-\tilde{H}\ d\lambda\wedge d\tau)$ (2.36)

or, in coordinates, as

$\displaystyle
S=\int\Omega_{ij}\partial_{\lambda}\gamma^{i}\partial_{\tau}\gamma^{j}d\lambda
d\tau-\int_{\partial m_{1}}Hd\tau$ (2.37)

Under variations, the point $P_{0}$ and the end paths $\partial m_{2}$ and
$\partial m_{3}$ are to be held fixed. Equations of motion derived by varying
the paths $\gamma$ then become

$\displaystyle 0=\delta S=\int_{\partial
m_{1}}\Omega_{ij}d\gamma^{i}\delta\gamma^{j}-\int_{\partial
m_{1}}\partial_{\gamma^{i}}H\delta\gamma^{i}d\tau$ (2.38)

where $d\Omega=0$ is used. This recovers Hamilton’s equations on $\partial
m_{1}=\\{\xi(\tau)\\}=AB$

$\displaystyle\Omega_{ij}\dot{\gamma}^{i}=\partial_{\gamma^{i}}H$ (2.39)

####  2.2.2 Geometric Actions on Coadjoint Orbits

We choose to consider theories with vanishing Hamiltonian. The symplectic
structure is the Kirillov form on coadjoint orbits of the infinite-dimensional
Lie algebras, Kac-Moody and Virasoro. The Kirillov form is non-exact in each
case, so we will employ the results of the previous section. Then according to
(2.36), the action functional (called the geometric action) is given simply by
the integral of the Kirillov form on an orbit

$\displaystyle S=\int_{\text{Orb}}\Omega$ (2.40)

The orbit is parametrized as a two-surface $\\{(\lambda,\tau)\\}$ so that we
need to construct adjoint vectors $u_{\tau},u_{\lambda}$ and coadjoint
(tangent) vectors $b_{\tau}=\text{ad}^{*}_{u_{\tau}}(b),\
b_{\lambda}=\text{ad}^{*}_{u_{\lambda}}(b)$ describing changes in $\tau$ and
$\lambda$ directions for a suitably chosen coadjoint vector
$b=b(\tau,\lambda)$. Then using (2.23) the action can be explicitly written as

$\displaystyle S=\int_{\text{Orb}(b)}d\lambda\ d\tau\
\Omega(b_{\tau},b_{\lambda})=\int_{\text{Orb}(b)}d\lambda\ d\tau\
\left<b|[u_{\tau},u_{\lambda}]\right>$ (2.41)

This will be done in the next chapter. The central part of the constructed
geometric action will turn out to be the WZW action in the KM case and P2DG
action in LCG in the Virasoro case.

###  2.3 Kac-Moody Algebra

For our purposes Kac-Moody (KM) algebra and the geometric action on its
coadjoint orbits play secondary roles. Therefore, we will not get into detail
as much as we do for the Virasoro algebra. The main references for this
section are [goddardolive86], [delius90] and [WZNW].

####  2.3.1 Loop Group, Loop Algebra and Its Central Extension

Let $G$ be a compact, connected, semi-simple Lie group. Then its Lie algebra
$\mathfrak{g}$ is semi-simple with Killing form $\delta^{ab}$ so that the
structure constants with fully upper indices $f^{abc}$ are defined and are
fully antisymmetric. The commutation relations for $\mathfrak{g}$ can then be
written in a basis $\\{T^{a}\\}$ as

$\displaystyle[T^{a},T^{b}]=if^{abc}T^{c}$ (2.42)

Since $G$ is connected, any element $g$ of $G$ can be obtained by
exponentiation of an algebra element, i.e. $g=\exp(-iT^{a}\theta_{a})$ with
parameters $\theta_{a}$.

A smooth map $\gamma$ from circle $S^{1}=\\{z\in\mathbb{C}\,:\,|z|=1\\}$ to
$G$ is called a loop in $G$. The set of loops forms a Lie group, called the
loop group of $G$, denoted $LG$, with the group multiplication defined by

$\displaystyle(\gamma_{1}\cdot\gamma_{2})(z)\equiv\gamma_{1}(z)\gamma_{2}(z)$
(2.43)

On the right, group multiplication of $G$ is implied.

To obtain the Lie algebra $L\mathfrak{g}$ of $LG$ consider its connected
component consisting of maps $\gamma:S^{1}\rightarrow G$ that can be
continuously deformed to the constant map $\gamma(z)=1$. Then any element of
this subset of $LG$ can be obtained using functions $\theta_{a}(z)$ defined on
the unit circle as $\gamma(z)=\exp(-iT^{a}\theta_{a}(z))$.

For elements near the identity map we have $\gamma(z)\approx
1-iT^{a}\theta_{a}(z)$. Making a Laurent expansion,
$\theta_{a}(z)=\theta_{a}^{n}z^{n}$, we see that the composite objects

$\displaystyle J^{a}_{n}\equiv T^{a}z^{n}$ (2.44)

are generators for the loop group. Indeed for elements near the identity we
have $\gamma(z)\approx 1-iJ^{a}_{n}\theta^{n}_{a}$. Using (2.42) and (2.44) we
get the commutation relations

$\displaystyle[J^{a}_{m},J^{b}_{n}]=if^{abc}J^{c}_{m+n}$ (2.45)

for the loop algebra $L\mathfrak{g}$. Note that $\\{J^{a}_{0}\\}$ generate a
subgroup of $LG$ isomorphic to its base group $G$.

Since $G$ is compact, picking up a Hermitian basis of $G$-generators,
$T^{a\dagger}=T^{a}$, the loop generators satisfy
$J^{a\dagger}_{n}=J^{a}_{-n}$, where $z^{*}=z^{-1}$ is used. Such a
representation of loop algebra basis generates unitary loops $\gamma(z)$ for
real $\theta^{n}_{a}$ and $|z|=1$.

The Kac-Moody algebra (or, more explicitly, the untwisted affine Kac-Moody
algebra) associated with a compact finite-dimensional Lie algebra
$\mathfrak{g}$ is the central extension of the loop algebra $L\mathfrak{g}$,
defined by the commutation relations

$\displaystyle[J^{a}_{m},J^{b}_{n}]=if^{abc}J^{c}_{m+n}+km\delta^{ab}\delta_{m+n}$
(2.46)

For the detailed arguments leading to this form of the central extension see
[goddardolive86]. For a mathematically more precise way of expressing central
extension, see (2.3.3).

####  2.3.2 Current Algebra

The loop algebra and its central extension appear in a variety of physical
theories. Here, we analyze one such theory222Later we are going to see the
same current algebra appearing in the WZW theory., namely, that of free
massless quarks in 2D. These quarks can be represented by $N$ massless
Majorana fermions $\psi^{i}$ ($i=1,\cdots,N$). The action reads

$\displaystyle S=\frac{i}{2}\int d^{2}x\
\overline{\psi}_{k}\gamma^{\mu}\partial_{\mu}\psi^{k}$ (2.47)

where $\overline{\psi}=\psi^{T}\gamma^{0}$. This action leads to the Dirac
equation

$\displaystyle\gamma^{\mu}\partial_{\mu}\psi^{k}=0$ (2.48)

Using the chirality matrix333See Section B.1 for the conventions in 2D.
$\gamma_{c}$ we can decompose $\psi=(\psi_{-}\ \psi_{+})^{T}$ with
$\gamma_{c}\psi_{\pm}=\mp\psi_{\pm}$. Then the Dirac equation yields the Weyl
equations for the chiral components

$\displaystyle\partial_{+}\psi_{-}=0=\partial_{-}\psi_{+}$ (2.49)

Therefore, we have $\psi_{-}=\psi_{-}(x^{-})$ and $\psi_{+}=\psi_{+}(x^{+})$,
and the equations for $\psi_{-}$ and $\psi_{+}$ are decoupled in the massless
case.

Upon quantization we get the anticommutation relations which can be written in
terms of the chiral components as

$\displaystyle\\{\psi^{i}_{\pm}(x),\psi^{j}_{\pm}(y)\\}$
$\displaystyle=\hbar\delta(x-y)\delta^{ij}$ (2.50)
$\displaystyle\\{\psi^{i}_{+}(x),\psi^{j}_{-}(y)\\}$ $\displaystyle=0$ (2.51)

Since dynamics of the chiral components are decoupled in the massless case,
the theory described has an internal O$(N)\times$O$(N)$ symmetry at the
classical level. Generators of O$(N)$ can be taken as $T^{a}=iM^{a}$, where
$M^{a}$ are $N\times N$ real, antisymmetric matrices satisfying

$\displaystyle[M^{a},M^{b}]=f^{abc}M^{c}$ (2.52)

Associated with this symmetry are the (classically) conserved chiral currents

$\displaystyle J^{a}_{\pm}=\frac{1}{2}\psi_{\pm}^{T}T^{a}\psi_{\pm}$ (2.53)

Their conservation read

$\displaystyle\partial_{-}J_{+}=0=\partial_{+}J_{-}$ (2.54)

so that $J^{a}_{+}$ ($J^{a}_{-}$) is a function of $x^{+}$ ($x^{-}$) only, as
implied by (2.49).

Quantization yields the following commutation relations

$\displaystyle[J^{a}_{\pm}(x),J^{b}_{\pm}(y)]$ $\displaystyle=i\hbar
f^{abc}J^{c}_{\pm}(x)\delta(x-y)+\frac{i\hbar^{2}}{2\pi}k\delta^{ab}\delta^{\prime}(x-y)$
(2.55a) $\displaystyle[J^{a}_{\pm}(x),J^{b}_{\mp}(y)]$ $\displaystyle=0$
(2.55b)

This algebra is none other than the KM algebra (2.46). The c-number term,
appearing upon quantization, is also known as the Schwinger term. Presence of
the $\hbar^{2}$ coefficient emphasizes that this is a second-order quantum
effect (i.e. corresponds to a 1-loop Feynman diagram).

The energy momentum tensor for this theory is given by

$\displaystyle
T^{\mu\nu}=\frac{i}{4}\left(\overline{\psi}\gamma^{\mu}\overset{\leftrightarrow}{\partial}_{\nu}\psi+\overline{\psi}\gamma^{\nu}\overset{\leftrightarrow}{\partial}_{\mu}\psi\right)$
(2.56)

It is traceless so the theory is classically conformally invariant. Upon
quantization its Laurent modes $\\{L_{n}\\}$ satisfy

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}m(m^{2}-1)\delta_{m,-n}$
(2.57)

with $c=N/2$. This is the Virasoro algebra. We will discuss the Virasoro
algebra in detail in Section 2.4.

####  2.3.3 Coadjoint Action of Kac-Moody Algebra

For the purposes of construction of the geometric action on the coadjoint
orbits of the Kac-Moody (KM) algebra we are going to use the conventions set
in [delius90].

The KM algebra is defined by

$\displaystyle[J^{a}_{m},J^{b}_{n}]$
$\displaystyle=f^{abc}J^{c}_{m+n}+km\delta_{m+n}\delta^{ab}I$
$\displaystyle[J^{a}_{m},I]$ $\displaystyle=[I,I]=0$ (2.58)

where $f^{abc}$ are the structure constants of the semi-simple Lie algebra
$\mathfrak{g}$ underlying KM algebra, $\delta^{ab}$ is the Killing metric, $I$
is the generator for the central charge, and $k$ is a constant. Structure
constants are taken real and $(J^{a}_{m})^{\dagger}=-J^{a}_{-m}$, so that $k$
is also real. (See Section 2.3.1 and for more details [goddardolive86].)

A general Lie algebra element is written as

$\displaystyle\Lambda+\alpha kI$
$\displaystyle=\Lambda^{n}_{a}J^{a}_{n}+\alpha kI=\oint\frac{dz}{2\pi i}\
\text{Tr}\ \Lambda(z)J(z)+\alpha kI$ $\displaystyle\equiv(\Lambda(z),\alpha)$
(2.59)

where $\Lambda(z)=\Lambda^{a}(z)T^{a}$ , $J(z)=J^{a}(z)T^{a}$ , $\text{Tr}\
T^{a}T^{b}=\delta^{ab}$ , $\Lambda^{a}(z)=z^{n}\Lambda^{a}_{n}$ and
$J^{a}(z)=z^{-n-1}J^{a}_{n}$. The contour integral is around the origin in the
complex $z-$plane and the following is used:

$\displaystyle\oint\frac{dz}{2\pi i}\ z^{p-1}=\delta_{p}$ (2.60)

The commutator between two general elements with central charges becomes

$\displaystyle[\Lambda+\alpha kI$ $\displaystyle,\Sigma+\beta kI]$
$\displaystyle=\oint\frac{dz}{2\pi i}\ \text{Tr}\
[\Lambda(z),\Sigma(z)]J(z)+k\oint\frac{dz}{2\pi i}\ \text{Tr}\
\partial_{z}\Lambda(z)\Sigma(z)I$ $\displaystyle\equiv\left([\Lambda,\Sigma]\
,\ \oint\frac{dz}{2\pi i}\ \text{Tr}\ \partial_{z}\Lambda(z)\Sigma(z)\right)$
(2.61)

This defines the action of KM algebra on itself, i.e., the ad-action. The
center of the commutator is defined through the so-called two-cocyle

$\displaystyle\omega((\Lambda,\alpha),(\Sigma,\beta))\equiv\oint\frac{dz}{2\pi
i}\ \text{Tr}\ \partial_{z}\Lambda(z)\Sigma(z)$ (2.62)

A finite group element,

$\displaystyle g=\exp(\Sigma+\beta kI)=\exp\left(\oint\frac{dz}{2\pi i}\
\text{Tr}\ \Sigma(z)J(z)+\beta kI\right)\equiv\exp\Sigma(z),$ (2.63)

acts on the algebra as
$(\Lambda(z),\alpha)\mapsto(\Lambda_{g}(z),\alpha_{g})\equiv\text{Ad}_{g}(\Lambda(z),\alpha)$
where

$\displaystyle(\Lambda_{g}(z),\alpha_{g})=\left(g(z)\Lambda(z)g^{-1}(z),\alpha+\oint\frac{dz}{2\pi
i}\ \text{Tr}\ \partial_{z}g(z)\Lambda(z)g^{-1}(z)\right)$ (2.64)

Infinitesimal reduction of (2.64) yields back (2.3.3).

We will denote a coadjoint vector by $(A(z),a)$. The pairing $\left<\ |\
\right>:\mathcal{G}^{*}\times\mathcal{G}\rightarrow\mathbb{R}$ is chosen as

$\displaystyle\left<(A(z),a)|(\Lambda(z),\alpha)\right>=\oint\frac{dz}{2\pi
i}\ \text{Tr}\ A(z)\Lambda(z)+a\alpha$ (2.65)

The coadjoint action
$(A(z),a)\mapsto(A_{g}(z),a_{g})\equiv\text{Ad}^{*}_{g}(A(z),a)$ is then
defined by invariance of the pairing under the action of $g$ :

$\displaystyle\left<(A_{g}(z),a_{g})|(\Lambda_{g}(z),\alpha_{g})\right>\overset{!}{=}\left<(A(z),a)|(\Lambda(z),\alpha)\right>$
(2.66)

Using (2.64) and (2.65), this condition yields

$\displaystyle A_{g}(z)$
$\displaystyle=g(z)A(z)g^{-1}(z)-a\partial_{z}g(z)g^{-1}(z)$ $\displaystyle
a_{g}$ $\displaystyle=a$ (2.67)

Notice that the coadjoint element $A$ can be identified as a gauge field in
1D. Alternatively, (2.3.3) can be identified as a time-independent gauge
transformation of $A_{1}=A$ component of a Yang-Mills field $A_{\mu}$ in 2D.

###  2.4 Virasoro Algebra

####  2.4.1 Diffemorphism Algebra in 1D

In any dimensions the Lie derivative of a vector field $\eta$ along another
vector field $\xi$ can be written as

$\displaystyle\mathscr{L}_{\xi}\eta^{a}=\xi^{b}\partial_{b}\eta^{a}-\eta^{b}\partial_{b}\xi^{a}\equiv(\xi\circ\eta)^{a}$
(2.68)

and it satisfies

$\displaystyle[\mathscr{L}_{\xi},\mathscr{L}_{\eta}]=\mathscr{L}_{\xi\circ\eta}$
(2.69)

This defines the diffeomorphism algebra [courant].

