发表机构
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对光滑黎曼曲面上的全纯向量丛，构造了秩2费米顶点算子超代数对应的手征克利福德代数上的迹映射，证明其为满足广义量子主方程的拟同构，应用该迹映射纯代数得到了费米子解析挠率变化的经典公式。
AI 中文摘要
对于光滑黎曼曲面X上秩为r的全纯向量丛F，我们在纯奇丛E=Π(F⊕F∨⊗ω_X)对应的手征克利福德代数𝓒的手征同调上构造了一个迹映射，它是秩2费米顶点算子超代数的手征代数实现。该自由费米子（bc型）共形场论由一对奇场β_i∈F、γ^j∈F∨⊗ω_X构成的对偶对建立。我们完整构造了该顶点算子超代数、其关联的顶点超代数丛，以及后者的手征代数与手征包络𝓒之间的同构。利用巴塔林-维尔科夫斯基（BV）形式化与费曼图，我们证明所得迹映射Trch: (𝓒̃^ch(X,𝓒)_𝓠, d^ch_𝓒)→(𝓞_BV, -d_BV)是满足广义量子主方程的链映射，且为拟同构，将Gui针对辛玻色子构造的手征外尔代数上的迹映射推广到奇/克利福德情形。我们确立了该迹映射的存在性、同伦唯一性与函子性（包括链同伦意义下显式度量无关性），证明了相关超迹的循环性，验证了d_BV的幂零性、分次莱布尼茨法则及所有引入微分的d²=0。作为应用，我们计算了修改仿射流与修改能量-动量张量上的迹映射，仅通过手征链复形纯代数地得到了费米子（Ray-Singer）解析挠率沿丛F的模空间及曲线X的模空间变化的Fay经典公式。
英文摘要
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 $E=Π(F\oplus F^\vee\otimesω_X)$. 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$, $γ^j\in F^\vee\otimesω_X$. 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 \[ \Trch : \bigl(\widetilde\sC^{\ch}(X,\CE)_{\sQ},\, \dch_{\CE}\bigr)\longrightarrow (\OBV,-\DBV) \] 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 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 $X$.
