Local Anomaly Cancellation and Equivariant Cohomology of Jet Bundles

Abstract

We study the problem, suggested by Singer in [17], and consisting in determining a notion of “local cohomology” adequate to deal with the problem of locality in those approaches to local anomalies based on the Atiyah–Singer index theorem.

Dedicated to Jaime Muñoz Masqué on the occasion of his 65th birthday, with my best wishes.

1 Introduction

An anomaly appears in a theory when a classical symmetry is broken at the quantum level. As we consider only local anomalies, we can assume that the group $\mathcal {G}$ is connected. Let $\mathcal {L}(\psi ,s)$ be a $\mathcal {G}$-invariant Lagrangian density depending on bosonic fields $s\in \varGamma (E)$ and fermionic fields $\psi$. At the quantum level, the corresponding effective action W(s), defined in terms of the fermionic path integral by $\exp (-W(s))=\int \mathcal {D}\psi \mathcal {D}\bar{\psi }\exp \left( -\int _{M}\mathcal {L}(\psi ,s)\right)$ could fail to be $\mathcal {G}$-invariant. We define a form $\mathcal {A}\in \varOmega ^{1}(\mathrm {Lie\,} \mathcal {G},\varOmega ^{0}(\varGamma (E)))$ by $\mathcal {A}=\delta W$, i.e.$\ \mathcal {A}(X)(s)=L_{X}W(s)$ for $X\in \mathrm {Lie\,}\mathcal {G}$, $s\in \varGamma (E)$. Although W is clearly a non-local functional, $\mathcal {A}$ is local in X and s, i.e. we have $\mathcal {A}\in \varOmega _{\mathrm {loc}} ^{1}(\mathrm {Lie\,}\mathcal {G}, \varOmega _{\mathrm {loc}}^{0}(\varGamma (E)))$. It is clear that $\mathcal {A}$ satisfies the condition $\delta \mathcal {A}=0$ (the Wess–Zumino consistency condition). Moreover, if $\mathcal {A}=\delta \Lambda$ for a local functional $\Lambda =\int _{M} \lambda \in \varOmega _{\mathrm {loc}}^{0}(\varGamma (E))$ then we can define a new lagrangian density $\mathcal {\hat{L}}=\mathcal {L}+\lambda$, such that the new effective action $\hat{W}$ is $\mathcal {G}$-invariant, and in that case the anomaly cancels. If $\mathcal {A}\ne \delta \Lambda$ for every $\Lambda \in \varOmega _{\mathrm {loc}} ^{0}(\varGamma (E))$ then we say that there exists an anomaly in the theory. Hence the anomaly is measured by the cohomology class of $\mathcal {A}$ in the BRST cohomology $H_{\mathrm {loc}}^{1}(\mathrm {Lie\,}\mathcal {G}, \varOmega _{\mathrm {loc}}^{0}(\varGamma (E)))$ (e.g. see [4, 6–8, 16]).

Local anomalies also admit a nice geometrical interpretation in terms of the Atiyah–Singer index theorem for families of elliptic operators (see [1, 2, 4, 17]). The first Chern class $c_{1}\left( \mathrm {det\,Ind}D\right) \in H^{2}(\varGamma (E)/\mathcal {G})$ of the determinant line bundle $\mathrm {det\,Ind}D\rightarrow \varGamma (E)/\mathcal {G}$ represents an obstruction for anomaly cancellation. However, the condition $c_{1}\left( \mathrm {det\,Ind}D\right) =0$ is a necessary but not a sufficient condition for local anomaly cancellation due to the problem of locality. In [17] (see also [1]) Singer proposes the problem of defining a notion of “local cohomology of $\varGamma (E)/\mathcal {G}$”, $H_{\mathrm {loc}}^{2}(\varGamma (E)/\mathcal {G})$, adequate to study local anomaly cancellation. The principal difficulty is the fact that the expression of the curvature of $\mathrm {det\,Ind}D$ itself contains non-local terms (Green operators).