In 1D we can write the bracket above, explicitly, as444One often uses the
shorthand notation, $[\xi,\eta]=\xi\eta^{\prime}-\eta\xi^{\prime}$.

$\displaystyle\left[\xi\,\frac{d}{d\theta},\eta\,\frac{d}{d\theta}\right]=(\xi\eta^{\prime}-\xi^{\prime}\eta)\frac{d}{d\theta}$
(2.70)

The Witt algebra (whose central extension is the Virasoro algebra) is a
realization of 1D diffeomorphism algebra. Indeed on a circle, the
realizations,

$\displaystyle\xi=ie^{im\theta}\partial_{\theta}=L_{m}$
$\displaystyle\eta=ie^{in\theta}\partial_{\theta}=L_{n}$ (2.71)
$\displaystyle\xi=-z^{m+1}\partial_{z}=L_{m}$
$\displaystyle\eta=-z^{n+1}\partial_{z}=L_{n}$ (2.72)

yield

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}$ (2.73)

Two copies of Witt algebra, with bases $\\{L_{m}\\}$ and $\\{\bar{L}_{n}\\}$
such that, for all $m,n\in\mathbb{Z}$, $[L_{m},\bar{L}_{n}]=0$, generate
conformal symmetry in 2D at the classical level.

####  2.4.2 Virasoro Algebra

We can centrally extend the 1D algebra by introducing a coordinate-invariant
cocycle $c(\ ,\ ):\mathfrak{g}\times\mathfrak{g}\rightarrow C$ with which the
bracket (2.69) is modified to

$\displaystyle[(\mathscr{L}_{\xi};a),(\mathscr{L}_{\eta};b)]=(\mathscr{L}_{\xi\circ\eta};c(\xi,\eta))$
(2.74)

In order that the Jacobi identity is satisfied, cocyle must be antisymmetric
and must satisfy the condition

$\displaystyle([\xi,\eta],\zeta)+([\eta,\zeta],\xi)+([\zeta,\xi],\eta)=0$
(2.75)

There are two commonly used conventions for the central extension of the
Virasoro algebra. The Gelfand-Fuchs cocyle is defined by [gelfand]

$\displaystyle
c(\xi,\eta)\equiv\int\frac{d\theta}{2\pi}\xi^{\prime}(\theta)\eta^{\prime\prime}(\theta)=-\frac{1}{2}\int\frac{d\theta}{2\pi}(\xi(\theta)\eta^{\prime\prime\prime}(\theta)-\xi^{\prime\prime\prime}(\theta)\eta(\theta))$
(2.76)

where the second equality follows by partial integrations and using the fact
that $\xi$ and $\eta$ are smooth vector fields on the circle. With this choice
we get

$\displaystyle\left[\xi\frac{d}{d\theta},\eta\frac{d}{d\theta}\right]=(\xi\eta^{\prime}-\xi^{\prime}\eta)\frac{d}{d\theta}-\frac{ic}{48\pi}\int_{0}^{2\pi}d\theta\
(\xi\eta^{\prime\prime\prime}-\xi^{\prime\prime\prime}\eta)$ (2.77)

where the additional factor of $ic/12$ is introduced for conventional purposes
[witten88]. Then in terms of the basis elements $\\{L_{m}\\}$ in (2.71) the
commutation relations become

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}m^{3}\delta_{m+n}$ (2.78)

In string theory (see e.g. [beckers] Section 2.4), the commonly used
convention differs by adding a constant to $L_{0}$ to replace (2.78) by

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n}+\frac{c}{12}(m^{3}-m)\delta_{m+n}$
(2.79)

Then the subset $\\{L_{-1},L_{0},L_{1}\\}$ is ”preserved”, i.e. does not
receive a contribution from the central extension. This subset generates the
Lie group SL$(2,\mathbb{R})$ or SU$(1,1)$. These are $2\times 2$ real matrices
with unit determinant.

####  2.4.3 Coadjoint Action of Virasoro Algebra

In this section, we follow the conventions set in [delius90]. Smooth vector
fields $\xi\in\text{Vect}(S^{1})$ on a circle generate orientation preserving
diffeomorphisms $F\in\text{Diff}(S^{1})$. The adjoint action of $F$ on a
smooth vector field $\xi(\theta)$ reads

$\displaystyle F:\xi(\theta)\mapsto\text{Ad}_{F}\xi\equiv\xi_{F}(\theta)$
(2.80)

such that

$\displaystyle\xi_{F}(F(\theta))=F^{\prime}(\theta)\xi(\theta)$ (2.81)

Indeed, for an infinitesimal diffeomorphism $F(\theta)=\theta-\eta(\theta)$
this reduces to (2.70)

$\displaystyle\delta\xi\equiv\xi_{\theta-\eta(\theta)}-\xi_{\theta}=\eta\xi^{\prime}-\eta^{\prime}\xi=[\eta,\xi]=\text{ad}_{\eta}\xi$
(2.82)

The central charge transforms as

$\displaystyle F:a\mapsto
a_{F}=a+\int\frac{d\theta}{2\pi}S(\theta,F)\xi(\theta)$ (2.83)

where $S(\theta,F)\equiv SF(\theta)$ is the Schwarzian derivative (B.27).
Indeed under an infinitesimal transformation $F(\theta)=\theta-\eta(\theta)$
we can compute

$\displaystyle
S(\theta,\theta-\eta(\theta))=-\eta^{\prime\prime\prime}(\theta)$ (2.84)

so that (2.83) reduces to (2.76)

$\displaystyle\delta
a=a_{\theta-\eta}-a_{\theta}=-\int\frac{d\theta}{2\pi}\,\eta^{\prime\prime\prime}\,\xi=\int\frac{d\theta}{2\pi}\,\eta\,\xi^{\prime\prime\prime}$
(2.85)

The coadjoint action is introduced by the invariant pairing555Note that the
dual space $\mathcal{G}^{*}$ considered here is not the set of all linear
functionals on $\mathcal{G}$. This is sometimes called the regular dual
[ovsienkobook] or the smooth dual [segal]. , $\left<\ |\
\right>:\mathcal{G}^{*}\times\mathcal{G}\rightarrow\mathbb{R}$ chosen to be

$\displaystyle\left<(u,b^{*})|(\xi,a)\right>=b^{*}a+\int\frac{d\theta}{2\pi}\,u(\theta)\,\xi(\theta)$
(2.86)

Invariance of the pairing under the action of a diffeomorphism $F$ means that

$\displaystyle\left<(u_{F},b_{F}^{*})|(\xi_{F},a_{F})\right>=\left<(u,b^{*})|(\xi,a)\right>$
(2.87)

Then the coadjoint action of $F$ on a coadjoint vector $(u,b^{*})$ is obtained
as

$\displaystyle u_{F}(F(\theta))\equiv\text{Ad}^{*}_{F}u$
$\displaystyle=(F^{\prime}(\theta))^{-2}(u(\theta)-b^{*}S(\theta,F))$ (2.88)
$\displaystyle b_{F}^{*}$ $\displaystyle=b^{*}$ (2.89)

Indeed using the given adjoint and coadjoint transformations we can verify the
invariance condition

$\displaystyle\left<(u_{F},b^{*}_{F})|(\xi_{F},a_{F})\right>$
$\displaystyle=\int\frac{df}{2\pi}u_{F}(f)\xi_{F}(f)+a_{F}b^{*}_{F}$
$\displaystyle=\int\frac{df}{2\pi}(F^{\prime}(\theta))^{-1}(u(\theta)-b^{*}SF(\theta))\xi(\theta)+b^{*}\int\frac{d\theta}{2\pi}SF(\theta)\xi(\theta)+b^{*}a$
$\displaystyle=\int\frac{d\theta}{2\pi}u(\theta)\xi(\theta)-b^{*}\int\frac{d\theta}{2\pi}SF(\theta)\xi(\theta)+b^{*}\int\frac{d\theta}{2\pi}SF(\theta)\xi(\theta)+b^{*}a$
$\displaystyle=b^{*}a+\int\frac{d\theta}{2\pi}u(\theta)\xi(\theta)$
$\displaystyle=\left<(u,b^{*}),(\xi,a)\right>$ (2.90)

where $f\equiv F(\theta)$.

For an infinitesimal diffeomorphism $F(\theta)=\theta-\xi(\theta)$ the
coadjoint action (2.88) reduces to

$\displaystyle\text{ad}^{*}_{\xi}\equiv\delta u\equiv
u_{\theta-\xi}-u_{\theta}=\xi
u^{\prime}+2\xi^{\prime}u+b^{*}\xi^{\prime\prime\prime}$ (2.91)

Note that the transformation (2.88) corresponds to a passive transformation.
The active version can be obtained by inverting (2.88) using
$\theta=F^{-1}(f)$ and using the Schwarzian identity (B.32). This yields

$\displaystyle
u_{F}(\theta)=(F^{\prime}(\theta))^{2}u(F(\theta))+b^{*}S(\theta,F)$ (2.92)

####  2.4.4 Coadjoint Element $\Sigma$ Formed from Affine Connection

Under a coordinate transformation $x\mapsto\bar{x}(x)$, affine connection
coefficients $\Gamma_{ab}^{c}$ transform as

$\displaystyle\bar{\Gamma}^{c}_{ab}(\bar{x})=\frac{\partial\bar{x}^{c}}{\partial
x^{d}}\frac{\partial x^{e}}{\partial\bar{x}^{a}}\frac{\partial
x^{f}}{\partial\bar{x}^{b}}\Gamma^{d}_{ef}(x)-\frac{\partial
x^{d}}{\partial\bar{x}^{a}}\frac{\partial
x^{e}}{\partial\bar{x}^{b}}\left(\frac{\partial^{2}\bar{x}^{c}}{\partial
x^{d}\partial x^{e}}\right)$ (2.93)

This deviates from the transformation of a (1,2)-tensor by the last term. In
1D it reduces to

$\displaystyle\bar{\Gamma}(\bar{x})=\frac{\partial
x}{\partial\bar{x}}\Gamma(x)-\left(\frac{\partial
x}{\partial\bar{x}}\right)^{2}\frac{\partial^{2}\bar{x}}{\partial x^{2}}$
(2.94)

For an infinitesimal coordinate transformation $\bar{x}=x-\xi(x)$ we have
$\partial\bar{x}/\partial x=1-\xi^{\prime}$ and $\partial
x/\partial\bar{x}=1+\xi^{\prime}$ (to first order in $\xi$). We shall use the
convention that if the argument of a field is suppressed, it is $x$ i.e. the
original coordinate. Plugging these into (2.94) we get

$\displaystyle\bar{\Gamma}(\bar{x})=\Gamma+\Gamma\xi^{\prime}+\xi^{\prime\prime}$
(2.95)

On the other hand, we also have (by Taylor expansion)

$\displaystyle\bar{\Gamma}(\bar{x})=\bar{\Gamma}(x-\xi)=\bar{\Gamma}-\xi\bar{\Gamma}^{\prime}=\bar{\Gamma}-\xi\Gamma^{\prime}$
(2.96)

Combining the two expressions we get

$\displaystyle\delta\Gamma:=\bar{\Gamma}-\Gamma=\xi\Gamma^{\prime}+\Gamma\xi^{\prime}+\xi^{\prime\prime}$
(2.97)

where $\delta\Gamma(x)$ is the Lie variation of $\Gamma(x)$ with respect to
the vector field $\xi(x)$.

Using $[\partial,\delta]=0$ we can compute

$\displaystyle\delta\Gamma^{\prime}=(\delta\Gamma)^{\prime}=\xi\Gamma^{\prime\prime}+2\xi^{\prime}\Gamma^{\prime}+\xi^{\prime\prime}\Gamma+\xi^{\prime\prime\prime}$
(2.98)

Since $\delta$ is a derivation we also have

$\displaystyle\delta(\Gamma^{2}/2)=\Gamma\delta\Gamma$ (2.99)

Combining the two results we can compute

$\displaystyle\delta(\Gamma^{\prime}-\Gamma^{2}/2)$
$\displaystyle=(\xi\Gamma^{\prime\prime}+2\xi^{\prime}\Gamma^{\prime}+\xi^{\prime\prime}\Gamma+\xi^{\prime\prime\prime})-\Gamma(\xi\Gamma^{\prime}+\Gamma\xi^{\prime}+\xi^{\prime\prime})$
$\displaystyle=\xi(\Gamma^{\prime}-\Gamma^{2}/2)^{\prime}+2\xi^{\prime}(\Gamma^{\prime}-\Gamma^{2}/2)+\xi^{\prime\prime\prime}$
(2.100)

In other words, the object

$\displaystyle\Sigma\equiv\Gamma^{\prime}-\Gamma^{2}/2$ (2.101)

transforms as a Virasoro coadjoint element with central charge one

$\displaystyle\delta\Sigma=\xi\Sigma^{\prime}+2\xi^{\prime}\Sigma+\xi^{\prime\prime\prime}$
(2.102)

Let us define the object $\Sigma_{k}\equiv\Gamma^{\prime}+k\Gamma^{2}$. Then
we can compute

$\displaystyle\delta\Sigma_{k}=\xi\Sigma_{k}^{\prime}+2\xi^{\prime}\Sigma_{k}+\xi^{\prime\prime}\Gamma(1+2k)+\xi^{\prime\prime\prime}$
(2.103)

So for $k\neq-1/2$ the object $\Sigma_{k}$ almost transforms as a Virasoro
coadjoint element, but there is an additional $\xi^{\prime\prime}$ center
which breaks the invariance of the pairing (2.86).

Now consider the object $c\Sigma$. Using (2.4.4) it is easy to see that
$c\Sigma$ transforms as a Virasoro coadjoint element of central charge $c$.

Next consider a rank-two tensor $S_{ab}$. It transforms under
$x\mapsto\bar{x}(x)$ as

$\displaystyle\bar{S}_{ab}(\bar{x})=\frac{\partial
x^{c}}{\partial\bar{x}^{a}}\frac{\partial
x^{d}}{\partial\bar{x}^{b}}S_{cd}(x)$ (2.104)

Following a similar analysis as in above, in 1D, we get

$\displaystyle\delta S\equiv\bar{S}-S=\xi S^{\prime}+2\xi^{\prime}S$ (2.105)

Adding a rank-two tensor to a rank-two object that transform as a Virasoro
coadjoint element does yield another object that transform as a Virasoro
coadjoint element with the same central charge. Indeed for an object $D$
defined by,

$\displaystyle D=S+c\Sigma$ (2.106)

we get the following transformation

$\displaystyle\delta D$ $\displaystyle=\delta
S+c\delta(\Gamma^{\prime}-\Gamma^{2}/2)$
$\displaystyle=\xi[S^{\prime}+c(\Gamma^{\prime\prime}-\Gamma\Gamma^{\prime})]+2\xi^{\prime}[S+c(\Gamma^{\prime}-\Gamma^{2}/2)]+c\xi^{\prime\prime\prime}$
$\displaystyle=\xi D^{\prime}+2\xi^{\prime}D+c\xi^{\prime\prime\prime}$
(2.107)

This calculation shows that we can use
$c\Sigma=c(\Gamma^{\prime}-\Gamma^{2}/2)$ as a core to build arbitrary
Virasoro coadjoint elements of central charge $c$ by adding arbitrary rank-two
tensors to it. In particular, we can build one from the spacetime metric $g$,
$D_{g}\equiv g+c(\Gamma^{\prime}-\Gamma^{2}/2)$.

Finally consider two objects that transform as Virasoro coadjoint elements,
$D_{1},D_{2}$ with the same central charge. Then we can compute

$\displaystyle\delta(D_{1}-D_{2})=\xi(D_{1}-D_{2})^{\prime}+2\xi^{\prime}(D_{1}-D_{2})$
(2.108)

Thus, the difference transforms as a rank-two tensor. This shows that using a
multiple of $(\Gamma^{\prime}-\Gamma^{2}/2)$ we can always extract a rank-two
tensor out of a Virasoro coadjoint element.

