[Paper Review] Topological sigma-models with H-flux and twisted generalized complex manifolds

Anton Kapustin, Yi Li|ArXiv.org|Jul 28, 2004

Black Holes and Theoretical PhysicsPhysics and Astronomy12 references60 citations

TL;DR

This paper establishes that topological A and B-models in N=2 sigma-models with H-flux are governed by twisted generalized complex geometry, showing that observables arise from the cohomology of a Lie algebroid associated to one twisted generalized complex structure. The key result is that correlators depend only on one such structure, providing a geometric framework for mirror symmetry in non-Kähler settings.

ABSTRACT

We study the topological sector of N=2 sigma-models with H-flux. It has been known for a long time that the target-space geometry of these theories is not Kahler and can be described in terms of a pair of complex structures, which do not commute, in general, and are parallel with respect to two different connections with torsion. Recently an alternative description of this geometry was found, which involves a pair of commuting twisted generalized complex structures on the target space. In this paper we define and study the analogues of A and B-models for N=2 sigma-models with H-flux and show that the results are naturally expressed in the language of twisted generalized complex geometry. For example, the space of topological observables is given by the cohomology of a Lie algebroid associated to one of the two twisted generalized complex structures. We determine the topological scalar product, which endows the algebra of observables with the structure of a Frobenius algebra. We also discuss mirror symmetry for twisted generalized Calabi-Yau manifolds.

Motivation & Objective

• To extend the geometric description of topological A and B-models in N=2 sigma-models to cases with non-zero H-flux.
• To show that the space of topological observables and the topological metric depend only on one of two commuting twisted generalized complex structures on the target manifold.
• To formulate mirror symmetry for twisted generalized Calabi-Yau manifolds, even in the absence of compact examples with non-zero H-flux.
• To provide a geometric realization of Frobenius algebra structures on observables using Lie algebroid cohomology.
• To establish a correspondence between deformations of twisted generalized complex structures and Frobenius manifolds, enabling a generalization of mirror symmetry beyond Kähler geometry.

Proposed method

• The authors use the twisted Dorfman bracket on the direct sum bundle $ T M \oplus T^* M $ to define twisted generalized complex structures (TGC-structures), which generalize complex and symplectic structures under H-flux.
• They show that the target-space geometry of N=2 sigma-models with H-flux is encoded by a pair of commuting TGC-structures, arising from the left- and right-moving complex structures $ I_\pm $ and torsionful connections.
• The space of topological observables is identified with the cohomology of a Lie algebroid associated to one of the two TGC-structures, providing a geometric realization of the observable algebra.
• The topological scalar product is constructed using the H-twisted differential $ d_H $, endowing the observable algebra with a Frobenius algebra structure.
• The authors prove that the integrability of a twisted generalized almost complex structure is equivalent to the decomposition $ d_H = \partial_H + \bar{\partial}_H $, linking algebraic and geometric conditions.
• Mirror symmetry is proposed as an isomorphism between Frobenius manifolds associated to deformations of TGC-structures on mirror manifolds $ M $ and $ M' $.

Experimental results

Research questions

• RQ1 How can the A and B-models in N=2 sigma-models with H-flux be geometrically described when the target space is not Kähler?
• RQ2 What is the role of twisted generalized complex structures in encoding the topological observables and scalar product of these models?
• RQ3 Can mirror symmetry be generalized to non-Kähler, H-flux backgrounds using twisted generalized complex geometry?
• RQ4 How do the Frobenius algebra structures of observables relate to deformations of twisted generalized complex structures?
• RQ5 What are the implications of the non-existence of compact twisted generalized Calabi-Yau manifolds with non-zero H-flux for mirror symmetry constructions?

Key findings

• The space of topological observables in the A- and B-models is isomorphic to the cohomology of a Lie algebroid associated to one of the two commuting twisted generalized complex structures on the target manifold.
• The topological scalar product on the observable algebra is given by integration over the manifold, endowing the cohomology with a Frobenius algebra structure.
• The Frobenius manifold corresponding to deformations of one twisted generalized complex structure on $ M $ is isomorphic to the Frobenius manifold of deformations of the other structure on the mirror manifold $ M' $, generalizing mirror symmetry.
• The integrability of a twisted generalized almost complex structure is equivalent to the decomposition $ d_H = \partial_H + \bar{\partial}_H $, which is essential for defining the cohomology of the observable algebra.
• The paper shows that the A-model correlators depend only on the symplectic-type data encoded in one TGC-structure, while B-model correlators depend only on the complex-type data from the other, confirming the expected duality.
• Despite the absence of compact examples of twisted generalized Calabi-Yau manifolds with $ H \neq 0 $, the framework still allows for a consistent formulation of mirror symmetry in non-Kähler settings.