Moreover, we recall (see [2, 5, 15]) that the BRST and index theory approaches are related by means of the transgression map t (see Sect. 2), i.e., we have $[\mathcal {A}]=t(c_{1}\left( \mathrm {det\,Ind}D\right) )$. As t is injective, the condition $c_{1}\left( \mathrm {det\,Ind}D\right) =0$ on $H^{2}(\varGamma (E)/\mathcal {G})$ is equivalent to $[\mathcal {A}]=0$ on $H^{1}(\mathrm {Lie\,}\mathcal {G} ,\varOmega ^{0}(\varGamma (E)))$. However, the condition for local anomaly cancellation is $[\mathcal {A}]=0$ on the BRST cohomology $H_{\mathrm {loc}} ^{1}(\mathrm {Lie\,}\mathcal {G}, \varOmega _{\mathrm {loc}}^{0}(\varGamma (E)))$. We define $H_{\mathrm {loc}}^{\bullet }(\varGamma (E)/\mathcal {G})$ in such a way that the preceding condition is equivalent to the vanishing of the class of $c_{1}\left( \mathrm {det\,Ind}D\right)$ on $H_{\mathrm {loc}}^{2} (\varGamma (E)/\mathcal {G})$, hence solving Singer’s problem.

2 The Transgression Maps

First we recall the definition of equivariant cohomology in the Cartan model (e.g. see [3]). We consider a left action of a connected Lie group $\mathcal {G}$ on a manifold $\mathcal {N}$. We have an induced Lie algebra homomorphism $\mathrm {Lie\,}\mathcal {G}\rightarrow \mathfrak {X} (\mathcal {N})$, $X\mapsto X_{\mathcal {N}}=\left. \frac{d}{dt}\right| _{t=0}\rho (\exp (-tX))$. We denote by $\mathcal {P}^{k}(\mathrm {Lie\,} \mathcal {G}, \varOmega ^{r}(\mathcal {N}))^{\mathcal {G}}$ the space of degree k $\mathcal {G}$-invariant polynomials on $\mathrm {Lie\,}\mathcal {G}$ with values in $\varOmega ^{r}(\mathcal {N})$. We recall that $\alpha \in \mathcal {P} ^{k}(\mathrm {Lie\,}\mathcal {G}, \varOmega ^{r}(\mathcal {N}))^{\mathcal {G}}$ if and only if $\alpha (\mathrm {Ad}_{g}X) =\rho (g^{-1})^{*}(\alpha (X))$ $\forall X\in \mathrm {Lie\,}\mathcal {G}$, $\forall g\in \mathcal {G}$.

The space of $\mathcal {G}$-equivariant differential q-forms is defined by

$$\begin{aligned} \varOmega _{\mathcal {G}}^{q}(\mathcal {N})= \bigoplus _{2k+r=q}(\mathcal {P} ^{k}(\mathrm {Lie\,}\mathcal {G},\varOmega ^{r} (\mathcal {N})))^{\mathcal {G}}. \end{aligned}$$

(1)

The Cartan differential $d_{c}:\varOmega _{\mathcal {G}}^{q}(\mathcal {N} )\rightarrow \varOmega _{\mathcal {G}}^{q+1}(\mathcal {N})$, $(d_{c}\alpha )(X)=d(\alpha (X))-\iota _{X_{\mathcal {N}}}\alpha (X)$ for $X\in \mathrm {Lie\,} \mathcal {G}$, satisfies $(d_{c})^{2}=0$, and the $\mathcal {G}$-equivariant cohomology of $\mathcal {N}$, $H_{\mathcal {G}}^{q}(\mathcal {N})$, is the cohomology of this complex.