####  2.4.5 Higher Dimensional Lift of $\Sigma$

Although an affine connection does not transform as a tensor, its Lie
derivative does. To see this one first computes the pullback of the connection
coefficients (i.e. the usual coordinate transformation) and applies the formal
definition of the Lie derivative to get

$\displaystyle\mathcal{L}_{\xi}\Gamma^{\mu}_{\nu\lambda}=\xi^{\rho}\partial_{\rho}\Gamma^{\mu}_{\nu\lambda}-\partial_{\rho}\xi^{\mu}\Gamma^{\rho}_{\nu\lambda}+\partial_{\nu}\xi^{\rho}\Gamma^{\mu}_{\rho\lambda}+\partial_{\lambda}\xi^{\rho}\Gamma^{\mu}_{\nu\rho}+\partial_{\lambda}\partial_{\nu}\xi^{\mu}$
(2.109)

The first four terms are what you would expect from a (1,2)-tensor and the
last term is the inhomogeneous term representing the nontensoriality of
$\Gamma$. The last term can be rewritten as part of
$\nabla_{\lambda}\nabla_{\nu}\xi^{\mu}$, then one can show that

$\displaystyle\mathcal{L}_{\xi}\Gamma^{\mu}_{\nu\lambda}=\nabla_{\lambda}\nabla_{\nu}\xi^{\mu}-\xi^{\rho}R^{\mu}_{\
\ \nu\lambda\rho}$ (2.110)

The expression on the right is a tensor, so the Lie derivative of the
connection coefficients form a tensor.

As discussed in the previous section the object
$\Sigma=\Gamma^{\prime}-\Gamma^{2}/2$ in 1D is a Virasoro coadjoint element of
central charge one, so we can extract a pure tensor out of a diff field $D$ of
central charge $c$ as $D-c\Sigma$. We lift the diff field to a rank-two object
$D_{\mu\nu}$ in higher dimensions since this is the most natural lift that
follows from the coadjoint action666It is also possible to lift the diff field
to pseudotensor densities with appropriate weight. See for instance
[rodgers19942d].. Then $\Gamma^{\prime}$ and $\Gamma^{2}$ should also have two
free indices.

For the $\Gamma^{\prime}$ the possible lifts are
$\Gamma^{\prime}\sim\partial_{\lambda}\Gamma^{\lambda}_{\mu\nu},\partial_{\mu}\Gamma^{\lambda}_{\lambda\nu}$
and $\partial_{\nu}\Gamma^{\lambda}_{\mu\lambda}$, whereas for the
$\Gamma^{2}$ we have
$\Gamma^{\lambda}_{\mu\nu}\Gamma^{\sigma}_{\sigma\lambda},\Gamma^{\lambda}_{\mu\lambda}\Gamma^{\sigma}_{\nu\sigma}$
and $\Gamma^{\lambda}_{\mu\sigma}\Gamma^{\sigma}_{\nu\lambda}$. Hence we can
form a general combination

$\displaystyle\Sigma_{\mu\nu}\equiv
a\partial_{\lambda}\Gamma^{\lambda}_{\mu\nu}+b\partial_{\mu}\Gamma^{\lambda}_{\lambda\nu}+c\partial_{\nu}\Gamma^{\lambda}_{\mu\lambda}+d\Gamma^{\lambda}_{\mu\nu}\Gamma^{\sigma}_{\sigma\lambda}+e\Gamma^{\lambda}_{\mu\sigma}\Gamma^{\sigma}_{\nu\lambda}+f\Gamma^{\lambda}_{\mu\lambda}\Gamma^{\sigma}_{\nu\sigma}$
(2.111)

and subject it to the condition

$\displaystyle q\equiv a+b+c=-2(d+e+f)$ (2.112)

This yields a higher-dimensional, rank-two lift of a Virasoro coadjoint
element of central charge $q$.

We can show by direct computation

$\displaystyle\partial_{\mu}\Gamma^{\lambda}_{\lambda\nu}=\partial_{\nu}\Gamma^{\lambda}_{\mu\lambda}$
(2.113)

So there is a redundancy in (2.111). However, we intentionally introduced
these two terms to keep symmetry manifest. Also note that these are related to
the metric determinant by

$\displaystyle\partial_{\mu}\Gamma^{\lambda}_{\lambda\nu}=\partial_{\nu}\left(\frac{1}{\sqrt{g}}\partial_{\mu}\sqrt{g}\right)$
(2.114)

The most natural lift of the Virasoro coadjoint transformation (with linear
center term ignored) (2.107) reads

$\displaystyle\delta
D_{\mu\nu}=\xi^{\lambda}\partial_{\lambda}D_{\mu\nu}+\partial_{\mu}\xi^{\lambda}D_{\lambda\nu}+\partial_{\nu}\xi^{\lambda}D_{\mu\lambda}+q\partial_{\mu}\partial_{\nu}\partial_{\lambda}\xi^{\lambda}$
(2.115)

which we use for building the transverse action for the diff field in Section
4.3. So the question is ”Does the object $\Sigma_{\mu\nu}$ defined in (2.111)
satisfy

$\displaystyle\delta\Sigma_{\mu\nu}=\xi^{\lambda}\partial_{\lambda}\Sigma_{\mu\nu}+\partial_{\mu}\xi^{\lambda}\Sigma_{\lambda\nu}+\partial_{\nu}\xi^{\lambda}\Sigma_{\mu\lambda}+q\partial_{\mu}\partial_{\nu}\partial_{\lambda}\xi^{\lambda}$
(2.116)

given the Lie derivative $\delta\Gamma^{\mu}_{\nu\lambda}$ (2.109)?” The
answer turns out to be negative. Here are the results : We are going to denote
the true Lie variation, i.e. one obtained using (2.109) by
$\delta_{\text{tr}}\Sigma_{\mu\nu}$, and the Lie variation obtained from the
ansatz (2.116) by $\delta_{\text{an}}\Sigma_{\mu\nu}$. Then we define the
difference

$\displaystyle\Delta_{\mu\nu}\equiv\delta_{\text{tr}}\Sigma_{\mu\nu}-\delta_{\text{an}}\Sigma_{\mu\nu}$
(2.117)

We find

$\displaystyle\Delta_{\mu\nu}$
$\displaystyle=(-a+d)\Gamma^{\rho}_{\mu\nu}\partial_{\rho}\partial_{\sigma}\xi^{\sigma}+(a+e)(\Gamma^{\rho}_{\sigma\nu}\partial_{\mu}\partial_{\rho}\xi^{\sigma}+\Gamma^{\rho}_{\sigma\mu}\partial_{\rho}\partial_{\nu}\xi^{\sigma})$
$\displaystyle+f(\Gamma^{\rho}_{\rho\nu}\partial_{\mu}\partial_{\sigma}\xi^{\sigma}+\Gamma^{\rho}_{\rho\mu}\partial_{\nu}\partial_{\sigma}\xi^{\sigma})+(b+c+d)\Gamma^{\rho}_{\rho\sigma}\partial_{\mu}\partial_{\nu}\xi^{\sigma}$
(2.118)

Notice that the difference is only made up of terms of order
$\xi^{\prime\prime}$, and 1D reduction of $\Delta$ vanishes with the condition
(2.112) as expected.

The next question is whether a subcase (with some of the terms set to zero)
subject to condition (2.112) yields a vanishing $\Delta_{\mu\nu}$. The answer
turns out to be negative again. There are simple cases which come close to the
goal. For instance for the case $b=1=c=-d$ and $a=0=e=f$ we get

$\displaystyle\Sigma_{\mu\nu}=\partial_{\mu}\Gamma^{\lambda}_{\lambda\nu}+\partial_{\nu}\Gamma^{\lambda}_{\mu\lambda}-\Gamma^{\lambda}_{\mu\nu}\Gamma^{\sigma}_{\sigma\lambda}$
(2.119)

and

$\displaystyle\Delta_{\mu\nu}=-\Gamma^{\rho}_{\mu\nu}\partial_{\sigma}\partial_{\rho}\xi^{\sigma}+\Gamma^{\rho}_{\rho\sigma}\partial_{\mu}\partial_{\nu}\xi^{\sigma}$
(2.120)

In Section 4.3, we are going to obtain (2.107) from (2.115) by some gauge
fixing arguments instead of the direct dimensional lifting.

####  2.4.6 Covariant Cocyle

Consider the Gelfand-Fuchs cocyle introduced before777For simplicity, $c$ is
scaled by $2\pi$.

$\displaystyle c(\xi,\eta)=-\frac{1}{2}\int d\theta\
(\xi\eta^{\prime\prime\prime}-\xi^{\prime\prime\prime}\eta)$ (2.121)

If we place an affine connection $\nabla$ on circle this cocyle can be
extended to

$\displaystyle c_{\Gamma}(\xi,\eta)=\frac{1}{2}\int
dx^{a}(\xi^{b}\nabla_{a}\nabla_{b}\nabla_{c}\eta^{c})-(\xi\leftrightarrow\eta)$
(2.122)

covariantly, in higher dimensions. Expanding the derivatives, in 1D, we get

$\displaystyle c_{\Gamma}(\xi,\eta)$ $\displaystyle=\frac{1}{2}\int dx\
\xi(\partial-\Gamma)\partial(\partial+\Gamma)\eta-(\xi\leftrightarrow\eta)$
$\displaystyle=\frac{1}{2}\int dx\
\xi(\eta^{\prime\prime\prime}+(2\Gamma^{\prime}-\Gamma^{2})\eta^{\prime})-(\xi\leftrightarrow\eta)$
(2.123)

Therefore we obtain [courant]

$\displaystyle c_{\Gamma}(\xi,\eta)=\int dx\
\frac{1}{2}(\xi\eta^{\prime\prime\prime}-\xi^{\prime\prime\prime}\eta)+\int
dx\ (\xi\eta^{\prime}-\xi^{\prime}\eta)(\Gamma^{\prime}-\Gamma^{2}/2)$ (2.124)

We can rewrite this in terms of the pairing (2.86) as

$\displaystyle
c_{\Gamma}(\xi,\eta)=c(\xi,\eta)+\left<\Gamma^{\prime}-\Gamma^{2}/2\,|\,[\xi,\eta]\right>$
(2.125)

Since we have shown that $\Sigma=\Gamma^{\prime}-\Gamma^{2}/2$ transforms as a
Virasoro coadjoint element of central charge one we can interpret the last
term as the Kirillov form $\Omega$ on the coadjoint orbit of $\Sigma$,
evaluated on two tangent vectors $\Sigma_{\xi},\Sigma_{\eta}$ obtained by the
action of the adjoint vectors $\xi,\eta$ (equation (2.23)). That is,

$\displaystyle c_{\Gamma}(\xi,\eta)-c(\xi,\eta)$
$\displaystyle=\Omega_{\Sigma}(\Sigma_{\xi},\Sigma_{\eta})$ (2.126)
$\displaystyle\Sigma_{\xi}\equiv\text{ad}^{*}_{\xi}\Sigma\ \ ,\ \ $
$\displaystyle\Sigma_{\eta}\equiv\text{ad}^{*}_{\eta}\Sigma$ (2.127)

####  2.4.7 Chiral Splitting of Curvature

In this section we would like to investigate an interesting possibility
related to the $N$D generalization of the diff field-affine connection
relationship. Consider the Ricci curvature tensor

$\displaystyle R_{\alpha\beta}\equiv R^{\rho}_{\ \
\alpha\rho\beta}=\partial_{\rho}\Gamma^{\rho}_{\beta\alpha}-\partial_{\beta}\Gamma^{\rho}_{\rho\alpha}+\Gamma^{\rho}_{\rho\lambda}\Gamma^{\lambda}_{\beta\alpha}-\Gamma^{\rho}_{\beta\lambda}\Gamma^{\lambda}_{\rho\alpha}$
(2.128)

Adding and subtracting the term
$a\partial_{\rho}\Gamma^{\rho}_{\beta\alpha}+(1-a)\partial_{\beta}\Gamma^{\rho}_{\rho\alpha}$
we can rewrite this as

$\displaystyle R_{\alpha\beta}$
$\displaystyle=\Big{(}(1+a)\partial_{\rho}\Gamma^{\rho}_{\beta\alpha}+(1-a)\partial_{\alpha}\Gamma^{\rho}_{\rho\beta}-\Gamma^{\rho}_{\beta\lambda}\Gamma^{\lambda}_{\rho\alpha}\Big{)}-\Big{(}a\partial_{\beta}\Gamma^{\rho}_{\rho\alpha}+(2-a)\partial_{\alpha}\Gamma^{\rho}_{\rho\beta}-\Gamma^{\rho}_{\rho\lambda}\Gamma^{\lambda}_{\beta\alpha}\Big{)}$
$\displaystyle\equiv R^{+}_{\alpha\beta}-R^{-}_{\alpha\beta}$ (2.129)

The point of this definition is that the 1D reduction of
$R^{\pm}_{\alpha\beta}$ are each given by

$\displaystyle R^{\pm}=2\Gamma^{\prime}-\Gamma^{2}$ (2.130)

Each transforms as a Virasoro coadjoint element with central charge two. Note
that although the Ricci tensor vanishes in 1D, $R^{\pm}$ do not. Note also
that we have
$\partial_{\beta}\Gamma^{\rho}_{\rho\alpha}=\partial_{\alpha}\Gamma^{\rho}_{\rho\beta}$
so that $R^{\pm}_{\alpha\beta}$ are each symmetric. We could also add and
subtract possible $\Gamma^{2}$ terms to obtain a more generic splitting as
long as we keep the ratio of coefficients of $\Gamma^{\prime}$ and
$\Gamma^{2}$ as in (2.130).

Now, if we introduce two copies of the diff field, $D^{\pm}$, each with
central charge $c$, it is possible to obtain two pure tensors out of the diff
field in 1D :

$\displaystyle c\tilde{R}^{\pm}\equiv D^{\pm}-cR^{\pm}$ (2.131)

Here $R^{\pm}$ are postulated to represent two ”chiral components” of the
Ricci tensor and $\tilde{R}^{\pm}$ are the tensorial chiral components of the
diff-corrected curvature tensor. Explicitly we have

$\displaystyle c\tilde{R}$ $\displaystyle=cR-(D^{+}-D^{-})$
$\displaystyle=(cR^{+}-D^{+})-(cR^{-}-D^{-})$
$\displaystyle=c\tilde{R}^{+}-c\tilde{R}^{-}$ (2.132)

Therefore, at least in 1D, we can construct a curvature tensor $\tilde{R}$
whose chiral components $\tilde{R}^{\pm}$ are tensorial with the chiral diff
corrections. Whether this result would be extended to $N$D is an interesting
mathematical quest to pursue.

####  2.4.8 Schwarzian Chain

Consider the active transformation of a Virasoro coadjoint element under the
action of a diffeomorphism $g$,

$\displaystyle\tilde{u}(x)=\left(\frac{dg}{dx}\right)^{2}u(g(x))+c\,Sg(x)$
(2.133)

For a coadjoint element made of central charge $c$ only, this implies

$\displaystyle(0,c)\overset{g}{\mapsto}(c\,Sg(x),c)$ (2.134)

That is, for $u=0$ we have $\tilde{u}(x)=c\,Sg(x)$ under the map
$\text{Ad}_{g}^{*}$. Then using the identity (B.30), under the action of a
second diffeomorphism $h$ we get

$\displaystyle(0,c)\overset{g}{\mapsto}(c\,Sg(x),c)\overset{h}{\mapsto}(c\,S(g\circ
h)(x),c)$ (2.135)

Therefore, the Schwarzian derivative operator is an invariant of the Virasoro
coadjoint action connecting a zero element to an infinite chain of nonzero
elements obtained by diffeomorphisms.

Now use the identity (B.30) again, but with the second transformation made
infinitesimal888The computation here is in the active picture, so the
transformation is taken to be $x\mapsto x+\xi$., $x\mapsto h(x)=x+\xi(x)$,

$\displaystyle S(g\circ(x+\xi))(x)$
$\displaystyle=Sg(x+\xi)(1+2\xi^{\prime})+\xi^{\prime\prime\prime}$
$\displaystyle(Sg)_{\xi}(x)$
$\displaystyle=(1+2\xi^{\prime})Sg(x)+\xi(Sg)^{\prime}(x)+\xi^{\prime\prime\prime}$
$\displaystyle(Sg)_{\xi}(x)$
$\displaystyle=Sg(x)+\xi(Sg)^{\prime}+2\xi^{\prime}(Sg)+\xi^{\prime\prime\prime}$
(2.136)

where we defined $S(g\circ(x+\xi))=S(g+g\circ\xi)\equiv(Sg)_{\xi}$. Hence, we
get

$\displaystyle\delta(Sg)(x)\equiv(Sg)_{\xi}(x)-(Sg)(x)=\xi(Sg)^{\prime}+2\xi^{\prime}(Sg)+\xi^{\prime\prime\prime}$
(2.137)

Therefore, we see that the Schwarzian derivative of a diffeomorphism
transforms infinitesimally as a Virasoro coadjoint element of central charge
one.

