Geometric quantization and the metric dependence of the self-dual field theory

Abstract

We investigate the metric dependence of the partition function of the self-dual p-form gauge field on an arbitrary Riemannian manifold. Using geometric quantization of the space of middle-dimensional forms, we derive a projectively flat connection on its space of polarizations. This connection governs metric dependence of the partition function of the self-dual field. We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold. We compute the local gravitational anomaly and show how our derivation relates to the classical computation based on index theory. As an application, we show that the one-loop determinant of the (2,0) multiplet on a Calabi-Yau threefold coincides with the square root of the one-loop determinant of the B-model.

32 pages. v2: Added an important clarification in section 4 and fixed some typos

References in corpus (7)

• Heisenberg Groups and Noncommutative Fluxes
• Holographic Action for the Self-Dual Field
• On the partition sum of the NS five-brane
• Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces
• The Picard group of the moduli space of curves with level structures
• Topological Strings and (Almost) Modular Forms
• The anomaly line bundle of the self-dual field theory

Cited by in corpus (10)

• The global anomaly of the self-dual field in general backgrounds
• Anomaly inflow and $p$-form gauge theories
• The anomaly line bundle of the self-dual field theory
• Automorphic Instanton Partition Functions on Calabi-Yau Threefolds
• The global gravitational anomaly of the self-dual field theory
• Non-uniqueness results for critical metrics of regularized determinants in four dimensions
• Canonical quadratic refinements of cohomological pairings from functorial lifts of the Wu class
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• Some comments on 6d global gauge anomalies
• Global anomalies and chiral p-forms