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\begin{document}
\begin{titlepage}
\vspace*{1cm}
\begin{center}
{\LARGE Faddeev's anomaly and bundle gerbes.}
\vspace*{2cm}
\end{center}
\begin{center}
{\Large A.L. Carey\footnote{acarey@maths.adelaide.edu.au},
M.K. Murray\footnote{mmurray@maths.adelaide.edu.au}}
\end{center}
\vspace*{1cm}
\begin{center}
Department of Pure Mathematics,\\
University of Adelaide,\\
South Australia
5005,\\ AUSTRALIA.
\end{center}
\vfill
\noindent{\bf Abstract:} We show that
transgression of the Dixmier-Douady class coming
from the natural
gerbe arising in the discussion of Faddeev's anomaly
is the Faddeev-Mickelsson two cocycle.
\vskip 1cm
\end{titlepage}
\bigskip
\section{Introduction}
\label{sec:intro}
In \cite{CarMur} we indicated how Segal's discussion of
Faddeev's anomaly \cite{Seg} could be phrased in the
language of bundle gerbes and conjectured that the
Faddeev anomaly was the transgression of the Dixmier-Douady class of the
corresponding bundle gerbe. In this note we
prove this conjecture.
We utilise the results of \cite{MicRaj} on the Faddeev-Mickelsson
2-cocycle on the gauge group to determine the Dixmier-Douady class via our
notion of transgression (defined in section 2).
This proof is of interest for several reasons. First, Segal's
work in \cite{Seg}
remains unpublished. Second, the discussion of this issue in
\cite{Mic} gives the impression that there is no connection between
Segal's approach and that of \cite{MicRaj}. The reason for this is
that the notion of bundle gerbe was invented recently and is essential
for the method we adopt here of relating the two.
Third, the notion of bundle gerbe provides a way of realising
degree three integral cohomology of a manifold
geometrically that is naturally adapted to the classes
arising in the study of anomalies
(see \cite {WanCarMur}).
The example discussed in this paper lends further weight to this
last assertion.
\section{Transgression}
\label{sec:trans}
If $M$ is a compact three-manifold with
Riemannian structure and spin-structure, denote by $\A$
the affine space of connections on a complex
vector bundle over $M$ and by $\G$ the
group of gauge transformations of this bundle.
We define the {\em transgression map} \cite{CarMur}
as follows. If $\omega$ is a three form on $\A/\G$
representing a cohomology class pull it back to a
closed three form $\pi^*(\omega)$ on $\A$.
If $A \in \A$
and $X$ and $Y$ are elements of the Lie algebra of $\G$
they define vectors $X$ and $Y$ in the tangent space at $A$ and
we define
$$
c(X, Y) (A) = \mu_A(X, Y)
$$
a co-cycle which defines a class in $H^2(L\G, {\rm Map}(\A,\C))$
the Lie algebra cohomology of the Lie algebra of the
gauge group with values in the module ${\rm Map}(\A,\C)$.
The resulting map
$$
H^3(\A/\G, \C) \to H^2(L\G, {\rm Map}(\A,\C))
$$
we call the transgression.
\section{Bundle gerbes}
\label{sec:gerbes}
Let $\pi \colon Y \to P$ be a submersion. That is $Y$ and $P$
are manifolds
and $\pi$ is onto and has onto derivative. It follows from the
submersion theorem that $Y \to P$ admits local sections. This
will be obvious in the example we consider below. Define the fibre
product of $Y$ with itself to be
$$
Y^{[2]} = \{ (x, y) \vert \pi(x) = \pi(y) \} \subset Y\times Y.
$$
Notice that the diagonal $\triangle = \{(y, y) \vert y \in Y\}$ is
a subset of $Y^{[2]}$.
A bundle gerbe over $P$ is a pair $(J, Y)$ where $Y$ is as above
and $J$ is a complex line bundle over $Y^{[2]}$ satisfying
three conditions:
\begin{enumerate}
\item restricted to the diagonal $J$ is trivial,
\item for any $x$, $y$ in the same fibre of $\pi \colon Y \to P$
there is an isomorphism $J_{(x, y)} = J_{(y, x)}^*$,
\item for any $x$, $y$ and $z$ in the same fibre
there is an isomorphism
$$
J_{(x, y)} \otimes J_{(y, z)} \to J_{(x, z)}
$$
\end{enumerate}
where $J_{(x,y)}$ denotes the fibre of $J$
over $(x, y)$. Moreover all these isomorphisms are globally defined and
commute with each other.
