Giovanni Molinari | Searcharxiv

arXiv2026

Milnor metric for Morse--Smale flows from field theory

We study the partition function of Abelian $BF$ theory with an axial gauge fixing condition given by a Morse--Smale vector field, and we show that it recovers the Milnor metric on the determinant line on the twisted cohomology of the manifold. To do this we use the Batalin--Vilkovisky formalism and, in particular, a BV pushforward to cohomology in two steps. In this context, the Milnor metric serves as the generalisation of the Ruelle dynamical zeta function evaluated at zero for systems containing both closed orbits and critical points. This, together with a classic result due to Schwarz on the partition function of $BF$ theory in the Lorenz gauge, provides a field-theoretic realisation of Fried's conjecture, which is true for Morse--Smale flows.

math-ph↗

arXiv2026

The Fried Conjecture for Morse-Smale Flows: A Survey on Ray-Singer and Milnor Metrics

This paper reviews the generalised Fried conjecture for Morse-Smale flows on compact Riemannian manifolds. We first establish the proper framework by constructing the twisted de Rham complex and deriving the Hodge decomposition, which underpins the definition of the Ray-Singer torsion. On the dynamical side, we characterise Morse-Smale vector fields, introducing the Ruelle Zeta function to encode the spectral data of closed orbits and constructing the Thom-Smale complex to include the contribution of fixed points. These invariants are synthesised into the definition of the Milnor metric on the determinant line of the twisted cohomology. Finally, we present the theorem proving the conjecture: the Ray-Singer metric coincides with the Milnor metric, identifying the analytic torsion with the product of the Thom-Smale combinatorial torsion and the value at zero of the Ruelle Zeta function.

math.DG↗