Ruelle Zeta Function from Field Theory

Area of Science:

• Mathematical Physics
• Number Theory
• Differential Geometry

Background:

• The Ruelle zeta function is a key object in dynamical systems and number theory.
• Analytic torsion is a topological invariant related to the spectrum of the Laplacian on differential forms.
• A conjecture by Fried proposes an equivalence between Ruelle zeta functions and analytic torsion.

Purpose of the Study:

• To provide a field-theoretic interpretation of the Ruelle zeta function.
• To establish a connection between the Ruelle zeta function and BF theory.
• To rephrase Fried's conjecture in a new mathematical framework.

Main Methods:

• Utilizing field-theoretic techniques to interpret the Ruelle zeta function.
• Employing BF theory as a model system.
• Imposing a specific gauge-fixing condition on contact manifolds.

Main Results:

• The Ruelle zeta function is shown to be equivalent to the partition function of BF theory under a novel gauge-fixing condition.
• This interpretation is valid for contact manifolds.
• The study suggests a reformulation of Fried's conjecture.

Conclusions:

• A novel field-theoretic perspective on the Ruelle zeta function is established.
• The connection to BF theory and contact manifolds opens new avenues for research.
• The proposed rephrasing of Fried's conjecture offers a geometric interpretation in terms of homotopies of Lagrangian submanifolds.