In the example we are interested in below it will be straightforward
to check that these conditions hold.
An easy way to construct a bundle gerbe $J$ is to
choose a line bundle
$L \to Y$ and define $J_{(x, y)} = L_x^* \otimes L_y$
for all $x$ and $y$ where $L_x$ denotes the fibre of $L$ over
$x$. We say that a bundle gerbe is
{\em trivial} if can be shown to arise in this way. A bundle gerbe has a
characteristic
class called its Dixmier-Douady class which is a three
class in the integral cohomology of $P$ and
is the obstruction to the bundle
gerbe being trivial. More details are in \cite{Mur}.
The definition we have given here differs from that in \cite{Mur}
in two respects. The first is that we consider here
complex rank one vector bundles instead of
$\C^{\times}$ principal bundles. This is
just the usual relationship between vector
bundles and their frame bundles and gives
equivalent definitions. More significant is that in \cite{Mur}
the map $Y \to X$ is required to be a fibration. The requirement
that it be a submersion is weaker but necesary for the example
below and still sufficient to
recover the theory developed in \cite{Mur}.
\section{Faddeev's anomaly}
\label{sec:anomaly}
Following Segal \cite{Seg} we consider the Dirac operator $D_A$
coupled to a connection $A \in \A$.
This is an operator on the Hilbert space $H$ of spinors.
Define a subset $\A_0 \subset \A \times \R$ to be all
pairs $(A, s)$ where $s$ is not in the spectrum of the Dirac operator
$D_A$. Given such a pair $(A, s)$ we can decompose the
Hilbert space of spinors into the direct sum of $H^+_{(A, s)}$
and $H^-_{(A, s)}$; the sums of the eigenspaces for eigenvalues
greater and less than $s$ respectively. We can then form the Fock
bundle
$$
F_{(A, s)} = \bigwedge H^+_{(A, s)} \otimes \bigwedge (H^-_{(A, s)})^*.
$$
If we leave $A$ fixed and consider another $t< s$ not in
the spectrum of $D_A$ then we have
$$
H = H^-_{(A, t)}\oplus V_{(A,t,s)} \oplus H^+_{(A, s)}
$$
where $V_{(A,t,s)}$ is the sum of all the eigenspaces for eigenvalues
between $t$ and $s$. Moreover
$$
H^+_{(A, t)} = V_{(A,t,s)} \oplus H^+_{(A, s)} \quad\mbox{and}\quad
H^-_{(A, s)} = H^-_{(A, t)} \oplus V_{(A,t,s)}.
$$
As $\bigwedge V^*_{(A,s,t)}\otimes\det V_{(A,s,t)} $
is canonically isomorphic to $\bigwedge V_{(A,s,t)}$ it follows that
$$
F_{(A,s)} = F_{(A,t)}\otimes \det V_{(A, s,t)}.
$$
Hence the projective spaces
$P(F_{(A,s)})$ and $P(F_{(A,t)})$ can be identified and descend to a
projective bundle $\P $ on $\A$.
We are interested in whether or not there is a Hilbert bundle $\H \to
\A/\G$
whose projective bundle is isomorphic to $\P/\G$. This question
can be phrased in two equivalent ways. Firstly we note that $\P \to \A$
is always the projective bundle of a Hilbert bundle $\H$ over $\A$
because $\A$ is contractible. However to make $\H$ a bundle
on $\A/\G$ we need to lift the group action of $\G$ to $\H$ and the
obstruction to that is a two-cocycle with values in ${\rm Map}(\A,
\C^{\times})$.
Equivalently we can tackle the problem directly on $\A/\G$.
There it is well-known that the obstruction to a projective
bundle over $\A/\G$ being the projectivisation of a Hilbert bundle is a
three class in the cohomology of $\A/\G$.
It was shown in \cite{CarMur}
that these two approaches are related by the
transgression defined in Sec. \ref{sec:trans}.
More precisely if we transgress the three class
in question we get a $L\G$ cocycle which
is the derivative of the group cocycle.
We will show that there is a naturally
defined bundle gerbe whose Dixmier-Douady
class is the three cocycle and such that the
transgressed Lie algebra two-cocycle is that of
Faddeev and Mickelsson \cite{Mic}.
(Somewhat confusingly, the occurrence of the
latter in chiral gauge theories is often referred to
as Faddeev's anomaly).
