Ruelle Zeta Function from Field Theory
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Annales Henri Poincar´ e
Ruelle Zeta Function from Field Theory Charles Hadfield, Santosh Kandel and Michele Schiavina 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.
Contents Introduction What to Expect from This Paper 1. Lagrangian Field Theory, the Batalin–Vilkovisky Formalism and Regularised Determinants 1.1. Classical Field Theory, Symmetries and Quantisation 1.2. Cohomological Approach and the BV Complex 1.3. Partition Functions 2. Geometric Setting 2.1. Flat Vector Bundle 2.2. Riemannian Structure and Analytic Torsion 2.3. Contact Structure 2.3.1. Contact Riemannian Structure 2.4. Anosov Dynamics 2.4.1. A Guiding Example 3. Ruelle Zeta Function 3.1. Differential Forms Decomposition 3.2. Ruelle Zeta Function as Regularised Determinant 3.3. Meromorphic Extension of the Resolvent 4. BF Theory on Contact Manifolds 4.1. Analytic Torsion from Resolutions of de Rham Differential 4.2. BV Interpretation: Metric Gauge for BF Theory
C. Hadfield et al.
5.
Ann. Henri Poincar´e
4.3. Contact Gauge Fixing for BF Theory Lagrangian Homotopies, Fried’s Conjecture and Gauge-fixing Independence
5.1. A Sphere Bundle Construction Acknowledgements References
Introduction 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. A positive example of this is the interpretation by Schwarz of the Ray– Singer analytic torsion in terms of a partition function for a degenerate functional [56,57]. The main ingredient in Schwarz’s construction is a topological field theory involving differential forms, which enjoys a symmetry given by the shift of closed forms by exact ones [8,19,28]. This is known nowadays with the name of BF theory. From a field-theoretic point of view, such symmetry needs to be removed, or gauge fixed, as it represents a fundamental redundancy in the description. 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. There are several ways of encoding a choice of gauge fixing within the framework of field theory, starting from the original ide
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