text:
\chapter{K-theory and Dirac operators}\label{chap:ktheory}

In order to get a proper handle on the topological aspects of gauge symmetries we need to discuss the topology of Dirac operators. To this end we introduce the rudiments of K-theory as a means to deal with homotopic families of Dirac operators, and show how this connects to the cohomology classification of bundle gerbes.

K-theory and operator topology is a vast landscape in its own right with many applications. One of special importance in gauge theory is the celebrated Atiyah-Singer index theorem: that the index of a Fredholm operator $D$ defined as a map
\[
	\ind D = \dim( \ker D) - \dim( \cker D) \in \Z
\]
in fact gives a map to a K-theory group, and thus is a topological invariant which can be related to suitable characteristic classes. We will briefly discuss the index theorem as applied to Dirac operators. In the next chapter we will see how this relates to the gauge anomalies; namely, how the families index theorem yields the symmetry-breaking cocycle terms in the Hamiltonian anomaly. This chapter is meant largely to give the general background for this particular application.

The origin of K-theory in quantisation anomalies is quite natural. The Dirac
operators are usually by construction self-adjoint and have polarised essential spectrum. Homotopy classes of such operators can be used as a definition for the elements in the group $K^1(X)$, where $X$ is the parameter space of the operator family. The characterisation of these elements -- ignoring possible torsion -- can be done via cohomology, where we also find the Dixmier-Douady class of the associated bundle gerbe. This then is also the topological source for the anomalous terms in the current algebra commutators we will explore in Chapter~\ref{chap:hanomaly}.

\section{Some fundamentals from operator theory}

Let us begin by recalling some of the basic definitions from operator theory. Let $\Hs$ be a complex Hilbert space and consider a linear operator $D:\Hs \to \Hs$. By the phrase \laina{operator $D$ on $\Hs$} we mean that the operator $D$ is defined on some (dense) subset of $\Hs$. For composite operators we expect that the intersection of their domains will also be dense.

\begin{definition}
	An operator $D$ on $\Hs$ is \termd{closed} if its graph is closed in $\Hs\oplus \Hs$. By \termd{graph} we mean the set
	\[
		\G_D = \{ (v,w) ~\mid~ Dv = w \} \subset \Hs\oplus \Hs .
	\]
	The \termd{closure} $\bar{D}$ of $D$ is its smallest closed extension: a closed operator $\bar{D}$ such that $\bar{D} = D$ on $\dom(D)$ and $\dom(D) \subset \dom(\bar{D})$.
\end{definition}

\begin{definition}
		\idx{operator!bounded}

	An operator $D$ on $\Hs$ is \termd{bounded} with respect to a given norm in $\Hs$ if there is some constant $m\geq 0$ such that
	\[
  	\norm{Dv} \leq m\norm{v} \quad \forall v \in \Hs \, .
	\]
	The least upper bound of $\norm{Dv}$ is called the \termd{norm of the operator $D$}.
\end{definition}
The \term{set of bounded operators on $\Hs$} is denoted by $\bnd{\Hs}$. If $\Hs$ is a Banach space (complete in the metric induced by the norm), the algebra generated by $\bnd{\Hs}$ with respect to the operator composition is a \term{Banach algebra}. The Banach algebra $\bnd{\Hs}$ is \term{unital}\idx{algebra!unital Banach} if the identity operator $\idoperator$ on $\Hs$ has the norm of $1$.

\begin{definition}
	An operator $D$ on $\Hs$ is \termd{compact} if there exists a neighbourhood $\U$ of the origin $0 \in \Hs$ and a compact subset $\V \subset \Hs$ such that $D(\U) \subset \V$.
\end{definition}

\begin{definition}
		\idx{operator!self-adjoint}

	An operator $D$ on $\Hs$ is \termd{self-adjoint} with respect to the inner product on $\Hs$ if $\dom(D) = \dom(D^*)$ and
	\[
		\ip{Du,v} = \ip{u,Dv} \quad \forall u,v \in \Hs \, .
	\]
	The operator is \termd{essentially self-adjoint} if its closure is self-adjoint.
\end{definition}

Let $D$ be a closed operator, and either bounded or unbounded but self-adjoint. The \term{resolvent set} $\rho(D) \subset \C$ constitutes of complex numbers $\alpha$ for which the operator $\alpha \idoperator - D$ is a bijection of the domain $\dom(D)$ and has a bounded inverse on $\Hs$. This inverse is then called the \term{resolvent} of the operator $D$. Note that the \term{spectrum} of the operator $D$ can now be defined as the complement of the resolvent set: $\spec(D) \defeq \C \setminus \rho(D)$. From the definitions it follows that the spectrum of a self-adjoint operator is always a real subset $\spec(D) \subset \R$.
\footnote{Note that the spectrum in this definition is not necessarily exactly the set of eigenvalues of $D$, but can be larger.}

Is the resolvent itself a compact operator? For \term{positive} operators, that is if $\ip{Dv,v} \geq 0$ for all $v\in \Hs$, the compactness can be solved by studying the operator $\inv{(\idoperator + D)}$.\footnote{The compactness of the resolvent does not depend on the choice of the number $\alpha$, and so in the case of positive operators we can use $\alpha=-1$, see \cite[p.~273]{GVF2001}.} Failing the positivity condition, we can consider the \term{Laplacian} $D^2$ and give the following definition.
\begin{definition}
	An operator $D$ on $\Hs$ has a \termd{compact resolvent} if $\inv{(\idoperator + D^2)}$ is compact.
\end{definition}

\begin{remark}
	The idea of the resolvent is linked to the \term{spectral theorem}. In essence, this means that self-adjoint operators can be realised as multiplication operators through unitary equivalence; to discuss this theorem properly we would have to introduce spectral measures of the operator and so we will not follow this further (see \cite{RS1980} for an introduction to the topic). Nevertheless, as a result, the underlying Hilbert space admits a spectral decomposition.

		\idx{observable}

	The usefulness of this in physics comes from the idea of representing quantum mechanical observables as self-adjoint operators on a Hilbert space, the state space of the system. One can describe the physical dynamics with unitary operators depending continuously on some external parameter (often the time variable). The self-adjoint generator of this group of unitary operators is called the \term{Hamiltonian} of the system, and changes in the system state can be linked to the spectrum of this operator.
	However, it is not always clear if this setting can be built in a fashion that is mathematically sound, see for instance \cite[Sec.~VII.11]{RS1980} for a discussion on possible problems.
\end{remark}

\idx{operator!Fredholm}

There are two important classes of operators we are interested in: Fredholm operators and elliptic operators.
\begin{definition}
	A bounded operator $D$ on $\Hs$ is \termd{Fredholm} if its kernel and cokernel are finite-dimensional, and its range is closed.
\end{definition}
Fredholm operators have a well-defined \term{analytical index}:
	\idx{index!of Fredholm operator}
	\idx{index!analytical}
	\index{index|seealso {index theorem}}

\[
	\ind D =  \dim( \ker D) - \dim( \cker D) .
\]
Note that these definitions extend easily to operators for which the domain and codomain are different Hilbert spaces.

