高亏格下的首个手征同调群
The first chiral homology group in higher genus
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中文总结 AI 辅助
该研究将顶点代数的首个手征同调群理论从椭圆曲线扩展到任意亏格紧致黎曼曲面，构造显式复形计算手征同调群，证明相关恒等式，回答了有限性问题，并通过退化和定理联系不同同调，得出特定顶点代数在各亏格下首个手征同调群消失的结论。
中文摘要 AI 辅助
我们将由范·埃克伦和赫卢阿尼为椭圆曲线发展的顶点代数的首个手征同调群理论扩展到任意亏格的紧致黎曼曲面。我们的方法通过将\(g\)个柄迭代自缝合到黎曼球面上实现亏格\(g\)的曲面，每个柄由穿孔圆盘内的缝合参数\(\rho_i\)控制，从而恢复范·埃克伦 - 赫卢阿尼的构造。我们构造了一个显式复形来计算具有\(n\)个标记点的亏格\(g\)曲面上顶点代数\(V\)的手征同调群\(H^{\mathrm{ch}}_0\)和\(H^{\mathrm{ch}}_1\)，在缝合参数的\(g\)维空间上为其配备射影平坦联络以及显式的中心荷反常，并证明了相关修正顶点算子的亏格\(g\)傅里叶空间博切尔兹恒等式。我们表明，范·埃克伦和赫卢阿尼在亏格\(1\)中分离出的相同两个有限性假设——朱\(C_2\) - 代数的首个泊松同调\(\HP_1(R_V)\)的有限维性以及相关分次代数的某个科祖尔同调的有限生成——意味着对于每个亏格\(g\)和每个顶点代数\(V\)，\(H^{\mathrm{ch}}_1(X,V)\)是有限维的，回答了他们工作中留下的一个问题。利用退化\(\rho_i \to 0\)以及达米奥利尼 - 吉布尼 - 塔拉萨的因式分解定理，我们将\(H_1^{\mathrm{ch}}\)的完全退化极限与朱代数上迭代构造的霍赫希尔德同调联系起来，并推导在亏格\(1\)中处理的相同经典自由、有理顶点代数在每个亏格下首个手征同调群的消失。
英文摘要
We extend the theory of the first chiral homology group of vertex algebras, developed by van Ekeren and Heluani for elliptic curves, to compact Riemann surfaces of arbitrary genus. Our approach realizes a genus $g$ surface by iterated self-sewing of $g$ handles onto the Riemann sphere, each governed by a sewing parameter $ρ_i$ in a punctured disc, so that the construction of van Ekeren-Heluani is recovered. We construct an explicit complex computing the chiral homology groups $H^{\mathrm{ch}}_0$ and $H^{\mathrm{ch}}_1$ of a vertex algebra $V$ on a genus $g$ surface with $n$ marked points, equip it with a projectively flat connection, with an explicit central-charge anomaly, over the $g$-dimensional space of sewing parameters, and prove a genus $g$ Fourier-space Borcherds identity for the associated modified vertex operators. We show that the same two finiteness hypotheses isolated by van Ekeren and Heluani in genus $1$ - finite dimensionality of the first Poisson homology $\HP_1(R_V)$ of the Zhu $C_2$- algebra, and finite generation of a certain Koszul homology of the associated graded algebra - imply finite dimensionality of $H^{\mathrm{ch}}_1(X,V)$ for every genus $g$ and every vertex algebra $V$, answering a question left open in their work. Using the degeneration $ρ_i \to 0$ together with the factorization theorem of Damiolini- Gibney-Tarasca, we relate the totally degenerate limit of $H_1^{\mathrm{ch}}$ to the Hochschild homology of an iterated construction on the Zhu algebra, and deduce vanishing of the first chiral homology group in every genus for the same classically free, rational vertex algebras treated in genus one.
