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D-Branes in Landau-Ginzburg Models and Algebraic Geometry

Anton Kapustin, Yi Li

TL;DR

The paper provides substantial evidence for Kontsevich's proposal that B-branes in massive N=2 Landau-Ginzburg models are described by twisted complexes, which in practice reduce to modules over Clifford algebras. By explicit analysis of LG models with quadratic superpotentials, it shows that the endomorphism algebras of branes are Clifford algebras and that the full B-brane category is controlled by a Clifford-module/GKoszul-duality framework, enabling concrete computations of open-string spectra. It also connects these LG constructions to Homological Mirror Symmetry, computing the A-brane categories for certain Fano varieties (e.g., CP^2 and CP^1×CP^1) via their LG mirrors and discussing the corresponding Floer-theoretic objects. The work highlights both the power and subtlety of the Kontsevich proposal, including a mild counterexample to naive uniqueness of D-brane sets, and argues for a broader program to classify D-branes in non-Calabi–Yau backgrounds through algebraic dualities.

Abstract

We study topological D-branes of type B in N=2 Landau-Ginzburg models, focusing on the case where all vacua have a mass gap. In general, tree-level topological string theory in the presence of topological D-branes is described mathematically in terms of a triangulated category. For example, it has been argued that B-branes for an N=2 sigma-model with a Calabi-Yau target space are described by the derived category of coherent sheaves on this space. M. Kontsevich previously proposed a candidate category for B-branes in N=2 Landau-Ginzburg models, and our computations confirm this proposal. We also give a heuristic physical derivation of the proposal. Assuming its validity, we can completely describe the category of B-branes in an arbitrary massive Landau-Ginzburg model in terms of modules over a Clifford algebra. Assuming in addition Homological Mirror Symmetry, our results enable one to compute the Fukaya category for a large class of Fano varieties. We also provide a (somewhat trivial) counter-example to the hypothesis that given a closed string background there is a unique set of D-branes consistent with it.

D-Branes in Landau-Ginzburg Models and Algebraic Geometry

TL;DR

The paper provides substantial evidence for Kontsevich's proposal that B-branes in massive N=2 Landau-Ginzburg models are described by twisted complexes, which in practice reduce to modules over Clifford algebras. By explicit analysis of LG models with quadratic superpotentials, it shows that the endomorphism algebras of branes are Clifford algebras and that the full B-brane category is controlled by a Clifford-module/GKoszul-duality framework, enabling concrete computations of open-string spectra. It also connects these LG constructions to Homological Mirror Symmetry, computing the A-brane categories for certain Fano varieties (e.g., CP^2 and CP^1×CP^1) via their LG mirrors and discussing the corresponding Floer-theoretic objects. The work highlights both the power and subtlety of the Kontsevich proposal, including a mild counterexample to naive uniqueness of D-brane sets, and argues for a broader program to classify D-branes in non-Calabi–Yau backgrounds through algebraic dualities.

Abstract

We study topological D-branes of type B in N=2 Landau-Ginzburg models, focusing on the case where all vacua have a mass gap. In general, tree-level topological string theory in the presence of topological D-branes is described mathematically in terms of a triangulated category. For example, it has been argued that B-branes for an N=2 sigma-model with a Calabi-Yau target space are described by the derived category of coherent sheaves on this space. M. Kontsevich previously proposed a candidate category for B-branes in N=2 Landau-Ginzburg models, and our computations confirm this proposal. We also give a heuristic physical derivation of the proposal. Assuming its validity, we can completely describe the category of B-branes in an arbitrary massive Landau-Ginzburg model in terms of modules over a Clifford algebra. Assuming in addition Homological Mirror Symmetry, our results enable one to compute the Fukaya category for a large class of Fano varieties. We also provide a (somewhat trivial) counter-example to the hypothesis that given a closed string background there is a unique set of D-branes consistent with it.

Paper Structure

This paper contains 27 sections, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mirror Symmetry for Fano varieties and LG models
  3. General properties of B-branes in LG models
  4. B-Branes in the LG model with $W=xy$
  5. Preliminaries
  6. Equations of motion and SUSY transformations
  7. Spectrum of D0-D0 strings
  8. Spectrum of D2-D2 strings
  9. Spectrum of D0-D2 strings
  10. Topological correlators in the LG model $W=xy$
  11. Topological B-twist
  12. Boundary Observables Associated with the D0-brane
  13. Boundary Observables Associated with the D2-brane
  14. The Boundary Operator Product Algebra
  15. B-branes in general massive LG models
  16. ...and 12 more sections

Figures (2)

  • Figure 1:
  • Figure 2: