Trace maps on chiral Clifford algebras for the rank two fermionic vertex operator superalgebra

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 $\text{CE}$ attached to the purely odd bundle $E=\mathrm{\Pi }(F\oplus F^{\vee }\otimes {\omega }_X)$. 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 ${\beta }_i\in F$, ${\gamma }^j\in F^{\vee }\otimes {\omega }_X$. 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 $\text{CE}$. Using the Batalin-Vilkovisky (BV) formalism together with Feynman diagrams we prove that the resulting trace map $$\text{Trch}:({\widetilde{\text{sC}}}^{\mathrm{ch}}(X,\text{CE}{)}_{\text{sQ}},{\text{dch}}_{\text{CE}})⟶(\text{OBV},-\text{DBV})$$ 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 metric-independence up to chain homotopy) of the trace map, prove cyclicity of the relevant supertrace, and verify nilpotency of $\text{DBV}$, the graded Leibniz rule, $d^2=0$ 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 $X$.