2026/07/27 by Giovanni Molinari, Michele Schiavina
#math-ph #math.AT #math.DG #math.MP
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.