We recall (e.g. see [3]) that if $\mathcal {N}\rightarrow \mathcal {N} /\mathcal {G}$ is a principal $\mathcal {G}$-bundle we have the (generalized) Chern–Weil homomorphism $\mathrm {ChW}:H_{\mathcal {G}}^{\bullet }(\mathcal {N})\rightarrow H^{\bullet }(\mathcal {N}/\mathcal {G})$. If A is an arbitrary connection on $\mathcal {N}\rightarrow \mathcal {N}/\mathcal {G}$ with curvature $F_{A}$, and $\alpha \in \varOmega _{\mathcal {G}}^{q}(\mathcal {N})$, then we have $\mathrm {ChW}([\alpha ])=[\mathrm {hor}_{A}(\alpha (F_{A}))]$, where $\mathrm {hor}_{A}$ is the horizontalization with respect to the connection A. We also use the notation $\underline{\alpha }=\mathrm {ChW}(\alpha )$.

If $\omega \in \varOmega _{\mathcal {G}}^{2}(\mathcal {N})$ is a closed $\mathcal {G}$-equivariant 2-form, then we have $\omega =\omega _{0}+\mu$ where $\omega _{0}\in \varOmega ^{2}(\mathcal {N})$ is closed, and $\mu :\mathrm {Lie\,} \mathcal {G}\rightarrow C^{\infty }(\mathcal {N})$, is a $\mathcal {G}$-equivariant moment map for $\omega _{0}$, i.e., $\iota _{X_{\mathcal {N}} }\omega _{0}=d(\mu (X))$ for $X\in \mathrm {Lie\,}\mathcal {G}$. A direct computation shows that we have the following

Proposition 1

Assume that $\mathcal {N}\rightarrow \mathcal {N}/\mathcal {G}$ is a principal $\mathcal {G}$-bundle, and let $A\in \varOmega ^{1}(\mathcal {N} ,\mathrm {Lie\,}\mathcal {G})$ be a connection form. If $\omega =\omega _{0}+ \mu \in \varOmega _{\mathcal {G}}^{2}(\mathcal {N})$ is a closed $\mathcal {G}$-equivariant 2-form and we define $\alpha \in \varOmega ^{1}(\mathcal {N} )^{\mathcal {G}}$ by $\alpha (X)=\mu (A(X))(x)$ for $X\in T_{x}\mathcal {N}$, then we have $\mathrm {ChW}_{A}(\omega )=\omega +d_{c}\alpha$.

Corollary 1

The map $\mathrm {ChW}:H_{\mathcal {G}}^{2}(\mathcal {N})\rightarrow H^{2}(\mathcal {N}/\mathcal {G})$ is an isomorphism.

Let us assume now that $H^{1}(\mathcal {N})=H^{2}(\mathcal {N})=0$. The cohomology of the Lie algebra $\mathrm {Lie\,}\mathcal {G}$ with values in $\varOmega ^{0}(\mathcal {N})$ is denoted by $H^{\bullet }(\mathrm {Lie\,} \mathcal {G},\varOmega ^{0}(\mathcal {N}))$.

Proposition 2

Let $\omega =\omega _{0}+\mu \in \varOmega _{\mathcal {G}}^{2} (\mathcal {N})$ be a closed $\mathcal {G}$-equivariant form. If $\rho \in \varOmega ^{1}(\mathcal {N})$ satisfies $\omega _{0}=d\rho$, then the map $\tau _{\rho }\in \varOmega ^{1}(\mathrm {Lie\,}\mathcal {G}, \varOmega ^{0}(\mathcal {N}))$ given by $\tau _{\rho }(X)\!=\!\rho (X_{\mathcal {N}})+\mu (X)$ determines a linear map $\tau :\! H_{\mathcal {G}}^{2}(\mathcal {N})\rightarrow H^{1} (\mathrm {Lie\,}\mathcal {G},\varOmega ^{0}(\mathcal {N}))$ which is independent of the form $\rho$ chosen, and that we call the transgression map $\tau$. If $\mathcal {G}$ is connected, then $\tau$ is injective.

