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Topological strings in generalized complex space

Vasily Pestun

TL;DR

This work constructs a two-dimensional topological string theory on a generalized Calabi–Yau target by deriving a BV–AKSZ formalism based on a single generalized complex structure $\mathcal{J}$ and a pure spinor $\rho$. The model's observables, extended moduli space, and deformation theory are encoded in a differential BV algebra $\mathcal{A}=C^{\infty}(\Pi L)$ with $Q=\bar{\partial}$ and $\Delta=\partial$, tied together by a trace defined from $\rho$; a symplectic realization $M$ ensures a nondegenerate BV bracket where necessary. In the closed sector, the theory yields a Frobenius-manifold structure on the extended moduli space, which in complex dimension three becomes a special Kähler manifold, with three-point functions governed by a generating potential. In the open sector, deforming the B-model by a holomorphic Poisson bivector $\beta^{ij}$ leads, through disk perturbation theory, to a holomorphic Kontsevich $*$-product on boundary observables. The framework unifies the A- and B-model viewpoints under generalized complex geometry, extends topological string theory to generalized CY spaces, and connects deformation quantization with open-string algebras in a BV–AKSZ setting.

Abstract

A two-dimensional topological sigma-model on a generalized Calabi-Yau target space $X$ is defined. The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure $J$ and a pure spinor $ρ$ on $X$. In the present construction the algebra of $Q$-transformations automatically closes off-shell, the model transparently depends only on $J$, the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N=2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector $β$ and recover holomorphic noncommutative Kontsevich $*$-product.

Topological strings in generalized complex space

TL;DR

This work constructs a two-dimensional topological string theory on a generalized Calabi–Yau target by deriving a BV–AKSZ formalism based on a single generalized complex structure and a pure spinor . The model's observables, extended moduli space, and deformation theory are encoded in a differential BV algebra with and , tied together by a trace defined from ; a symplectic realization ensures a nondegenerate BV bracket where necessary. In the closed sector, the theory yields a Frobenius-manifold structure on the extended moduli space, which in complex dimension three becomes a special Kähler manifold, with three-point functions governed by a generating potential. In the open sector, deforming the B-model by a holomorphic Poisson bivector leads, through disk perturbation theory, to a holomorphic Kontsevich -product on boundary observables. The framework unifies the A- and B-model viewpoints under generalized complex geometry, extends topological string theory to generalized CY spaces, and connects deformation quantization with open-string algebras in a BV–AKSZ setting.

Abstract

A two-dimensional topological sigma-model on a generalized Calabi-Yau target space is defined. The model is constructed in Batalin-Vilkovisky formalism using only a generalized complex structure and a pure spinor on . In the present construction the algebra of -transformations automatically closes off-shell, the model transparently depends only on , the algebra of observables and correlation functions for topologically trivial maps in genus zero are easily defined. The extended moduli space appears naturally. The familiar action of the twisted N=2 CFT can be recovered after a gauge fixing. In the open case, we consider an example of generalized deformation of complex structure by a holomorphic Poisson bivector and recover holomorphic noncommutative Kontsevich -product.

Paper Structure

This paper contains 12 sections, 97 equations.

Table of Contents

  1. Introduction
  2. A differential BV algebra $\mathcal{A}$ of a generalized CY manifold
  3. The two-dimensional topological $\mathcal{J}$-model
  4. The AKSZ construction of sigma-model for $\mathop{\mathrm{Maps}}\nolimits(\hat{\Sigma}, M)$ for a target space $M$ with a $PQ$-structure
  5. Construction of target space with $PQ$-structure for a generalized CY manifold
  6. The BV gauge fixing.
  7. Observables and deformations
  8. The extended moduli space
  9. The topologically trivial maps in genus zero
  10. The $\dim_{\mathbb C}X = 3$ case
  11. The Kontsevich $*$-product as a $(-2,0)$ deformation of the open topological B-model by a holomorphic Poisson bivector
  12. Conclusion