A Topological Field Theory with a Finite Number of Connected Feynman Diagrams
Franco Ferrari
A new topological field theory is constructed, which is characterized by cubic interactions similar to those of non-abelian Chern-Simons field theories, but still retains the simplicity of the abelian case. The perturbative expansion of this theory contains in fact only two connected Feynman diagrams, the propagator and a three vertex. Apart from the Gauss linking number, the Wilson loop amplitudes generate a further topological invariant, whose physical and mathematical meaning is investigated.
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\begin{document}
\title{A Topological Field Theory with a Finite Number of Connected
Feynman Diagrams}
\author{Franco Ferrari\\
{\it Institute of Physics, University of Szczecin, ul. Wielkopolska 15,}\\
{\it 70-451 Szczecin, Poland}\thanks{e-mail:
fferrari@univ.szczecin.pl}.}

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\maketitle

\abstract{
A new topological field theory is constructed, which
 is characterized by cubic interactions similar to those of
non-abelian Chern-Simons field theories, but still retains the
simplicity of the abelian case. The perturbative expansion of this
theory contains in fact only two connected Feynman diagrams, the
propagator and a three vertex. Apart from the Gauss linking number, the
Wilson loop amplitudes generate a further topological invariant, whose
physical and mathematical meaning is investigated.
}
%\vfill\eject
%\phantom{aaa}
\vfill\eject
\pagestyle{plain}
\section{Foreword}
In several situations it has been experimentally observed that the
topological properties of certain physical systems may influence their
behavior to a relevant extent. This is for instance the case of vortex
structures in nematic liquid crystals \cite{nemliq}
and in $^3He$ superfluids \cite{supflu}. Other
examples are provided by polymers \cite{wasserman}
 or by the lowest lying excitations of
two-dimensional electron gases, which have topological non-trivial
configurations at some filling fractions \cite{quahaleff}.
In the investigation of phenomena
related to the presence of topological constraints in the system, the
use of
quantum or statistical mechanical models
coupled to abelian Chern--Simons (C-S) field
theories \cite{chesim}
has been particularly successful.
One reason of this success is the fact that abelian models
do not require a complex mathematical
treatment as their non-abelian counterparts and thus
their physical meaning is more transparent.
%without obscuring its physical meaning due to the necessity
%of a cumbersome analytic treatment.

Motivated by possible applications in physics,
the aim of
this work is the construction
of a topological field theory
with non-trivial cubic interactions similar to those of
non-abelian C-S field theories, but which still retains
the
simplicity of the abelian case.
As a result of this effort an exactly solvable
 topological field theory is obtained, called hereafter truncated
topological field theory or briefly TTFT, 
which contains only
two connected Feynman diagrams in its perturbative expansion.
%of connected Feynman diagrams does not consist just in the
%propagators, as it happens in a pure abelian field theory,
%but contains only a finite number of graphs.
%Such theory, called hereafter truncated
%topological field theory or TTFT, is also exactly solvable.
The name of the theory is owing to the fact that any further expansion of
the perturbative series, which could in principle generate new 
diagrams, has been truncated by the introduction of suitable constraints.
From the computation of the Wilson loop amplitudes it turns out
that, apart from the Gauss linking number that is already present in
the abelian C-S field theory, the TTFT delivers a further
topological invariant, which can be interpreted as an Hopf term.

The material presented in this paper is divided as follows.
A naive topological field theory
consisting of three BF--models \cite{bfmod} coupled together by
cubic interaction terms is investigated in Section 2. Since this theory
is topological, it is convenient to choose as its observables the
so-called Wilson loops. Unfortunately, after the insertion of the
Wilson loops in the partition function in order to compute their amplitudes,
one observes that the constraints generated by the
longitudinal components of the fields become inconsistent with the
rest of the equations of motion.
This problem is solved in Section 3 by enlarging the gauge group of
the naive model via the addition of suitable topological terms to its
action. In this way a well defined TTFT is obtained, in which the
longitudinal components of the fields are harmless, because they
correspond to pure gauge field configurations and are thus irrelevant.
In Section 4 the Wilson loops amplitudes of the TTFT are computed in
the Lorentz gauge, showing that they contain a single topological
invariant apart from the Gauss linking number.
The physical and mathematical meaning of this invariant is
investigated. Finally, the Conclusions and a possible extension of the TTFT
are presented in Section 5.
%
%like Chern-Simons field theories or even simpler, so that they
% may be applied to p
%$\BF\Omega$
\section{Problems with Cubic Interactions in Abelian BF--Models}
In the quest for a topological field theory which generates only a finite
number of topological invariants, it is natural to start from the naive
action:
\beq{S=
\int d^3x\left[\frac\kappa{4\pi}
\tilde{\BF\Omega}^i\cdot(\nablab\times{\BF\Omega}^i)+
\Lambda{\BF\Omega}^1\cdot
({\BF\Omega}^2\times{\BF\Omega}^3)+
\bJ_i\cdot\tilde{\BF\Omega}^i\right]}{snaivevector}
For simplicity, $S$ has been defined here on a three dimensional
Euclidean space.
In Eq.~\ceq{snaivevector} $\kappa$ and $\Lambda$ denote
real coupling constants, while
$\tilde{\BF\Omega}^i$ and ${\BF\Omega}^i$, $i=1,2,3$, form a set of
six abelian
vector fields. The $\bJ_i$'s are assumed to be
conserved external currents, i.e. such
that $\nablab\cdot\bJ_i=0$.
Analogous sources coupled to the fields ${\BF\Omega}^i$ have been
omitted for a reason which will be clear below.
Summation over repeated indices is everywhere
understood.
The action $S$ describes a BF--model \cite{bfmod} with the addition
of a cubic interaction term.
In components, Eq.~\ceq{snaivevector} becomes
\beq{S=
\int d^3x\left[\epsilon^{\mu\nu\rho}\left(\frac\kappa{4\pi}
\tilde\Omega^i_\mu\partial_\nu\Omega^i_\rho+
\Lambda\Omega^1_\mu
\Omega^2_\nu\Omega^3_\rho\right)+J^\mu_i\tilde\Omega^i_\mu\right]}{snaive}
where, as a convention, greek letters label space indices, while
roman letters distinguish different vector fields. Finally, 
$\epsilon^{\mu\nu\rho}$ is the Levi-Civita tensor density
defined so that  $\epsilon^{123}=1$.

%We will see that, without
%the addition of some more ingredients, the classical
%equations of motion of the above theory
%are  inconsistent due to the presence of undamped longitudinal
%components of the vector fields. For the moment, however, this problem
%will be ignored since it is irrelevant for the following discussion.


Let us note that the fields
$\tilde{\BF\Omega}^i$ play in \ceq{snaivevector}
the role of pure Lagrange multipliers, which
constrain the fields ${\BF\Omega}^i$ and neutralize possible radiative
corrections.
%Other constraints are introduced by the equations of motion of the
%longitudinal components of the vector fields ${\BF\Omega}^i$.
%We will see below that these further constraints, if untreated,
%lead to inconsistent equations of motion for the fields, but for the
%moment this problem will be ignored since it is irrelevant for the following
%discussion.
Moreover, at the classical level there are only two
connected Feynman diagrams, which are shown in Fig.~\ref{feyrul}.
\begin{figure}[h]
\vspace{1 truein}
\special{psfile=feyrul.eps hscale=60 vscale=60 hoffset=100 voffset=0}
\vspace{0.2in}
\caption{Feynman rules corresponding to the action \ceq{snaivevector}.
Dashed lines propagate $\tilde{\BF\Omega}^i$ fields while solid lines
are associated to ${\BF\Omega}^i$ fields.}
\label{feyrul}
\end{figure}
They correspond to the field
propagators and to the three-vertex associated to the
cubic interaction term present in
Eq.~\ceq{snaivevector}. Higher order tree diagrams, which  could
in principle be generated by contracting together
the legs of many three-vertices, 
are actually ruled out due to the off-diagonal structure of the
propagators, which forbids any self-interaction among
the fields ${\BF\Omega}^i$.
For the same reason,
the source term $\int d^3x\tilde{\bJ}_i\cdot{\BF\Omega}^i$
for the fields $\tilde{\BF\Omega}^i$,
where the $\tilde{\bJ}_i$'s, $i=1,2,3$, are conserved external currents,
 has been
omitted from Eq.~\ceq{snaivevector}. As a matter of fact, the addition of
such term
would not change the dynamics of the fields ${\BF\Omega}^i$ and, besides,
 it is easy to see that it can be eliminated
by a shift of the Lagrange multipliers $\tilde{\BF\Omega}^i$.
%is not necessary to introduce
%in the action $S$ a source term
%of the kind $\int d^3x\tilde{\bJ}_i{\BF\Omega}^i$, which expresses
%the interaction of the fields ${\BF\Omega}^i$ with external
%currents $\tilde{\bJ}_i$, 
%may be eliminated
%by means of a simple redefinition of the fields $\tilde{\BF\Omega}^i$
%and thus it has not be written in Eq.~\ceq{snaivevector}.

