2026/08/03 by A. Zuevsky
Mathematics · #math.FA
arxiv created 2026/08/03 · arxiv updated 2026/08/04
For a holomorphic vector bundle F of rank r on a smooth Riemann surface X we construct a trace map on the chiral homology of the chiral Clifford algebra \CE attached to the purely odd bundle E=Π(F⊕ F^\vee⊗ωX). It is the chiral-algebraic realization of the rank two fermionic vertex operator superalgebra. The free-fermion (bc-type) conformal field theory built from a dual pair of odd fields βi∈ F, γj∈ F^\vee⊗ωX. We give a complete construction of this vertex operator superalgebra, its associated vertex superalgebra bundle, and the isomorphism between the latter's chiral algebra and the chiral envelope \CE. Using the Batalin-Vilkovisky (BV) formalism together with Feynman diagrams we prove that the resulting trace map \Trch : (\widetilde\sC\ch(X,\CE)\sQ, \dch\CE)\longrightarrow (\OBV,-\DBV) is a chain map satisfying a generalized quantum master equation and is a quasi-isomorphism, generalizing to the odd/Clifford setting the trace map on chiral Weyl algebras constructed by Gui for symplectic bosons. We establish existence, homotopy uniqueness, and functoriality (including explicit metric-independence up to chain homotopy) of the trace map, prove cyclicity of the relevant supertrace, and verify nilpotency of \DBV, the graded Leibniz rule, d2=0 for every differential introduced. As an application we compute the trace map on a modified affine current and on a modified energy-momentum tensor, recovering, purely algebraically from the chiral chain complex, Fay's classical formulas for the variation of the fermionic (Ray-Singer) analytic torsion along the moduli of the bundle F and along the moduli of the curve X.