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
for higher abelian targets
for symplectic Lie n-algebroid targets
FQFT and cohomology
Types of quantum field thories
Witten introduced two topological twists of a supersymmtric nonlinear sigma model, which is a certain N=2 superconformal field theory attached to a compact Calabi–Yau variety . One of them is the B-model topological string; it is a 2-dimensional topological N=1 superconformal field theory. In Kontsevich’s version, instead of SCFT with Hilbert space, one assembles all the needed data in terms of Calabi–Yau A-infinity-category which is the A-infinity-category of coherent sheaves on the underlying variety. In fact only the structure of a derived category is sufficient (and usually quoted): it is now known (by the results of Dmitri Orlov and Valery Lunts) that under mild assumptions on the variety, a derived category of coherent sheaves has a unique enhancement.
The B-model arose in considerations of superstring-propagation on Calabi–Yau varieties: it may be motivated by considering the vertex operator algebra of the 2dSCFT given by the N=2 supersymmetric nonlinear sigma-model with target and then changing the fields so 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 a B-model for Landau–Ginzburg models. The category of D-branes for the open string theory is given by the category of matrix factorizations (this was proposed by Kontsevich and elaborated by Kapustin-Li; see also papers of Orlov). For the genus 0 closed string theory, see the work of Saito.
The B-model was first conceived in
An early review is in
The motivation from the point of view of string theory with an eye towards the appearance of the Calabi-Yau categories is reviewed for instance in
A summary of these two reviews is in
Overflow discussion: higher-genus-closed-string-b-model