Clearly, the action $S$ is metric independent and its topological properties
are not spoiled by quantum corrections, since the latter vanish identically
as previously remarked. Thus, we are in presence of a topological
field theory, which is also invariant under the following abelian
gauge transformations
\beq{\tilde{\BF\Omega}^i(x)\rightarrow\tilde{\BF\Omega}^i(x)+\nablab
\tilde\lambda^i(x)}
{gftilfie}
%corresponding to an abelian group $U(1)\otimes U(1)\otimes U(1)$.
As a consequence, one may choose as observables
metric independent and gauge invariant operators like
the Wilson loops:
\beq{W_i(C)={\rm exp} \left[i\oint_\Gamma d\bx\cdot{\BF\Omega}^i\right]}{wldef}
$\Gamma$ is defined here as a superposition of closed non-intersecting
paths $\gamma,\gamma',\gamma'',\ldots$,
i.~e. $\Gamma=\gamma+\gamma'+\gamma''+\ldots$, so that a generic
correlation function of Wilson loops is given
by\footnotemark{}\footnotetext{Actually, we will see later that, due
to the simplicity of the theory under consideration, the only relevant
correlation function of Wilson loops occurs when $\Gamma_i=\gamma_i$,
$i=1,2,3$.}:
%where $C$ is a closed trajectory in the space.
%The insertion of Wilson loops in the theory requires the introduction
%of the new Feynman rule given in Fig.~\ref{feyrulwl}.
%\begin{figure}[h]
%\vspace{1.3 truein}
%\special{psfile=feyrulwl.eps hscale=30 vscale=30 hoffset=150 voffset=-10}
%\vspace{0.2in}
%\caption{Feynman rules corresponding to the insertion of a Wilson loop
%coupled to a ${\BF\Omega}^i$ vector field (on the left) or to a
%$\tilde{\BF\Omega}^i$ vector field (on the right).}
%\label{feyrulwl}
%\end{figure}
%Due to the simplicity of the action~\ceq{snaivevector}, it is possible to
%verify that a general correlation function of Wilson loops can be
%decomposed into a product of correlation functions containing only three
%Wilson loops.
%Therefore, it is sufficient to consider the following amplitude:
%Looking at Eq.~\ceq{snaivevector}, it is easy to realize that
%the most general Wilson loop amplitude can be expressed in terms of
%products of the expectations value of three Wilson loops:
\beq{\langle W_1(\Gamma_1) W_2(\Gamma_2) W_3(\Gamma_3)
\rangle=\int\left[\prod_{i=1}^3{\cal D}{\BF\Omega}^i
{\cal D}\tilde{\BF\Omega}^i\right]e^{-iS}}{wlampl}
In writing the above equation, the insertion of Wilson loops has been
taken into account by making the special choice of external currents:
\beq{J_i^\mu=\oint_{\Gamma_i}dx_i^\mu\delta(x-x_i)}{currdef}
$\delta(x)$ being the Dirac $\delta-$function.

Unfortunately, any attempt to compute the amplitude \ceq{wlampl} runs
into troubles due to the
%where the $\Gamma_i$, $i=1,2,3$ are closed non-intersecting trajectories.
%As a matter of fact,
%the fields $\tilde{\BF\Omega}^i$ play the role of Lagrange
%multipliers, which constrain the fields ${\BF\Omega}^i$ and neutralize
%possible radiative corrections. 
%On the other side, the Feynman rules associated
%to the action \ceq{snaivevector} are very simple, see Fig.~\ref{feyrul}.
%Additional Feynman rules to take into account the insertion of a Wilson
%loop in which a closed path $C$ is coupled with a vector field
%${\BF\Omega}^i$ or $\tilde{\BF\Omega}^i$ are displayed in Fig.~\ref{feyrulwl}.
%Solid lines and dashed lines propagate the fields ${\BF\Omega}^i$ and
%$\tilde{\BF\Omega}^i$ respectively.
%In writing
%Let us also notice that the legs of the three vertex in
%Fig.~\ref{feyrul}
%corresponding to the
%cubic interaction in \ceq{snaivevector} cannot be contracted together
%to build more complicated diagrams. As a consequence, the only way
%in which the vector fields may interact with a Wilson loop is that shown
%in Fig.~\ref{wlvertexint}.
%\begin{figure}[h]
%\vspace{1.3 truein}
%\special{psfile=wlvertexint.eps hscale=30 vscale=30 hoffset=150 voffset=-10}
%\vspace{0.2in}
%\caption{This figure shows the only possible way in which the legs of
%the three vertex displayed in Fig.~\ref{feyrul} may be saturated using
%the contributions coming from Wilson loop insertions.}
%\label{wlvertexint}
%\end{figure}
%
%in the action $S$ the fields $\tilde{\BF\Omega}^i$ are pure Lagrange
%multipliers imposing constraints on the fields ${\BF\Omega}^i$.
%
%
%
%Each vector field $\tilde\Omega^i_\mu$ is coupled to an external current
%of the kind:
%
%the $\Gamma_i$'s represents closed trajectories in the space.
%
%
longitudinal components of the fields, whose role has not
been discussed so far. In the case of the $\tilde{\BF\Omega}^i$-fields, it is possible
to get rid of them by fixing the gauge in such a way that:
\beq{\nablab\cdot\tilde{\BF\Omega}^i=0}{otgaufix}
Yet, the undamped
 longitudinal components of the fields ${\BF\Omega}^i$ remain
in the cubic interaction term of Eq.~(\ref{snaivevector}) and introduce
new constraints which, without any treatment, lead to inconsistences
in the theory.
To see how the problem arises, we investigate the
classical equations of motion
associated to the action \ceq{snaivevector}.
%To this purpose, let us denote with ${\BF\omega}^i$ and
%$\tilde{\BF\omega}^i$ the classical counterparts of the fields
%${\BF\Omega}^i$ and
%$\tilde{\BF\Omega}^i$.
A variation of $S$ with respect
to the fields $\tilde\Omega^i_\mu$ produces the constraints:
%\beq{\frac\kappa{4\pi}\epsilon^{\mu\nu\rho}\partial_\nu\Omega^i_\rho+J^\mu_i=
%0}{confromot}
\beq{\frac\kappa{4\pi}\nablab\times{\BF\Omega}^i+\bJ_i=0}{confromot}
An analogous variation with respect to the fields $\Omega^i_\mu$
yields as a result the following relations:
\begin{eqnarray}
\frac\kappa{4\pi}\nablab\times\tilde{\BF\Omega}^1
& + &\Lambda{\BF\Omega}^2\times
{\BF\Omega}^3=0\label{Aeom}\\
\frac\kappa{4\pi}\nablab\times\tilde{\BF\Omega}^2
& - &\Lambda{\BF\Omega}^1\times
{\BF\Omega}^3=0 \\
\frac\kappa{4\pi}\nablab\times\tilde{\BF\Omega}^3
 & + & \Lambda{\BF\Omega}^1\times
{\BF\Omega}^2=0\label{Ceom}
\end{eqnarray}
It is easy to check that the general solutions of the constraints
\ceq{confromot} are:
\beq{\Omega^i_\mu=b_\mu^i+\partial_\mu\omega^i}{gensolcon}
In the above equation we have put
\beq{b_\mu^i(x)=\frac 1\kappa\epsilon_{\mu\alpha\beta}\int d^3y
\frac{(x-y)^\alpha}{|x-y|^3}J^\beta_i(y)=\frac
1\kappa\epsilon_{\mu\alpha\beta}
\oint_{\Gamma_i}dx_i^\beta\frac{(x-x_i)^\alpha} {|x-x_i|^3}}
{consol}
while the $\omega^i(x)$ represent
differentiable functions, which take into
account the longitudinal components of the vectors $\Omega_\mu^i(x)$.
The form of the $\omega^i(x)$'s cannot be determined from Eqs.~\ceq{confromot}.
At this point, it is possible to solve also Eqs.~(\ref{Aeom}--\ref{Ceom})
exactly with respect to the $\tilde{\BF\Omega}^i$.
However, this is not the end of the story, because there are further
constraints which can be obtained by 
%considering the longitudinal
%components
%of both members in
%Eqs.~(\ref{Aeom}--\ref{Ceom}).
applying the differential operator $\partial_\mu$
to Eqs.~(\ref{Aeom}--\ref{Ceom}). Exploiting
Eqs.~\ceq{confromot} in order to evaluate the curls of the
${\BF\Omega}^i$--fields, one finds the following  relations:
\begin{eqnarray}
{\BF\Omega}^1\cdot\bJ_2-{\BF\Omega}^2\cdot\bJ_1&=&0\label{Aextcon}\\
{\BF\Omega}^2\cdot\bJ_3-{\BF\Omega}^3\cdot\bJ_2&=&0\label{Bextcon}\\
{\BF\Omega}^3\cdot\bJ_1-{\BF\Omega}^1\cdot\bJ_3&=&0\label{Cextcon}
\end{eqnarray}
The solutions of Eqs.~(\ref{Aextcon}--\ref{Cextcon}):
\beq{{\BF\Omega}^i=\bJ_i}{solextcon}
are inconsistent with Eqs.~\ceq{confromot} if $\bJ_i\ne0$.
In fact, since the currents $\bJ_i$ are
conserved by assumption, Eqs.~\ceq{consol} and \ceq{solextcon}
 give two different and clearly incompatible
expressions for the transverse components of the ${\BF\Omega}^i$ fields.