\section{The bundle gerbe}
The space $\A_0$ introduced in Sec. \ref{sec:anomaly}
projects onto the space $\A$ of connections. This projection
is clearly onto and a submersion because locally $\A_0$ is a
product of an open set in $\A$ and an open set in $\R$.
However the map $\A_0 \to \A$ is not a fibration
because it fails to be locally trivial near points of $\A$
for which $D_A$ has degenerate eigenvalues.
The fibre product $\A_0^{[2]}$ is just the set of all
triples $(A, t, s)$ where neither of the real
numbers $s$ and $t$ are in the spectrum of $D_A$.
We define a line bundle $J$ over $\A_0^{[2]}$ by defining
its fibre at $(A, t, s)$ to be $\det V_{(A,t, s)}$ if
$t \leq s$ and $\det V_{(A,t,s)}^*$ if $t \geq s$. The first two
conditions for a bundle gerbe follow naturally. For the
third consider $t < s < r$ then
$$
V_{(A, t, s)} \oplus V_{(A,s, r)} = V_{ (A, t, r)}
$$
so that
$$
\det(V_{(A, t, s)}) \otimes \det( V_{(A, s, r)}) = \det(V_{(A, t, r)})
$$
as required.
In the notation of Sec. \ref{sec:anomaly} we have
$$
F_{(A, s)} = F_{(A, t)} \otimes J_{(A, s, t)}.
$$
Consider what happens if $J$ is trivial. That means there is a
line bundle $L$ over $\A_0$ such that
$$
J_{(A, s, t)} = L_{(A, s)}^* \otimes L_{(A, t)}.
$$
Then we have
$$
F_{(A, s)}\otimes L_{(A, s)} = F_{(A, t)} \otimes L_{(A, t)}
$$
and we can define a Hilbert bundle $\H$ on $\A$
by taking its fibre over $A$ to be any
of the $F_{(A, s)}\otimes L_{(A, s)}$ which are now all identified.
Clearly we have $P(\H) \simeq \P$.
So if the bundle gerbe $J$ is trivial then the
projective bundle $\P$ is the
projectivisation of a Hilbert bundle. The converse
is also true. Assume that we have identified $\P = P(H)$
for some $H \to \A$. Then define $L_{(A, s)}$
to be all linear isomorphisms from $F_{(A, s)} $ to
$H_A$ whose projectivisation is the identification
map $\P_A \to P(H_A)$. Notice that elements of $J_{(A, s, t)}$
define linear isomorphisms from $
F_{(A, s)} $ to $ F_{(A, t)} $ which induce the identity
map $ P(F_{(A, s) }) = \P_A $ to $ P(F_{(A, t)}) = \P_A$
because that is how $\P$ is defined. It follows
that composing these maps with elements
of $L_{(A, t)}$ defines an isomorphism
\begin{equation}
J_{(A, s, t)} = L_{(A, s)}^* \otimes L_{(A, t)} \label{eq:triv}
\end{equation}
and hence the bundle gerbe is trivial.
So far everything we have said is happening on $\A$ where
the bundle gerbe is trivial. However $\G$
clearly acts on $J$ and it descends to a bundle
gerbe on $\A/ \G$ where the same discussion holds.
A trivialisation of the bundle gerbe on $\A/\G$ is
therefore equivalent to a $\G$ equivariant trivialisation
of the bundle gerbe over $\A$.
Notice that if $J$ has two trivialisations
$L$ and $\hat L$ then we must have
$$
L_{(A, s)}^* \otimes \hat L_{(A, s)} =
L_{(A, t)}^* \otimes \hat L_{(A, t)}
$$
and hence $ L^*\otimes \hat L$ descends to
a line bundle on $\A$. As $\A$ is contractible
any line bundle is trivial so we conclude that
any two trivialisations of $J$ over $\A$ are isomorphic
as line bundles. Moreover because $\A$ is
contractible $J$ does have a trivialisation
which is therefore essentially unique.
We shall see how to construct it in the next section.
We conclude that the bundle $\P/\G$ is
a Hilbert bundle if the trivialisation $L$
of $J$ admits a lift of the gauge group action,
compatible with (\ref{eq:triv}).
Given that there exists a bundle $L$ and
a trivialisation as in (\ref{eq:triv}) consider
what it means for
$L$ to admit a lift of the
gauge group action compatible with (\ref{eq:triv}).