###  2.5 Semi-direct Product of Virasoro and Kac-Moody Algebras

Commutation relations of the semi-direct product of Virasoro and Kac-Moody
algebras are given by

$\displaystyle[L_{m},L_{n}]$
$\displaystyle=(m-n)L_{m+n}+(cm^{3}+hm)\delta_{m+n}I_{\text{Vir}}$
$\displaystyle[J_{m}^{a},J_{n}^{b}]$
$\displaystyle=if^{abc}J_{m+n}^{c}+km\delta_{m+n}\delta^{ab}I_{\text{KM}}$
$\displaystyle[L_{m},J^{a}_{n}]$ $\displaystyle=-nJ^{a}_{m+n}$
$\displaystyle[I_{\text{Vir}},\text{all}]$
$\displaystyle=0=[I_{\text{KM}},\text{all}]$ (2.138)

where we introduced generators of centers $I_{\text{Vir}},I_{\text{KM}}$ for
each of the algebras, and we did not fix the Virasoro cocyle as in (2.79).
Realization of the basis elements in angular and complex coordinates are given
by

$\displaystyle L_{m}(\theta)$
$\displaystyle=ie^{im\theta}\partial_{\theta}\hskip 14.45377pt$ $\displaystyle
J^{a}_{m}(\theta)=T^{a}e^{im\theta}$ angular (2.139) $\displaystyle L_{m}(z)$
$\displaystyle=-z^{m+1}\partial_{z}$ $\displaystyle J^{a}_{m}(z)=T^{a}z^{m}$
complex (2.140)

These are related by $z=e^{i\theta}$. Note that these realizations satisfy
only the non-central part of (2.138).

Dual elements will be denoted with tildes $\tilde{L}_{m}$ and
$\tilde{J}^{a}_{m}$. They are defined through the individual invariant
pairings of the algebras without central extension

$\displaystyle\left<\tilde{L}_{m}\Big{|}L_{n}\right>=\delta_{mn}\ \ \ ,\ \ \
\left<\tilde{J}^{a}_{m}\Big{|}J^{b}_{n}\right>=\delta_{mn}\delta^{ab}$ (2.141)

For the semi-direct product algebra we form an adjoint basis element
$(L_{m},J_{n}^{a},\mu)$ and a coadjoint basis element
$(\tilde{L}_{m},\tilde{J}_{n}^{a},\tilde{\mu})$. Then we introduce the
invariant pairing

$\displaystyle\left<(\tilde{L}_{m},\tilde{J}^{a}_{n},\tilde{\mu})\Big{|}(L_{m^{\prime}},J^{a^{\prime}}_{n^{\prime}},\mu\right>=\delta_{mm^{\prime}}+\delta_{nn^{\prime}}\delta^{aa^{\prime}}+\tilde{\mu}\mu$
(2.142)

Recall the infinitesimal form of invariance of the pairing : if $u,v$ denote
two adjoint elements and $\alpha$ denote a coadjoint element then invariance
reads

$\displaystyle
0=u*\left<\alpha|v\right>=\left<u*\alpha|v\right>+\left<\alpha|u*v\right>$
(2.143)

The commutation relations (2.138) yield $u*v$ so that using (2.142) one can
compute $\left<\alpha|u*v\right>$. Then again using (2.142) one can deduce
$u*\alpha$, namely, the infinitesimal coadjoint acton for the semi-direct
product algebra. We will state the result for generic adjoint and coadjoint
elements below.

From the basis elements of the Virasoro and KM algebras and their duals we can
construct generic adjoint and coadjoint elements of the algebras as999The
negative sign in the Virasoro adjoint element is introduced to avoid negative
signs in the Lie derivative by switching the passive $x\mapsto x-\xi$ and
active $x\mapsto x+\xi$ transformations.

$\displaystyle\xi(\theta)=-\xi^{n}L_{n}(\theta)$
$\displaystyle\Lambda(\theta)=\Lambda_{a}^{n}J^{a}_{n}(\theta)$ adjoint
(2.144) $\displaystyle D(\theta)=D^{n}\tilde{L}_{n}(\theta)$ $\displaystyle
A(\theta)=A_{a}^{n}\tilde{J}^{a}_{n}(\theta)$ coadjoint (2.145)

Then generic elements of the semi-direct product algebra become

adjoint: $\displaystyle\mathcal{F}=(\xi(\theta),\Lambda(\theta),a)$ (2.146)
coadjoint: $\displaystyle B=(D(\theta),A(\theta),\mu)$ (2.147)

Finally, we can write the (infinitesimal) coadjoint action as

$\displaystyle\delta_{\mathcal{F}}B\equiv ad^{*}_{\mathcal{F}}B=(\delta
D(\theta),\delta A(\theta),0)$ (2.148)

By the procedure described following equation (2.143), one can compute
[lano92]

$\displaystyle\delta D(\theta)$
$\displaystyle=2\xi^{\prime}D+D^{\prime}\xi+\frac{c\mu}{2\pi}\xi^{\prime\prime\prime}+\frac{h\mu}{2\pi}\xi^{\prime}-\text{Tr}\
(A\Lambda^{\prime})$ (2.149a) $\displaystyle\delta A(\theta)$
$\displaystyle=A^{\prime}\xi+\xi^{\prime}A-[\Lambda,A]+k\mu\Lambda^{\prime}$
(2.149b)

Setting $\delta D$ and $\delta A$ to zero we get the isotropy equations for
the semi-direct product algebra.

##  Chapter 3 GEOMETRIC ACTIONS

###  3.1 WZW Action

Wess-Zumino-Witten (WZW) model arises in a variety of phenomena in 2D field
theories. First, we review its motivation. Namely, it is a closed form
solution in 2D to the Wess-Zumino (WZ) functional, describing the low-energy
effective action of QCD, and encoding the chiral anomaly. Then we discuss
bosonization, namely the equivalence between WZW model and the theory of 2D
chiral fermions. Finally we review Polyakov and Wiegmann’s treatment which
further clarifies its relation to chiral anomaly and motivates the
correspondence between WZW theory and the P2DG theory in LCG. In the following
we mainly follow [divecchia], [WZNW], [polyakovwiegmann] and
[polyakovwiegmann84].

####  3.1.1 WZ Functional in 2D

Reconsider the massless, free theory of fermions in 2D having a
U$(N)\times\text{U}(N)$ chiral flavor symmetry, introduced in section 2.3.2.
If we couple this theory to a background gauge field (i.e. its dynamics can be
ignored for the discussion) $A_{\mu}$ the chiral symmetry is broken by the
axial anomaly [adleraxial], [belljackiwaxial]. With the gauge coupling the
action reads

$\displaystyle S_{F}[\psi,\overline{\psi},A_{\mu}]=\frac{i}{2}\int d^{2}x\
\overline{\psi}_{k}\not{D}\psi^{k}$ (3.1)

where $\not{D}=i\gamma^{\mu}(\partial_{\mu}+A_{\mu})$ and
$A_{\mu}=v_{\mu}+\gamma_{5}a_{\mu}$. Here $v_{\mu}$ is the vector gauge field
and $a_{\mu}$ is the axial gauge field. One way to express the anomaly is
through the effective action111There is a closely related but distinct notion
of effective action, denoted by $\Gamma$ in the literature. For the
distinction and the relationship between the two, see [bilalanomaly] Section
3.6. which is obtained by path integration over the fermionic degrees of
freedom. The result is formally written as

$\displaystyle W[A_{\mu}]=\log\text{Det}\not{D}=\text{Tr}\log\not{D}$ (3.2)

It is more convenient to work with the chiral components of the gauge field,
$A^{L}_{\mu}$, $A^{R}_{\mu}$ which transform under the action of
$(g_{L},g_{R})\in$ U$(N)\times$U$(N)$ as

$\displaystyle A^{L}_{\mu}\rightarrow
g^{-1}_{L}(\partial_{\mu}+A_{\mu}^{L})g_{L}\ ,\ \ A^{R}_{\mu}\rightarrow
g^{-1}_{R}(\partial_{\mu}+A_{\mu}^{R})g_{R}$ (3.3)

The anomaly is exposed through the evaluation of the formal fermion
determinant by a choice of regularization. The problem is in the measure of
the path integral [fujikawa]. There one sees that there is no regulator that
preserves both the vector symmetry and the axial symmetry simultaneously.
Thus, one is forced to choose preserving one of the symmetries, losing the
other. Anomalous gauge symmetry is catastrophic for a theory since it leads to
nonrenormalizability and states of negative norm, thereby to the violation of
unitarity[bilalanomaly]. Hence, one evaluates the determinant by a regulator
preserving the vector symmetry, which forms a subgroup of the gauge group,
sacrificing the axial symmetry. In other words, under a vector transformation
$(g_{L},g_{R})$ with $g_{L}=g_{R}$, $W[A_{\mu}]^{\text{reg}}$ is invariant
whereas under a chiral transformation, i.e. $(g_{L},g_{R})$ with
$g_{R}=g_{L}^{-1}\equiv g$ it changes by

$\displaystyle
W[A^{g}_{\mu}]^{\text{reg}}=W[A_{\mu}]^{\text{reg}}+\text{WZ}(g^{2},A_{\mu})$
(3.4)

This defines the Wess-Zumino functional, WZ, encoding the chiral anomaly.

The WZ functional has been evaluated explicitly in 2D by Witten [WZNW], taking
the name WZW. For this purpose, consider the following complexified
parametrization of the 2D gauge field :

$\displaystyle A_{+}$ $\displaystyle=A_{0}+iA_{1}\equiv B^{-1}\partial_{+}B$
(3.5) $\displaystyle A_{-}$ $\displaystyle=A_{0}-iA_{1}\equiv
C^{-1}\partial_{-}C$ (3.6)

Here, $A_{+}=A_{-}^{\dagger}$ implies $B^{-1}=C^{\dagger}$. Then the effective
action is given by

$\displaystyle W[A_{\mu}]=I[BC^{-1}]$ (3.7)

where $I$ is the WZW functional given by

$\displaystyle I[G]=\frac{1}{8\pi}\int_{\partial Q}d^{2}x\
\text{tr}(\partial_{\mu}G\partial_{\mu}G^{-1})-\frac{i}{12\pi}\int_{Q}d^{3}x\
\epsilon^{ABC}\
\text{tr}(G^{-1}\partial_{A}GG^{-1}\partial_{B}GG^{-1}\partial_{C}G)$ (3.8)

Here $Q$ is a 3D hemisphere with compactified 2D space as its boundary.
Alternatively $\partial Q=S^{2}$ and $Q$ is a three-ball [WZNW]. The map $G$
originally defined on the two-dimensional boundary has topologically (more
precisely homotopically) distinct possible extensions to the three-space $Q$.
The last term above can be evaluated to be $2\pi n$, where $n\in\mathbb{Z}$
where $n$ is the winding number specifying the homotopy class of the extended
map222To make this distinction apparent, Witten [WZNW] denotes the extended
map by a hat, so that all the $G$’s in the second integrand are hatted. We
will use the same symbol for the original and extended maps throughout for
simplicity.. For more details, see [WZNW], or [NairQFT] Section 17.6.

Under a vector gauge transformation $g\in\text{U}(N)$ we have $B\rightarrow
Bg$ and $C\rightarrow Cg$ so that the effective action $I(BC^{-1})$ is vector
gauge invariant as desired. On the other hand, under a chiral transformation,
$C\rightarrow Cg$ and $B\rightarrow Bg^{-1}$ we get

$\displaystyle
W[A^{g}_{\mu}]=I[Bg^{2}C^{-1}]=I[BC^{-1}]+\text{WZ}[g^{2},A_{\mu}]$ (3.9)

or

$\displaystyle\text{WZ}[U,A_{\mu}]=I[BUC^{-1}]-I[BC^{-1}]$ (3.10)

A straightforward calculation shows that

$\displaystyle\text{WZ}[U,A_{\mu}]=I[U]+\frac{1}{4\pi}\int d^{2}x\
\text{tr}(A_{+}U\partial_{-}U^{-1}$ $\displaystyle+A_{-}U^{-1}\partial_{+}U$
$\displaystyle+A_{+}UA_{-}U^{-1}-A_{+}A_{-})$ (3.11)

Under a chiral transformation defined by

$\displaystyle U\rightarrow g^{-1}Ug^{-1}\ \ ,\ \ B\rightarrow Bg\ \ ,\ \
C\rightarrow Cg^{-1}$ (3.12)

we get

$\displaystyle-\text{WZ}[U^{g},A^{g}_{\mu}]$
$\displaystyle=I[Bg^{2}C^{-1}]-I[BUC^{-1}]$
$\displaystyle=-\text{WZ}[U,A_{\mu}]+\text{WZ}[g^{2},A_{\mu}]$ (3.13)

Comparing with (3.4) we see that $-\text{WZ}$ can be taken as the effective
action $W$. Next, we discuss the bosonization, namely the (quantum)
equivalence of the bosonic $\text{WZ}[U,A_{\mu}]$ theory and the original
Fermi theory in the background field $A_{\mu}$.

####  3.1.2 Bosonization of Chiral Fermion Theory

The action (3.1) of chiral fermions coupled to a background gauge field
$A_{\mu}$ can be rewritten as

$\displaystyle S_{F}[\psi,\bar{\psi},A_{\mu}]=\int d^{2}x\
[\bar{\psi}i\not{\partial}\psi+\text{tr}(J_{+}A_{-}+J_{-}A_{+})]$ (3.14)

where $J_{\pm}$ are the chiral currents. Equivalence of the WZW functional to
the fermion theory is an example of bosonization, and it can be formally
expressed as

$\displaystyle\int\mathcal{D}\psi\ \mathcal{D}\bar{\psi}\
\exp(-S_{F}[\psi,\bar{\psi},A_{\mu}])=\text{const}\times\int\mathcal{D}U\
\exp(-\text{WZ}[U,A_{\mu}])$ (3.15)

where the bosonic functional measure $\mathcal{D}U$ is formally a product of
Haar measures on U$(N)$. Unlike the Fermi theory, the Bose theory is assumed
not to have any anomalies. Instead the lack of chiral invariance is explicit
in the bosonic action, whereas the quantum measure $\mathcal{D}U$ is taken
chirally invariant. Indeed one uses an identity obtained from the chiral
invariance of the Haar measure to reach the bosonization result (3.15).

Then for the abelian case, $U=\exp(i\varphi)$ with a scalar field $\varphi$,
we get $\mathcal{D}U=\mathcal{D}\varphi$, and using (3.1.1) the bosonization
formula (3.15) simplifies to

$\displaystyle\int\mathcal{D}$ $\displaystyle\psi\ \mathcal{D}\bar{\psi}\
\exp\left(\int d^{2}x\
(\bar{\psi}i\not{\partial}\psi+J_{+}A_{-}+J_{-}A_{+})\right)$
$\displaystyle=\text{const}\int\mathcal{D}\varphi\ \exp\left(-\int d^{2}x\
\left[\frac{1}{8\pi}\partial_{\mu}\varphi\partial^{\mu}\varphi+\frac{i}{4\pi}(A_{+}\partial_{-}\varphi+A_{-}\partial_{+}\varphi\right]\right)$
(3.16)

By taking functional derivatives with respect to $A_{+}$ and $A_{-}$ we see
that

$\displaystyle\left<J_{+}(x_{-})\cdots J_{+}(x_{n})J_{-}(y_{1})\cdots
J_{-}(y_{m})\right>_{F}$ $\displaystyle\hskip
14.45377pt=\left(-\frac{i}{4\pi}\right)^{n+m}\left<\partial_{+}\varphi(x_{1})\cdots\partial_{+}\varphi(x_{n})\partial_{-}\varphi(y_{1})\cdots\partial_{-}\varphi(y_{m})\right>_{B}$
(3.17)

where $\left<\cdots\right>_{F}$ and $\left<\cdots\right>_{B}$ denote
expectation values in the fermionic and bosonic theory, respectively, thereby
expressing the usual result of the bosonization prescription i.e.
$J_{\pm}\leftrightarrow\partial_{\pm}\varphi$ [colemanbosonization],
[mandelslambosonization].