In the next Section it will be shown how to solve this problem.
% may be 
%is the addition of other terms to the action $S$ which make the resulting
%theory gauge invariant with respect to both sets of fields
%${\BF\Omega}^i$ and 
%$\tilde{\BF\Omega}^i$.
\section{Solving The Problems: The Truncated Topological Field Theory}
We have seen in the previous Section that the abelian
BF--model with a cubic interaction term defined in Eq.~\ceq{snaivevector}
is inconsistent if the external currents $\bJ_i$ are different from zero.
The difficulties come from the undamped longitudinal components
of the ${\BF\Omega}^i$--fields. In fact, these components
are responsible for the
constraints (\ref{Aextcon}--\ref{Cextcon}), which are incompatible
with the other classical
equations of motion of the theory.
A possible strategy to overcome this problem is to
add suitable terms
to the action $S$, so that the new action is
invariant with respect to gauge
transformations of the kind:
\beq{{\BF\Omega}^i(x)\longrightarrow{\BF\Omega}^i(x)+\nablab\lambda^i(x)}
{gautraome}
The idea behind this strategy is that,
once gauge invariance is established, the longitudinal components
of the ${\BF\Omega}^i$'s become irrelevant. For instance,
they may be easily eliminated
choosing 
a gauge
condition in which the  fields are purely transverse.

To start with, we compute first of all the variations
$\delta_{\lambda^i}S$ of the action $S$ under the gauge
transformations \ceq{gautraome}.
%For instance, taking $i=1$ in Eq.~\ceq{gautraome}, one obtains:
After a few calculations one finds:
\begin{eqnarray}
\delta_{\lambda^1}S&=&\Lambda\int d^3x\nablab\lambda^1\cdot
({\BF\Omega}^2\times{\BF\Omega}^3)\label{Afvar}\\
\delta_{\lambda^2}S&=&-\Lambda\int d^3x\nablab\lambda^2\cdot
({\BF\Omega}^1\times{\BF\Omega}^3)\label{Bfvar}\\
\delta_{\lambda^3}S&=&\Lambda\int d^3x\nablab\lambda^3\cdot
({\BF\Omega}^1\times{\BF\Omega}^2)\label{Cfvar}
\end{eqnarray}
%Analogous relations can be written in the case of $\delta_{\lambda^2}S$ and
%$\delta_{\lambda^3}S$.
%Apparently the right hand side of
%Eq.~\ceq{lonevar} is non-linear in the fields
%${\BF\Omega}^i$.
In deriving the above equations
it has not been
%render exactly the idea of the way in which gauge invariance is
%violated. The reason is that
%Eqs.~(\ref{Afvar}--\ref{Cfvar}) 
%do not
taken into account
the fact that there is no real dynamics
in our theory and, for this reason, the expression of the variations
$\delta_{\lambda^i}S$ is unnecessarily complicated. Indeed, in the
right hand sides of 
Eqs.~(\ref{Afvar}--\ref{Cfvar})
there is a
linear dependence% of the
%$\delta_{\lambda^i}$'s
 on the fields ${\BF\Omega}^i$ which is hidden.
To show that, we remember that
the transverse components
of the ${\BF\Omega}^i$'s are bounded to live in the
subspace of all classical field configurations determined by the constraints
\ceq{confromot} and coincide with the
vectors $\bb^i$ given in Eq.~\ceq{consol}.
Exploiting these constraints, it is possible to rewrite
the variations $\delta_{\lambda^i}S$ in the form:
\begin{eqnarray}
\delta_{\lambda^1}S&=&\frac{4\pi\Lambda}\kappa\int d^3x
\lambda^1\left[
\bJ_2\cdot{\BF\Omega}^3_c-\bJ_3\cdot{\BF\Omega}^2_c\right]\label{Agninv}\\
\delta_{\lambda^2}S&=&\frac{4\pi\Lambda}\kappa\int d^3x
\lambda^2\left[
\bJ_3\cdot{\BF\Omega}^1_c-\bJ_1\cdot{\BF\Omega}^3_c\right]\\
\delta_{\lambda^3}S&=&\frac{4\pi\Lambda}\kappa\int d^3x
\lambda^3\left[
\bJ_1\cdot{\BF\Omega}^2_c-\bJ_2\cdot{\BF\Omega}^1_c\right]\label{Bgninv}
\end{eqnarray}
Here the symbols ${\BF\Omega}^i_c$ have been introduced to remember
that in Eqs.~(\ref{Agninv}--\ref{Bgninv})
the transverse degrees of freedom of the fields ${\BF\Omega}^i$
have been fixed by
means of Eq.~\ceq{confromot}.
%Since we are just interested in the evaluation of Wilson loop amplitudes,
%%Recalling that the currents $\bJ_i$ 
%%are of the form given in Eq.~\ceq{currdef},
%it is possible to rewrite the above equations in a metric independent
%form, as it should be in a topological field theory.
%In fact, using the expressions of the currents $\bJ_i$ given in
%Eq.~\ceq{currdef}, one finds:
%\begin{eqnarray}
%\delta_{\lambda^1}S&=&\frac{4\pi\Lambda}\kappa\left[\oint_{\Gamma_2}dx_2^\mu
%\lambda^1(x_2)\Omega^3_\mu(x_2)-
%\oint_{\Gamma_3}dx_3^\mu
%\lambda^1(x_3)\Omega^2_\mu(x_3)\right]\label{Amiforgauvar}\\
%\delta_{\lambda^2}S&=&\frac{4\pi\Lambda}\kappa\left[\oint_{\Gamma_3}dx_3^\mu
%\lambda^2(x_3)\Omega^1_\mu(x_3)-
%\oint_{\Gamma_1}dx_1^\mu
%\lambda^2(x_1)\Omega^3_\mu(x_1)\right]\label{Bmiforgauvar}\\
%\delta_{\lambda^3}S&=&\frac{4\pi\Lambda}\kappa\left[\oint_{\Gamma_1}dx_1^\mu
%\lambda^3(x_1)\Omega^2_\mu(x_1)-
%\oint_{\Gamma_2}dx_2^\mu
%\lambda^3(x_2)\Omega^1_\mu(x_2)\right]\label{Cmiforgauvar}
%\end{eqnarray}
%%the right hand sides of Eqs.~(\ref{Agninv}--\ref{Bgninv})
%%contain the metric. This is not in contradiction with the fact that
%%the action $S$ describes a topological field theory.
%%In fact, i