The group of all automorphisms of $L$ covering
the gauge group action, ${\rm Aut}_\G(L) $
is an extension of $\G$ by the group ${\rm Map}(\A_0, \C^\times)$
where the latter is the subgroup of automorphisms that
cover the identity in $\G$. We are interested
in the subgroup $\hat\G$ of automorphisms
compatible with (\ref{eq:triv}).
The intersection of this with
${\rm Map}(\A_0, \C^\times)$ is easily seen to
be the space all functions $f(A, s)$
such that
$$
f(A, s)^{-1}f(A, t) = 1
$$
that is $f(A, s)$ is independent of $s$.
Hence we have an extension
\begin{equation}
0 \to {\rm Map}(\A, \C^\times) \to \hat\G \to \G \to 0 \label{eq:ext}
\end{equation}
of groups. A lift of the $\G$ action to an action
on $L$ compatible with (\ref{eq:triv}) is a
splitting of (\ref{eq:ext}).
\section{The Grassmanian}
\label{sec:Grass}
To construct the trivialisation of $J$ we use the
results of \cite{LanMic} and \cite{MicRaj}. There it is shown that
if $(A, s) \in \A_0$ then $H_{(A, s)}^+$
defines an element of a Grassmanian
$\Gr_2$ of subspaces satisfying a Schatten class condition.
This defines a map
\begin{equation}
p \colon \A_0 \to \Gr_2. \label{eq:embed}
\end{equation}
Notice that the embedding $\A \to \Gr_2$
defined in \cite{MicRaj} is actually singular.
In \cite{MicRaj} \cite{Mic} a determinant line bundle $\L$
is defined over $\Gr_2$. If we denote by $L$ the pull-back
of this under $p$, that is
$$
L_{(A, s)} = \L_{p(A, s)},
$$
we shall show in the Appendix
that this trivialises the gerbe $J$.
It is also shown in \cite{MicRaj} and \cite{LanMic}
that the obstruction to lifting the gauge group
action to $L$ is a cocycle cohomologous
to the Faddeev-Mickelsson 2-cocycle. Moreover that cocycle
has values in ${\rm Map}(\A, \C^{\times})$
not just ${\rm Map}(\A_0, \C^{\times})$
so that it must be the obstruction to a lift
compatible with (\ref{eq:triv}).
We conclude that the obstruction to the
bundle $\P/ \G$ being the projectivisation of
a bundle of Hilbert spaces over $\A/ \G$ is the
Fadeev cocycle. It is a straightforward calculation \cite{WanCarMur}
to check that this is the transgression of the
three form
$$
\omega_A(X, Y, Z) = \int_M \tr(X \wedge Y \wedge Z)
$$
on $\A/\G$.
This is therefore the Dixmier-Douady class of the gerbe
$J/\G$.
\section{Appendix}
This calculation relies on definitions and
techniques from \cite{Mic}.
We want to construct an isomorphism
\begin{equation}
L_{(A, s)}^* \otimes L_{(A, t)} \to J_{(A, s, t)} \label{eq:iso}
\end{equation}
Recall from \cite{MicRaj}
that we begin by choosing a reference subspace $H_0$ with an orthonormal
basis $e_i, i\geq 0$,
of the Hilbert space of sections of the spin bundle.
We assume that $e_i, i\in \Z$ forms a basis of all $L^2$ sections. Then
the spaces $H_k = {\rm span}\{ e_i \mid i = k, k+1, \dots \}$
are in the Grassmanian $Gr_2$ of \cite{MicRaj}. In fact
each $H_k$ will be in a
different connected component of $\Gr_2$.
If our reference subspace $H_0$ is one of the spaces
$H^+_{(A, s)}$ then $H^+_{(A,s)}\in Gr_2$ for all $A$ and $s$.
The element $H^+_{(A, s)}$ in the Grassmanian and its orthogonal
complement define a splitting of $H$ as does
$H_0$ and its orthogonal complement. There is
therefore a map
$$
\pmatrix{ a & b \cr c & d\cr}
$$
mapping the former splitting to the latter.
The index of $H^+_{(A, s)}$ is defined to
be the index of the operator $a$.
If $H^+_{(A, s)}$ has index $d$ then an admissible basis is a basis
$$
w_{-d}, w_{-d+1} , \dots
$$
of $H^+_{(A, s)}$ which can be reached from an orthonormal
basis by a linear isomorphism and such that the matrix
$w_+$ defined by
$$
pr_{-d}( w_j) = \sum_{k=-d}^\infty (w_+)_{kj} e_k,
$$
where $pr_{-d}$ is the orthogonal
projection onto $H_{-d}$, is a map in $GL^2$.
