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The closed state space of affine Landau-Ginzburg B-models

Ed Segal

TL;DR

The paper develops a rigorous bridge between the Hochschild-type invariants of the category of B-branes in affine Landau-Ginzburg models and the geometric closed state-space given by differential forms with the $dW$-twisted differential. It constructs an explicit chain map and proves a quasi-isomorphism at the level of Borel-Moore Hochschild complexes, enabling a concrete realization of the closed sector from the open sector. The Kapustin-Li disc-correlator formula is recovered as the leading term of this chain map, linking worldsheet correlators to residues. The results extend to orbifolds via twisted group algebras and fixed-locus state spaces, providing a coherent framework for open-closed TCFTs in affine LG settings with and without singularities.

Abstract

We study the category of perfect cdg-modules over a curved algebra, and in particular the category of B-branes in an affine Landau-Ginzburg model. We construct an explicit chain map from the Hochschild complex of the category to the closed state space of the model, and prove that this is a quasi-isomorphism from the Borel-Moore Hochschild complex. Using the lowest-order term of our map we derive Kapustin and Li's formula for the correlator of an open-string state over a disc.

The closed state space of affine Landau-Ginzburg B-models

TL;DR

The paper develops a rigorous bridge between the Hochschild-type invariants of the category of B-branes in affine Landau-Ginzburg models and the geometric closed state-space given by differential forms with the -twisted differential. It constructs an explicit chain map and proves a quasi-isomorphism at the level of Borel-Moore Hochschild complexes, enabling a concrete realization of the closed sector from the open sector. The Kapustin-Li disc-correlator formula is recovered as the leading term of this chain map, linking worldsheet correlators to residues. The results extend to orbifolds via twisted group algebras and fixed-locus state spaces, providing a coherent framework for open-closed TCFTs in affine LG settings with and without singularities.

Abstract

We study the category of perfect cdg-modules over a curved algebra, and in particular the category of B-branes in an affine Landau-Ginzburg model. We construct an explicit chain map from the Hochschild complex of the category to the closed state space of the model, and prove that this is a quasi-isomorphism from the Borel-Moore Hochschild complex. Using the lowest-order term of our map we derive Kapustin and Li's formula for the correlator of an open-string state over a disc.

Paper Structure

This paper contains 16 sections, 7 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Curved algebras
  3. Curved algebras and cdg-modules
  4. Some non-commutative geometry
  5. Curved $A_\infty$-structures and polynomial curved $A_\infty$-structures
  6. Differential forms
  7. The infinite-dimensional case
  8. The Hochschild and Borel-Moore Hochschild complexes
  9. Curved $A_\infty$-categories
  10. Translation maps
  11. Borel-Moore Hochschild complexes of categories of cdg-modules
  12. The generalized trace map
  13. Landau-Ginzburg B-models
  14. The closed state-space
  15. The Kapustin-Li formula
  16. ...and 1 more sections

Key Result

Lemma 2.10

We have an isomorphism between the Borel-Moore Hochschild complexes of $\mathcal{P}$ and $(\tilde{\mathcal{P}}, -W)$.

Figures (1)

  • Figure 1: Correlators over discs

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Lemma 2.10
  • ...and 15 more