At this point we denote with the symbol $\bcas$ the gauge invariant
extension of the action $S$ and we try for it the ansatz:
\beq{\bcas=S+S^1_b+S^2_b}{ansatz}
where $S^1_b$ and $S^2_b$ contain respectively terms which are linear
and quadratic in the fields ${\BF\Omega}^i$:
\begin{eqnarray}
S^1_b&=&\nonumber\\
&&\frac\Lambda\kappa \int d^3x\Omega^1_\mu(x)
\left[\oint_{\Gamma_3} dx^\nu_3
\frac{(x-x_3)^\mu}{|x-x_3|^3}b_\nu^2(x_3)
-\oint_{\Gamma_2}dx^\nu_2
\frac{(x-x_2)^\mu}{|x-x_2|^3}b_\nu^3(x_2)\right]\nonumber\\
&+&\frac\Lambda\kappa \int d^3x\Omega^2_\mu(x)
\left[\oint_{\Gamma_1} dx^\nu_1
\frac{(x-x_1)^\mu}{|x-x_1|^3}b_\nu^3(x_1)
-\oint_{\Gamma_3}dx^\nu_3
\frac{(x-x_3)^\mu}{|x-x_3|^3}b_\nu^3(x_3)\right]\nonumber\\
&+&\frac\Lambda\kappa \int d^3x\Omega^3_\mu(x)
\left[\oint_{\Gamma_2} dx^\nu_2
\frac{(x-x_2)^\mu}{|x-x_2|^3}b_\nu^1(x_2)
-\oint_{\Gamma_1}dx^\nu_1
\frac{(x-x_1)^\mu}{|x-x_1|^3}b_\nu^2(x_1)\right]\nonumber\\
\label{couone}
\end{eqnarray}
\begin{eqnarray}
S^2_b&=&\frac\Lambda{4\pi\kappa}\int d^3x\partial^\mu\Omega^1_\mu(x)\int d^3y
\partial^\nu\Omega^2_\nu(y)\oint_{\Gamma_3}dx_3^\rho\frac1{|x-x_3|}
\partial^{x_3}_\rho\frac1{|y-x_3|}\nonumber\\
&-& \frac\Lambda{4\pi\kappa}\int d^3x\partial^\mu\Omega^1_\mu(x)\int d^3y
\partial^\nu\Omega^3_\nu(y)\oint_{\Gamma_2}dx_2^\rho\frac1{|x-x_2|}
\partial^{x_2}_\rho\frac1{|y-x_2|}\nonumber\\
&+&\frac\Lambda{4\pi\kappa}\int d^3x\partial^\mu\Omega^2_\mu(x)\int d^3y
\partial^\nu\Omega^3_\nu(y)\oint_{\Gamma_1}dx_1^\rho\frac1{|x-x_1|}
\partial^{x_1}_\rho\frac1{|y-x_1|}\nonumber\\
\label{coutwo}
\end{eqnarray}
Since we are interested in the computation of Wilson loop amplitudes,
%Eqs.~(\ref{couone}--\ref{coutwo})
 the expressions of $S_b^1$ and $S_b^2$
have been written directly for the special
case in which the currents $\bJ_i$ are given by Eq.~\ceq{currdef}. The
generalization to currents of general form is straightforward.
The action $\bcas$ of Eq.~\ceq{ansatz} defines what we call here
truncated topological field theory or TTFT.

It is now possible to check that
$S^1_b$ and $S^2_b$
satisfy the conditions listed
below:
\begin{description}
%\item {i)\phantom{ii}} $S^1_b$
%and $S^2_b$ contain terms which are respectively linear and
%quadratic in the fields ${\BF\Omega}^i$.
\item {i)\phantom{ii}} The variations of $S^1_b$ and $S^2_b$ under the
gauge transformations \ceq{gautraome} satisfy the relations:
\beq{\delta_{\lambda^i}(S^1_b+S^2_b)+\delta_{\lambda^i}S=0}{sbtra}
for $i=1,2,3$.
This condition guarantees the gauge invariance of the action $\bcas$.
\item {ii)\phantom{i}} The addition of the counterterms $S^1_b$ and $S^2_b$
to the action $S$ does not affect the
equations of motion \ceq{confromot} and (\ref{Aeom}--\ref{Ceom})
for what is concerning the transverse components of the fields
${\BF\Omega}^i$ and $\tilde{\BF\Omega}^i$.
\item {iii)} $S_b^1$ and $S^2_b$
consists of topological terms, so that the
topological properties of the action $\bcas$ are not spoiled.
\end{description}
%
%The explicit expressions of the counterterms $S^1_b$ and $S^2_b$
%can be computed and read as follows:
%
%\beq{counterterm\enskip 1}{couone}
%\beq{counterterm\enskip 2}{coutwo}
To verify the validity of Eq.~\ceq{sbtra}, it is sufficient to compute
the variations
$\delta_{\lambda^i}(S^1_b+S^2_b)$.
A straightforward calculation yields the following result:
\begin{eqnarray}
\delta_{\lambda^1}(S^1_b+S^2_b)&=&\frac{4\pi\Lambda}\kappa
\left[\oint_{\Gamma_2}d\bx_2\cdot
\lambda^1(x_2){\BF\Omega}^3_c(x_2)-
\oint_{\Gamma_3}d\bx_3\cdot
\lambda^1(x_3){\BF\Omega}^2_c(x_3)\right]\nonumber\\
\delta_{\lambda^2}(S^1_b+S^2_b)
&=&\frac{4\pi\Lambda}\kappa\left[\oint_{\Gamma_3}d\bx_3\cdot
\lambda^2(x_3){\BF\Omega}^1_c(x_3)-
\oint_{\Gamma_1}d\bx_1\cdot
\lambda^2(x_1){\BF\Omega}^3_c(x_1)\right]\nonumber\\
\delta_{\lambda^3}(S^1_b+S^2_b)
&=&\frac{4\pi\Lambda}\kappa\left[\oint_{\Gamma_1}d\bx_1\cdot
\lambda^3(x_1){\BF\Omega}^2_\mu(x_1)-
\oint_{\Gamma_2}dx_2\cdot
\lambda^3(x_2){\BF\Omega}^1_c(x_2)\right]\nonumber\\
\label{Cmiforgauvar}
\end{eqnarray}
It is easy to realize that the right hand sides of
Eqs.~\ceq{Cmiforgauvar} coincide exactly, apart from
a sign, with the gauge variations $\delta_{\lambda^i}S$ of
Eqs.~(\ref{Agninv}--\ref{Bgninv}) if 
the external currents $\bJ_i$ are given by Eq.~\ceq{currdef}.
This proves condition i)  and thus the gauge invariance of the action
$\bcas$\footnotemark{}\footnotetext{Let us notice however that,
similarly to what happens in the usual Chern-Simons field theories,
the gauge invariance of $\bcas$ is realized only up to terms of the
kind $\int d^3x\nablab\left[{\BF\lambda}^i
\cdot(\nablab\times\tilde{\BF\Omega}^i)\right]$, which can be discarded
only if the theory is defined on manifolds without boundary.}.

At this point we note that $S^1_b$ and $S^2_b$ contain only the
longitudinal components ${\BF\Omega}^i_L$ of the fields
${\BF\Omega}^i$.
Due to this fact, condition ii) is automatically satisfied and,
moreover, it is possible to perform the following substitutions in
Eqs.~(\ref{couone}--\ref{coutwo}):
\beq{{\BF\Omega}^i(x)\equiv
{\BF\Omega}^i_L(x)=\nablab(\Omega^i(x)-\Omega^i(x_0))}{subone}
where the $\Omega^i(x)$ are singlevalued scalar fields defined by the
relations:
\beq{\Omega^i(x)-\Omega^i(x_0)=\int^{\sbx}_{\sbx_0}
d\bx^{i\prime}\cdot
{\BF\Omega}^i_L(x')}{omegaidef}
In terms of the $\Omega^i(x)$'s, $S^1_b$ and $S^2_b$ can be written in
a form which is explicitly
metric independent in agreement with condition iii):
\begin{eqnarray}
S^1_b&=&\frac{4\pi\Lambda}\kappa\left[\oint_{\Gamma_3}dx_3^\rho\Omega^1(x_3)
b^2_\rho(x_3)-\oint_{\Gamma_2}dx_2^\rho\Omega^1(x_2)
b^3_\rho(x_2)\right]\nonumber\\
&+&\frac{4\pi\Lambda}\kappa\left[\oint_{\Gamma_1}dx_1^\rho\Omega^2(x_1)
b^3_\rho(x_1)-\oint_{\Gamma_3}dx_3^\rho\Omega^2(x_3)
b^1_\rho(x_3)\right]\nonumber\\
&+&\frac{4\pi\Lambda}\kappa\left[\oint_{\Gamma_3}dx_3^\rho\Omega^1(x_3)
b^2_\rho(x_3)-\oint_{\Gamma_2}dx_2^\rho\Omega^1(x_2)
b^3_\rho(x_2)\right]\label{Cindmetfor}
\end{eqnarray}
\begin{eqnarray}
S^2_b&=&\frac{4\pi\Lambda}\kappa\left[
\oint_{\Gamma_3}dx_3^\rho\Omega^2(x_3)\partial_\rho\Omega^1(x_3)
-\oint_{\Gamma_2}dx_2^\rho\Omega^3(x_2)\partial_\rho\Omega^1(x_2)\right.\nonumber\\
&+&\left.\oint_{\Gamma_1}dx_1^\rho\Omega^3(x_1)\partial_\rho\Omega^2(x_1)\right]
\label{sbtmetind}
\end{eqnarray}
We stress the fact that in the above formulas the undesired presence
of the variable $\bx_0$ introduced in Eqs.~(\ref{subone}) and
\ceq{omegaidef} has disappeared.