Proof

The first part of the Proposition easily follows using that $L_{Y_{\mathcal {N} }}\mu (X)=\mu ([Y,X])$ by the invariance of $\mu$. We restrict ourselves to prove that $\tau$ is injective. By definition $[\tau _{\rho }]=0$ on $H^{1}(\mathrm {Lie\,}\mathcal {G},\varOmega ^{0}(\mathcal {N}))$ if and only if there exists $\beta \in \varOmega ^{0}(\mathcal {N})$ such that for every $X\in \mathrm {Lie\,}\mathcal {G}$ we have $\tau _{\rho }(X)=L_{X_{\mathcal {N}} }\beta = \iota _{X_{\mathcal {N}}}d\beta$. If we set $\rho ^{\prime }=\rho -d\beta$ then for every $X\in \mathrm {Lie\,}\mathcal {G}$ we have $d\rho ^{\prime } =\omega _{0}$, $\iota _{X_{\mathcal {N}}}\rho ^{\prime }=-\mu (X)$, $L_{X_{\mathcal {N}}}\rho ^{\prime }=0$, i.e., $\rho ^{\prime }\in \varOmega ^{1}(\mathcal {N})^{\mathcal {G}}$ and $d_{c}\rho ^{\prime }=\omega$.

Now we assume that $\pi :\mathcal {N}\rightarrow \mathcal {N}/\mathcal {G}$ is a principal $\mathcal {G}$-bundle. Then we can consider the more familiar transgression map defined as follows

Proposition 3

Let $\underline{\omega }\in \varOmega ^{2}(\mathcal {N}/\mathcal {G})$ be a closed 2-form. If $\eta \in \varOmega ^{1}(\mathcal {N})$ is a form such that $\pi ^{*}\underline{\omega }=d\eta$, then the map $t_{\eta }:\mathrm {Lie\,} \mathcal {G}\rightarrow \varOmega ^{0}(\mathcal {N})$, $t_{\eta }(X)=\eta (X_{\mathcal {N}})$ determines a linear map $t:H^{2}(\mathcal {N} /\mathcal {G})\rightarrow H^{1} (\mathrm {Lie\,}\mathcal {G},\varOmega ^{0}(\mathcal {N}))$, which is independent of the form $\eta$ chosen, and that we call the transgression map t. If $\mathcal {G}$ is connected, then t is injective.

Proof

Again we only prove that t is injective. If $t_{\eta }=\delta \upsilon$ for certain $\upsilon \in \varOmega ^{0}(\mathcal {N})$, then $\eta (X_{\mathcal {N}}) =L_{X_{\mathcal {N}}}\upsilon$. We define $\eta ^{\prime }=\eta -d\upsilon$, and we have $d\eta ^{\prime }=\pi ^{*}\underline{\omega }$, $\iota _{X_{\mathcal {N} }}\eta ^{\prime }=0$, $L_{X_{\mathcal {N}}}\eta ^{\prime }=0$. Hence $\eta ^{\prime }$ is projectable onto a form $\underline{\eta ^{\prime }}\in \varOmega ^{1}(\mathcal {N}/\mathcal {G})$ and $d\underline{\eta ^{\prime }}=\underline{\omega }$.

Proposition 4

If $\omega \in H_{\mathcal {G}}^{2}(\mathcal {N})$ and $\underline{\omega }=\mathrm {ChW}(\omega )$ then we have $\tau (\omega )=t(\underline{\omega })$.

Proof

If $\omega =\omega _{0}+\mu$, by Proposition 1 we have $\omega =\pi ^{*}\underline{\omega }+d_{c}\alpha$ for some $\alpha \in \varOmega _{\mathcal {G}}^{1}(\mathcal {N})=\varOmega ^{1}(\mathcal {N})^{\mathcal {G}}$, i.e. $\omega _{0}=\pi ^{*}\underline{\omega }+d\alpha$ and $\mu (X)=-\alpha (X_{\mathcal {N}})$.