Let $D: \sct{E} \to \sct{F}$ be a \term{differential operator}
	\idx{operator!differential}

on vector bundles $E,F \to X$ over a manifold $X$. In local coordinates we can write
\[
	D = \sum_{|\alpha| \leq m} A_{\alpha}(x) D^{|\alpha|} ,
\]
where $\alpha$ is the multi-index $(\alpha_1,\dots,\alpha_n)$ with $|\alpha| = \sum_k \alpha_k$, and
\[
	D^{|\alpha|} = \partial^{|\alpha|}/\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n} .
\]
The coefficients $A_{\alpha}(x)$ are matrices of smooth functions on $X$ fulfilling $A_{\alpha} \neq 0$ for some $\alpha$ with $|\alpha| = m$. The integer $m$ is then called the \term{order} of the operator. In the following we will abbreviate the local expression by $D = A_{\alpha}(x) D^{\alpha}$.

Let $\xi$ be a cotangent vector at a point $x\in X$ such that locally $\xi = \xi_i \ext x^i$ for some scalar coefficients $\xi_i$. The \term{principal symbol}
	\idx{symbol!differential operator}

of the operator $D$ is a map $\sigma_{\xi}(D):E_x \to F_x$ defined locally by
\[
	\sigma_{\xi}(D) = i^m \sum_{|\alpha| = m} A_{\alpha}(x) \xi^{\alpha} .
\]
The principal symbol is a section of the bundle $(\odot^mTX)\otimes\Hom(E,F)$, represented by the coefficients $\{i^m A_{\alpha}\}_{|\alpha|=m}$.
\begin{definition}
		\idx{operator!elliptic}

	A differential operator $D$ is \termd{elliptic} if its principal symbol $\sigma_{\xi}(D)$ is an isomorphism for all non-zero cotangent vectors $\xi \in T^{*}X$ .
\end{definition}

Any elliptic operator can be extended to a Fredholm operator with a matching analytical index. Moreover, on compact spaces $X$ all elliptic operators are Fredholm.~\cite[Sec.~III.5]{LM1989} The Hilbert spaces for these Fredholm extensions are the Sobolev space completions of the sections $\sct{E}$ and $\sct{F}$.
	\idx{operator!Fredholm extension of}

\subsection[{$C^*$-algebras}]{\mots{\boldsymbol{C^*}}-algebras}\label{sec:cstar}

\idxalku{C@$C^*$-algebra}

\idx{algebra!$^*$-}

We introduce one additional operator in order to define $^*$-algebras: Let $\A$ be an associative algebra over $\C$. The operator $(\cdot)^*$ on $\A$ is called an \term{involution} if it satisfies the following conditions for all $a,b \in \A$:
\begin{enumerate}
	\item $(ab)^* = b^*a^*$.
	\item $(a^*)^* = a$.
\end{enumerate}
If furthermore the conditions $(a+b)^* = a^* + b^*$ and $(\lambda a)^* = \bar{\lambda}a^*$ are satisfied for all $a,b \in \A$ and $\lambda \in \C$, we call the algebra $(\A,^*)$ a \term{$*$-algebra}.
\begin{definition}
	A Banach algebra $\A$ over $\C$ is a \termd{$\boldsymbol{C^*}$-algebra} if it is a $^*$-algebra $\A$ and for all $a\in \A$ the following condition holds:
	\[
		\norm{a^*a} = \norm{a}\norm{a^*} = \norm{a}^2 .
	\]
\end{definition}

\begin{example}[Algebra of continuous functions]
	Let $X$ be a compact Hausdorff topological space. The set of all continuous complex functions $C(X)$ on $X$ has a natural Abelian $C^*$-algebra structure. Its unit is the constant function $f_1: x \mapsto 1$, and the Banach-algebra structure is given by the sup-norm
	\[
		\norm{f} \defeq \sup_{x\in X} \abs{f(x)} , \quad f \in C(X) .
	\]
	Involution can be defined as the complex conjugate $f^*(x) \defeq \overline{f(x)}$.
\end{example}

The algebra of bounded operators $\bnd{\Hs}$ is a $C^*$-algebra, when the involution is defined to be the operator adjoint. Indeed, by \emph{Gelfand-Naimark theorem},
	\idx{theorem!Gelfand-Naimark}

any given $C^*$-algebra is isometrically $^*$-isomorphic to a $C^*$-algebra $\bnd{\Hs}$ of some Hilbert space $\Hs$; that is, the norm and the involution are preserved under the isomorphism.

\begin{remark}
	The Gelfand-Naimark equivalence of $C^*$-algebras with operators on a Hilbert spaces demands that the operators are bounded. This is somewhat problematic in quantum physics, where the relevant operators are usually unbounded. Various refinements to remedy this shortcoming have been proposed; for a \emph{higher perspective}, see \cite{MITCHENER2018} in which the problem is addressed by introducing an algebroid structure.
\end{remark}

\subsection*{CAR-algebra}\idx{canonical anticommutation relations (CAR)!algebra}

Recall that in Chapter~\ref{chap:gauge} we sketched the basic idea for transforming classical fields into quantum operators. For quantum field theories on Minkowskian manifolds this leads to $C^*$-algebras generated by operators subject to the \emph{Haag-Kastler axioms}~\cite{HK1964}. The $C^*$-algebra structure is necessary in order to include proper causal relations into the algebra of observables. For fermionic fields this can be extended to include globally hyperbolic Lorentzian manifolds~\cite{DIMOCK1982} and, as the end result, one recovers a functorial isomorphism linking the space-time structures and the algebraic $C^*$-structures.

