{Three-Dimensional Chern-Simons and BF Theories}
{A. Borowiec$^{*}$ and M. Francaviglia$^{**}$}
Our aim in this note is to clarify a relationship between covariant Chern-Simons 3-dimensional theory and Schwartz type topological field theory known also as BF theory.\\ Keywords: Chern-Simons theory, BF theory, covariance, N\"other conservation laws. PACS: 02.40.-k, 11.15.-q
math-ph0506068.tex
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\author {{\bf A. Borowiec$^{*}$ and M. Francaviglia$^{**}$}}
\title{{\bf Three-Dimensional Chern-Simons and BF Theories}}
\maketitle
\date{ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $ $\ $
AIP Conf.Proc.751:165-167,2005  \\
$^{\ \,*\ }$Institute for Theoretical Physics, University of Wroc{\l}aw, Poland \\
$^{**}$ Dipartimento di Matematica, Universit\`a di Torino, Italy}
%Via C. Alberto 10\\ 10123 TORINO, ITALY }}
%LPTL, Universit\'e Paris-VI, CNRS UMR 7600, 4 Place Jussieu, Paris, France}
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 \begin{abstract}
Our aim in this note is to clarify a relationship between covariant
Chern-Simons 3-dimensional theory and Schwartz type topological
field theory known also as BF theory.\\
Keywords: Chern-Simons theory, BF theory, covariance, N\"other conservation laws.
 PACS: 02.40.-k, 11.15.-q
\end{abstract}
\maketitle

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\section{Introduction.}

Chern-Simons theory gives an interesting example of topological
field theory. Its Lagrangian 3-form lives on a principal $G$-bundle
and after pulling back to space-time (base) manifold provides, in
general, a family of local, non-covariant 
Lagrangian densities
\cite{BFF2,BFFP}. \footnote{However, the corresponding
Euler-Lagrange equations of motion have well-defined global
meaning.} Because of this, it is also more difficult to analyze, in
this case, N\"oether conserved quantities \cite{BFF1}. A more standard
approach to the problem of symmetries and conservation laws has been
applied in the so called covariant formalism \cite{BFF2}. It
exploits the transgression 3-form  as a global and covariant
Chern-Simons Lagrangian with two dynamical gauge fields. This
formalism has been used for the calculation of N\"other currents and
their identically vanishing parts - superpotentials. Augmented
variational principle and relative conservation laws have been
recently proposed in \cite{FFF}. Our aim in the present note, 
which can be viewed as an appendix to \cite{BFF2}, is to explain a 
link between covariant Chern-Simons theory and the so called BF
theories \cite{BBRT,CCFM,Hor}.


\section{Change of variables.}

Let us consider a principal bundle $P(M,G)$ over a three-dimensional
manifold $M$ with a (semisimple) structure group $G$. Let
$\omega_{i}$ \ ($i=0,1$)  be two principal connection 1-forms with the
corresponding curvature 2-forms
\begin{eqnarray}\label{a1}
\Omega_{i}=d\omega_{i}+\omega_{i}^{2}=d\omega_{i}+\frac{1}{2}[\omega
_{i},\omega_{i}] .\end{eqnarray}
Denote by $\alpha=\omega_{1}-\omega_{0}$, a  tensorial 1-form.

The transgression 3-form is given by the well known formula
\begin{eqnarray}\label{a2}
Q\left( \omega_{1},\omega_{0}\right) = -Q\left(\omega_{0},\omega_{1}\right) =tr\left(2\Omega_{0}\wedge\alpha+D_{0}\alpha\wedge\alpha+\frac{2}{3}%
\alpha^{3}\right)\nonumber\\%\lefteqn
=tr\left(  2\Omega_{1}\wedge\alpha-D_{1}\alpha\wedge\alpha+\frac{2}{3}\alpha^{3}\right)
\end{eqnarray}
where $D_i\alpha=d\alpha+[\omega_i,\alpha]$ denotes the covariant
derivative of $\alpha$ with respect to the connection $\omega_i$.
Thus $Q(\omega_{1},\omega_{0})$ is a tensorial (covariant) object
which well-defines the corresponding global 3-form on $M$. It
undergoes a non-covariant splitting as a difference of two
Chern-Simons Lagrangians
\begin{eqnarray}\label{a3}
Q\left(  \omega_{1},\omega_{0}\right)  =CS(\omega
_{1})-CS(\omega_{0})+dtr\left(  \omega_{0}\wedge\omega_{1}\right)\end{eqnarray}
where
\begin{eqnarray}\label{a4}
CS(\omega)=tr\left(\Omega\wedge\omega-\frac{1}{3}\omega^{3}\right)\end{eqnarray}
stands for non-covariant Chern-Simons Lagrangian.

