Abstract
We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function for BF theory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds.
Highlights
Quantum field theory is a useful tool in many areas of pure and applied mathematics
One possible way to do this is by choosing a reference metric g and enforcing a g-dependent condition on fields.1. It allows to compute the partition function of the theory—the starting point for quantum considerations on the system—and one is left to show that the choice of metric is immaterial. The proof that such choice of metric is irrelevant was given by Schwarz for the partition function of abelian BF theory, and it is tantamount to the statement of independence of the analytic torsion on the metric used to define a Laplacian on the underlying manifold
Conjecture 21, due to Fried [37], proposes that when M = Sg∗Σ, the Ruelle zeta function exactly computes the analytic torsion of the associated sphere bundle. To connect this to field theory, we observe that this means Ruelle zeta function is expected to compute—in Schwarz’s terms—the partition function of BF theory in a given gauge fixing
Summary
Quantum field theory is a useful tool in many areas of pure and applied mathematics. It provides a number of precise answers, often involving insight coming from statements that are theorems in finite dimensions, and that need to be appropriately checked and generalised in infinite dimensions. Conjecture 21, due to Fried [37], proposes that when M = Sg∗Σ, the Ruelle zeta function (evaluated at zero) exactly computes the analytic torsion of the associated sphere bundle To connect this to field theory, we observe that this means Ruelle zeta function is expected to compute—in Schwarz’s terms—the partition function of BF theory in a given (metric dependent) gauge fixing. We believe that the field-theoretic presentation of the Ruelle zeta function provided in this paper will allow the problem to be tackled from a different angle: by means of homotopies of Lagrangian submanifolds To this aim, we set up a convenient construction to compare Anosov vector fields that are related to a choice of a metric on a base manifold Σ 5 we interpret Fried’s conjecture in terms of gauge-fixing independence of BF theory in the BV formalism and suggest a construction for sphere bundles that allows to present explicit homotopies between Lagrangian submanifolds. We choose to deal with this problem by means of the Batalin–Vilkovisky (BV) formalism
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