Let $(V,\ip{\cdot,\cdot})$ be a topological vector space endowed with a nondegenerate symmetric bilinear form with values in $\R$.
The elements of $V$ generate a (unital) $^*$-algebra by the introduction of a bilinear multiplication and a linear involution $^*: V \to V$.
	\footnote{The identity map always yields a trivial involution; for the multiplication we demand \emph{associativity} in light of the preceding definition.}

Using the symmetric form we can then define the CAR-algebra subject to the following conditions:
\begin{enumerate}
	\item $\acomm{f}{g} = \ip{f,g} \idoperator $.
	\item $f^* = f$.
\end{enumerate}
A standard example is given by a suitable algebra of $n\times n$-matrices with (conjugate) transpose as the involution, such as the subspace of symmetric or Hermitian matrices. More generally one introduces a linear map $\phi:V\to \bnd{\Hs}$ to bounded self-adjoint operators on a Hilbert space $\Hs$, and demands the relation
\[
	\acomm{\phi(f)}{\phi(g)} = \ip{f,g} \idoperator , \quad \text{for } f,g \in V .
\]
The map $\phi$ is then called the \term{representation of the CAR over $V$ in $\Hs$}; sometimes this is also called the Clifford system or Clifford relation on $\Hs$, reflecting the history of these relations in Clifford algebras.

For operators on a Hilbert space $\Hs$ we can define the analogous \term{$\boldsymbol{C^*}$ CAR-algebra} as the completion of the above unital $^*$-algebra. This algebra is generated by the symbols $\{B(f),B^*(f) : f \in \Hs \}$ and fulfills
\begin{enumerate}
	\item $\acomm{B(f)}{B^*(g)} = \ip{f,g} \idoperator $.
	\item $\acomm{B(f)}{B(g)} = \acomm{B^*(f)}{B^*(g)} = 0$.
	\item $(B(f))^* = B^*(f)$.
\end{enumerate}
Recall from Chapter~\ref{chap:gauge} that these are the relations for the creation and annihilation operators on the particle states. A representation of the CAR-algebra can be then built on the fermionic Fock space; for details, see \cite{ARAKI1987}.

\begin{remark}
	For the bosonic theory we would consider canonical \emph{commutation} relations, for which the anticommutator in the above conditions is changed into a commutator. Mixtures of these two then lead to free-particle states of mixed particle theory (and by perturbation, to interacting systems).
\end{remark}

\idxloppu{C@$C^*$-algebra}

\section{Rudiments of K-theory}\label{sec:ktheory}

In this section we will briefly introduce the K-theories of topological vector bundles and of operators on a Hilbert space. These two concepts are closely linked, and together they will give us useful tools for studying the topological properties of families of Dirac operators, such as the operator index theorem introduced in a later section. We rely mostly on the introductory materials provided by \cite{ATIYAH1967,HATCHER2017} (topological K-theory) and \cite{BLACKADAR1986,GVF2001} (operator K-theory).

\subsection{Topological K-theory}
\idxalku{K-theory!topological}

Let $X$ be a compact topological manifold. We consider complex vector bundles for which globally the fibre dimension may vary, as long as it is locally constant. Now for any two topological vector bundles $E \to X$, $F \to X$ we can take the direct sum $E\oplus F \to X$, itself a topological vector bundle of (local) dimension $\dim(E) + \dim(F)$. Thus the sum respects isomorphism classes of vector bundles so that
\[
	[E] \oplus [F] = [E \oplus F] ,
\]
and these classes form a monoid $V(X)$ under the sum operation, with the $0$-dimensional trivial vector bundle as the unit called the \term{zero bundle}. We can extend this to a group structure as follows.

First we define vector bundles $E$ and $F$ to be \term{stably isomorphic} if there are trivial vector bundles such that
\[
	E \oplus I^k \isom F \oplus I^l .
\]
We denote the stable isomorphism by $E \sim_S F$. This is an equivalence relation in $V(X)$, and the quotient set $V(X)/{\sim_S}$ is called the \term{reduced $K$-group of $X$} denoted by $\redK(X)$: the group structure is natural, since for every vector bundle we can find a complement bundle with which the direct sum is stably isomorphic to the zero bundle.

We can further extend the reduced $K$-group to a \term{$K$-group of $X$}
	\idx{K-group!topological}
denoted by $K(X)$ by defining elements as virtual differences
\[
	E - F \in K(X) ,
\]
with respect to the equivalence relation
\[
	E_1 - F_1 = E_2 - F_2 \quad \Leftrightarrow \quad E_1 \oplus F_2 \sim_S E_2 \oplus F_1 .
\]
The group addition in $K(X)$ is defined by the taking the direct sum of the minuends and subtrahends, and the unit element is the class given by $E - E$. The group $K(X)$ is obviously Abelian. We sometimes also use the term \term{K-theory} of $X$ for the group, if the context is clear.

This procedure of constructing groups from monoids is general and such groups go by the name of Grothendieck groups (of commutative monoids). In the case of vector bundles, the contravariant functor assigning Abelian groups to topological spaces is called the \term{K-theory functor} or the Grothendieck functor.
	\idx{K-theory!functor}
	\idx{Groethendieck functor}

\begin{remark}
	The fibrewise tensor product of vector bundles introduces a multiplication on $K(X)$, and thus the topological K-theory is endowed with a natural \emph{ring structure} as well. In general, this property does not extend to noncommutative operator K-theory.
\end{remark}

\subsubsection[{Groups $K^n$}]{Groups \mots{\boldsymbol{K^n}}}\label{sec:kn-groups}
\idx{K-group!$K^n$}

The elementary topological K-theory group introduced above is often denoted by $K^0(X)$ -- the degree may be omitted if the context is clear. The degree is in relation to general $K^n(X)$ groups defined as (reduced) suspensions of the base.
	\footnote{Reduced suspension is analogous to suspension in \emph{pointed spaces}. For $X$ compact Hausdorff, both suspension and reduced suspension will yield the same K-theory, see~\cite[Ch.~I]{BLACKADAR1986}.}

We define
\[
	K^{-n}(X) \defeq K(\Sigma^n X) .
\]

\begin{remark}
	The use of \emph{negative} indexing is purely a convention in reference to the cohomology aspect of sequences of $K$-groups. The idea is that there is a coboundary-like operator mapping from $K^{n}$ to $K^{n+1}$, and that the $K$-groups represent a (periodic) cohomology theory in this sense; see Section~\ref{sec:bott} for an example of the Bott periodicity. In the complex case the period is $2$, so we usually use just $K^0$ and $K^1$, since all the other groups are isomorphic to these two.
\end{remark}

\idxloppu{K-theory!topological}

\subsection{Families of operators and K-theory}

Topological K-theory can be generalised to a homotopy classification of operators acting on vector bundles. Thus we have a K-theory functor not only on topological spaces, but also on $C^*$-algebras.