Let $\eta \in \varOmega ^{1}(\mathcal {N})$ be a form such that $\pi ^{*} \underline{\omega }=d\eta$. If we set $\rho =\eta +\alpha$ then $\omega _{0} =d\rho$ and for every $X\in \mathrm {Lie\,}\mathcal {G}$ we have $\tau _{\rho }(X)=t_{\eta }(X)$.

3 Local Equivariant Cohomology

Let $p:E\rightarrow M$ be a bundle over a compact, oriented n-manifold M without boundary. We denote by $J^{r}E$ its r-jet bundle, by $J^{\infty }E$ the infinite jet bundle and by $\varGamma (E)$ be the manifold of global sections of E (assumed to be not empty).

We denote by $\mathrm {Proj}E$ the space of projectable diffeomorphism of E, and by $\mathrm {Proj}^{+}E$ the subgroup of elements preserving the orientation of M. The space of projectable vector fields on E is denoted by $\mathrm {proj}E$. We consider the natural actions of $\mathrm {Proj}E$ on $J^{\infty }E$ and $\varGamma (E)$.

Let $\mathrm {j}^{\infty }:M\times \varGamma (E)\rightarrow J^{\infty }E$, $\mathrm {j}^{\infty }(x,s)=j_{x}^{\infty }s$ be the evaluation map. We define a map $\mathfrak {I}:\varOmega ^{n+k}(J^{\infty }E)\rightarrow \varOmega ^{k}(\varGamma (E))$, by $\mathfrak {I}[\alpha ]=\int _{M}\left( \mathrm {j}^{\infty }\right) ^{*}\alpha$, for $\alpha \in \varOmega ^{n+k}(J^{\infty }E)$. The map $\mathfrak {I}$ commutes with the exterior differential and is $\mathrm {Proj}^{+}E$-equivariant (see [9]). We define the space of local k-forms on $\varGamma (E)$, as the image of the map $\mathfrak {I}$, i.e. $\varOmega _{\mathrm {loc}}^{k}(\varGamma (E)) = \mathfrak {I}(\varOmega ^{n+k}(J^{\infty }E))\subset \varOmega ^{k}(\varGamma (E))$. The cohomology $H_{\mathrm {loc}}^{\bullet }(\varGamma (E))$ of the complex $(\varOmega _{\mathrm {loc} }^{\bullet }(\varGamma (E)),d)$ is called the local cohomology of $\varGamma (E)$. We have $H_{\mathrm {loc}}^{k}(\varGamma (E))\cong H^{n+k}(E)$ for $k>0$ (see [10]).

Let $\mathcal {G}$ be a Lie group acting on E by elements $\mathrm {Proj} ^{+}E$. In order to define an adequate notion of local equivariant cohomology we made the following

Assumption 1

We assume that $\mathrm {Lie\,}\mathcal {G}$ is isomorphic to the space of sections of a Lie algebroid $V\rightarrow M$, i.e. $\mathrm {Lie\,}\mathcal {G}\cong \varGamma (V)$. We also assume that the map $\mathrm {Lie\,}\mathcal {G}\cong \varGamma (V)\rightarrow \mathrm {proj}E$, $X\mapsto X_{E}$ is a differential operator. Finally, in the definition of $\mathcal {G}$-equivariant cohomology $H_{\mathcal {G}}^{n+k}(J^{\infty }E)$, we assume that the polynomial maps $\alpha :\mathrm {Lie\,}\mathcal {G}\rightarrow \varOmega ^{\bullet }(J^{\infty }E)$ are differential operators.