Before concluding this Section, let us show that the TTFT of
 Eq.~\ceq{ansatz}
is free from the inconsistences which
affected the naive BF--model with cubic interactions of
Eq.~\ceq{snaivevector}.
To this purpose, we need to study the classical equations of motion of
the fields. First of all, the variation of $\bcas$ with respect to
$\tilde{\BF\Omega}^i$ produces again the constraints \ceq{confromot}.
The solution of these equations has been already given in
Eq.~\ceq{consol}.
Varying instead $\bcas$ with respect to the
$\Omega^i_\mu(x)$'s 
%$S_b^1$ and $S^2_b$ and read as follows:
one obtains the following relations:
\begin{eqnarray}
&&\frac\kappa{4\pi}\epsilon^{\mu\nu\rho}\partial_\nu
\tilde\Omega^1_\rho(x)+\Lambda
\epsilon^{\mu\nu\rho}\Omega^2_\nu(x)\Omega^3_\rho(x)\nonumber\\
&+&\frac\Lambda\kappa\left[\oint_{\Gamma_3}dx_3^\nu b^2_\nu(x_3)
\partial_x^\mu\frac1{|x-x_3|}-
\oint_{\Gamma_2}dx_2^\nu b^3_\nu(x_2)
\partial_x^\mu\frac1{|x-x_2|}\right]\nonumber\\
&+&\frac\Lambda{4\pi\kappa}\int
d^3y\partial^\nu\Omega^3_\nu(y)\oint_{\Gamma_2}dx_2^\rho\partial_x^\mu\frac
1{|x-x_2|} \partial_\rho^{x_2}\frac 1{|y-x_2|}\nonumber\\
&-&\frac\Lambda{4\pi\kappa}\int
d^3y\partial^\nu\Omega^2_\nu(y)\oint_{\Gamma_3}dx_3^\rho\partial_x^\mu\frac
1{|x-x_3|} \partial_\rho^{x_3}\frac 1{|y-x_3|}=0\label{corequone}
\end{eqnarray}
\begin{eqnarray}
&&\frac\kappa{4\pi}\epsilon^{\mu\nu\rho}\partial_\nu
\tilde\Omega^2_\rho(x)-\Lambda
\epsilon^{\mu\nu\rho}\Omega^1_\nu(x)\Omega^3_\rho(x)\nonumber\\
&+&\frac\Lambda\kappa\left[\oint_{\Gamma_1}dx_1^\nu b^3_\nu(x_1)
\partial_x^\mu\frac1{|x-x_1|}-
\oint_{\Gamma_3}dx_3^\nu b^1_\nu(x_3)
\partial_x^\mu\frac1{|x-x_3|}\right]\nonumber\\
&-&\frac\Lambda{4\pi\kappa}\int
d^3y\partial^\nu\Omega^1_\nu(y)\oint_{\Gamma_3}dx_3^\rho\frac
1{|y-x_3|} \partial_x^\mu\partial_\rho^{x_3}\frac 1{|x-x_3|}\nonumber\\
&-&\frac\Lambda{4\pi\kappa}\int
d^3y\partial^\nu\Omega^3_\nu(y)\oint_{\Gamma_1}dx_1^\rho\partial_x^\mu\frac
1{|x-x_1|} \partial_\rho^{x_1}\frac 1{|y-x_1|}=0\label{corequtwo}
\end{eqnarray}
\begin{eqnarray}
&&\frac\kappa{4\pi}\epsilon^{\mu\nu\rho}\partial_\nu
\tilde\Omega^3_\rho(x)-\Lambda
\epsilon^{\mu\nu\rho}\Omega^1_\nu(x)\Omega^2_\rho(x)\nonumber\\
&+&\frac\Lambda\kappa\left[\oint_{\Gamma_2}dx_2^\nu b^1_\nu(x_2)
\partial_x^\mu\frac1{|x-x_2|}-
\oint_{\Gamma_1}dx_1^\nu b^2_\nu(x_1)
\partial_x^\mu\frac1{|x-x_1|}\right]\nonumber\\
&+&\frac\Lambda{4\pi\kappa}\int
d^3y\partial^\nu\Omega^1_\nu(y)\oint_{\Gamma_2}dx_2^\rho\frac
1{|y-x_2|} \partial_x^\mu\partial_\rho^{x_2}\frac 1{|x-x_2|}\nonumber\\
&-&\frac\Lambda{4\pi\kappa}\int
d^3y\partial^\nu\Omega^2_\nu(y)\oint_{\Gamma_1}dx_1^\rho\frac
1{|y-x_1|} \partial_x^\mu
\partial_\rho^{x_1}\frac 1{|x-x_1|}=0\label{corequthree}
\end{eqnarray}
We note that the differences between
Eqs.~(\ref{corequone}--\ref{corequthree}) and
Eqs.\ceq{Cmiforgauvar} are limited to purely longitudinal terms, so
that there is no effect
on the transverse
components of the fields, in agreement with condition ii).
Moreover,
the
longitudinal
components of the fields ${\BF\Omega}^i$ do not generate further
constraints, contrarily to what
%if one applies the operator $\partial_\mu$ to both sides of
%Eqs.~(\ref{corequone}--\ref{corequthree}), no further constraints are
%generated, contrarily to what
%Moreover, the projections of
%Eqs.~(\ref{corequone}--\ref{corequthree}) on the longitudinal
%directions do not generate any further constraints among the fields
%as it
happens
in the naive BF--model of the previous Section.
As a matter of fact, if
one applies the operator $\partial_\mu$ to both sides of
Eqs.~(\ref{corequone}--\ref{corequthree}), it is easy to
%if the operator $\partial_\mu$ is applied
%to both sides of Eqs.~(\ref{corequone}--\ref{corequthree}),
realize that the contributions %of the kind $\nablab\cdot
%({\BF\Omega}^i\times{\BF\Omega}^j)$
coming from the cubic
interactions present in the action $S$
cancel exactly against the new contributions coming
from $S_b^1$ and $S^2_b$. The reason is that now only the transverse components
of the fields are physical, while the longitudinal components are
associated to gauge degrees of freedom and remain thus undetermined by
the equations of motion.
In this way, the extension of the gauge symmetry to include the
transformations \ceq{gautraome} has solved
 the consistency problems discussed in Section 2.
\section{The Wilson Loop Amplitudes of the TTFT}
%Summarizing the results of the previous Section,
%the action $\bcas$ describes a consistent topological
%field
%theory which is gauge invariant under the transformations
%\ceq{gftilfie} and \ceq{gautraome}.
Summarizing the results of the previous Section, the action $\bcas$
describes a well defined topological field theory coupled to a set of
Wilson loops. Since the inconsistencies of the original action $S$
have been eliminated by the introduction of the terms $S^1_b$ and
$S^2_b$, we are now ready to compute the Wilson loop amplitudes of the
TTFT, which are given by:
\beq{\langle W_1(\Gamma_1) W_2(\Gamma_2) W_3(\Gamma_3)
\rangle_b=\int\left[\prod_{i=1}^3{\cal D}{\BF\Omega}^i
{\cal D}\tilde{\BF\Omega}^i\right]e^{-i\sbcas}}{wlcorr}
Here the average with respect to the fields ${\BF\Omega}^i$ and
$\tilde{\BF\Omega}^i$ has been written with the symbol
$\langle\ldots\rangle_b$ to distinguish it from the analogous average
of Eq.~\ceq{wlcorr}, in which the fields' behavior is governed by the
action $S$. Eq.~\ceq{wlcorr} represents the most general correlator of
Wilson loops.
In the following, we suppose that none of the Wilson loop operators is
trivial, i.~e.
\beq{W_i(\Gamma_i)\ne 1\qquad\qquad\qquad i=1,2,3}{conwls}
This condition is useful to rule out simpler subcases
which
are not of particular interest in the present context.
As a matter of fact, it is easy to
see
that, in the computation of
 amplitudes of the kind $\langle W_i(\Gamma_i) W_j(\Gamma_j)
\rangle_b$, $1\le i\ne j\le 3$, the
contributions coming from the cubic interaction terms present in
$\bcas$ are irrelevant. Thus, the TTFT behaves as a standard abelian
BF--model and delivers as topological invariants only the Gauss
linking numbers of the set of trajectories $\Gamma_i$ and $\Gamma_j$.

