nLab
A-model

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

String theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

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 XX. This is is effectively the Gromov–Witten theory of XX.

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 XX and then deforming it such that one of the super-Virasoro generators squares to 00. 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.

See the references on Lagrangian formulation.

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.

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 4S^4compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
\;\;\;\;\downarrow topological sector
7-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
\;\;\;\; \downarrow KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
\;\;\;\;\; \downarrow topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G, Donaldson theory

\,

gauge theory induced via AdS5-CFT4
type II string theory
\;\;\;\;\downarrow Kaluza-Klein compactification on S 5S^5
\;\;\;\; \downarrow topological sector
5-dimensional Chern-Simons theory
\;\;\;\;\downarrow AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
\;\;\;\;\; \downarrow topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
\;\;\;\; \downarrow KK-compactification on Riemann surface
A-model on Bun GBun_G and B-model on Loc GLoc_G, geometric Langlands correspondence

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

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, 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

Revised on March 21, 2014 07:56:17 by Urs Schreiber (89.204.138.115)