Similarly, in the nonabelian case, for a generic $U$ in (3.1.1), varying
(3.15) with respect to $A_{-}$ and setting $A_{\pm}=0$ we get

$\displaystyle\left<J_{+}^{mn}(x)J_{+}^{m^{\prime}n^{\prime}}(x^{\prime})\cdots\right>_{F}$
$\displaystyle\hskip
21.68121pt=\left<(-1/4\pi)(U^{-1}\partial_{+}U)^{mn}(x)(-1/4\pi)(U^{-1}\partial_{+}U)^{m^{\prime}n^{\prime}}(x^{\prime})\cdots\right>_{B}$
(3.18)

Varying (3.15) with respect to $A_{+}$ and setting $A_{\pm}=0$ we get

$\displaystyle\left<J_{-}^{mn}(x)J_{-}^{m^{\prime}n^{\prime}}(x^{\prime})\cdots\right>_{F}$
$\displaystyle\hskip
21.68121pt=\left<(-1/4\pi)(U\partial_{+}U^{-1})^{mn}(x)(-1/4\pi)(U\partial_{+}U^{-1})^{m^{\prime}n^{\prime}}(x^{\prime})\cdots\right>_{B}$
(3.19)

These verify333See [divecchia] for the mixed correlators
$\left<J_{+}J_{-}\cdots\right>$.  the bosonization rules introduced by Witten
[WZNW]

$\displaystyle J^{mn}_{+}\leftrightarrow(-1/4\pi)(U^{-1}\partial_{+}U)^{mn}\ \
,\ \ J^{mn}_{-}\leftrightarrow(-1/4\pi)(U\partial_{+}U^{-1})^{mn}$ (3.20)

These satisfy [WZNW]

$\displaystyle[\text{Tr}AJ_{-}(x),\text{Tr}BJ_{-}(y)]$
$\displaystyle=2i\delta(x-y)\text{Tr}[A,B]J_{-}(x)+\frac{iN}{\pi}\delta^{\prime}(x-y)\text{Tr}AB$
(3.21) $\displaystyle[\text{Tr}AJ_{+}(x),\text{Tr}BJ_{+}(y)]$
$\displaystyle=2i\delta(x-y)\text{Tr}[A,B]J_{+}(x)-\frac{iN}{\pi}\delta^{\prime}(x-y)\text{Tr}AB$
(3.22) $\displaystyle[\text{Tr}AJ_{-}(x),\text{Tr}BJ_{+}(y)]$
$\displaystyle=0$ (3.23)

where $A$ and $B$ are arbitrary antisymmetric matrices. These are equivalent
to (2.55). To see this take $A=M^{a}$, $B=M^{b}$ with the $O(N)$ basis
$\\{M^{a}\\}$ satisfying (2.52) and the normalization condition
$\text{Tr}(M^{a},M^{b})=2\delta^{ab}$.

####  3.1.3 Polyakov-Wiegmann’s treatment

Here is another calculation [polyakovwiegmann], [polyakovwiegmann84] which
better shows that the WZW functional is the integrated anomaly, and the
correspondence between the WZW action and the P2DG action in LCG (to be
discussed in the next section).

Consider the formal Dirac determinant in 2D

$\displaystyle
W[A_{\mu}]=\log\text{Det}(\gamma^{\mu}(i\partial_{\mu}+A_{\mu}))$ (3.24)

The quantum current can be defined through

$\displaystyle J_{\mu}=\frac{\delta W}{\delta A_{\mu}}$ (3.25)

One can choose a regularization such that the following quantum relations hold

$\displaystyle\partial_{\mu}J^{\mu}+[A_{\mu},J^{\mu}]$ $\displaystyle=0$
(3.26a)
$\displaystyle\epsilon^{\mu\nu}(\partial_{\mu}J_{\nu}+[A_{\mu},J_{\nu}])$
$\displaystyle=\frac{1}{2\pi}\epsilon^{\mu\nu}F_{\mu\nu}$ (3.26b)

The first equation simply states the conservation of the vector current
$J^{\mu}$. Note that due to the identity
$\overline{\psi}\gamma^{\mu}\gamma_{5}\psi=\epsilon^{\mu\nu}\overline{\psi}\gamma_{5}\psi$
the left-hand side of the second equation is equivalent to
$\partial_{\mu}J_{5}^{\mu}+[A_{\mu},J^{\mu}_{5}]$ so that this equation states
the chiral anomaly i.e. the nonconservation of the chiral current
$J^{\mu}_{5}$ with the anomaly function given on the right-hand side.

If we switch to the LCC taken in this section as $x^{\pm}=x^{0}\pm x^{1}$ and
introduce the parametrizations

$\displaystyle A_{+}=g^{-1}\partial_{+}g\ \ ,\ \ A_{-}=h^{-1}\partial_{-}h$
(3.27)

for the gauge field we see that equations (3.26) are solved for the chiral
currents by

$\displaystyle J_{+}=g^{-1}\partial_{+}g-h^{-1}\partial_{+}h$ (3.28a)
$\displaystyle J_{-}=h^{-1}\partial_{-}h-g^{-1}\partial_{-}g$ (3.28b)

Now let us restrict our attention to the axial gauge $A_{-}=0$, $h=I$ we get
the following variation for the effective action

$\displaystyle\delta W$ $\displaystyle=\int d^{2}x\ \text{Tr}(J_{-}\delta
A_{+})$ (3.29) $\displaystyle=\int d^{2}x\
\text{Tr}(\partial_{-}(g^{-1}\partial_{+}g)\delta gg^{-1})$ (3.30)

The solution to this equation is none other than the WZW functional

$\displaystyle W[g]$ $\displaystyle=\frac{1}{2}\int_{\partial Q=S^{2}}d^{2}x\
\text{Tr}(\partial_{\mu}g^{-1}\partial^{\mu}g)$ $\displaystyle\ \ \ \
+\frac{i}{8\pi^{2}}\int_{Q}d^{3}y\
\epsilon^{ABC}\,\text{Tr}(g^{-1}\partial_{A}gg^{-1}\partial_{B}gg^{-1}\partial_{C}g)$
(3.31)

If the $A_{-}=0$ gauge is turned off then the effective action becomes
[polyakovwiegmann84]

$\displaystyle W[A_{\mu}]=W_{+}(A_{+})+W_{-}(A_{-})+\text{Tr}(A_{+}A_{-})$
(3.32)

where the last term is a dimensionless counterterm added to make $W[A_{\mu}]$
gauge invariant. In terms of the parametrizations (3.27) this reads

$\displaystyle W[gh^{-1}]=W[g]+W[h^{-1}]+\int d^{2}x\
\text{Tr}((g^{-1}\partial_{+}g)(h^{-1}\partial_{-}h))$ (3.33)

###  3.2 Polyakov’s 2D Gravity

In this section we are going to introduce the Polyakov 2D quantum gravity
action (P2DG) from a number of perspectives. In doing so, we are aiming to
show in what sense it is a quantum gravity action in 2D, and its relation to
WZW theory and to chiral fermion theories.

In 2D, the classical theory of gravity described by the Einstein-Hilbert
action does not provide any dynamics as the Einstein equations reduce to
$0=0$. The Einstein-Hilbert action reduces (using the Gauss-Bonnet theorem) to
$2\pi\chi$ where $\chi$ is the Euler characteristic which is an invariant
under homeomorphisms. Hence, in 2D, Einstein’s gravity provides only
topological information about the spacetime.

Upon quantization, however, theories can pick up contributions from anomalies
(forming the one-loop quantum effective action) in case the symmetries of
classical theory fails to hold in the quantum theory. In particular, anomalies
can provide dynamics to the spacetime metric. The main references for this
section are [polyakov81bosonic], [polyakov2Dgravity], [knizhnik].

####  3.2.1 P2DG as Integrated Conformal Anomaly

P2DG action arises as the effective action for the conformal anomaly. It is
introduced in [polyakov81bosonic], in the context of bosonic string theory,
but it is relevant to any 2D conformal field theory. Classical implication of
conformal invariance is the vanishing of the trace of the energy momentum
tensor. Hence, breaking of the conformal symmetry at the quantum level, namely
the conformal anomaly, arises as the nonvanishing of the trace and turns out
to be given by

$\displaystyle g^{ab}\frac{\delta W}{\delta
g^{ab}}=g^{ab}\left<T_{ab}\right>=\frac{D}{48\pi}(R+\text{const})$ (3.34)

Here $D$ is a constant, $T_{ab}$ the energy momentum tensor of the theory and
$R$ is the Ricci scalar of the 2D spacetime underlying the theory.

Equation (3.34) can be solved for the effective action $W$ in covariant but
nonlocal form,

$\displaystyle W[g_{ab}]=$ $\displaystyle-\frac{D}{96\pi}\int
d^{2}x\,d^{2}x^{\prime}\,\sqrt{g(x)}\,R(x)K(x,x^{\prime})R(x^{\prime})\,\sqrt{g(x^{\prime})}$
$\displaystyle+\text{const}\int d^{2}x\,\sqrt{g}$ (3.35)

where $K$ is the kernel of the Laplacian

$\displaystyle\partial_{a}(\sqrt{g}g^{ab}\partial_{b})K(x,x^{\prime})=\delta(x-x^{\prime})$
(3.36)

For computational purposes one chooses a gauge for the
metric444Reparametrization invariance combined with the symmetry of the metric
tensor, reduce its number of independent degrees of freedom to one., in which
$W$ becomes local. In the same paper $W$ is introduced in the conformal gauge
$g_{ab}=\rho\delta_{ab}$, $R=\rho^{-1}\partial^{2}\rho$. In this case we get
the 2D Liouville gravity

$\displaystyle W[\rho]=-\frac{D}{96\pi}\int d^{2}x\
\left[\frac{1}{2}(\partial_{a}\log\rho)^{2}+\mu^{2}\rho\right]$ (3.37)

####  3.2.2 P2DG in Lightcone Gauge

In [polyakov2Dgravity], Polyakov chooses a different gauge for the metric to
evaluate the effective action for the conformal anomaly, namely, the lightcone
gauge (LCG)

$\displaystyle ds^{2}=dx^{+}dx^{-}+h_{++}(x^{+},x^{-})(dx^{+})^{2}$ (3.38)

The metric underlying this line element is given by

$\displaystyle g^{(+-)}_{\mu\nu}=\left(\begin{array}[]{cc}h_{++}&1/2\\\
1/2&0\end{array}\right)$ (3.41)

Instead of substituting this metric into (3.2.1) he introduces the conformal
anomaly operator relations, analogous to (3.26) and (3.29), in LCC, namely,

$\displaystyle\delta W$ $\displaystyle=\int T_{--}\delta h_{++}$ (3.42)
$\displaystyle\nabla_{+}T_{--}$
$\displaystyle\equiv\partial_{+}T_{--}-h_{++}\partial_{-}T_{--}-2(\partial_{-}h_{++})T_{--}=\frac{d}{24\pi}\partial_{-}R$
(3.43)

The Ricci scalar for (3.41) becomes $R=\partial_{-}^{2}h_{++}$. Recall the
mentioned analogy between WZW theory and the P2DG theory in LCG. In the stated
analogy we have the correspondences $A_{+}\leftrightarrow h_{++}$,
$J_{-}\leftrightarrow T_{--}$. We will have more to say about it below.

Polyakov states that it is possible to work out $W[h_{++}]$ perturbatively and
even in closed form. However, he chooses to work with an action that is
obtained from it by a field redefinition. Namely he introduces a field $f$
defined by

$\displaystyle\partial_{+}f=h_{++}\partial_{-}f$ (3.44)

He points out to the analogy between (3.44) and (3.27). The former redefines
the $h_{++}$ component in terms of a field $f$ and the latter redefines the
$A_{+}$ component in terms of a field $g$. That’s the first reason why P2DG
action in LCG is considered as the gravitational WZW model.

The effective action can then be written as

$\displaystyle W[f]\propto\int d^{2}x\
\left[\frac{(\partial^{2}_{-}f)(\partial_{+}\partial_{-}f)}{(\partial_{-}f)^{2}}-\frac{(\partial^{2}_{-}f)^{2}(\partial_{+}f)}{(\partial_{-}f)^{3}}\right]$
(3.45)

In analogy with (3.32), if the $h_{--}=0$ gauge is turned off, we would have
[polyakovBook2D]

$\displaystyle
W[h_{++},h_{--}]=W_{+}[h_{++}]+W_{-}[h_{--}]+\Lambda(h_{++},h_{--})$ (3.46)

where $\Lambda$ is a counterterm. In particular, if we introduce a Polyakov
field $\bar{f}$ for $W_{-}$ as well, in terms of the fields $f$ and $\bar{f}$
the effective action would read

$\displaystyle W[f,\bar{f}]=W_{+}[f]+W_{-}[\bar{f}]+\Lambda(f,\bar{f})$ (3.47)

where $W_{-}[\bar{f}]$ is of the same form as (3.45) with
$f\leftrightarrow\bar{f}$, $x^{+}\leftrightarrow x^{-}$. We can obtain
$W_{\pm}$ as parts of geometric action on the orbits of the direct product of
two Virasoro algebras (Section 3.6). The counterterm may arise from
quantization conditions entangling the diff field components $D_{++}$,
$D_{--}$.

####  3.2.3 Chiral Fermions Coupled to Gravity in 2D

The approach followed here to introduce the P2DG action in LCG is from
[knizhnik]. We differ in our LCC conventions (Section B.1).

The Dirac Lagrangian in a curved spacetime is defined as

$\displaystyle\mathcal{L}$ $\displaystyle=\sqrt{-g}\
\overline{\psi}\gamma^{\mu}\nabla_{\mu}\psi$ $\displaystyle=-(\det v)\
\overline{\psi}\gamma^{a}e_{\
a}^{\mu}\left(\partial_{\mu}-\frac{1}{2}\omega_{\mu
ab}\gamma^{a}\gamma^{b}\right)\psi$ (3.48)

Here $v^{a}_{\ \mu}$ are the vielbein components, $\det v$ is the determinant
of the vielbein, $e^{\mu}_{\ a}$ are the inverse vielbein components and
$\omega_{\mu ab}$ are the spin connection components. In 2D the spin
connection $\omega_{\mu}$ vanishes (see e.g. [nakahara] Section 7.10.3 ) thus
this reduces to

$\displaystyle\mathcal{L}=-(\det v)\ \overline{\psi}\gamma^{a}e_{\
a}^{\mu}\partial_{\mu}\psi$ (3.49)

We use different letters for the vielbein and its inverse to avoid confusion.

Using the LCC toolbox developed in Section B.1 we can do the sum

$\displaystyle\mathcal{L}/\sqrt{2}=\psi_{-}(v^{-}_{\ +}\partial_{-}-v^{-}_{\
-}\partial_{+})\psi_{-}+\psi_{+}(-v^{+}_{\ +}\partial_{-}-v^{+}_{\
-}\partial_{+})\psi_{+}$ (3.50)

Note that in obtaining this result we haven’t used any gauge fixing
conditions.

Next, we introduce the field redefinitions $\phi_{-}\equiv\sqrt{v^{-}_{\
+}}\psi_{-}$ and $\phi_{+}\equiv\sqrt{v^{+}_{\ -}}\psi_{+}$. Upon action of
the derivatives we get terms of the form $\phi_{-}\phi_{-}$ and
$\phi_{+}\phi_{+}$. These vanish by the Grassmann nature of the $\phi_{\pm}$.
Therefore, in the end, we are left with

$\displaystyle\mathcal{L}/\sqrt{2}=\phi_{-}\left(\partial_{-}-\frac{v^{-}_{\
-}}{v^{-}_{\ +}}\partial_{+}\right)\phi_{-}\ +\
\phi_{+}\left(\partial_{+}-\frac{v^{+}_{\ +}}{v^{+}_{\
-}}\partial_{-}\right)\phi_{+}$ (3.51)

Polyakov chooses the following vielbein gauge fixing conditions

$\displaystyle v^{-}_{\ -}=0\ \ \ ,\ \ \ v^{-}_{\ +}=1/v^{+}_{\ -}$ (3.52)

Now, under a coordinate transformation $x\mapsto x^{\prime}$ metric transforms
with the inverse Jacobian, $\partial x/\partial x^{\prime}$, i.e.