To begin with, we look at the index map of Fredholm operators $\F(\Hs_1,\Hs_2)$ between Hilbert spaces $\Hs_1$ and $\Hs_2$:
	\idx{index!of Fredholm operator}

\[
	\ind: \F \to \Z : D \mapsto \dim(\ker D) - \dim(\cker D) .
\]
The index map is continuous and thus locally constant with respect to the operator norm topology~\cite[Sec.~III.7]{AS1968, LM1989}. In fact, the index defines a bijection between integers and the connected components of the operator space. If we have Fredholm operators $\F(\Hs)$ on a Hilbert space $\Hs$, we can use the composition of operators to give the space of connected components $\pi_0 \F$ a semigroup structure. The index map then becomes a group isomorphism
\[
	\ind : \pi_0 \F(\Hs) \to \Z .
\]
In general, the index of an elliptic operator is defined as the index of its Fredholm extension~\cite[Cor.~III.5.3]{LM1989}.

\idx{operator!elliptic}

Let $E$ and $F$ be vector bundles over $X$ and consider a continuous family of elliptic operators $D_t : \sct{E} \to \sct{F}$, where the parametrisation is over the unit interval, $t \in I$; a family $D_t$ is \term{continuous} if the operators
\[
	D_t = A^{\alpha}(x,t)D^{\alpha}
\]
are continuous both in $x \in X$ and $t \in I$. This implies that the order of $P_t$ is constant and that the Fredholm extension can be made continuous.
	\footnote{We can use the norm topology for bounded operators. In the unbounded case one needs to apply additional conditions, for instance by introducing the Riesz map $D \mapsto D/(\norm{D}+1)$ to bounded operators. See \cite[Sec.~2.6]{BB2013} for a more detailed account.}

Hence $\ind P_t$ is constant in $I$ as the index map of Fredholm operators.

\idxalku{operator!homotopy class of}
Now, two elliptic operators $P_1,P_2$ over $(X;E,F)$ are \term{homotopic} if they can be joined by a continuous family $P_t$ of elliptic operators. Thus we get that the index of the elliptic operator on a compact manifold depends only on its homotopy class, which in turn is isomorphic to the symbol class. Moreover, since the principal symbol of the operator defines an element of the K-theory $K(TX)$ (where $TX$ is the tangent bundle) and the index depends only on the symbol class of the operator, the index map induces a homomorphism
\[
	K(TX) \to \Z .
\]
The index theorem states that the analytical and topological indices coincide as such homomorphisms.~\cite[Sect. III.13]{LM1989}

\idxalku{operator family}

The homotopy parametrisation of operators gives us a simple first instance of families of operators: that of a product family, where for the continuous family $P_t$ of operators the parameter $t$ lies not in the unit interval $I$ but in a more general Hausdorff space $A$. If $X$ is a compact manifold and constant in $A$, we consider the product space $X \times A$.
This way we get an {index bundle} for Fredholm operators.

More generally, letting the parameter space $A$ be homotopically nontrivial, we need to consider twisting of the manifold and the operators over $A$. Instead of a product space $X \times A$, we have a bundle of manifolds $X \to A$ over the parameter space. The analytical index is then a map
\[
	K(TX) \to K(A) ,
\]
where $TX$ is the tangent bundle along the fibres, and the symbol class of the operator is an element of the K-theory $K(TX)$.~\cite{AS1971}.

Let $A \to \F(\Hs_1,\Hs_2)$ be a continuous family of Fredholm operators $T:\Hs_1 \to \Hs_2$, where $\Hs_i$ are Hilbert spaces and the parameter space $A$ is a compact topological space. We assume that the Hilbert spaces are fixed over $A$.
	\footnote{Cf. Remark 3.31 in \cite[p. 85]{BB2013}.}

If $A$ is connected, we have
\[
	\ind T_a = \ind T_{a'}	\quad \textrm{for all } a,a' \in A ,
\]
and as above, each Fredholm operator is assigned an integer and for the connected parameter space there is an isomorphism
\[
	\ind: [A,\F] \to \Z
\]
on the homotopy set $[A,\F]$~\cite[Sec.~3.7]{BB2013}.

For a continuous family with \emph{constant kernel dimension}, there are well-defined vector bundles $\ker T$ and $\cker T$ over $A$, and we can consider the index via topological K-theory.
For each continuous family $A \to \F$ of Fredholm operators in a Hilbert space $\Hs$, we can then assign an \term{index bundle}
	\idx{index!of operator family}

\[
	\ind T \coloneqq [\ker T ] - [\cker T] \in K(A) .
\]

Note that this definition relies on the fact that the dimensions of the nullspaces do not vary. However, we emphasise the following: it can be shown that the K-theoretic classification has sufficient stability \emph{even in the general case where the dimensions of the operator nullspaces may vary with a topologically nontrivial parameter space $A$}~\cites[Sec.~III.8]{LM1989}[p. 84]{BB2013}. In this case the K-theory does not map to integers, but the definition is nevertheless formally meaningful.

In the case of $\Hs_1 = \Hs_2$, the homotopy invariance generalises to the statement that the space of Fredholm operators $\F(\Hs)$ actually constitute a classifying space for the K-theory functor: there is a natural isomorphism
\[
	\ind : [A,\F] \to K(A)
\]
such that for any continuous $f:A' \to A$, where $A'$ and $A$ are compact Hausdorff,
\[
	\ind \circ \, \pull{f} = \pull{f} \circ \ind ,
\]
where $\pi:\F \to A$ is the projection.

\idxloppu{operator family}

\subsubsection[{Group $K^1$}]{Group \mots{\boldsymbol{K^1}}}
\idx{K-theory!operator}
\idx{K-group!$K^1$}

The index map of homotopies is a degree $0$ K-theory:
\[
	[A,\F] \isom K^0(A) .
\]
This isomorphism gives a complete classification of the homotopy type of the space of Fredholm operators on a Hilbert space $\Hs$. In anticipation of families of Dirac operators (Section~\ref{sec:dirac_families}), we present the refinement into the group $K^1(A)$ as follows:
\begin{theorem}\label{theorem:k1-group}
	Let $\Hs$ be a separable Hilbert space, and let $\F^*$ be the space of self-adjoint Fredholm operators on $\Hs$ with polarised essential spectrum. Then for any compact Hausdorff space $A$, the space $\F^*$ is a classifying space for the $K^1$-functor, that is
	\[
	  K^1(A) \isom [A,\F^*] .
	\]
\end{theorem}
\begin{proof}
For the original result, see \cite{AS1969}.
\end{proof}

The importance of this classification is that it enables us to work with different families of Dirac operators up to homotopy. Later in Section \ref{sec:families_cohomology} we connect the homotopy classes to the third cohomology of the parameter space, which will be useful in the analysis of the Hamiltonian anomaly.

