For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
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 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.
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.
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
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-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 $Log_G$, geometric Langlands correspondence |
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
Discussion of how the A-model Lagrangian arises in AKSZ theory:
around page 19 in
section 5.3 of
Also
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