$\displaystyle\tilde{\mathbf{g}}=(J^{-1})^{T}\mathbf{g}J^{-1}$ (3.53)

where we used boldface letter for the metric tensor matrix, to avoid confusion
with the metric determinant $g$. In the special case of the transformation
from a flat metric, the vielbein matrix is the same as the inverse Jacobian
matrix. Therefore, the ligthcone flat metric

$\displaystyle\eta=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right)$
(3.56)

transforms to (coordinates are in order of $(x^{+},x^{-})$ )

$\displaystyle\mathbf{g}=\left(\begin{array}[]{cc}2v^{-}_{\ +}v^{+}_{\
+}&v^{-}_{\ -}v^{+}_{\ +}+v^{-}_{\ +}v^{+}_{\ -}\\\ v^{-}_{\ -}v^{+}_{\
+}+v^{-}_{\ +}v^{+}_{\ -}&2v^{-}_{\ -}v^{+}_{\ +-}\end{array}\right)$ (3.59)

Under the conditions (3.52) this yields the Polyakov metric

$\displaystyle\mathbf{g}=\left(\begin{array}[]{cc}\frac{2v^{+}_{\ +}}{v^{+}_{\
-}}&1\\\ 1&0\end{array}\right)$ (3.62)

For the Polyakov metric, the Lagrangian (3.51) reduces to

$\displaystyle\mathcal{L}/\sqrt{2}=\phi_{+}\left(\partial_{+}-\frac{g_{++}}{2}\partial_{-}\right)\phi_{+}\
\ \ +\ \ \ \phi_{-}\partial_{-}\phi_{-}$ (3.63)

Dropping the $-$ chiral mode we get (ignoring the factor of $\sqrt{2}$)

$\displaystyle\mathcal{L}_{+}=\phi_{+}\left(\partial_{+}-\frac{g_{++}}{2}\partial_{-}\right)\phi_{+}$
(3.64)

Integrating over the fermionic modes we get the P2DG action in LCG555Note that
Polyakov [polyakov2Dgravity], [knizhnik] used the LCC definitions
$x^{\pm}=(t\pm x)$ so is off by a factor of $1/2$ from our conventions. As a
result the corresponding Lagrangian reads
$\mathcal{L}_{+}\propto\phi_{+}(\partial_{+}-g_{++}\partial_{-})\phi_{+}$.

$\displaystyle
W[g_{++}]\propto\log\text{Det}\left(\partial_{+}-\frac{g_{++}}{2}\partial_{-}\right)=\text{Tr}\log\left(\partial_{+}-\frac{g_{++}}{2}\partial_{-}\right)$
(3.65)

Since WZW action arises as the Dirac determinant with gauge coupling (3.2), we
again see that P2DG action in LCG is the gravitational analog of the WZW
action. That’s why it is also called the gravitational WZW model.

###  3.3 Kac-Moody Geometric Action

In this section we review the construction of the geometric action on Kac-
Moody (KM) coadjoint orbits[delius90]. Using the conventions set in Section
2.3 we first need to construct adjoint and coadjoint vectors that are suitable
for $(\tau,\lambda)$-parametrized coadjoint orbit $m$. For this purpose we
introduce group elements $g(z,\tau,\lambda)$ i.e. for each point
$(\tau,\lambda)$ on the orbit we have a group element $g(z)$. Then as adjoint
vectors we can take

$\displaystyle u_{\tau}\equiv
g(z,\tau,\lambda)\partial_{\tau}g^{-1}(z,\tau,\lambda)\hskip
14.45377pt\text{and}\hskip 14.45377ptu_{\lambda}\equiv
g(z,\tau,\lambda)\partial_{\lambda}g^{-1}(z,\tau,\lambda)$ (3.66)

To construct the geometric action we need the commutator of $u_{\tau}$ and
$u_{\lambda}$

$\displaystyle(\Lambda,a)\leftrightarrow[(u_{\tau},c_{\tau}),(u_{\lambda},c_{\lambda})]=\left([u_{\tau},u_{\lambda}],\oint\frac{dz}{2\pi
i}\ \text{Tr}\ \partial_{z}u_{\tau}u_{\lambda}\right)$ (3.67)

As a coadjoint vector on the orbit we pick a fixed coadjoint vector $(A(z),a)$
and act on it by $g(z,\tau,\lambda)$ :

$\displaystyle(A(z),a)_{g}=\Big{(}g(z,\tau,\lambda)A(z)g^{-1}(z,\tau,\lambda)-a\partial_{z}g(z,\tau,\lambda)g^{-1}(z,\tau,\lambda)\,,\,a\Big{)}$
(3.68)

Forming the pairing between the constructed coadjoint and adjoint vectors, and
integrating it over the orbit $m$, we get the action

$\displaystyle S_{\text{KM}}$ $\displaystyle\equiv\int_{m}\Omega$
$\displaystyle=\int d\lambda\ d\tau\ \left<(A,a)_{g}\
|[(u_{\tau},c_{\tau}),(u_{\lambda},c_{\lambda})]\right>$ $\displaystyle=k\int
d\lambda d\tau\left(\oint\frac{dz}{2\pi i}\
\text{Tr}(gAg^{-1}[g\partial_{\tau}g^{-1},g\partial_{\lambda}g^{-1}g])\right.$
$\displaystyle\hskip 93.95122pt-a\oint\frac{dz}{2\pi i}\
\text{Tr}(\partial_{z}gg^{-1}[g\partial_{\tau}g^{-1},g\partial_{\lambda}g^{-1}])$
$\displaystyle\hskip 130.08621pt\left.+a\oint\frac{dz}{2\pi i}\ \text{Tr}\
\partial_{z}(g\partial_{\tau}g^{-1})g\partial_{\lambda}g^{-1}\right)$ (3.69)

where $g=g(z,\tau,\lambda)$. We first write this as far as possible in terms
of total $z,\tau$ and $\lambda$ derivatives. Total $z$ derivative terms
vanish. For $\tau$ and $\lambda$ dependence, we choose to impose boundary
conditions such that total $\tau$ derivatives vanish, and $g$ is
$\tau$-independent at $\lambda=0$. This is equivalent to the requirement that
the $g(z,\tau,\lambda)$ describing embedding of a 3-ball into the group
manifold with $\lambda$ as the radial coordinate and $z$ and $\tau$, the
coordinates on $S_{2}$. With these we arrive at the WZW action plus a
background field $A(z)$ interacting with the WZW field $g(z,\tau,\lambda)$ :

$\displaystyle S$ $\displaystyle=\frac{ka}{2}\int d\tau\oint\frac{dz}{2\pi i}\
\text{Tr}\bigg{(}\partial_{\tau}g(\lambda=1)\,\partial_{z}g^{-1}(\lambda=1)$
$\displaystyle\left.\hskip 115.63243pt+\frac{1}{3}\int d\lambda\
\epsilon^{\alpha\beta\gamma}\
\partial_{\alpha}gg^{-1}\partial_{\beta}gg^{-1}\partial_{\gamma}gg^{-1}\right)$
$\displaystyle\ \ \ \ \ +k\int d\tau\oint\frac{dz}{2\pi i}\
\text{Tr}(A\,g^{-1}(\lambda=1)\,\partial_{\tau}g(\lambda=1))$ (3.70)

where $\epsilon^{z\lambda\tau}=1$. Exact correspondence with the original form
of WZW action (as provided in [WZNW]) is achieved via $g\leftrightarrow
g(z,\tau,\lambda=1)$ , $\bar{g}\leftrightarrow g(z,\tau,\lambda)$ ,
$d^{2}x\leftrightarrow dzd\tau$ and $d^{3}y\leftrightarrow dzd\tau d\lambda$.

In the angular coordinate $\theta$ the same analysis leads to the action,
[rairodgers90] :

$\displaystyle S$ $\displaystyle=\frac{ka}{4\pi}\int d\tau\ d\theta\
\text{Tr}\
\left(\partial_{\tau}g\partial_{\theta}g^{-1}\Big{|}_{\lambda=1}+\frac{1}{3}\int
d\lambda\ \epsilon^{\alpha\beta\gamma}\
\partial_{\alpha}gg^{-1}\partial_{\beta}gg^{-1}\partial_{\gamma}gg^{-1}\right)$
$\displaystyle+\frac{k}{2\pi}\int_{\lambda=1}d\tau\ d\theta\ \text{Tr}\
(A\,g^{-1}\partial_{\tau}g)$ (3.71)

where $A=A(\theta)$ and $g=g(\theta,\tau,\lambda)$.

The coupling of WZW field to a background gauge field $A_{\mu}$ is given by
the second term of (3.1.1) (with $U\rightarrow g,\,A_{+}\rightarrow
A_{\tau},\,A_{-}\rightarrow A_{\theta}$),

$\displaystyle S_{gA_{\mu}}=\frac{1}{4\pi}\text{Tr}\ \int d\tau\ d\theta\
(A_{\tau}g\partial_{\theta}g^{-1}+A_{\theta}g^{-1}\partial_{\tau}g+A_{\tau}gA_{\theta}g^{-1}-A_{\tau}A_{\theta})$
(3.72)

In temporal gauge, $A_{\tau}=0$, this reduces to

$\displaystyle S_{gA_{\theta}}$ $\displaystyle=\frac{1}{4\pi}\text{Tr}\ \int
d\tau\ d\theta\ (A_{\theta}g^{-1}\partial_{\tau}g)$ (3.73)

Comparing this with the last term of the geometric action (3.3) we see that
the background field $A$ can be identified with the $A_{\theta}$ component of
a YM field $A_{\mu}$ in temporal gauge. This identification will be further
motivated by analyzing the infinitesimal coadjoint action of the KM coadjoint
element $A$. In fact, using this we are going to obtain YM action from KM
algebra by the transverse prescription in Section 4.2.

###  3.4 Virasoro Geometric Action

In this section, we review the geometric action on Virasoro coadjoint orbits
[delius90]. Again we first need to construct adjoint and coadjoint vectors
from the group elements parametrized on the orbit $m=\\{(\lambda,\tau)\\}$
i.e. from group elements of the form $s(z,\lambda,\tau)\equiv
s(\lambda,\tau)\cdot z$ where $s(\lambda,\tau)$ is the group element, and
$s(z,\lambda,\tau)$ is the diffeomorphism formed by its action on $z$.

Let us begin by the adjoint element in the direction of $\tau$. We will use
the differential operator representation666Here minus sign is needed for
consistency with the commutation relations.,
$-u_{\tau}\partial_{\tau}=s\partial_{\tau}s^{-1}$. Using $s\cdot
z=s(z)\equiv\bar{z}$ we get
$u_{\tau}(\bar{z})=\partial_{\tau}s(z,\lambda,\tau)$ or
$u_{\tau}(z)=\partial_{\tau}s(s^{-1}(z,\lambda,\tau),\lambda,\tau)$. The
analogous expression can be found for $u_{\lambda}$. Next we evaluate the
commutator $[(u_{\tau},c_{\tau}),(u_{\lambda},c_{\lambda})]$ which using chain
rule becomes

$\displaystyle\xi_{s}(\bar{z})=[u_{\tau},u_{\lambda}](\bar{z})=\partial_{\tau}s(z,\lambda,\tau)\frac{\partial_{\lambda}\partial_{z}s(z,\lambda,\tau)}{\partial_{z}s(z,\lambda,\tau)}-\partial_{\lambda}s(z,\lambda,\tau)\frac{\partial_{\tau}\partial_{z}s(z,\lambda,\tau)}{\partial_{z}s(z,\lambda,\tau))}$
(3.74)

For the center, as well, it is more convenient to do the computation at
$\tilde{z}$

$\displaystyle\mu_{s}\equiv c(u_{\tau},u_{\lambda})$
$\displaystyle=\oint\frac{dz}{2\pi i}\
\partial^{3}_{z}u_{\tau}(z)u_{\lambda}(z)$
$\displaystyle=\oint\frac{d\bar{z}}{2\pi i}\
\partial_{\bar{z}}^{3}u_{\tau}(\bar{z})u_{\lambda}(\bar{z})$
$\displaystyle=\oint\frac{dz}{2\pi i}\
\partial_{z}s\left[((\partial_{z}s)^{-1}\partial_{z})^{3}\partial_{\tau}s\right]\partial_{\lambda}s$
(3.75)

where $s=s(z,\lambda,\tau)$.

To get the coadjoint vector on the orbit $m$, we act on a fixed element
$(D(z),b^{*})$ by a parametrized group element $s(z,\lambda,\tau)$, i.e.

$\displaystyle(D_{s}(\bar{z}),b^{*}_{s})$
$\displaystyle=\Big{(}(D(z)-b^{*}S(z,s))\left(\partial_{z}s\right)^{-2},b^{*}\Big{)}$
(3.76)

With all these ingredients we obtain the action

$\displaystyle S$ $\displaystyle=\int d\tau\,d\lambda\oint\frac{d\bar{z}}{2\pi
i}\left(D_{s}(\bar{z})\xi_{s}(\bar{z})+b^{*}_{s}\mu_{s}\right)$
$\displaystyle=\int d\tau\,d\lambda\oint\frac{dz}{2\pi
i}\,(\partial_{z}s)^{-2}(D(\bar{z})-b^{*}S(z,s))(\partial_{z}\partial_{\tau}s\,\partial_{\lambda}s-\partial_{z}\partial_{\lambda}s\,\partial_{\tau}s)$
$\displaystyle\ \ \ \ +b^{*}\int d\tau\,d\lambda\,\oint\frac{dz}{2\pi i}\
\partial_{z}s\left[((\partial_{z}s)^{-1}\partial_{z})^{3}\partial_{\tau}s\right]\partial_{\lambda}s$
(3.77)

where $s=s(z,\tau,\lambda)$. The same boundary conditions as in the case of KM
are assumed. Namely, those that make $z$ total derivatives vanish and make
$s$, $\tau$-independent for $\lambda=0$. After a fairly long calculation one
reaches the following action

$\displaystyle S=\int d\tau\ \oint\frac{dz}{2\pi i}\
\frac{\partial_{\tau}s}{\partial_{z}s}\ D(z)-\frac{b^{*}}{2}\int
d\tau\oint\frac{dz}{2\pi i}\ F_{\tau}(z,\tau,\lambda=1)$ (3.78)

where

$\displaystyle
F_{\tau}=\frac{(\partial_{z}^{2}s)^{2}(\partial_{z}s)}{(\partial_{z}s)^{3}}-\frac{(\partial_{z}^{2}s)(\partial_{z}\partial_{\tau}s)}{(\partial_{z}s)^{2}}$
(3.79)

If we change the notation as $z\rightarrow x^{-}$, $\tau\rightarrow x^{+}$,
$s\rightarrow f$, the second term in the action (3.78) is identical to P2DG
action in LCG (3.45). Explicitly, in Polyakov’s notation we have

$\displaystyle S=\int d^{2}x\ D(x^{-})\
\frac{\partial_{+}f}{\partial_{-}f}-\frac{b^{*}}{2}\int d^{2}x\
\left[\frac{(\partial^{2}_{-}f)(\partial_{+}\partial_{-}f)}{(\partial_{-}f)^{2}}-\frac{(\partial^{2}_{-}f)^{2}(\partial_{+}f)}{(\partial_{-}f)^{3}}\right]$
(3.80)

Here the noncentral Virasoro coadjoint element $D(x^{-})$ couples as a
background field to the lightcone Polyakov field $f(x^{+},x^{-})$ in analogy
with a background YM field $A$ coupling to the WZW field $g$ in the case of KM
geometric action.

Consider the first term in (3.80). The coadjoint field $D(x^{-})$ corresponds
to $D_{zz}\leftrightarrow D_{\theta\theta}\leftrightarrow D_{--}$ component of
a rank-two object $D_{\mu\nu}$ called the diffeomorphism field or the ”diff
field” in short. Using (3.44) the interaction term can be written as

$\displaystyle S_{\text{int}}=\int d^{2}x\ D_{--}g_{++}$ (3.81)

We can rewrite the integrand in the covariant form

$\displaystyle D_{\mu\nu}h^{\mu\nu}=D_{-+}-D_{--}g_{++}$ (3.82)

given the temporal gauge $D_{-+}=0$ for the diff field accompanied with
Polyakov’s LCG (3.41) for the metric. This shows that the first term in (3.80)
is the coupling of the diff field to the metric. We can’t, on the other hand,
say anything about $D_{++}$ component since in LCG we have $g_{--}=0$. Hence,
$D_{++}$ is simply invisible on coadjoint orbits.