\idxloppu{operator!homotopy class of}

\subsection{K-theory of operator algebras}
\idx{K-theory!of operator algebra}
\idx{C@$C^*$-algebra}

Let $C(X)$ be the $C^*$-algebra of a topological space $X$ defined by continuous complex functions. By the Gelfand-Naimark theorem

\idx{theorem!Gelfand-Naimark}

there is an identification
\[
	\begin{tikzcd}
		\mlnode{unital Abelian $C^*$-algebra $\A \defeq C(X)$} \arrow{r} & \arrow{l} \mlnode{compact Hausdorff space $X$} ,
	\end{tikzcd}
\]
where from the algebra-perspective the space $X$ is the space of characters of $\A$, and from the topology-perspective the algebra consists of evaluation maps at points of $X$.

Consider a vector bundle $E\to X$, and define another vector bundle $F\to X$ with respect to $E$ by setting $E\oplus F \isom X \times \C^n$; such a bundle always exists~\cite[Prop.~2.2,~p.~52]{GVF2001}. Then besides the algebra $\A = C(X)$ we can also define modules $M = C(X,E)$ and $N = C(X,F)$ so that $M \oplus N \isom \A^n$. In other words, the vector bundle $E\to X$ defines a canonical finitely generated projective module $M$ over $\A$. Conversely, any such module over $\A$ is the space of all continuous sections of some vector bundle over $X$.
All in all, we have the \term{Serre-Swan theorem}: the category of vector bundles over the topological space $X$ is equivalent to the category of finitely generated projective modules over the algebra $\A$.~\cite[Ch.~2]{GVF2001}

Therefore we can define $K$-groups of $C^*$-algebras similarly to the topological K-theory. The set of isomorphism classes of finitely generated projective modules over $\A$ is an Abelian semigroup with respect to the direct sum. By the Grothendieck construction
	\idx{Groethendieck functor}

we get a group $K_0(\A)$ by defining the equivalence relation
\[
	(M,N) \sim (M',N') \Leftrightarrow M \oplus N' \oplus B \isom M' \oplus N \oplus B ,
\]
for some projective module $B$ over $\A$. Then~\cite[Sec.~3.2]{GVF2001}
\[
	K_0(C(X)) \isom K^0(X) .
\]

\subsection{Bott periodicity}\label{sec:bott}
\idxalku{Bott periodicity}

Lastly, we make note of the (complex) K-theoretic formulation of the \term{Bott periodicity}. Its application to group cohomology is often useful in gauge theory; the \emph{real} K-theory version on the other hand has applications in condensed matter physics, where it is used to classify topological phases of matter for free fermionic systems~\cite{THIANG2016}.

Let $X$ be a locally compact Hausdorff space, $Y$ a closed subspace of $X$, and define $Z=X\setminus Y$ as the complement. Let $q: X^+ \to Z^+$ be the map between one-point compactification of the spaces $X$ and $Z$ such that $q(x) = x$ for $x\in Z^+$ and $q: x \mapsto \{+\}$ for $x\in X^+\setminus Z$. This induces an isomorphism between the relative group $K(X,Y)$ and the group $K(Z)$. Hence the following sequence is exact:
\[
	K(Z) \overset{q^*}{\longrightarrow} K(X) \overset{\iota^*}{\longrightarrow} K(Y) ,
\]
where $q^*$ and $\iota^*$ are pullbacks to the commutative monoid of vector bundles over the spaces. Here we have identified the groups $K(X)$ and $K(X^+)$ with each other, as usual.~\cite[Ch.~I]{BLACKADAR1986}

\begin{theorem}[Bott periodicity]
	The diagram of $K$-groups

	\[
	\begin{tikzcd}
		K^0(Z) \arrow[r,"q^*"] & K^0(X) \arrow[r,"\iota^*"] & K^0(Y) \arrow[d,"\bdr"] \\
		\arrow[u,"\bdr\,"] K^{-1}(Y) & \arrow[l,"\iota^*"] K^{-1}(X) & \arrow[l,"q^*"] K^{-1}(Z)
	\end{tikzcd}
	\]

	is a cyclic exact sequence. For real K-theory there is a similar cycle with the period of $8$ instead of $2$.
\end{theorem}
\begin{proof}
See \cite[Ch.~IV]{BLACKADAR1986} or \cite[Sec.~I.9]{LM1989}.
\end{proof}

\begin{remark}\label{rem:bottcohomology}
	The Bott periodicity is quite a general phenomenon. It applies to the groups of unitaries $U(p)$ and $SU(p)$ as follows:~\cite{BOTT1957}
	\[
		\pi_k(U(p)) \isom  \pi_k(SU(p)) = \begin{cases} 0, & k \text{ even;}\\ \Z, & k \text{ odd.} \end{cases}
	\]
	assuming $k>1$ and $p \geq (k+1)/2$.
	From this one also gains generators of the integral cohomology groups $\coh^{2m-1}(SU(p))$ in the odd degrees for $m\leq p$; by the Bott periodicity they are one-dimensional and can thus be generated by a single element. This will be useful when computing the anomalous Dixmier-Douady class in Section~\ref{sec:general_comm}.
\end{remark}

\idxloppu{Bott periodicity}

\section{Atiyah-Singer index theorem}
\idxalku{index theorem}
\idxsee{theorem!Atiyah-Singer}{index theorem}

The Atiyah-Singer index theorem states that the analytical index of an elliptic operator $P$ on a compact Riemannian manifold $X$
	\idx{index!analytical}

\[
	\ind_a P = \ker P - \cker P
\]
coincides with the \emph{topological index}
	\idx{index!topological}

$\ind_t P$ as a map
\[
	K(TX) \to \Z .
\]
The gist of the theorem is in connecting the functional analysis of an elliptic operator and the homotopy theory related to the topological properties of the operator symbol. It is then clear that the operator index is a \emph{topological invariant} and, by cohomology formulas for the topological index, it can be expressed with characteristic classes for which there are well-known computation methods.

\subsection{The topological index}
\idxalku{index!topological}

Our aim is to sketch the construction of the topological index as a mapping
\[
	K(TX) \to \Z .
\]
As usual, on a Riemannian manifold we can canonically identify the tangent and cotangent bundles, and in the following we do not bother with being too pedantic about the distinction.

