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An open-closed string analogue of Hochschild cohomology

Hang Yuan

Abstract

We prove that every open-closed homotopy algebra, introduced by Kajiura and Stasheff (arXiv: archive/0410291), naturally gives rise to an open-closed version of Hochschild cochain complex whose cohomology admits a canonical Gerstenhaber algebra structure. We also develop the open-closed brace relations, provide a concise description of OCHAs, and establish an A-infinity structure that extends the open-closed Hochschild differential.

An open-closed string analogue of Hochschild cohomology

Abstract

We prove that every open-closed homotopy algebra, introduced by Kajiura and Stasheff (arXiv: archive/0410291), naturally gives rise to an open-closed version of Hochschild cochain complex whose cohomology admits a canonical Gerstenhaber algebra structure. We also develop the open-closed brace relations, provide a concise description of OCHAs, and establish an A-infinity structure that extends the open-closed Hochschild differential.

Paper Structure

This paper contains 13 sections, 15 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $A_\infty$ algebras
  4. $L_\infty$ algebras
  5. Open-closed brace operations and closed string actions
  6. Actions from closed strings
  7. Higher brace operations
  8. A key lemma
  9. Hochschild cohomology via OCHA
  10. A concise formulation of OCHA
  11. An open-closed analogue of Hochschild cohomology
  12. Higher structures
  13. Gerstenhaber algebra structure

Key Result

Theorem 1.1

Every open-closed homotopy algebra $(B, A, \mathfrak l, \mathfrak q)$ naturally induces a differential $\delta=\delta_{(\mathfrak l,\mathfrak q)}$ on $C^{\bullet,\bullet}(B; A, A)$. Moreover, its cohomology has a canonical Gerstenhaber algebra structure.

Theorems & Definitions (34)

  • Theorem 1.1
  • Proposition 1.2
  • Conjecture 1.3
  • Proposition 3.1
  • proof
  • Lemma 3.2: Brace Relation
  • proof
  • Lemma 3.3
  • proof
  • Remark 4.1
  • ...and 24 more