The argument that the Virasoro coadjoint element $D$ can be identified with
the space-space component of a rank-two object will be further supported in
Section 4.3, where we will construct a covariant action governing the dynamics
of the diff field.

###  3.5 Semi-direct Product Geometric Action

####  3.5.1 Action

We have previously constructed the geometric actions for Virasoro and KM
algebras separately. In the KM sector we obtained the WZW action plus the
interaction of the WZW field $g$ with a background YM field $A_{\mu}$ in
temporal gauge. In the Virasoro sector we obtained the P2DG theory plus the
interaction of the Polyakov field $s$ with a background diff field tensor
$D_{\mu\nu}$ in temporal gauge.

When we consider the semi-direct product algebra we get the sum of the
previous geometric actions plus corrections in the $A_{\mu}-g$ interaction
term involving the Polyakov field $s$. This correction follows from the
nontrivial action of the Virasoro generators on the KM generators. Let us
review the main steps [LR95].

The two cocyle of the semi-direct product algebra is the sum of the cocycles
of the algebras given in (2.62) and (2.76)

$\displaystyle\omega\Big{(}(\xi_{1},\Lambda_{1},\mu_{1})$
$\displaystyle,(\xi_{2},\Lambda_{2},\mu_{2})\Big{)}$
$\displaystyle=\frac{c}{48\pi i}\int_{0}^{2\pi}d\theta\
(\xi^{\prime\prime\prime}_{1}\xi_{2}-\xi_{1}\xi^{\prime\prime\prime}_{2})+\frac{k}{2\pi}\int_{0}^{2\pi}d\theta\
\text{Tr}(\Lambda_{1}\Lambda^{\prime}_{2})$ (3.83)

For convenience, we took the Gelfand-Fuchs cocyle on the Virasoro sector, i.e.
we did not include the linear center.

As before ($z\leftrightarrow\theta$) we denote the KM group element by
$g(\lambda,\tau,\theta)$ and the Virasoro group element by
$s(\lambda,\tau,\theta)$. In analogy with the notation of the previous two
sections, adjoint elements that describe the changes in $\lambda$ and $\tau$
directions can be taken as

$\displaystyle U_{\lambda}$
$\displaystyle=(\partial_{\lambda}s,\partial_{\lambda}gg^{-1},0)$
$\displaystyle U_{\tau}$
$\displaystyle=(\partial_{\tau}s,\partial_{\tau}gg^{-1},0)$ (3.84)

Denoting a composite group element by $\tilde{g}\equiv(g,s)$, the adjoint
action $\text{Ad}_{\tilde{g}}U$ become

$\displaystyle\tilde{U}_{\lambda}\equiv\tilde{g}^{-1}U_{\lambda}\tilde{g}$
$\displaystyle=(\partial_{\lambda}s/\partial_{\theta}s,g^{-1}\partial_{\lambda}g,0)$
$\displaystyle\tilde{U}_{\tau}\equiv\tilde{g}^{-1}U_{\tau}\tilde{g}$
$\displaystyle=(\partial_{\tau}s/\partial_{\theta}s,g^{-1}\partial_{\tau}g,0)$
(3.85)

Let us also denote $\tilde{B}_{\lambda}\equiv\text{ad}_{\tilde{U}_{\lambda}}B$
where $B=(D(\theta),A(\theta),\mu)$ is a fixed coadjoint element and
$B_{0}=(D(\theta),A(\theta),0)$, its non-central part. Then the geometric
action becomes

$\displaystyle S_{B}$ $\displaystyle=\int d\lambda\ d\tau\
\Omega(\tilde{B}_{\lambda},\tilde{B}_{\tau})$ $\displaystyle=\int d\lambda\
d\tau\ \left<B|[\tilde{U}_{\lambda},\tilde{U}_{\tau}]\right>$
$\displaystyle=\int d\lambda\ d\tau\
\left(\left<B_{0}|[\tilde{U}_{\lambda},\tilde{U}_{\tau}]\right>+\mu\omega(\tilde{U}_{\lambda},\tilde{U}_{\tau})\right)$
(3.86)

Using (2.138), (2.144), (2.142) and (3.5.1), and performing partial
integrations with the same boundary conditions as in the individual geometric
actions one reaches,

$\displaystyle S$ $\displaystyle=\underbrace{\frac{1}{2\pi}\int d\tau\
d\theta\left(\frac{\partial_{\tau}s}{\partial_{\theta}s}\right)D}_{\text{coupling
to background diff field}}$ $\displaystyle\underbrace{+\frac{1}{2\pi}\int
d\lambda\ d\theta\ d\tau\
\text{Tr}\left(A\left(\frac{\partial_{\lambda}s}{\partial_{\theta}s}\partial_{\theta}(g^{-1}\partial_{\tau}g)-\frac{\partial_{\tau}s}{\partial_{\theta}s}\partial_{\theta}(g^{-1}\partial_{\lambda}g)+[g^{-1}\partial_{\lambda}g,g^{-1}\partial_{\tau}g]\right)\right)}_{\text{coupling
to background gauge field}}$ $\displaystyle\underbrace{-\frac{\mu
c}{48\pi}\int d\tau\ d\theta\
\left(\frac{\partial^{2}_{\theta}s}{(\partial_{\theta}s)^{2}}\partial_{\tau}\partial_{\theta}s-\frac{(\partial^{2}_{\theta}s)^{2}}{(\partial_{\theta}s)^{3}}\partial_{\tau}s\right)}_{\text{Polyakov
gravity}}$ $\displaystyle\underbrace{-\frac{\mu k}{4\pi}\int d\tau\ d\theta\
\text{Tr}\ (g^{-1}\partial_{\theta}gg^{-1}\partial_{\tau}g)+\frac{\mu
k}{4\pi}\int d\lambda\ d\tau\ d\theta\ \text{Tr}\
([g^{-1}\partial_{\theta}g,g^{-1}\partial_{\lambda}g]g^{-1}\partial_{\tau}g)}_{\text{WZW}}$
(3.87)

The KM and Virasoro geometric actions obtained in the previous two sections
can be recovered from this action by setting $s(\theta)=0$ and $g(\theta)=0$,
respectively.

####  3.5.2 Equations of Motion

The equations of motion that follow [LR95] from the geometric action (3.5.1)
are

$\displaystyle 0$
$\displaystyle=(\partial_{\theta}D)\frac{\partial_{\tau}s}{\partial_{\theta}s}+2D\
\partial_{\theta}\left(\frac{\partial_{\tau}s}{\partial_{\theta}s}\right)+\frac{c\mu}{24\pi}\partial^{3}_{\theta}\left(\frac{\partial_{\tau}s}{\partial_{\theta}s}\right)-\text{Tr}\\{A\
\partial_{\theta}(g^{-1}\partial_{\tau}g)\\}$ (3.88) $\displaystyle 0$
$\displaystyle=A\
\partial_{\theta}\left(\frac{\partial_{\tau}s}{\partial_{\theta}s}\right)+\left(\partial_{\theta}A\right)\frac{\partial_{\tau}s}{\partial_{\theta}s}-[g^{-1}\partial_{\tau}g,A]+k\mu\
\partial_{\theta}(g^{-1}\partial_{\tau}g)$ (3.89)

Let us simplify the notation a bit. First we will denote $\tau$-derivative
with a dot and $\theta$-derivative with a prime. We will also take
$(\partial_{\tau}s/\partial_{\theta}s)\equiv\xi$,
$g^{-1}\partial_{\tau}g\equiv\Lambda$, $k\mu\equiv e^{-1}$ and
$(c\mu/24\pi)\equiv q$. With all these the equations read

$\displaystyle 0$
$\displaystyle=D^{\prime}\xi+2D\xi^{\prime}+q\xi^{\prime\prime\prime}-\text{Tr}\\{A\Lambda^{\prime}\\}$
(3.90) $\displaystyle 0$
$\displaystyle=\xi^{\prime}A+A^{\prime}\xi-[\Lambda,A]+e^{-1}\Lambda^{\prime}$
(3.91)

These are equations (2.149), with the linear center of the Virasoro algebra
ignored (Section B.2), and with the left-hand sides set to zero, i.e. $\delta
D=0=\delta A$. In other words, the equations of motions turn out to be the
isotropy equations.

In the absence of diffeomorphisms ($\xi=0$), equation (3.91) has solutions

$\displaystyle g(\theta,\tau)=L(\theta)R(\tau)$ (3.92)

where $L(\theta)\in G$ is arbitrary and the generators of $R(\tau)$ commute
with $A$. This implies that $\Lambda=R^{-1}\dot{R}$ commutes with $A$.

In the presence of diffeomorphisms the solution is modified to

$\displaystyle g(\theta,\tau)=L(\theta)M(\theta,\tau)R(\tau)$ (3.93)

where $L$ and $R$ the same as above, and

$\displaystyle M(\theta,\tau)=\text{T}\exp\left(-e\int_{-\infty}^{\tau}dt\ \xi
A\right)$ (3.94)

with the boundary condition $g(\theta,\tau\rightarrow-\infty)=1$.

Now, let us consider the Virasoro equation $0=\delta D$. Using (3.93), we get

$\displaystyle\Lambda=g^{-1}\partial_{\tau}g=R^{-1}\dot{R}-e\xi A$ (3.95)

Inserting this into $0=\delta D$ we obtain

$\displaystyle
0=D^{\prime}\xi+2D\xi^{\prime}+q\xi^{\prime\prime\prime}+e\text{Tr}\
(A\xi^{\prime}A+A\xi A^{\prime})$ (3.96)

Introducing a new field

$\displaystyle\tilde{D}\equiv D+\frac{e}{2}\ \text{Tr}\ (AA)$ (3.97)

equation (3.96) simplifies to

$\displaystyle
0=\tilde{D}^{\prime}\xi+2\tilde{D}\xi^{\prime}+q\xi^{\prime\prime\prime}$
(3.98)

so that the KM variables $A,\Lambda$ disappear from the equation. Thus
$\tilde{D}$ is invariant under gauge transformations contrary to $D$ :

$\displaystyle\delta_{\text{gauge}}D$ $\displaystyle=-\text{Tr}\
(A\Lambda^{\prime})\neq 0$ (3.99)
$\displaystyle\delta_{\text{gauge}}\tilde{D}$ $\displaystyle=0$ (3.100)

We will use this result to obtain the gauge-invariant extension of the diff
field $D_{\mu\nu}$.

###  3.6 Direct Product of Two Virasoro Algebras

The 2D algebra of generators of conformal transformations turns out to be (see
e.g. [ketov]) direct product of two copies of Witt algebras

$\displaystyle[\ell_{m},\ell_{n}]$ $\displaystyle=(m-n)l_{m+n}$ (3.101)
$\displaystyle[\bar{\ell}_{m},\bar{\ell}_{n}]$
$\displaystyle=(m-n)\bar{\ell}_{m+n}$ (3.102)
$\displaystyle[\ell_{m},\bar{\ell}_{n}]$ $\displaystyle=0$ (3.103)

Upon quantization (of the underlying conformal field theory) these commutation
relations pick up central extensions yielding the direct product of two
Virasoro algebras. Quantization, however, may put further restrictions on the
generators.

Commutation relations for the direct product of two Virasoro algebras, with
central charges left arbitrary, are given by

$\displaystyle[L_{m},L_{n}]$
$\displaystyle=(m-n)L_{m+n}+(cm^{3}+hm)I\delta_{m+n}$
$\displaystyle[\bar{L}_{m},\bar{L}_{n}]$
$\displaystyle=(m-n)\bar{L}_{m+n}+(\bar{c}m^{3}+\bar{h}m)\bar{I}\delta_{m+n}$
$\displaystyle[L_{m},\bar{L}_{n}]$ $\displaystyle=0$ (3.104)

where $c,h,\bar{c},\bar{h}$ are constants and $I$ and $\bar{I}$ are the
generators of centers.

Firstly, notice that, in the geometric actions constructed in the previous
sections, whenever two parts of an algebra did commute (like the non-central
and the central parts of the algebra) they yielded separate terms summed in
the geometric action. The situation here is the same. Since the two copies of
the Virasoro algebras commute with each other, the geometric action of the
direct product algebra will be given by the sum of the geometric actions
coming from each copy.

Secondly, the role of the LCCs $x^{\pm}$ are switched once we switch from
$\\{L\\}$ algebra to $\\{\bar{L}\\}$ algebra. Hence, we can get the geometric
action for the latter, from the first simply by $x^{+}\leftrightarrow x^{-}$.
We will denote the noncentral coadjoint element coming from the first algebra
by $D$ (with central charge $\mu$), and from the second one by $\bar{D}$ (with
central charge $\bar{\mu}$). Similarly, the corresponding Polyakov fields
(circle diffeomorphisms) will be denoted by $f$ and $\bar{f}$, respectively.

Recall the geometric action obtained from a single Virasoro algebra

$\displaystyle S=\int d^{2}x\ D(x^{-})\
\frac{\partial_{+}f}{\partial_{-}f}-\frac{\mu}{2}\int d^{2}x\
\left[\frac{(\partial^{2}_{-}f)(\partial_{+}\partial_{-}f)}{(\partial_{-}f)^{2}}-\frac{(\partial^{2}_{-}f)^{2}(\partial_{+}f)}{(\partial_{-}f)^{3}}\right]$
(3.105)

where the coadjoint vector used in building the geometric action is
$\text{Ad}^{*}_{f}(D,\mu)$.

Then, the geometric action for the direct-product algebra becomes

$\displaystyle S$ $\displaystyle=\int d^{2}x\ D(x^{-})\
\frac{\partial_{+}f}{\partial_{-}f}-\frac{\mu}{2}\int d^{2}x\
\left[\frac{(\partial^{2}_{-}f)(\partial_{+}\partial_{-}f)}{(\partial_{-}f)^{2}}-\frac{(\partial^{2}_{-}f)^{2}(\partial_{+}f)}{(\partial_{-}f)^{3}}\right]$
$\displaystyle+\int d^{2}x\ \bar{D}(x^{+})\
\frac{\partial_{-}\bar{f}}{\partial_{+}\bar{f}}-\frac{\bar{\mu}}{2}\int
d^{2}x\
\left[\frac{(\partial^{2}_{+}\bar{f})(\partial_{-}\partial_{+}\bar{f})}{(\partial_{+}\bar{f})^{2}}-\frac{(\partial^{2}_{+}\bar{f})^{2}(\partial_{-}\bar{f})}{(\partial_{+}\bar{f})^{3}}\right]$
(3.106)

Notice that at this point there is no relationship between $D,\bar{D}$, their
central charges $\mu,\bar{\mu}$ or the diffeomorphisms $f,\bar{f}$ acting on
the coadjoint elements. It is just denotation of distinct variables.

Upon quantization of a conformal theory with classical algebra given by direct
product of two copies of Witt algebras $\\{l_{n}\\},\\{\bar{l}_{m}\\}$ a
complication arises (see [GSWsuperstring] Section 2.2). The energy momentum
tensor of the 2D conformal theory does vanish classically (constraint
equations of the classical conformal field theory). Since Virasoro generators
are the Laurent modes of the energy momentum tensor they do need to vanish
classically as well. At the quantum level, however, vanishing of $L_{n}$ for
all $n\in\mathbb{Z}$ lead to inconsistencies (due to the central extension
terms). Fortunately, by correspondence principle one only needs to have the
expectation values of the quantum operators corresponding to Virasoro
generators to vanish which can be achieved without setting all of them to
zero, only half is sufficient. So one imposes the following conditions to be
satisfied by the physical states of the theory

$\displaystyle L_{n}\left|\phi\right>=0=\bar{L}_{n}\left|\phi\right>\ \
\text{for $n>0$}\ \ \ \text{and}\ \ \ L_{0}=\bar{L}_{0}$ (3.107)

How do these translate into a condition in terms of the noncentral coadjoint
elements $D,\bar{D}$? The Virasoro generators (adjoint elements) interact with
the coadjoint elements through the pairing between them and the coadjoint
action. Since our main case of interest is the simplest type orbits, namely
Diff$S^{1}/S^{1}$ whose isotropy algebra is generated by $L_{0}$ (and
$\bar{L}_{0}$ for the second copy) the latter quantization condition implies
in this case that the coadjoint elements $D,\bar{D}$ are fixed by the same
element in their corresponding orbits.

