nLab A-model

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.

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

• quantization via the A-model.

and

• holographic principle.

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.

Related concepts

• ∞-Chern-Simons theory

• sigma-model

  • AKSZ sigma-model

    • Poisson sigma-model

      • A-model, B-model

      • half-twisted model

    • Courant sigma-model

      • Chern-Simons theory

  • topological membrane

• topologically twisted D=4 super Yang-Mills theory

• Landau-Ginzburg model

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6	F-theory perspective
11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$	compactificationon 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 invariance	M5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface	double dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondence	D3-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 $Bun_G$, Donaldson theory

$\,$

gauge theory induced via AdS5-CFT4
type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ 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 $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence

References

General

The A-model was first conceived in

• Edward Witten, Topological sigma models, Commun. Math. Phys. 118 (1988) 411–449, euclid, MR90b:81080

An early review is in

• Edward Witten. Mirror manifolds and topological field theory, in: Essays on mirror manifolds, pp. 120–-158. Int. Press, Hong Kong, 1992. (arXiv:hep-th/9112056).

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

• Paul Aspinwall, D-Branes on Calabi-Yau Manifolds (arXiv:hep-th/0403166)

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

• Francesco Bonechi, Alberto Cattaneo, Riccardo Iraso, Comparing Poisson Sigma Model with A-model (arXiv:1607.03411)

Review and further discussion includes

• Francesco Bonechi, Maxim Zabzine, section 5.3 of Poisson sigma model on the sphere (arXiv:0706.3164)

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

• Edward Witten, Chern-Simons Gauge Theory As A String Theory, Prog.Math. 133 (1995) 637-678 (arXiv:hep-th/9207094)

• Kevin Costello, Topological conformal field theories and gauge theories (arXiv:math/0605647)

formalizing at least aspects of the observations in