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The Global Sections of Chiral de Rham Complexes on Closed Complex Curves

Bailin Song, Wujie Xie

TL;DR

The paper determines the global sections $H^0(X,\Omega^{ch}_X)$ of the chiral de Rham complex on a closed complex curve $X$ of genus $g\ge 2$ by translating to antiholomorphic sections of $SW(\bar{T}^{*}X)$ via a canonical isomorphism and then analyzing the differential $\bar{D}$ in graded pieces. It shows that, for Hermitian locally symmetric $X$, higher curvature corrections vanish beyond $F_2$ and reduces the calculation to holomorphic data controlled by $F_1$; in particular, the space of global sections decomposes into contributions from $\ker F_1$ on the $(k,l,s)=(k,s,s)$ sector and from all sectors with $l>s$, with the former tied to $\mathfrak{sl}_2$-invariants of the $\beta\gamma-bc$ system. The results provide an explicit vertex algebra description of $H^0(X,\Omega^{ch}_X)$ for high-genus curves and connect chiral de Rham theory to classical geometric structures via the curvature and Chern connection. Overall, the work extends the S3 framework to closed curves, yielding a precise, structure-rich account of global chiral de Rham sections on genus $g\ge 2$ curves and highlighting the role of $\mathfrak{sl}_2$-invariance in the holomorphic sector.

Abstract

The space of global sections of the chiral de Rham complex on any closed complex curve with genus $g \ge2$ is calculated.

The Global Sections of Chiral de Rham Complexes on Closed Complex Curves

TL;DR

The paper determines the global sections of the chiral de Rham complex on a closed complex curve of genus by translating to antiholomorphic sections of via a canonical isomorphism and then analyzing the differential in graded pieces. It shows that, for Hermitian locally symmetric , higher curvature corrections vanish beyond and reduces the calculation to holomorphic data controlled by ; in particular, the space of global sections decomposes into contributions from on the sector and from all sectors with , with the former tied to -invariants of the system. The results provide an explicit vertex algebra description of for high-genus curves and connect chiral de Rham theory to classical geometric structures via the curvature and Chern connection. Overall, the work extends the S3 framework to closed curves, yielding a precise, structure-rich account of global chiral de Rham sections on genus curves and highlighting the role of -invariance in the holomorphic sector.

Abstract

The space of global sections of the chiral de Rham complex on any closed complex curve with genus is calculated.
Paper Structure (18 sections, 21 theorems, 99 equations)

This paper contains 18 sections, 21 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Chiral de Rham Complex
  3. Vertex algebras
  4. $\beta\gamma-bc$ system.
  5. Chiral de Rham Algebra.
  6. Antiholomorphic vector bundles corresponding to chiral de Rham complexes
  7. Vector bundles induced from chiral de Rham complex.
  8. Chern connection.
  9. A canonical isomorphism of sheaves.
  10. Hermitian locally symmetric space
  11. Global Sections on closed Complex Curves
  12. The curvature form on $SW(\overline{T}^{*}X)$.
  13. The equation $\bar{D}a=0$.
  14. Case 1: $l-s<0$
  15. Case 2: $l-s > 0$.
  16. ...and 3 more sections

Key Result

Theorem 2.1

(i) The chiral Hodge cohomology $H^{*}(X,\Omega^{ch}_X)$ is equal to the cohomology of the complex $(\Omega^{ch,*}_X,\bar{\partial})$. (ii) The chiral Hodge cohomology $H^{*}(X,\Omega^{ch}_X)$ is a vertex algebra. In particular, $\Gamma(X, \Omega^{ch}_{X})=H^{0}(X, \Omega^{ch}_{X})$ is a sub vertex

Theorems & Definitions (36)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.1
  • ...and 26 more