We extend the integration operator to a map $\mathfrak {I}:\varOmega _{\mathcal {G} }^{n+k}(J^{\infty }E)\rightarrow \varOmega _{\mathcal {G}}^{k}(\varGamma (E))$, by setting $(\mathfrak {I}[\alpha ])(X)=\mathfrak {I}[\alpha (X)]$ for every $\alpha \in \varOmega _{\mathcal {G}}^{n+k}(J^{\infty }E)$, $X\in \mathrm {Lie\,}\mathcal {G}$. The map $\mathfrak {I}$ commutes with the Cartan differential and induces a homomorphism in equivariant cohomology $\mathfrak {I}:H_{\mathcal {G}} ^{n+k}(J^{\infty }E)\rightarrow H_{\mathcal {G}}^{k}(\varGamma (E))$ (see [9]).

We define the space of local $\mathcal {G}$-equivariant k-forms by

$$\begin{aligned} \varOmega _{\mathcal {G},\mathrm {loc}}^{k}(\varGamma (E)) =\mathfrak {I}(\varOmega _{\mathcal {G} }^{n+k}(J^{\infty }E)) \subset \varOmega _{\mathcal {G}}^{k}(\varGamma (E)). \end{aligned}$$

(2)

The local $\mathcal {G}$-equivariant cohomology of $\varGamma (E)$, $H_{\mathcal {G} ,\mathrm {loc}}^{\bullet }(\varGamma (E))$, is defined as the cohomology of the complex $(\varOmega _{\mathcal {G},\mathrm {loc}}^{\bullet }(\varGamma (E)),d_{c})$.

4 Application to Local Anomaly Cancellation

Let us define the BRST cohomology (see [6, 16]). Recall (see Assumption 1) that we assume $\mathrm {Lie\,}\mathcal {G} \cong \varGamma (V)$ for some vector bundle $V\rightarrow M$. A map $\alpha :\bigwedge ^{k}\mathrm {Lie\,}\mathcal {G}\rightarrow \varOmega _{\mathrm {loc} }^{0}(\varGamma (E))$ is said to be local if there exists a differential operator $A:\bigwedge ^{k}\mathrm {Lie\,}\mathcal {G}\rightarrow \varOmega ^{n} (J^{\infty }E)$ such that $\alpha (X_{1},\ldots ,X_{k})=\mathfrak {I}[A(X_{1} ,\ldots ,X_{k})]$ for every $X_{1},\ldots ,X_{k}\in \mathrm {Lie\,}\mathcal {G}$. We denote by $\varOmega _{\mathrm {loc}}^{k}(\mathrm {Lie\,}\mathcal {G}, \varOmega _{\mathrm {loc}}^{0}(\varGamma (E))$ the space of local k-forms on $\mathrm {Lie\,}\mathcal {G}$ with values on $\varOmega _{\mathrm {loc}}^{0} (\varGamma (E))$. The differential $\delta$ on the complex $\varOmega ^{\bullet }(\mathrm {Lie\,}\mathcal {G},\varOmega _{\mathrm {loc}}^{0}(\varGamma (E))$ induces a differential on $\varOmega _{\mathrm {loc}}^{\bullet }(\mathrm {Lie\,}\mathcal {G}, \varOmega _{\mathrm {loc}}^{0}(\varGamma (E))$. The corresponding cohomology $H_{\mathrm {loc}}^{\bullet }(\mathrm {Lie\,}\mathcal {G}, \varOmega _{\mathrm {loc} }^{0}(\varGamma (E)))$ is called the BRST cohomology. We assume that $H^{2} (\varGamma (E))=H^{1}(\varGamma (E))=0$ and also that $H_{\mathrm {loc}}^{2} (\varGamma (E))=H_{\mathrm {loc}}^{1}(\varGamma (E))=0$.

