nLab
A-model

Context

\infty-Chern-Simons theory

Ingredients

Definition

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

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

Quantum field theory

Contents

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  • FQFT and

String theory

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Critical string models

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Extended objects

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Topological strings

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Backgrounds

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Phenomenology

Physics

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Surveys, textbooks and lecture notes

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    • Axiomatizations

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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,ω)(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 XX 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.

induced via

perspective via perspective
/
\;\;\;\;\downarrow on S 4S^4compactificationon 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 GBun_G,

\,

induced via
\;\;\;\;\downarrow on S 5S^5
\;\;\;\; \downarrow topological sector
\;\;\;\;\downarrow
\;\;\;\;\; \downarrow
\;\;\;\; \downarrow on
on Bun GBun_G and on Loc GLoc_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.