# nLab A-model

## Examples

• For targets

• For targets

• For discrete targets

• For targets

• coupled to
• For targets extending the

(such as the , the )

• Chern-Simons-

• for higher abelian targets

• for targets

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• for the $L_\infty$-structure on the of the closed :

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• FQFT and

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## Surveys, textbooks and lecture notes

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• Axiomatizations

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• Tools

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• Structural phenomena

• Types of quantum field thories

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# 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 $(X,\omega)$. This is the Poisson sigma-model of the underlying Poisson manifold after appropriate gauge fixing (AKSZ 97, p 19). The A-model on $X$ 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 space $X$ 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 corresponding open string theory is given by the Fukaya?Seidel category?.

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

## Properties

### Lagrangian

The action functional of the A-model is that associated by AKSZ theory to a Lagrangian submanifold 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

### Second quantization / effective background field theory

The second quantization effective background field theory defined by the perturbation series of the A-model string has been argued to be Chern-Simons theory. (Witten 92, Costello 06)

For more on this see at TCFT – Worldsheet and effective background theories. A related mechanism is that of world sheets for world sheets.

induced via

perspective via perspective
/
$\;\;\;\;\downarrow$ on $S^4$compactificationon followed by
$\;\;\;\;\downarrow$ topological sector
$\;\;\;\;\downarrow$
on the with worldvolume theory
$\;\;\;\; \downarrow$ on on /
with invariance; worldvolume theory with type IIB
$\;\;\;\;\; \downarrow$
$\;\;\;\; \downarrow$ on
on $Bun_G$,

$\,$

induced via
$\;\;\;\;\downarrow$ on $S^5$
$\;\;\;\; \downarrow$ topological sector
$\;\;\;\;\downarrow$
$\;\;\;\;\; \downarrow$
$\;\;\;\; \downarrow$ on
on $Bun_G$ and on $Loc_G$,

## References

### General

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

That the A-model Lagrangian arises in AKSZ theory by gauge fixing the Poisson sigma-model was observed in

• M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, around page 19 in The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997

with more details in

Review and further discussion includes

Also

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

Discussion of how the second quantization effective field theory given by the A-model perturbation series is Chern-Simons theory is in

formalizing at least aspects of the observations in

Last revised on July 13, 2016 at 05:01:39. See the history of this page for a list of all contributions to it.