Notice that in the case of two connections one has
\begin{eqnarray}\label{a5}
2\Omega_{0}+D_{0}\alpha=2\Omega_{1}-D_{1}\alpha=\nonumber\\
\Omega_{0}+\Omega_{1}%
+\frac{1}{2}\left(  D_{0}\alpha-D_{1}\alpha\right)=
\Omega_{0}+\Omega_{1}-\alpha^{2}
\end{eqnarray}
The last equality entitles us to rewrite
\begin{eqnarray}\label{a6}
Q\left(  \omega_{1},\omega_{0}\right)  =
2tr\left(\bar\Omega\alpha
+\frac{1}{12}\alpha^{3}\right)  \
\end{eqnarray}
where $\bar\omega=\frac{1}{2}\left(  \omega_{1}+\omega_{0}\right)  $ is a new (average)
connection and $\bar\Omega=d\bar\omega+\bar\omega^{2}$. 
%Hint: $\Omega=\frac{1}{2}\left(  \Omega_{1}+\Omega_{0}\right)  -\frac{1}%
%{4}\alpha^{2}$
Of course, one has $\omega_{1}=\bar\omega+\frac{1}{2}\alpha$ \ , \ \ $\omega_{0}=\bar\omega
-\frac{1}{2}\alpha$.

Thus the Lagrangian $Q\left(  \omega_{1},\omega_{0}\right)  $ can be treated in
three different (but equivalent) ways:
\begin{itemize}
\item with two (flat) connections $\omega_{0},\ \omega_{1}$ as dynamical variables; see (\ref{a3}). In this case $\Omega_{0}=0$  and $\ \Omega_{1}=0$ are  equations of motion. This point of view was presented in \cite{BFF2}.

\item with a (flat) connection $\omega_{1}$ and tensorial 1-form $\alpha$ as dynamical variables; see (\ref{a2}). In this case equations of motion are
$\Omega_{1}=0$ \ and $D_{1}\alpha=\alpha^{2}$.%\equiv\frac{1}{2}[\alpha,\alpha]$

\item with an "average' (non-flat) connection $\bar\omega=\frac{1}{2}\left(  \omega
_{1}+\omega_{2}\right)$ and tensorial 1-form $\alpha$ as independent dynamical
variables; see (\ref{a6}). In this case
$\bar \Omega=-\frac{1}{4}\alpha^{2}$ \ and \ $\bar D\alpha=0$ are equations of motion.
This is the so called BF theory with a cosmological constant $\Lambda=1$ (see e.g references
\cite{BBRT,CCFM,Mont,Sarda}).
\end{itemize}
\bigskip

More generally, one can define a new connection $\omega_{t}=t\omega_{1}+\left(
1-t\right)  \omega_{0}=\omega_{0}+t\alpha$ as a convex combination
of two connections  with parameter \ $0\leq t\leq1$. The inverse transformation is $\omega_{0}=\omega_t-t\alpha$ \ , $\omega_{1}=\omega_{t}+(1-t)\alpha$. In this case
$$\Omega_{t}=t\Omega_{1}+\left(  1-t\right)  \Omega_{0}-t\left(
1-t\right)  \alpha^{2}\ .$$
Now the equation (\ref{a5}) can be replaced by the more general one
\begin{eqnarray}\label{a7}
2\Omega_{1}-D_{1}\alpha=2\Omega_{0}+D_{0}\alpha=\\\nonumber
=2\Omega_{t}-2t\left(  1-t\right)  \alpha^{2}-\left(  2t-1\right)
D_{t}\alpha
\end{eqnarray}
(notice that $t\omega_{1}+\left(  t-1\right)  \omega_{0}=\left(  2t-1\right)
\omega_{t}+2t\left(  1-t\right)  \alpha$).

In this new variables $(\omega_t,\ \alpha)$ we  obtain
\begin{eqnarray}\label{a8}
Q\left(  \omega_{1},\omega_{0}\right)  =
2tr\left(  \Omega_{t}\wedge\alpha-\left(  t-\frac{1}{2}\right)  D_{t}\alpha
\wedge\alpha+(\frac{1}{3}-t+t^{2})\alpha^{3}\right)
\end{eqnarray}
The corresponding equations of motion are:
\begin{eqnarray}
\Omega_{t}=-t\left(  1-t\right)  \alpha^{2} \ \ \ \ , \ \ \ \ \ \ D_{t}%
\alpha=\left(  2t-1\right)  \alpha^{2}\end{eqnarray}
Thus the choices $t=\frac{1}{2}, 0, 1$ lead to the simplest formulae.

It is interesting to observe that the superpotential related to (infinitesimal) gauge
transformation $\chi$ remains independent of the choice of variables $(\omega_t,\ \alpha)$
(compare formula (19) in \cite{BFF2}), i.e.:
$$
U(\chi)=tr(\alpha\chi) .
$$
Instead, an explicite expression for the superpotential related to (infinitesimal) diffeomorphism transformation driven by a vectorfield $\xi$ does depend on the variables $(\omega_t,\ \alpha)$ and equals
to (compare formulae (23,25) in \cite{BFF2}):
$$
U(\xi)=tr[\alpha(2\omega_t(\xi)+(1-2t)\alpha(\xi))]\ .
$$
   
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%  Infandum, regina, iubes renovare dolorem, Troianas ut opes et
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\begin{thebibliography}{15}

\bibitem{BBRT}
D. ~Birmingham, M. ~Blau, M. ~Rakowski and G. ~Thompson,
\emph{Phys. Rep.}, \textbf{209}, 129 (1991).