Proposition 5

The restriction of the transgression map $\tau$ to $H_{\mathcal {G} ,\mathrm {loc}}^{2}(\varGamma (E))$ takes values on the BRST cohomology $H_{\mathrm {loc}}^{1}(\mathrm {Lie\,}\mathcal {G}, \varOmega _{\mathrm {loc}} ^{0}(\varGamma (E)))$. The map $\tau :H_{\mathcal {G},\mathrm {loc} }^{2} (\varGamma (E))\rightarrow H_{\mathrm {loc}}^{1} (\mathrm {Lie\,}\mathcal {G}, \varOmega _{\mathrm {loc}}^{0}(\varGamma (E)))$ is injective for $\mathcal {G}$ connected.

Proof

Let $\omega =\omega _{0}+\mu \in \varOmega _{\mathcal {G},\mathrm {loc}}^{2}(\varGamma (E))$ be a closed local $\mathcal {G}$-equivariant 2-form. As $H_{\mathrm {loc}} ^{2}(\varGamma (E))=0$, we have $\omega _{0}=d\rho$, for certain $\rho \in \varOmega _{\mathrm {loc}}^{1}(\varGamma (E))$. By our assumption in the definition of local equivariant cohomology (Assumption 1), the map $\tau _{\rho }:\mathrm {Lie\,}\mathcal {G}\rightarrow \varOmega _{\mathrm {loc} }^{0}(\varGamma (E))$, $\tau _{\rho }(X)=\rho (X_{\varGamma (E)})+\mu (X)\in \varOmega _{\mathrm {loc}}^{0}(\varGamma (E))$ is a local map. The injectiveness of $\tau$ follows from Proposition 2.

Assume that $\varGamma (E)\rightarrow \varGamma (E)/\mathcal {G}$ is a principal $\mathcal {G}$-bundle. Then we define the local cohomology by $H_{\mathrm {loc} }^{k}(\varGamma (E)/\mathcal {G}) =\mathrm {ChW} (H_{\mathcal {G},\mathrm {loc}} ^{k}(\varGamma (E)))$. By Proposition 4 we have the following

Proposition 6

Let $\omega \in \varOmega _{\mathcal {G},\mathrm {loc}}^{2}(\varGamma (E))$ be a closed local $\mathcal {G}$-equivariant 2-form and let $\underline{\omega }=\mathrm {ChW}(\omega )$. Then we have $\tau (\omega )=t(\underline{\omega })$, and in particular we conclude that $t(\underline{\omega })\in H_{\mathrm {loc} }^{1} (\mathrm {Lie\,}\mathcal {G},\varOmega _{\mathrm {loc}}^{0} (\varGamma (E)))$. Moreover, the following conditions are equivalent

(a) $[\omega ]=0$ on $H_{\mathcal {G},\mathrm {loc}}^{2} (\varGamma (E))$.

(b) $[\underline{\omega }]=0$ on $H_{\mathrm {loc}}^{2} (\varGamma (E)/\mathcal {G})$.

(c) $[\tau (\omega )]=[t(\underline{\omega })]=0$ on $H_{\mathrm {loc}}^{1} (\mathrm {Lie\,}\mathcal {G},\varOmega _{\mathrm {loc}}^{0} (\varGamma (E)))$.

Proposition 6 applied to $\underline{\omega }=c_{1} (\mathrm {det\,Ind} D)$ shows that our definition of $H_{\mathrm {loc}}^{2} (\varGamma (E)/\mathcal {G})$ solves Singer’s problem. We also note that if $\omega \in \varOmega _{\mathcal {G},\mathrm {loc}}^{2}(\varGamma (E))$ is closed, the form $\underline{\omega }\in \varOmega ^{2}(\varGamma (E)/\mathcal {G})$ determining the class $\mathrm {ChW}([\omega ])$ could contain non-local terms, as $\underline{\omega }$ depends on the curvature of a connection $\Theta$ on the principal $\mathcal {G}$-bundle $\varGamma (E)\rightarrow \varGamma (E)/\mathcal {G}$, and $\Theta$ usually contains non-local terms. This fact explains the appearance of non-local terms on the expression of the curvature of $\mathrm {det\,Ind}D$ commented on the Introduction.