Some Remarks about Duality, Analytic Torsion and Gaussian Integration in Antisymmetric Field Theories

From a path integral point of view (e.g. [Q98]) physicists have shown how duality in antisymmetric quantum field theories on a closed spacetime manifold M relies in a fundamental way on Fourier Transformations of formal infinite-dimensional volume measures. We first review these facts from a measure theoretical point of view, setting the importance of the Hodge decomposition theorem in the underlying geometric picture, ignoring the local symmetry which lead to degeneracies of the action. To handle these degeneracies we then apply Schwarz’s Ansatz showing how duality leads to a factorization of the analytic torsion of M in terms of the partition functions associated to degenerate “dual” actions, which in the even dimensional case corresponds to the identification of these partition functions. Introduction Antisymmetric field theories are generalizations of electromagnetic theory where the potential 1-form is replaced by a k-form. Some remarkable facts arising in electromagnetism are also observed in general antisymmetric theories, notably T-duality on which we will focus here. In electromagnetic theory this type of duality corresponds to the observation that electric and magnetic fields in the theory are interchanged under transformations taking solutions of field equations into solutions of the Bianchi identity, particles into topological defects, weak couplings into strong couplings, etc. (for a review see [O95]). Consider a theory of antisymmetric tensors on a n-dimensional space-time manifold M equpped with a Riemannian metric. Let ωi1i2...ik be a k-tensor field on