\bibitem{BFF1}
A. ~Borowiec, M. ~Ferraris and M. ~Francaviglia,
\emph{J. Phys. A: Math. Gen.}, \textbf{36}, 8823-8833 (1998).

\bibitem{BFF2}
A. ~Borowiec, M. ~Ferraris and M. ~Francaviglia,
\emph{J. Phys. A: Math. Gen.}, \textbf{31}, 2589-2598 (2003).

\bibitem{BFFP}
A. ~Borowiec, M. ~Ferraris, M. ~Francaviglia and M. Palese,
\emph{Universitatis Iagellonicae Acta Mathematica}, \textbf{XLI}, 319-331 (2003).

\bibitem{CCFM}
A ~.S. Cattaneo, P. Cotta-Ramusino, J. Fr\"ohlich and M. Martellini, \emph{J. Math. Phys.}, \textbf{36}, 6137-6160 (1995).

\bibitem{FFF}
L. ~Fatibene, M. ~Ferraris, and M. ~Francaviglia,
``Augmented Variational Principles and Relative Conservation Laws in
Classical Field Theory"
\emph{math-phys/0411029} (2004).

\bibitem{Hor}
G ~.T. Horowitz, \emph{Commun. Math. Phys.}, \textbf{125}, 417 (1989).

\bibitem{Mont}
~M. Montesinos,
\emph{Class. Quantum Grav.}, \textbf{20}, 3569-3575 (2003).

\bibitem{Sarda}
~G. Sardanashvily, \emph{Int.J. Geom. Meth. Mod. Phys.}, \textbf{1},
  4 (2004).

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\bibitem{Brown2000}
M.~P. Brown,  and K.~Austin, \emph{The New Physique}, Publisher Name,
  Publisher City, 2000, pp. 212--213.

\bibitem{BrownAustin:2000}
M.~P. Brown,  and K.~Austin, \emph{Appl. Phys. Letters}, \textbf{85},
  2503--2504 (2000).

\bibitem{Wang}
R.~Wang, ``Title of Chapter,'' in \emph{Classic Physiques}, edited by
  R.~B. Hamil, Publisher Name, Publisher City, 2000, pp. 212--213.

\bibitem{SJ:1999}
C.~D.~Smith and E.~F.~Jones,  ``Load-Cycling in Cubic Press,'' in
  \emph{Shock Compression of Condensed Matter-1999}, edited by M.~D.~F. et~al.,
  AIP Conference Proceedings 505, American Institute of Physics, New York,
  1999, pp. 651--654.

\end{thebibliography}
\end{document}
math-ph0506068.extracted_bib
\begin{thebibliography}{15}

\bibitem{BBRT}
D. ~Birmingham, M. ~Blau, M. ~Rakowski and G. ~Thompson,
\emph{Phys. Rep.}, \textbf{209}, 129 (1991).

\bibitem{BFF1}
A. ~Borowiec, M. ~Ferraris and M. ~Francaviglia,
\emph{J. Phys. A: Math. Gen.}, \textbf{36}, 8823-8833 (1998).

\bibitem{BFF2}
A. ~Borowiec, M. ~Ferraris and M. ~Francaviglia,
\emph{J. Phys. A: Math. Gen.}, \textbf{31}, 2589-2598 (2003).

\bibitem{BFFP}
A. ~Borowiec, M. ~Ferraris, M. ~Francaviglia and M. Palese,
\emph{Universitatis Iagellonicae Acta Mathematica}, \textbf{XLI}, 319-331 (2003).

\bibitem{CCFM}
A ~.S. Cattaneo, P. Cotta-Ramusino, J. Fr\"ohlich and M. Martellini, \emph{J. Math. Phys.}, \textbf{36}, 6137-6160 (1995).

\bibitem{FFF}
L. ~Fatibene, M. ~Ferraris, and M. ~Francaviglia,
``Augmented Variational Principles and Relative Conservation Laws in
Classical Field Theory"
\emph{math-phys/0411029} (2004).

\bibitem{Hor}
G ~.T. Horowitz, \emph{Commun. Math. Phys.}, \textbf{125}, 417 (1989).

\bibitem{Mont}
~M. Montesinos,
\emph{Class. Quantum Grav.}, \textbf{20}, 3569-3575 (2003).

\bibitem{Sarda}
~G. Sardanashvily, \emph{Int.J. Geom. Meth. Mod. Phys.}, \textbf{1},
  4 (2004).

\end{thebibliography}