A crucial component in defining the right mapping for the topological index is the \term{Thom isomorphism}. Let $\pi : E \to X$ be a vector bundle of rank $n$ over a compact manifold $X$. For any integer $k \geq 0$, there is an isomorphism
\[
	\psi: \coh^k(X,\Z_2) \to \tilde{\coh}^{k+n}(T(E),\Z_2) ,
\]
with $\tilde{\coh}$ referring to reduced cohomology and $T(E)$ is the \emph{Thom space} of the bundle. The K-theoretic variant of this map can be given as follows.~\cite[App. C]{LM1989}
\begin{proposition}[Thom isomorphism in K-theory]
	The mapping
	\[
		\psi: K(X) \to K(E) : x \mapsto (\pull{x}) \lambda_E
	\]
	is an isomorphism.
\end{proposition}
Here $\lambda_E$ is the canonical \emph{difference element} in the K-theory $K(E)$; see \cite[Sec.~I.9]{LM1989} or \cite[Sec.~12.2]{BB2013} for details.

Consider then the \term{principal symbol}
\[
	\sigma(D): T^*X \to \Hom(p^* E, p^* F)
\]
of an elliptic operator $D : \sct{E} \to \sct{F}$, where $p:T^*X \to X$. For an elliptic operator, the symbol defines an isomorphism away from the zero section. Hence we have a \term{symbol class}
\[
	[\sigma(P)] = [p^*E,p^*F;\sigma(P)] \in K(T^*X) .
\]

Take a smooth embedding $f: X \embedd Y$, with $Y = \R^k$,\footnote{By the Whitney embedding theorem, this is always possible.} and push forward the tangent bundle $f_*: TX \embedd TY$. Denote by $N$ the normal bundle of $X$ in $Y$. We also have the normal bundle $TN$ of $TX$ in $TY$, which we gain by pulling $N\oplus N$ back to $TX$. Thus we have a fibre bundle $TN \to TX$ with pointwise components in manifold and fibre directions in $Y$. Now the Thom isomorphism gives us a map
\[
	\psi_{TN \to TX} : K(TX) \to K(TN) .
\]

Consider now the extension homomorphism $h:K(TN) \to K(TY)$ given by the natural embedding $TN \embedd TY$. Composing this with the Thom isomorphism we obtain a homomorphism
\[
	f_{!} : K(TX) \to K(TY) .
\]

We defined $Y = \R^k$, so $TY = \R^{2k}$. Let $i$ denote the inclusion of the origin in $Y$, and we get the induced map
\[
	i_{!}: K(T\{0\}) \to K(\R^{2k}) ,
\]
which in fact is a Thom isomorphism. Now, obviously, $K(T\{0\}) \isom \Z$, so that
\[
	\inv{i_{!}} : K(TY) \to \Z .
\]
Let us wrap this all in the final definition:
\begin{definition}[Topological index] The \termd{topological index of an elliptic operator $D$} is the map
\[
	\ind_t(D) : K(TX) \to \Z : D \mapsto \inv{i_{!}} f_{!} ~(\sigma(D)) .
\]
\end{definition}

The index is independent of the choice of the embedding $f$: First take a linear inclusion $j: \R^k \embedd \R^{k+n}$ for some $n \in \N$, and consider the composition $\tilde{f} = j \circ f$. The induced $j_! : K(T\R^k) \to K(T\R^{k+n})$ is a Thom isomorphism, and thus $ \inv{i_{!}} f_{!} = \inv{\tilde{i}_{!}} \tilde{f}_{!}$. Let then
$g$ be another embedding $X \embedd \R^{n}$, and consider the inclusion $j_g:\R^{n} \embedd \R^{n+k}$. The composite embeddings $j \circ f: X \embedd \R^{n+k}$ and $j_g \circ g : X \embedd \R^{n+k}$ are isotopic, and in fact we have a smooth family of such embeddings. Since K-theory is homotopy invariant, it follows that the topological index is independent of the choice of embedding.

\idxloppu{index!topological}

\subsection{The index theorem and cohomology formulas}

We can now state the index theorem, although we will not attempt to give a proof here.
\begin{theorem}[Atiyah-Singer]
	Let $D$ be an elliptic operator on a compact Riemannian manifold $M$. Then
	\[
		\ind_a D = \ind_t D .
	\]
\end{theorem}
\begin{proof}
	Good expositions can be found in \cite[Ch.~III]{LM1989} and \cite[Pt.~III]{BB2013}. Consult \cite{AS1968} for the original proof relying on \emph{equivariant} K-theory.
\end{proof}

The K-theoretical index map can be realised also as a map between cohomologies, giving the common formulation for the index. The topological index can be written as
\[
	\ind_t D = \int_M \chern(E) \wedge \ahatgenus(M) ,
\]
where $\chern(E)$ is the Chern character of the vector bundle $E\to M$, and $\ahatgenus(M)$ is the $\ahatgenus$-genus of the base manifold. These invariants can be formulated in local terms: on one hand we have the Chern forms $\tr F^k$, where $F$ is the curvature $2$-form of \emph{some} connection on the bundle $E\to M$; and on the other hand we can write the $\ahatgenus$-genus as in terms of the curvature form $R$ on the tangent space of $M$.

\begin{remark}
	Not all topological information is preserved by the cohomology formula; K-theory does detect torsion, but this is lost when mapped to Chern characteristic class.~\cite[Sec.~III.17]{LM1989}
\end{remark}

\idxloppu{index theorem}

\section{Families of Dirac operators}\label{sec:dirac_families}
\idxalku{operator family}
\idxalku{Dirac operator}

Let $X$ be a closed smooth manifold, and consider a family $\{D_x\}$ of self-adjoint unbounded Fredholm operators with compact resolvent. Such a family defines an element in $K^1(X)$, as explained in Section~\ref{sec:ktheory}.

Denote by $\F_R^*$ the \term{space of regular self-adjoint unbounded Fredholm operators}. This means that if $F \in \F_R^*$, then
\begin{enumerate}
	\item $F$ is closed and self-adjoint,
	\item the resolvent $\inv{(\idoperator + F^2)}$ is compact, and
	\item $F$ has both positive and negative spectrum, and both are infinite.
\end{enumerate}

A \term{family of such operators} $D_X \defeq \{D_x\}$ parametrised by a closed smooth manifold $X$ is a mapping $X \to \F_R^*$. Given a family $\{D_x\} \subset X \times \F_R^*$, the mapping $\xi(D_x) = D_x(\idoperator+D_x^2)^{-1/2}$ makes each operator $D_x$ bounded and we can define the bounded family
\[
  D_X^b \defeq \{\xi(D_x)\} .
\]
Applying the mapping $\xi$ does not change the homotopy of the original family, and thus to each family $D_X$ we can pair an element in $K^1(X)$ through the homotopy class of its bounded equivalent.~\cite{DK2010}

