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Topological strings on noncommutative manifolds

Anton Kapustin

TL;DR

The paper develops a sigma-model framework in which noncommutative deformations of a Calabi–Yau X, encoded by H^2(O_X) and H^0(Λ^2 T_hol X) data, correspond to deformations of the B-brane category and interpolate between A- and B-models on hyperkähler X. It connects these deformations to generalized complex and generalized Kähler structures, showing that unequal left/right-moving complex structures naturally realize NC and twisted B-brane geometries, with localization on generalized holomorphic maps arising in the topological sigma-model. The authors establish a precise charge-constraint for topological D-branes, cast the brane geometry into GC-submanifolds, and illustrate the NC-torus case via Seiberg–Witten maps and the Fourier–Mukai transform, offering a unified picture that extends the derived-category viewpoint to noncommutative and GC geometries. The work provides a framework for understanding D-branes in noncommutative Calabi–Yau settings and points to generalized Floer-type theories and finite-α' physics as directions for future development.

Abstract

We identify a deformation of the N=2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorphic maps, whereas for the A-model and the B-model it is localized on holomorphic and constant maps, respectively. The geometry of topological D-branes is also best described using generalized complex structures. We also derive a constraint on the Chern character of topological D-branes, which includes A-branes and B-branes as special cases.

Topological strings on noncommutative manifolds

TL;DR

The paper develops a sigma-model framework in which noncommutative deformations of a Calabi–Yau X, encoded by H^2(O_X) and H^0(Λ^2 T_hol X) data, correspond to deformations of the B-brane category and interpolate between A- and B-models on hyperkähler X. It connects these deformations to generalized complex and generalized Kähler structures, showing that unequal left/right-moving complex structures naturally realize NC and twisted B-brane geometries, with localization on generalized holomorphic maps arising in the topological sigma-model. The authors establish a precise charge-constraint for topological D-branes, cast the brane geometry into GC-submanifolds, and illustrate the NC-torus case via Seiberg–Witten maps and the Fourier–Mukai transform, offering a unified picture that extends the derived-category viewpoint to noncommutative and GC geometries. The work provides a framework for understanding D-branes in noncommutative Calabi–Yau settings and points to generalized Floer-type theories and finite-α' physics as directions for future development.

Abstract

We identify a deformation of the N=2 supersymmetric sigma model on a Calabi-Yau manifold X which has the same effect on B-branes as a noncommutative deformation of X. We show that for hyperkahler X such deformations allow one to interpolate continuously between the A-model and the B-model. For generic values of the noncommutativity and the B-field, properties of the topologically twisted sigma-models can be described in terms of generalized complex structures introduced by N. Hitchin. For example, we show that the path integral for the deformed sigma-model is localized on generalized holomorphic maps, whereas for the A-model and the B-model it is localized on holomorphic and constant maps, respectively. The geometry of topological D-branes is also best described using generalized complex structures. We also derive a constraint on the Chern character of topological D-branes, which includes A-branes and B-branes as special cases.

Paper Structure

This paper contains 14 sections, 114 equations.

Table of Contents

  1. Introduction
  2. Fourier-Mukai transform and noncommutative complex tori
  3. T-duality and $N=2$ supersymmetry
  4. The proposal
  5. Uncertainty principle for topological D-branes
  6. Holomorphic line bundles on noncommutative tori as B-branes
  7. Generalized complex structures and topological D-branes
  8. Generalized complex structures
  9. Topological D-branes and generalized complex submanifolds
  10. Generalized Kähler structures and topological sigma-models
  11. Generalized Kähler structures
  12. Topological sigma-models
  13. Constraints on the charge vector of a topological D-brane
  14. Discussion