Trace maps on chiral Clifford algebras for the rank two fermionic vertex operator superalgebra
Abstract
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∨⊗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 $ β_i\in F$, γj∈ F∨⊗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 : (sC̃ch(X,CE)sQ, dchCE)⟶ (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 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.
Related
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BibTeX
@misc{zuevsky2026trace,
title = {{Trace maps on chiral Clifford algebras for the rank two fermionic vertex operator superalgebra}},
author = {A. Zuevsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02065},
eprint = {2608.02065},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02065},
note = {CC0},
}
RIS
TY - UNPB
TI - Trace maps on chiral Clifford algebras for the rank two fermionic vertex operator superalgebra
AU - A. Zuevsky
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2608.02065
UR - https://arxiv.org/abs/2608.02065
AB - 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^∨⊗ω_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 $ β_i\in F$, γ^j∈ F^∨⊗ω_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 CE. Using the Batalin-Vilkovisky (BV) formalism together with Feynman diagrams we prove that the resulting trace map Trch : (sC̃^ch(X,CE)_sQ, dch_CE)⟶ (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 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, 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.
ER -