# nLab A-model

### Context

#### $\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

What is called the A-model topological string is the 2-dimensional topological conformal field theory corresponding to the Calabi–Yau category called the Fukaya category of a symplectic manifold $X$. This is is effectively the Gromov–Witten theory of $X$.

The A-model arose in formal physics from considerations of superstring-propagation on Calabi-Yau spaces: it may be motivated by considering the vertex operator algebra of the 2dSCFT given by the supersymmetric sigma-model with target $X$ and then deforming it such that one of the super-Virasoro generators squares to $0$. The resulting “topologically twisted” algebra may then be read as being the BRST complex of a TCFT.

One can also define an A-model for Landau–Ginzburg models. The category of D-branes for the open string theory is given by the Fukaya–Seidel category.

By homological mirror symmetry, the A-model is dual to the B-model.

## Lagrangian

The action functional of the A-model is that associated by AKSZ theory to a Lagrangian sumbaifold in a target symplectic Lie n-algebroid which is the Poisson Lie algebroid of a symplectic manifold.

## Boundary theory / holography

On coisotropic branes in symplectic target manifolds that arise by complexification of phase spaces, the boundary path integral of the A-model computes the quantization of that phase space. For details see

and

## References

The A-model was first conceived in

An early review is in

The motivation from the point of view of string theory is reviewed for instance in

A summary of these two reviews is in

• H. Lee, Review of topological field theory and homological mirror symmetry (pdf)

### Action functional

Discussion of how the A-model Lagrangian arises in AKSZ theory:

around page 19 in

• M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997

section 5.3 of

Also

• Noriaki Ikeda, Deformation of graded (Batalin-Volkvisky) Structures in Dito, Lu, Maeda, Weinstein (eds.) Poisson geometry in mathematics and physics Contemp. Math. 450, AMS (2008)

Revised on December 16, 2012 17:52:10 by Urs Schreiber (71.195.68.239)