Assume that we are at a point $(A, s)$ where $H_{(A, s)}^+$ has
index $n$ and that at another point $(A,t)$ with $t < s$ we have
\begin{equation}
H_{(A, t)}^+ = V_{(A,t, s)} \oplus H_{(A, s)}^+ \label{eq:sum}
\end{equation}
with $\dim(V_{(A, t, s)}) = d$ so that $H_{(A, t)}^+$
has index $d+n$.
Then we can restrict attention to admissible bases $\tilde w$
for $H_{(A, t)}^+$
that respect the orthogonal decomposition in (\ref{eq:sum}). That
is
$\tilde w = (u, w)$ for
a $u$ a basis of $V_{(A, s, t)}$ and $w$ an admissible
basis for $H_{(A, s)}^+$. We then have that
$$
\tilde w_+ = \left( \begin{array}{cc}
\alpha & \beta \cr
\gamma & w_+ \cr
\end{array}
\right).
$$
The space of admissible bases is acted on upon the right
by elements of
the group $GL^2$. Again in the case of $\tilde w$ we
restrict attention to elements $\tilde t$ that preserve
the decomposition (\ref{eq:sum})
and hence we have
$$
\tilde t = \left( \begin{array}{cc}
T & 0 \cr
0 & t \cr
\end{array}
\right)
$$
where $T$ is a $d$ by $d$ matrix and $t$
an element of $GL^2$ acting on $w$.
The fibre of the pull-back line bundle $L$ over the point
$(A, s)$ is the set of all equivence classes consisting
of an admissible basis $w$ and a complex number $\lambda$
subject to the equivalence relation
$$
(w, \lambda) \simeq (wt, \lambda \omega_2(w_+, t)^{-1})
$$
where $\omega_2(w_+, t)$ is defined in \cite{Mic}.
Using square brackets to denote equivalence
classes we define the isomorphism in (\ref{eq:iso}) by
$$
[w, \lambda] \otimes [\tilde w , \mu] \mapsto {\lambda \over \mu}
\exp(\tr(\alpha - 1)) \det(u)
$$
where $\det(u)$ is the wedge product of the elements of $u$.
We need to check that this is well-defined. That reduces to
proving that
$$
\omega_2(\tilde w_+ , \tilde t) = \omega_2(w_+, t) \det(T)
\exp(-\tr(\alpha T - 1)) \exp(\tr(T-1)).
$$
For this kind of identity we can follow the standard
technique and use the fact that both sides
are continuous and that the various groups $GL^p$
are dense in each other. Hence it
suffices to prove it for elements in $GL^1$.
Applying the results of \cite{Mic}, namely
$$
\omega_2(A, B) = {\det}_2(B) \exp(-\tr(AB-1) + \tr(A-1) + \tr(B-1))
$$
and
$$
{\det}_2(A) = \det(A) \exp(-\tr(A-1))
$$
yields
$$
\omega_2(A, B) = \det(B) \exp(-\tr(AB-1) + \tr(A -1)).
$$
The result then follows by a calculation.
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and anomalies.} To appear in Int J. Mod. Phys. A. hep-th/9408141.
\bibitem{WanCarMur} B. Wang, A.L. Carey and M.K. Murray.:
{\sl Higher bundle gerbes, descent equations and 3-Cocycles},
preprint 1995.
\bibitem{LanMic} E. Langmann and J. Mickelsson.: {\sl (3+1)-Dimensional
Schwinger terms and non-commutative geometry.} Preprint. hep-th 9407193.
\bibitem{Mic} J. Mickelsson.: {\sl Current Algebras and Groups.}
Plenum Press, New York and London, 1989.
\bibitem{MicRaj} J. Mickelsson and S.G. Rajeev.:
{\sl Current algebras in d+1
dimensions and determinant line bundles over infinite-dimensional
Grassmanians.},
Commun. Math. Phys. {\bf 116}, 400, (1988).
\bibitem{Mur} M.K. Murray.: {\sl Bundle gerbes.}
dg-ga/9407015. To appear in Journal of
the London Mathematical Society.
\bibitem{Seg} G.B. Segal.: {\sl Faddeev's anomaly
in Gauss's law.} Preprint.
\end{thebibliography}
\end{document}
\bye