In particular in (3.6) we know that $D$ and $\bar{D}$ will be identified with
$D_{--}$ and $D_{++}$, respectively. Moreover, $\partial_{+}f/\partial_{-}f$
and $\partial_{-}\bar{f}/\partial_{+}\bar{f}$ will be identified with $h_{++}$
and $h_{--}$, respectively. Then we can rewrite (3.6) as

$\displaystyle S$ $\displaystyle=\int D_{--}h_{++}+\int
D_{++}h_{--}+W_{+}[f]+W_{-}[f]$ (3.108) $\displaystyle=\int
D_{\mu\nu}h^{\mu\nu}+W[f]-\Lambda(f,\bar{f})\ \ \ \text{with}\ \ \ D_{-+}=0$
(3.109)

where we used (3.47). Hence, we see that this is the extension of (3.47) with
$D_{++}$ turned on. The question that follows then is whether we can recover
the counterterm $\Lambda$ in (3.47) from (3.6) using the restrictions placed
by quantization discussed above. We will not continue this analysis here.

##  Chapter 4 TRANSVERSE ACTIONS

###  4.1 Introduction

In the previous chapter we obtained geometric actions on the coadjoint orbits
of Kac-Moody (KM) and Virasoro algebras. In each case, the non-central
coadjoint element was seen as a background field coupled to group-valued
bosonic fields.

In this chapter we are going to lift each of these background fields to a
dynamical field by providing a Hamiltonian for each and lifting the isotropy
equation of coadjoint orbits to a constraint field equation.

The isotropy equation (2.20) defines the set of adjoint vectors that fix the
coadjoint element on its orbit, i.e. that do not move it along the orbit. We
then visualize infinitely many copies of coadjoint orbits stacked as sheets
and interpret the isotropy equation as defining motion transverse to each
sheet, i.e. from one sheet to the other, but not along the sheet itself (see
Figure (4.1)). This is the motivation for the term transverse. Coadjoint
transformations on the orbits lift to local transformations of the constructed
field theory.

Figure 4.1: Geometric vs Transverse Actions

We are going to apply this method first to the KM coadjoint element $A$
reaching the Yang-Mills (YM) theory of a gauge field $A_{\mu}$, then to the
Virasoro coadjoint element $D$, reaching the diffeomorphism field theory, the
dynamical theory of $D_{\mu\nu}$.

The method of transverse actions provided here is originally from [BLR97],
[BLR00], [takeshithesis]. However, at the step where one picks up an ansatz
for the momentum to construct the action, the equation (4.64), we diverge. The
reason for changing the momentum ansatz was the observation that the action
for the diff field, obtained in [BLR00] does not yield the same momentum as in
the ansatz taken. In [BLR00], the YM form for the momentum is directly assumed
in the KM case. Here, we do not assume the YM form. Rather, we introduce a
fairly relaxed ansatz for the momentum. Then the YM form arises automatically
from the constructed action. This seems puzzling at first, but is essentially
due to the Gauss law constraint being introduced into the action. Applying the
same line of reasoning to the diff field we reached a different action than
the one obtained in [BLR00]. Moreover, the structures of the actions for the
YM field and the diff field become similar upon this modification.

In this chapter, $x$ denotes the spatial coordinate unless otherwise stated.

###  4.2 Yang-Mills Action from Kac-Moody Algebra

####  4.2.1 Kac-Moody Gauge-Fixing

Recall the coadjoint action (2.149) of the adjoint vector
$\mathcal{F}=(\xi(x),\Lambda(x),a)$ on the coadjoint vector
$B=(D(x),A(x),\mu)$ of the semi-direct product of Virasoro and KM algebras,
$\text{ad}^{*}_{\mathcal{F}}(B)=(\delta D(x),\delta A(x),0)$ with

$\displaystyle\delta D$
$\displaystyle=2\xi^{\prime}D+D^{\prime}\xi+q\xi^{\prime\prime\prime}+2\beta\xi^{\prime}-\text{Tr}\
(A\Lambda^{\prime})$ (4.1) $\displaystyle\delta A$
$\displaystyle=A^{\prime}\xi+\xi^{\prime}A-[\Lambda,A]+e^{-1}\Lambda^{\prime}$
(4.2)

where we introduced $q\equiv c\mu/(2\pi)$, $\beta\equiv h\mu/(4\pi)$ and
$e\equiv(k\mu)^{-1}$ for simplicity. $\delta D=0$ and $\delta A=0$ are the
isotropy equations.

We can separate $\delta A$ and $\delta D$ into their pure Virasoro and KM
sectors. The pure KM sector ($\xi=0$) of $\delta A$ is

$\displaystyle\delta A(x)$
$\displaystyle=-[\Lambda(x),A(x)]+e^{-1}\partial\Lambda(x)$
$\displaystyle=(-f_{abc}\Lambda_{b}(x)A_{c}(x)+e^{-1}\partial\Lambda_{a}(x))T_{a}$
(4.3)

where we introduced $\Lambda(x)=\Lambda_{b}(x)T_{b}$ and $A(x)=A_{c}(x)T_{c}$.
Here $\\{T_{a}\\}$ forms a basis for the Lie algebra of the base group of KM
group. If we identify the coadjoint field $A$ with the space component $A_{1}$
of the YM field $A_{\mu}$ in 2D, then the transformation above corresponds to
a time-independent gauge transformation of $A_{1}$ :

$\displaystyle A_{1}(t,x)\rightarrow
A^{g}_{1}(t,x)=g(x)A_{1}(t,x)g^{-1}(x)-e^{-1}(\partial_{1}g(x))g^{-1}(x)$
(4.4)

For an infinitesimal transformation, $g(x)=1-\Lambda_{a}(x)T_{a}$, (4.4)
yields

$\displaystyle\delta A_{1}=\delta A_{a1}T_{a}\equiv
A^{g}_{1}-A_{1}=(-[\Lambda(x),A_{1}(t,x)]_{a}+e^{-1}\partial_{1}\Lambda_{a}(x))T_{a}$
(4.5)

This is the same as (4.2.1) except that $A_{1}=A$ now also depends on time.

####  4.2.2 Gauss Law

We introduce the operator

$\displaystyle Q=\int dx\,e^{-1}G_{a}(x)\,\Lambda_{a}(x)$ (4.6)

(the coupling constant $e^{-1}$ is introduced for convenience) that generates
$\delta A_{1}$ :

$\displaystyle\delta A_{1a}(x)=\\{A_{1a}(x),Q\\}$ (4.7)

Here $\\{\ ,\ \\}$ is the standard Poisson bracket (PB) and introducing the
conjugate momentum $\pi^{1}_{a}$ to $A_{1a}$ it is explicitly given by the
spatial integral

$\displaystyle\\{F,G\\}$ $\displaystyle=\int dx\left[\frac{\delta F}{\delta
A_{1c}(x)}\frac{\delta G}{\delta\pi^{1}_{c}(x)}-\frac{\delta
F}{\delta\pi^{1}_{c}(x)}\frac{\delta G}{\delta A_{1c}(x)}\right]$ (4.8)

In particular, we have

$\displaystyle\\{A_{1a}(x),\pi^{1}_{b}(y)\\}=\delta_{ab}\delta(x-y)$ (4.9)

Combining (4.6), (4.5) and (4.10) equation (4.7) yields

$\displaystyle\delta A_{1a}(x)$ $\displaystyle=\frac{\delta
Q}{\delta\pi^{1}_{a}(x)}=f_{abc}\Lambda_{b}(x)A_{1c}(x)-e^{-1}\partial_{1}\Lambda_{a}(x)$
(4.10)

From this we deduce

$\displaystyle G_{a}(x)$
$\displaystyle=ef_{abc}A_{1b}(x)\pi^{1}_{c}(x)+\partial_{1}\pi^{1}_{a}(x)$
(4.11) $\displaystyle=(e[A_{1}(x),\pi^{1}(x)]+\partial_{1}\pi^{1}(x))_{a}$
(4.12)

$G_{a}=0$ is the well-known Gauss law constraint. In the context of YM theory,
Gauss law is the field equation for $A_{0}$ which is non-dynamical since
$\pi^{0}$ vanishes. It is encountered in Dirac’s constrained
formalism111Dirac’s formalism is reviewed in Section A.1.1.  as the secondary
constraint that follows from the consistency condition of the primary
constraint $\pi^{0}=0$.

We also calculate $\delta\pi^{1}_{a}(x)=\\{\pi^{1}_{a}(x),Q\\}$:

$\displaystyle\delta\pi^{1}_{a}(x)=-\frac{\delta Q}{\delta
A_{1a}(x)}=ef_{abe}\Lambda_{b}(x)\pi^{1}_{e}(x)=e[\Lambda(x),\pi^{1}(x)]_{a}$
(4.13)

This shows that $\pi^{1}$ is gauge-covariant as (4.13) is the infinitesimal
reduction of

$\displaystyle\pi^{1}_{g}=g^{-1}\pi^{1}g$ (4.14)

for a time-independent gauge transformation $g$.

####  4.2.3 Kac-Moody Transverse Action

Notice that the momentum has a hidden time index by its definition via the
action functional (to be constructed)

$\displaystyle\pi^{1}=\frac{\delta S}{\delta(\partial_{0}A_{1})}(=\pi^{01})$
(4.15)

We take the ansatz

$\displaystyle\pi^{1}=-\partial^{0}A^{1}+L^{10}$ (4.16)

where $L$ is a rank-two object that does not involve $\partial^{0}A^{1}$, but
may involve the fields $A_{1},A_{0}$ and their spatial derivatives.

Transverse action is constructed using the isotropy equation $\delta
A_{1a}=0$. This enforces the dynamics to be transverse to the orbits, i.e.,
evolution of the field does not move the field on the orbit. We would like to
define an action functional that enforces this condition. Since $\delta A_{1}$
is generated by $G_{a}$, the isotropy equation can be enforced by the Gauss
law constraint, $G_{a}=0$. Hence, we introduce the following prescription for
the transverse Lagrangian

$\displaystyle\mathcal{L}=\text{ST}-\mathcal{H}+\lambda C$ (4.17)

where ST stands for the symplectic term, $\mathcal{H}$ for Hamiltonian
density, $C$ for the constraint and $\lambda$ for the Lagrange multiplier of
$C$.

Since $G_{a}$ is linear in the momentum, it has a hidden upper time index.
Then, for the Lagrangian to be a GCT-scalar, we need to contract $G_{a}$ with
an object having a lower time index. Again if the only fields at hand are
$A_{0}$, $A_{1}$ and their derivatives then the simplest choice is a constant
times $A_{0}$. This is indeed a requirement to obtain the YM action.

For convenience, let us define $B_{\mu\nu}\equiv\partial_{\nu}A_{\mu}$. Then
the pieces become

ST
$\displaystyle=(\partial_{0}A_{1a})\pi^{1}_{a}=B_{10a}(-B^{10}_{a}+L^{10}_{a})$
$\displaystyle\mathcal{H}$
$\displaystyle=(1/2)\pi_{1a}\pi^{1}_{a}=(1/2)(-B_{10a}+L_{10a})(-B^{10}_{a}+L^{10}_{a})$
$\displaystyle\lambda C$
$\displaystyle=cA_{0a}(ef_{abc}A_{1b}(-B^{10}_{c}+L^{10}_{c})+\partial_{1}(-B^{10}_{a}+L^{10}_{a}))$
(4.18)

Combining we get

$\displaystyle\mathcal{L}$ $\displaystyle=\text{ST}-\mathcal{H}+\lambda C$
$\displaystyle=-\frac{1}{2}B_{10a}B^{10}_{a}+\frac{1}{2}L_{10a}L^{10}_{a}-cA_{0a}(\partial_{1}B^{10}_{a})+cA_{0a}(\partial_{1}L^{10}_{a})$
$\displaystyle\hskip 62.15224pt-
cef_{abc}A_{0a}A_{1b}B^{10}_{c}+cef_{abc}A_{0a}A_{1b}L^{10}_{c}$ (4.19)

Then we recompute the momentum

$\displaystyle\pi^{1}_{a}$ $\displaystyle\equiv\frac{\delta
S}{\delta(\partial_{0}A_{1a})}=\frac{\delta S}{\delta
B_{10a}}=\frac{\partial\mathcal{L}}{\partial
B_{10a}}-\partial_{1}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{1}B_{10a})}\right)$
$\displaystyle=-\partial^{0}A^{1}_{a}+c\partial^{1}A^{0}_{a}-cef_{abc}A^{0}_{b}A^{1}_{c}$
(4.20)

We see that this yields the YM momentum only when $c=1$ :

$\displaystyle\pi^{1}_{a}=\partial^{1}A^{0}_{a}-\partial^{0}A^{1}_{a}+ef_{abc}A^{1}_{b}A^{0}_{c}=F^{10}_{a}$
(4.21)

Setting $c$ to any real constant is legitimate as it amounts to rescaling
$A_{0}$ which does not affect the dynamical characteristics of the theory
since $A_{0}$ is nondynamical. The meaning of setting $c=1$, on the other
hand, is that the Gauss law is the constraint associated with $A_{0}$, not
with an arbitrary multiple of it.

Hence, the field equations force $L^{10}$ (thereby the momentum $\pi^{1}$) to
take the desired form; we did not need to enforce it (as in [BLR00]). The rest
is straightforward. Inserting the momentum (4.21) back into $\mathcal{L}$, and
writing $\pi^{1}=F^{10}$ we get

$\displaystyle\mathcal{L}$
$\displaystyle=\partial_{0}A_{1a}F^{10}_{a}-\frac{1}{2}F_{10a}F^{10}_{a}+A_{0a}(\partial_{1}F^{10}_{a}+ef_{abc}A_{1b}F^{10}_{c})$
$\displaystyle=\frac{1}{2}F_{10a}F^{10}_{a}+\partial_{1}(A_{0a}F^{10}_{a})$
(4.22)

where we applied partial integrations and rearranged the indices. Let $y$
denote the spacetime coordinate $(t,x)$. Then, the action reads

$\displaystyle S=\int d^{2}y\ \mathcal{L}=\frac{1}{2}\int d^{2}y\
F_{10a}F^{10}_{a}+\int d^{2}y\ \partial_{1}(A_{0a}F^{10}_{a})$ (4.23)

Using Stoke’s theorem the second term is converted to an integral on the
boundary :

$\displaystyle\int_{V}d^{2}y\ \partial_{1}(A_{0a}F^{10}_{a})=\int_{\partial
V}ds_{1}A_{0a}F^{10}_{a}$ (4.24)

We will assume the boundary conditions (such as $A_{0a}=0$ or $F^{10}_{a}=0$
on $\partial V$) that make this term vanish. Hence, we get

$\displaystyle S=\frac{1}{2}\int d^{2}y\ F_{10a}F^{10}_{a}$ (4.25)

Since in 2D we have $F_{\mu\nu a}F^{\mu\nu}_{a}=2F_{10a}F^{10}_{a}$ the action
can be written as

$\displaystyle S=\frac{1}{4}\int d^{2}y\ F_{\mu\nu a}F^{\mu\nu}_{a}$ (4.26)

Using the normalization $\text{Tr}\ (T_{a}T_{b})=\delta_{ab}$ we get

$\displaystyle\text{Tr}\ (F_{\mu\nu}F^{\mu\nu})=F_{\mu\nu a}F^{\mu\nu}_{b}\
\text{Tr}\ (T_{a}T_{b})=F_{\mu\nu a}F^{\mu\nu}_{a}$ (4.27)

Thus, the action becomes

$\displaystyle S=\frac{1}{4}\int d^{2}y\ \text{Tr}\ (F_{\mu\nu}F^{\mu\nu})$
(4.28)

We can covariantly extend this action to higher dimensions as

$\displaystyle S=\frac{1}{4}\int d^{N}y\ \sqrt{g}\ \text{Tr}\
(F_{\mu\nu}F^{\mu\nu})$ (4.29)

Since $F_{\mu\nu}$ are the components of a two-form they are not affected by
covariantization.

####  4.2.4 Virasoro Sector of Kac-Moody Coadjoint Transformation

Now, let us examine the Virasoro sector of (4.2) :

$\displaystyle\delta A=A^{\prime}\xi+\xi^{\prime}A$ (4.30)