Let us now come back to the evaluation of Eq.~\ceq{wlcorr} under the
assumption \ceq{conwls}.
To eliminate the gauge freedom with respect to the transformations 
\ceq{gftilfie} and \ceq{gautraome}, we choose the Lorentz gauge
fixing, in which the fields are purely transverse:
\beq{\nablab\cdot{\BF\Omega}^i=\nablab\cdot\tilde{\BF\Omega}^i=0}{lgcon}
An immediate consequence of the Lorentz gauge is that $S^1_b$ and $S_b^2$
vanish identically because they contain only the longitudinal
components of the fields.
Thus, the amplitude
\ceq{wlcorr} can be written as
follows\footnotemark{}\footnotetext{Since the gauge group is abelian,
the contribution of the ghost fields to the Wilson loop amplitude
\ceq{wlcorr} amounts to a trivial normalization constant which can be
omitted.}:
\beq{\langle W_1(\Gamma_1) W_2(\Gamma_2) W_3(\Gamma_3)
\rangle_b=\int\left[\prod_{i=1}^3{\cal D}{\BF\Omega}^i
{\cal D}\tilde{\BF\Omega}^i\right]e^{-i\sbcas_q}}{wlsimp}
where the ``quantum'' action $\bcas_q$ is given
by:
\beq{\bcas_q=S-iS_{gf}}{quaact}
$S_{gf}$ being the gauge fixing term\footnotemark{}\footnotetext{Let
us note that, as it happens in C-S field theories, the topological actions 
$\bcas$ and $S$ keep the complex factor $i$ in the Feynman path
integral even in spaces equipped with Euclidean metrics. For this
reason, in agreement with our conventions, the gauge fixing term
appears in Eq.~\ceq{quaact} with a $-i$ factor in front.}:
\beq{S_{gf}=\int d^3x\left(\nablab \varphi^i\cdot{\BF\Omega}^i+
\nablab \tilde \varphi^i \cdot\tilde{\BF\Omega}^i\right)}{gaufixact}
In the above equation we have introduced
 the scalar fields $\varphi^i,\tilde\varphi^i$, which are Lagrange
multipliers imposing the gauge constraint \ceq{lgcon}.

We stress the fact that the disastrous effects caused by the
longitudinal components of the fields if one uses the naive action $S$
are brilliantly removed by the presence of the gauge fixing term
$S_{gf}$.
Indeed, one may easily check that the classical equations of motion
coming from the action \ceq{quaact} are free of inconsistences and 
admit the non-trivial solutions given below:
\begin{eqnarray}
{\BF\Omega}^i(x)&=&\bb^i(x)\label{Asolclaeqn}\\
\tilde{\BF\Omega}^i(x)&=&\frac\Lambda{2\kappa}\epsilon^{ijk}\int d^3y
\left(\nablab\frac 1{|x-y|}\right)\times\left[
\bb^j(y)\times\bb^k(y)\right]\label{Bsolclaeqn}\\
\tilde\varphi^i(x)&=&0\label{Csolclaeqn}\\
\varphi^1(x)&=&\frac{i\Lambda}\kappa\int d^3y \frac
1{|x-y|}\left[\bb^2(y)\cdot\bJ^3(y) - \bb^3(y)\cdot\bJ^2(y)\right]
\label{Dsolclaeqn}\\
\varphi^2(x)&=&\frac{i\Lambda}\kappa\int d^3y \frac
1{|x-y|}\left[\bb^3(y)\cdot\bJ^1(y) - \bb^1(y)\cdot\bJ^3(y)\right]
\label{Esolclaeqn}\\
\varphi^3(x)&=&\frac{i\Lambda}\kappa\int d^3y \frac
1{|x-y|}\left[\bb^1(y)\cdot\bJ^2(y) - \bb^2(y)\cdot\bJ^1(y)\right]
\label{Fsolclaeqn}
\end{eqnarray}
At this point it is possible to compute the generic Wilson loop
amplitude \ceq{wlsimp}.
As in the case of the naive BF-model discussed in Section 2, there are
only two connected Feynman diagrams, which are represented in
Fig.~\ref{feyrul}. The path integrals
in Eq.~\ceq{wlsimp} may be easily evaluated  integrating first over
the $\tilde{\BF\Omega}^i$ fields and then exploiting the constraints
\ceq{confromot} obtained in this way to perform the integration over the
fields ${\BF\Omega}^i$. Alternatively, one can derive the analytic
expression of $\langle W_1(\Gamma_1) W_2(\Gamma_2) W_3(\Gamma_3)
\rangle_b$ by means of successive Gaussian integrations. In both cases
the result is:
\beq{\langle W_1(\Gamma_1) W_2(\Gamma_2) W_3(\Gamma_3)
\rangle_b=N\mbox{exp}\left[-i\Lambda\int
d^3x\bb^1(x)\cdot(\bb^2(x)\times\bb^3(x))\right]}{wlfin}
where the $\bb^i$'s have been defined in \ceq{consol} and $N$ is a
normalization constant given by:
\beq{N=\int\left[\prod_{i=1}^3{\cal D}{\BF\Omega}^i
{\cal D}\tilde{\BF\Omega}^i\right]e^{-i\int d^3x\frac\kappa{4\pi}
\epsilon^{\mu\nu\rho} \Omega_\mu^i\partial_\nu\tilde\Omega_\rho^i}}{normcon}
We recall that the symbols $\Gamma_i$ in Eq.~\ceq{wlfin}
denote an ensemble of closed, non-intersecting paths
$\gamma_i,\gamma_i',\ldots$:
$\Gamma_i=\gamma_i+\gamma_i'+\ldots$. Due to the linearity properties
of the exponent in the right hand side of Eq.~\ceq{wlfin}, however, it
is clear that the amplitude $\langle W_1(\Gamma_1) W_2(\Gamma_2) W_3(\Gamma_3)
\rangle_b$ can be decomposed into a product of correlation functions
of three Wilson loops. For this reason, it will be sufficient to
consider from now on only the fundamental three loop correlation
function $\langle W_1(\gamma_1) W_2(\gamma_2) W_3(\gamma_3)
\rangle_b$, putting $\Gamma_i=\gamma_i$, $i=1,2,3$ in Eq.~\ceq{wlfin}.
As an upshot, the TTFT \ceq{ansatz} contains in
practice a single topological invariant, which appears in the exponent
of the right hand side of Eq.~\ceq{wlfin} and it is given by:
\beq{{\cal H}=\frac 13\int
d^3x\epsilon^{\mu\nu\rho}\epsilon^{ijk}b_\mu^i(x)b_\nu^j(x)b_\rho^k(x)
}{tisymm} 

To conclude this Section, we study the topological term ${\cal
H}$. In the following, it will be
convenient to interpret
the vector fields $b_\mu^i(x)$ of Eq.~\ceq{consol} as magnetic
fields\cite{edwards}
\beq{\bb^i(x)=-\frac
1\kappa\oint_{\Gamma_i}d\bx_i\times\frac{(\bx-\bx_i)} {|x-x_i|^3}}{magfieint}
generated by the currents $\bj_i=-\frac 1\kappa\bJ_i$. Indeed, it is
possible to see that the
$b_\mu^i(x)$'s
satisfy
the relations:
\begin{eqnarray}
\nablab\times\bb^i=\bj_i&\qquad\qquad&\nablab\cdot\bb_i=0\nonumber\\
\nablab\times\ba^i=\bb_i&\qquad\qquad&\nablab\cdot\ba^i=0\label{emfieeqs}
\end{eqnarray}
where the $\ba^i$'s are their associated electromagnetic potentials:
\beq{\ba^i=-\frac 1\kappa\oint_{\Gamma_i}d\bx_i\frac
1{|x-x_i|}}{emvecpot}
Moreover, one can introduce
the multivalued magnetic potentials
$v^i(x)$:
\beq{v^i(x)-v^i(x_0)=-\int_{x_0}^x d\bx'\cdot\bb^i(x')}{elpot}
defined in such a way that $\bb^i(x)=-\nablab
v^i(x)$\footnotemark{}\footnotetext{Mathematically,
each $v^i(x)$ is the solid  angle under which the trajetory $\Gamma_i$
appears as seen from a point $x$ (see e.~g.\cite{klePI}, Ch. 16).}.