Observe that the common definition for the Dirac operator on a spin manifold fulfills these requirements. Let $M$ be a compact Riemannian spin manifold with an associated spinor bundle $S$ and let $E \to M$ be a complex vector bundle with Hermitian metric and a unitary connection $\conn^E$.
\begin{definition}\label{def:dirac_op}
	The \termd{Dirac operator} $D$ is the composition of maps
	\[
		\sct{S\otimes E} \overset{\conn^E}{\longrightarrow} \sct{T^*M \otimes S\otimes E} \overset{\gamma}{\longrightarrow} \sct{S\otimes E} ,
	\]
	so that locally we have the formula
	\[
		D = -i \gamma(\ext x^k) \conn^E_k ,
	\]
	where $\gamma$ is the Clifford multiplication in the fibre, and $k$ is the local coordinate index.
\end{definition}

\begin{remark}
	The factor $-i$ in the local formulation is due to the requirement for self-adjointness; see Proposition~\ref{prop:dirac_properties} below.
\end{remark}

\begin{remark}
	On even-dimensional manifolds the spinor bundle has a $\Z_2$-grading and so one gains Dirac operators $D^{\pm}$ mapping \emph{positive} spinors to \emph{negative}, and vice versa. This gives the often useful \term{Weyl spinor polarisation}, and the index of the Dirac operator can be defined as the difference of the nullspaces:
	\[
		\ind D = \dim(\ker D^+) - \dim(\ker D^-) .
	\]
\end{remark}

\begin{proposition}\label{prop:dirac_properties}
	The Dirac operator is regular, essentially self-adjoint, and Fredholm.
\end{proposition}
\begin{proof}
	The Dirac operator is \emph{elliptic}, and hence Fredholm on compact manifolds $M$.
	For regularity and self-adjointness, see for instance \cite[Ch.~9]{GVF2001}.
\end{proof}
In the following, we will call an operator \term{Dirac-like} if it fulfills these three requirements.

\begin{remark}
	\idx{Dirac operator!pseudo-Riemannian}
	\idx{space-time!pseudo-Riemannian}

	The operator in Definition~\ref{def:dirac_op} could be called \emph{mathematician's Dirac operator}. It differs somewhat from what Paul Dirac originally considered in~\cite{DIRAC1928}, most importantly in the requirement of a Riemannian metric. One can define Dirac operators in a pseudo-Riemannian metric too, only then the operator will not be elliptic but hyperbolic. Proper take on the pseudo-Riemannian Dirac operators leads one to consider essentially self-adjoint operators on \emph{Krein spaces} instead of Hilbert spaces~\cite{STROHMAIER2006}. Some properties of the elliptic case are nevertheless retained; for instance, there is a sound definition for the Fredholm index and a related index theorem on globally hyperbolic Lorentzian manifolds with boundary~\cite{BS2017}. It is still an open question if this pseudo-Riemannian index theorem extends to other variants such as the families index theorem.
\end{remark}

\idxloppu{Dirac operator}

\subsection{Families index theorem}
\idxalku{index theorem!of operator family}

The Atiyah-Singer index theorem has a natural extension to families of Fredholm operators. Recall from Section~\ref{sec:ktheory} that a family of Fredholm operators parametrised by a Hausdorff space $A$ induces an isomorphism
\[
	[A,\F] \to K(A) ,
\]
which we call the \term{index of the family of operators}.
	\idx{index!of operator family}

Given a family $D_A$ of operators $D_a$, with $a\in A$, we can define the analytical index as a formal difference
\[
	\ind D_A = \{ \ker D_a \} - \{ \cker D_a \} ,
\]
where on the right-hand side we have families of vector spaces. If the dimension of the null-space $\ker D_a$ is \emph{constant} over $A$, these spaces are true vector bundles over $A$, and the definition of their difference as an element in $K(A)$ is natural.

More generally, we can have elliptic operators over a family of compact manifolds $\M \defeq \{M_a\}$, parametrised by a compact space $A$ in such a way that we have a fibre bundle $\M \to A$ with $M$ as the typical fibre and the structure group is the diffeomorphisms of $M$.

The analytical index of the family is then a map
\[
	K(T\M) \to K(A) .
\]
The families index theorem then states the following:~\cite{AS1971}

\begin{theorem}[Families index theorem]
	Let $D_A$ be a family of elliptic operators on a compact Riemannian manifold $M$ parametrised by a compact Hausdorff space $A$. Then
	\[
		\ind_a D_A = \ind_t D_A .
	\]
\end{theorem}

The cohomology formula of the index map can be written similarly to the normal index. For our purposes it is not necessary for the manifold $M$ to vary with the parameter space, and then the index formula is simply
\[
	\ind D_A = \int_M \chern(E)\wedge \ahatgenus(M) ,
\]
where $E$ is now a vector bundle over the \emph{product space} $M\times A$.

\begin{remark}\label{rem:aps_index}
		\idx{index theorem!Atiyah-Patodi-Singer}
		\idx{$\eta$-invariant}

	In Chapter~\ref{chap:hanomaly} we discuss the index of a family over a manifold with boundary. In this case there is a boundary term given by the spectral $\eta$-invariant. Now boundary conditions apply: in the Atiyah-Patodi-Singer index theorem~\cite{APS1975}, one requires that the boundary values should lie in the subspace spanned by certain eigenfunctions under an orthogonal projection.
	The $\eta$-terms in the index formula are not local, since they reflect the spectral properties of the operators on the boundary. However, when applying the index theorem to the Hamiltonian anomaly, we see that these boundary parts do not actually contribute due to the gauge invariance.
\end{remark}

\idxloppu{index theorem!of operator family}
\idxloppu{operator family}

\section{Cohomology and gerbes}\label{sec:families_cohomology}

Let $D_X$ be a family of Dirac-like operators, parametrised over a closed smooth manifold $X$. We can define the \term{spectral graph} of the family as the closed subset
\[
	\mathcal{S}(D_X) = \{ (x,\lambda) ~\mid~ \lambda \in \spec(D_x) \} \subset X \times \R .
\]
If the spectral graph is disconnected, the family represents a trivial element of the K-theory $K^1(X)$. There are then topological obstructions to deforming a given family to a one which has a disconnected spectral graph, and these obstructions can be represented as the components of the related Chern character.~\cite{DK2010}

\idx{Dixmier-Douady class}

The first component of the Chern character can be realised as a $1$-cocycle and it defines the spectral flow of the family. The second component manifests as a $2$-cocycle and we have the following theorem connecting the K-theoretic classification of families and the Dixmier-Douady class of the bundle gerbe~\cite[Thm.~2.3]{DK2016}:

\begin{theorem}\label{theorem:family_kclass}
	Let $X$ be a closed smooth manifold and let $\{D_x\}_{x\in X}$ be a family of self-adjoint unbounded Fredholm operators with compact resolvent. Then the cohomology class $[D_X] \in \coh^3(X,\Z)$ equals the Dixmier-Douady class of the bundle gerbe of the family.
\end{theorem}

