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Cohomology of multipoint connections on complex curves

A. Zuevsky

Abstract

Assuming the conformal field theory correlation functions defined on complex curves satisfy the recursion formulas, we express the corresponding cohomology theory via the generalizations of holomorphic connections. The cohomology is explicitly found in terms of higher genus counterparts of elliptic functions as analytic continuations of solutions for functional equations. Explicit examples associated to correlation functions on various genera are provided.

Cohomology of multipoint connections on complex curves

Abstract

Assuming the conformal field theory correlation functions defined on complex curves satisfy the recursion formulas, we express the corresponding cohomology theory via the generalizations of holomorphic connections. The cohomology is explicitly found in terms of higher genus counterparts of elliptic functions as analytic continuations of solutions for functional equations. Explicit examples associated to correlation functions on various genera are provided.

Paper Structure

This paper contains 16 sections, 2 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. The chain complex and cohomology
  3. Chain complex spaces of $n$-variable correlation functions
  4. Recursion relations as mutipoint connections
  5. Computation of the recursion cohomology
  6. Motivating examples
  7. The rational case
  8. Modular and elliptic functions
  9. The elliptic case
  10. The case of deformed elliptic functions
  11. The recursion formulas for Jacobi functions
  12. Multiparameter Jacobi forms
  13. The genus two counterparts of Weierstrass functions
  14. The genus two case
  15. The genus $g$ generalizations of elliptic functions
  16. ...and 1 more sections

Key Result

Lemma 1

Correlation functions $\mathcal{Z}\left({\bf z}_n, \mu \right)$, $n \ge 0$, belong to the space of multipoint connections. The corresponding recursion cohomology is given by $H^n(\mu) = {\mathcal{C}on}^{n}/G^{n-1}$.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Proposition 1
  • proof