The most straightforward interpretation of ${\cal H}$ is that of an Hopf
invariant of the underlying gauge group $U(1)\otimes U(1)\otimes
U(1)\equiv[U(1)]^3$. To show that, we build the $[U(1)]^3$ group
element
\beq{g(x)=e^{-i\sum_{i=1}^3 v^i(x)}}{groele}
It is now easy to check that, apart from a proportionality factor,
${\cal H}$ has exactly the form of the desired Hopf term:
\beq{{\cal H}\propto\epsilon^{\mu\nu\rho}\int d^3x
\frac {\partial v^1(x)}{\partial x^\mu}
\frac {\partial v^2(x)}{\partial x^\nu}
\frac {\partial v^3(x)}{\partial x^\rho}
g^{-1}\frac{\partial g}{\partial v^1}
g^{-1}\frac{\partial g}{\partial v^2}
g^{-1}\frac{\partial g}{\partial v^3}}{firint}

Another  form of ${\cal H}$ may be derived
introducing the Pauli matrices $\sigma^i$ and the vector fields
\beq{b_\mu(x)=\sigma^ib^i_\mu(x)}{postwo}
The expression of ${\cal H}$ as a function of $b_\mu(x)$ becomes:
\beq{{\cal H}=\frac 16
\int d^3x
\mbox{Tr}\left[\bb(x)\cdot(\bb(x)\times\bb(x))\right]}{htlike}
Here the symbol Tr denotes trace over the Pauli matrices.
To go back to the original formulation of ${\cal H}$
given in Eq.~\ceq{tisymm} it is sufficient to
use the relation
$\mbox{Tr}[\sigma^i\sigma^j\sigma^k]=2\epsilon^{ijk}$.
Apparently, from the above equation
 ${\cal H}$ coincides with an Hopf term
for the group $SU(2)$, which is not a symmetry group of our theory,
but of course one should remember that
$\bb(x)$ is not a pure $SU(2)$ gauge field configuration.

Finally, one can give a physical meaning
to ${\cal H}$ exploiting the electromagnetic analogy established by
Eqs.~(\ref{magfieint}--\ref{emvecpot})
%is obtained by noting
%that
%the classical vector fields $b_\mu^i(x)$ of Eq.~\ceq{consol} may be
%interpreted as magnetic fields
%\beq{\bb^i(x)=-\frac
%1\kappa\oint_{\Gamma_i}d\bx_i\times\frac{(\bx-\bx_i)} {|x-x_i|^3}}{magfieint}
%%generated by the currents $\bj_i=-\frac 1\kappa\bJ_i$ and satisfying
%t%he relations:
%\begin{eqnarray}
%\nablab\times\bb^i=\bj_i&\qquad\qquad&\nablab\cdot\bb_i=0\nonumber\\
%\nablab\times\ba^i=\bb_i&\qquad\qquad&\nablab\cdot\ba^i=0\label{emfieeqs}
%\end{eqnarray}
%Here $\ba^i$ represents the electromagnetic potentials
%\beq{\ba^i=-\frac 1\kappa\oint_{\Gamma_i}d\bx_i\frac
%1{|x-x_i|}}{emvecpot}
and the fact that the $b_\mu^i(x)$'s
satisfy the classical equations of motion
(\ref{corequone}--\ref{corequthree}) in the Lorentz gauge. After some
calculations one finds:
\beq{{\cal H}=-\frac{4\pi}{3\kappa}\epsilon^{ijk}
\int\!\!\!\int_{\Sigma_i}
d\bS^i\cdot(\bb^j\times\bb^k)
}{forint}
Here
%In Eq.~\ceq{forint}
$\Sigma_i$ denotes an arbitrary surface whose
boundary is given by the contour
$\Gamma_i$, while $d\bS^i$ is the projection of the infinitesimal area
element of $\Sigma_i$ along the normal direction with respect to the
surface.
From Eq.~\ceq{forint} it turns out that ${\cal H}$ measures the sum
for $i=1,2,3$ of the fluxes of the vector fields
$\epsilon^{ijk}\bb^j\times\bb^k$ through the surfaces $\Sigma_i$.

%Looking at Eq.~\ceq{lonevar}, one could suppose that $S_b$ should be
%non-linear in the fields ${\bf\Omega}^i$, but this is not true.
%The reason is that the ${\bf\Omega}^i$'s have no real dynamics, but
%are bounded to live in the subspace of all field configurations
%determined by the constraints \ceq{confromot}.
%Exploiting these constraints and after an integration by parts, one
%finds that the variation of $S$ with respect to a gauge transformation
%are linear in the fields:
%To understand how it is possible to construct a gauge invariant extension
%of the theory under examination, we perform in the action $S$ a gauge
%transformation of the kind:
%\beq{{\BF\Omega}^i(x)\longrightarrow{\BF\Omega}^i(x)+\nablab\lambda^i(x)}
%{gautraome}
%Considering for simplicity only the case $i=1$, the result is the new
%action $S'=S+\delta_{\lambda^1}S$, where
%\beq{\delta_{\lambda^1}S=\Lambda\int d^3x\nablab\lambda^1\cdot
%({\BF\Omega}^2\times{\BF\Omega}^3)}{lonevar}
%Exploiting Eqs.~\ceq{confromot} one finds, after an integration by
%parts, that
%\beq{\delta_{\lambda^1}S=\frac{4\pi\Lambda}\kappa\int d^3x
%\lambda^1\left[
%\bJ_2\cdot{\BF\Omega}^3-\bJ_3\cdot{\BF\Omega}^2\right]}{modlonevar}
%Analogous relations may be derived starting from a gauge variation of
%the fields ${\BF\Omega}^2$ and ${\BF\Omega}^3$:


%
%At this point we construct a new action ${\cab S}$ 
%\beq{{\cab S}=S+S_b^1+S_b^2}{newactiondef}
%defined in such a way that
%\beq{\delta_{\lambda^i}{\cab S}=0}{newactgauinv}
%Clearly, this requires to find terms $S_b^1+S_b^2$ with the property that
%they transform as follows under the gauge transformations \ceq{gautraome}:
%counterterms which, when added to the
%action $S$, are able to cancel the terms
%(\ref{modlonevar}--\ref{Bgninv}) which spoil
%the gauge invariance of the theory under
%the
%transformations \ceq{gautraome}.
%Let us notice that the gauge invariance obtained in this way will not
%be ``on shell'' despite the fact that we used Eqs.~\ceq{confromot} in order to
%evaluate the gauge non-invariant contributions as we made in
%(\ref{modlonevar}--\ref{Bgninv}).
%As a matter of fact, the fields ${\BF\Omega}^i$ are bound
%anyway to live in the subspace of all field configurations determined
%by the constraints \ceq{confromot}.




\section{Conclusions}
In this work a new topological field theory has been constructed, the
TTFT of Eq.~\ceq{ansatz}, with the property that its perturbative
series contains only the finite set of Feynman diagrams given in
Fig.~\ref{feyrul}. The TTFT is exactly solvable and, besides the Gauss
link invariant which already appears in
abelian C-S field theories, it produces the further topological
invariant ${\cal H}$ of Eq.~\ceq{tisymm}. The latter has been
interpreted as an Hopf term in Eq.~\ceq{firint}. Another form of
${\cal H}$ has been given in \ceq{htlike}.
This equation suggests also an interesting generalization of
the TTFT, consisting in the replacement
of the cubic interaction present  in the naive action $S$ 
with a new interaction of the kind $\int d^3x
f^{ijk}{\BF\Omega}^i\cdot({\BF\Omega}^j\times{\BF\Omega}^k)$, where
$f^{ijk}$ denotes
the structure constants of a compact Lie group.