The cohomology classes are constructed as follows.
Let $\{\U_i\}$ be an open cover
of the space $X$, and assume that its element sets and their finite intersections are contractible. Given an open set $\U_i$ in the cover, there is a \emph{non-eigenvalue} $\lambda_i \in \R$ such that $\lambda_i \notin \spec(D_x)$, for any $x\in \U_i$.
Such non-eigenvalues indeed exist for any given $x\in X$ since the Dirac operator coupled to a potential $x\in X$ always has a spectral gap, and in particular on compact manifolds the spectrum is discrete~\cite[Thm.~III.5.8, p.~196]{LM1989}. This extends over the set $\U_i$ by Propositions~2.4--2.5 in \cite{DK2010}.

Recall from Chapter~\ref{chap:gauge} that then the eigenspace $\Hs$ of $D_x$ can be decomposed into a subspace $\Hs^+$ for eigenvalues greater than $\lambda_i$
and its orthogonal complement $\Hs^-$: this is the spectral decomposition $\Hs = \Hs^+ \oplus \Hs^-$, locally defining the Dirac sea of \laina{positive} and \laina{negative} particles with respect to the spectrum of the Dirac operator.

\idx{canonical anticommutation relations (CAR)}

Let then $\pi^+(x)$ be the local projection onto the space $\Hs^+$. It induces a quasi-free representation of the CAR-algebra $\Ca(\Hs)$ by the map
\[
	\alpha_i(x) : \Ca(\Hs) \to \bnd{\F_{i}(\Hs)} .
\]
Here $\F_{i}(\Hs)$ is the fermionic Fock space over a point $x\in X$ defined by
	\idx{fermionic Fock space}

\[
	\F_{i}(\Hs) = \F(\pi^+(x)\Hs)\otimes \F(\overline{(\idoperator - \pi^+(x))\Hs}) ,
\]
recall Chapter~\ref{chap:gauge} and see \cite{ARAKI1987} for the general idea.
By \term{quasi-free}
	\idx{quasi-free representation}

we mean that in each of the Fock spaces there is a vacuum vector $v_0$ satisfying
	\idx{vacuum vector}

\[
	\annih(v) v_0 = \creat(u) v_0 = 0
\]
for all $v\in \Hs^+$ and $u\in \Hs^-$. In general, the choice of a vacuum vector cannot always be made continuously, and the topology of the vacuum line bundle is physically significant. We will discuss this more closely in Chapter~\ref{chap:hanomaly}.

Now for $x$ in the intersection $\U_i \cap \U_j$, we can define two different but \emph{equivalent} representations $\alpha_i(x)$ and $\alpha_j(x)$. Between these there is an intertwining operator
\[
	S_{j,i}(x): \F_j(\Hs) \to \F_{i}(\Hs)
\]
such that
\[
	S_{j,i}(x) \alpha_i(x) S_{j,i}^*(x) = \alpha_j(x) .
\]
Let us then consider triple intersections: let $x\in\U_i\cap\U_j\cap\U_k$. We can define a circle-valued function $g(\U_i,\U_j,\U_k) : \U_i\cap\U_j\cap\U_k \to S^1$ induced by the composition
\[
	S_{j,i}(x) S_{i,k}(x) S_{k,j}(x) = g(\U_i,\U_j,\U_k)(x) .
\]
By Schur's lemma, this defines a scalar in $S^1$ since the composition is an intertwining operator of irreducible representations of the algebra $\Ca(\Hs)$.

A crucial property of the map $g$ is that it is a continuous function over $X$, since the intertwining operators can be made continuous; see \cite{DK2016} for details. Therefore the \emph{families of maps} $\{g:\U_i\cap\U_j\cap\U_k \to S^1\}$ define a $2$-cocycle over the cover $\{\U_i\}$ with values in the sheaf of $S^1$-valued functions, and thus we have a  cohomology class
\[
	[g] \in \coh^2(\{\U_i\},\sheaf{S^1}) .
\]

Furthermore, with a suitable choice of the cover,
this class maps to the second \v{C}ech cohomology by the canonical isomorphism (see Appendix~\ref{app:cohomology}).
For smooth and paracompact $X$ the groups in \v{C}ech cohomology and de Rham cohomology are equivalent~\cite[Sec.~1.4]{BRYLINSKI1993}, and we have an isomorphism with the third integral de Rham cohomology
\[
	\coh^2(X,\sheaf{S^1})  \isom \coh^3(X,\Z)
\]
induced by the exponential exact sequence~\cites[Ch.~23]{HJJS2007}[Sec.~5.2]{BRYLINSKI1993}
\[
	\exact{\Z}{\C}{S^1} .

\]

The cohomology class defined by the map $g$ depends on the family $D_X$ only up to homotopy: hence we can link the homotopy class of the operator family to the third de Rham cohomology. That is, by construction, we have the following useful result:

\begin{lemma}
	The cohomology class $[D_X] \in \coh^3(X,\Z)$ depends on the family up to homotopy.
\end{lemma}

This forms the basis for computing the topological invariants related to the anomalous current commutators, as one can apply a homotopy transformation to the moduli space and yet derive the Schwinger terms from the same de Rham representations of the cohomology class. We will come back to an example of this in Section~\ref{sec:general_comm}.

\idx{Dixmier-Douady class}

Furthermore, these classes are the Dixmier-Douady classes defining equivalent bundle gerbes related to the operator family.
The gerbe class used in \cite{DK2016} is that of an \emph{index gerbe} defined in \cite{LOTT2002}. This is a variant of the bundle gerbe we discussed in \ref{sec:gerbes}; see \cite{CW2006} for further details.

\begin{remark}
	An interesting question is when the element of $K^1(X)$ given by the family is trivial -- in \cite{DK2010,DK2016} it is shown that the obstruction to trivialisation can be expressed as relevant Chern classes, the first two relating to the \emph{spectral flow} and the \emph{index gerbe}. Vanishing Dixmier-Douady class leads to a trivial element under the assumption that the operator family has a spectral multiplicity bounded by $2$.
	For families with \emph{constant} spectral multiplicity, it is enough to trivialise the spectral flow~\cite{DK2010}.
	These results do not in general show up in the physics applications though, since the restriction to the spectral multiplicity is often too severe.
\end{remark}