[Paper Review] Local Anomalies, Local Equivariant Cohomology and the Variational Bicomplex

Roberto Ferreiro Pérez|arXiv (Cornell University)|Jul 4, 2006

Advanced Operator Algebra Research22 references5 citations

TL;DR

This paper establishes a geometric framework for local anomalies in quantum field theory using local equivariant cohomology and the variational bicomplex. It proves that anomaly cancellation conditions—particularly for gravitational, gauge, and mixed anomalies—reduce to vanishing of specific cohomology classes in the jet bundle, with explicit criteria derived from relative Gelfand-Fuchs cohomology and Weil polynomials.

ABSTRACT

The locality conditions for the vanishing of local anomalies in field theory are shown to admit a geometrical interpretation in terms of local equivariant cohomology, thus providing a method to deal with the problem of locality in the geometrical approaches to the study of local anomalies based on the Atiyah-Singer index theorem. The local cohomology is shown to be related to the cohomology of jet bundles by means of the variational bicomplex theory. Using these results and the techniques for the computation of the cohomology of invariant variational bicomplexes in terms of relative Gel'fand-Fuks cohomology introduced in [6], we obtain necessary and sufficient conditions for the cancellation of local gravitational and mixed anomalies.

Motivation & Objective

• To resolve the problem of defining a suitable notion of local cohomology for geometric approaches to anomalies, as posed by Singer.
• To provide a geometric interpretation of locality conditions for anomaly cancellation in terms of local equivariant cohomology.
• To derive necessary and sufficient conditions for cancellation of local gravitational and mixed anomalies using jet bundle cohomology.
• To establish isomorphisms between local cohomology of the space of sections and cohomology of the jet bundle via the variational bicomplex.
• To apply these tools to the case of Riemannian metrics under diffeomorphism action and connections on principal bundles under automorphism action.

Proposed method

• Uses the integration map $\Im: \Omega^{n+k}(J^\infty E) \to \Omega^k(\Gamma(E))$ to define local forms on the space of sections $\Gamma(E)$, lifting forms from the jet bundle.
• Applies the variational bicomplex formalism to relate the cohomology of $\Gamma(E)$ to that of $J^\infty E$, particularly via the interior Euler operator $I$.
• Introduces local $\mathcal{G}$-invariant cohomology for group actions on $E$, with isomorphisms $H_{\text{loc}}^k(\Gamma(E))^\mathcal{G} \cong H^k(\mathcal{F}^\bullet(J^\infty E))^\mathcal{G}$.
• Employs relative Gelfand-Fuchs cohomology techniques from [3] to compute the cohomology of invariant variational bicomplexes.
• Applies the Atiyah-Singer index theorem for families to compute the equivariant curvature of the determinant line bundle, yielding the anomaly form.
• Uses the isomorphism $H_{\text{loc}}^2(\mathfrak{Met}M \times \mathcal{A}_P)^{\mathrm{Aut}^+P} \cong H^{n+2}(J^\infty(\mathcal{M}_M \times_M C(P)))^{\mathrm{Aut}^+P}$ to translate anomaly conditions into cohomological vanishing.

Experimental results

Research questions

• RQ1 What is the correct geometric notion of local cohomology that captures the locality of anomalies in field theory?
• RQ2 How can the variational bicomplex and jet bundle formalism be used to characterize local forms on the space of sections?
• RQ3 What are the necessary and sufficient conditions for cancellation of local gravitational and mixed anomalies?
• RQ4 How does the action of diffeomorphisms on metrics and automorphisms on connections affect the structure of local anomalies?
• RQ5 Can the anomaly cancellation condition be expressed independently of the specific manifold or bundle, depending only on the group structure and dimension?

Key findings

• The local cohomology $H_{\text{loc}}^k(\Gamma(E))$ is isomorphic to $H^{n+k}(J^\infty E)$ for $k > 0$, enabling cohomological computation via the jet bundle.
• For the space of Riemannian metrics under diffeomorphism action, nontrivial local cohomology classes are constructed using the variational bicomplex and equivariant cohomology.
• The first obstruction to anomaly cancellation in the mixed gravitational-gauge case is given by the cohomology class $\Im\left[\left(\hat{A}(\mbox{\boldmath$\Omega$}) \wedge \mathrm{ch}^\rho(\mbox{\boldmath$\Omega$}) \wedge \mathrm{ch}^\beta(\mathbb{F})\right)_{n+2}\right]$.
• The mixed anomaly cancels if and only if the component $Q$ of degree $n/2+1$ of the Weil polynomial $\hat{A}\mathrm{ch}^\rho \otimes \mathrm{ch}^\beta$ vanishes in $I^{SO(n)} \otimes I^G$.
• The gauge and gravitational anomalies cannot cancel between each other, as the obstruction classes lie in distinct components of the cohomology.
• The anomaly cancellation condition depends only on the dimension $n$ of the manifold and the structure group $G$, not on the specific manifold or bundle.