As a final remark, let us note
 that the derivation of the action of the TTFT starting
from the naive BF--model of Eq.~\ceq{snaivevector}
% and then adding the
%terms $S_b^1$ and $S_b^2$ in order to cure its inconsistences via the
%enlargement of the gauge group of symmetry,
 has
some analogies with the way in which gauge invariance is implemented 
%C-S field theories defined on manifolds with boundary. For instance,
 in C-S based models of the quantum Hall effect\cite{qhe}.
% one has to add an
%edge action to the C-S action in order to make the whole theory gauge
%invariant.
In our case a fictitious one-dimensional ``boundary'', which lies on
the trajectories
$\Gamma_i$, appears due to the introduction of
the Wilson loops.
% there is a ``boundary'', which lies on the
%trajectories $\Gamma_i$ and it is due to
% the $\delta-$function
%distributions introduced in the theory by
% the insertion of the Wilson loops.
%(see Eq.~\ceq{currdef}).
 The inconsistent constraints 
(\ref{Aextcon}--\ref{Cextcon}) arising in the naive BF--model are all
concentrated along these trajectories because of the particular form of the
currents $\bJ_i$ defined in Eq.~\ceq{currdef}.
The analogue of the edge state action of the quantum Hall effect is
given here by the boundary terms 
$S_b^1$ and $S_b^2$, which restore gauge invariance in the fields
${\BF\Omega}^i$ and eliminate in this way the inconsistences of the action
\ceq{snaivevector}. 
\begin{thebibliography}{99}
\bibitem{nemliq} M.~J.~Bowick, L.~Chander, E.~A.~Schiff and
A.~M.~Srivastava, {\it Science} {\bf 263} (1994), 943.
\bibitem{supflu} C. B\"auerle et al., {\it Nature} {\bf 382} (1996),
332;
V. M. H. Ruutu et al., {\it Nature} {\bf 382} (1996), 334.
\bibitem{wasserman} E. Wasserman, {\it Jour. Am. Chem. Soc.} {\bf 82}
(1960), 4433.
\bibitem{quahaleff} D. R. Leadly et al., {\it Phys. Rev. Lett.} {\bf
79} (1997), 4246.
\bibitem{chesim} 
%\cite{Schwarz:1978cn}
A.~S.~Schwarz,
%``The Partition Function Of Degenerate Quadratic Functional And
%Ray-Singer Invariants,''
{\it Lett.\ Math.\ Phys.\ }  {\bf 2} (1978), 247;
%%CITATION = LMPHD,2,247;%%
%\cite{Deser:1982vy}
%\bibitem{Deser:1982vy}
S.~Deser, R.~Jackiw and S.~Templeton,
%``Three-Dimensional Massive Gauge Theories,''
{\it Phys.\ Rev.\ Lett.\ } {\bf 48} (1982), 975;
%%CITATION = PRLTA,48,975;%%
%\cite{Deser:1982wh}
%\bibitem{Deser:1982wh}
S.~Deser, R.~Jackiw and S.~Templeton,
%``Topologically Massive Gauge Theories,''
{\it Annals Phys.\ }  {\bf 140} (1982), 372;
%[Erratum-ibid.\  {\bf 185} (1982) 406.1988\ APNYA,281,409];
%%CITATION = APNYA,140,372;%%
%\cite{Hagen:1984rp}
%\bibitem{Hagen:1984rp}
C.~R.~Hagen,
%``A New Gauge Theory Without An Elementary Photon,''
{\it Annals Phys.\ }  {\bf 157} (1984), 342.
%%CITATION = APNYA,157,342;%%
\bibitem{bfmod} G. Horowitz, {\it Comm. Math. Phys.} {\bf 125} (1989),
417;
A. S. Schwartz, {\it Comm. Math. Phys.} {\bf 67}(1979), 1;
M. Blau and G. Thompson, {\it Ann. Phys.} (NY) {\bf 205} (1991), 130.
\bibitem{edwards}
S. F. Edwards, {\it Proc.~Phys.~ Soc.}
{\bf 91} (1967), 513;
J.~Phys.~{\bf A1} (1968), 15.
\bibitem{klePI}
H. Kleinert,
{\em Path Integrals}, (World Scientific Publishing, 2nd Ed.,
Singapore, 1995).
\bibitem{qhe}
%\cite{Wilczek:1990ik}
%\bibitem{Wilczek:1990ik}
F.~Wilczek,
%''Fractional Statistics And Anyon Superconductivity,''
{\it Fractional Statistics and Anyon
Superconductivity},
(World Scientific, New Jersey 1990), Ch. 9;
X. G. Wen, {\it Phys. Rev.} {\bf B43} (1991), 11025.
\end{thebibliography}
\end{document}

math-ph0108026.extracted_bib
\begin{thebibliography}{99}
\bibitem{nemliq} M.~J.~Bowick, L.~Chander, E.~A.~Schiff and
A.~M.~Srivastava, {\it Science} {\bf 263} (1994), 943.
\bibitem{supflu} C. B\"auerle et al., {\it Nature} {\bf 382} (1996),
332;
V. M. H. Ruutu et al., {\it Nature} {\bf 382} (1996), 334.
\bibitem{wasserman} E. Wasserman, {\it Jour. Am. Chem. Soc.} {\bf 82}
(1960), 4433.
\bibitem{quahaleff} D. R. Leadly et al., {\it Phys. Rev. Lett.} {\bf
79} (1997), 4246.
\bibitem{chesim} 
%\cite{Schwarz:1978cn}
A.~S.~Schwarz,
%``The Partition Function Of Degenerate Quadratic Functional And
%Ray-Singer Invariants,''
{\it Lett.\ Math.\ Phys.\ }  {\bf 2} (1978), 247;
%%CITATION = LMPHD,2,247;%%
%\cite{Deser:1982vy}
%\bibitem{Deser:1982vy}
S.~Deser, R.~Jackiw and S.~Templeton,
%``Three-Dimensional Massive Gauge Theories,''
{\it Phys.\ Rev.\ Lett.\ } {\bf 48} (1982), 975;
%%CITATION = PRLTA,48,975;%%
%\cite{Deser:1982wh}
%\bibitem{Deser:1982wh}
S.~Deser, R.~Jackiw and S.~Templeton,
%``Topologically Massive Gauge Theories,''
{\it Annals Phys.\ }  {\bf 140} (1982), 372;
%[Erratum-ibid.\  {\bf 185} (1982) 406.1988\ APNYA,281,409];
%%CITATION = APNYA,140,372;%%
%\cite{Hagen:1984rp}
%\bibitem{Hagen:1984rp}
C.~R.~Hagen,
%``A New Gauge Theory Without An Elementary Photon,''
{\it Annals Phys.\ }  {\bf 157} (1984), 342.
%%CITATION = APNYA,157,342;%%
\bibitem{bfmod} G. Horowitz, {\it Comm. Math. Phys.} {\bf 125} (1989),
417;
A. S. Schwartz, {\it Comm. Math. Phys.} {\bf 67}(1979), 1;
M. Blau and G. Thompson, {\it Ann. Phys.} (NY) {\bf 205} (1991), 130.
\bibitem{edwards}
S. F. Edwards, {\it Proc.~Phys.~ Soc.}
{\bf 91} (1967), 513;
J.~Phys.~{\bf A1} (1968), 15.
\bibitem{klePI}
H. Kleinert,
{\em Path Integrals}, (World Scientific Publishing, 2nd Ed.,
Singapore, 1995).
\bibitem{qhe}
%\cite{Wilczek:1990ik}
%\bibitem{Wilczek:1990ik}
F.~Wilczek,
%''Fractional Statistics And Anyon Superconductivity,''
{\it Fractional Statistics and Anyon
Superconductivity},
(World Scientific, New Jersey 1990), Ch. 9;
X. G. Wen, {\it Phys. Rev.} {\bf B43} (1991), 11025.
\end{thebibliography}