In the broad sense of the word, a topological string is a 2-dimensional TQFT. In its refined form this goes by the name TCFT. The “C” standing for conformal field theory points to what historically was the main inspiration and still is the default meaning of topological strings: the A-model and B-model 2d TQFTs, which are each obtained by a “topological twisting” of 2d SCFTs.
Accordingly, much of “physical” string theory has its analogs in topological string theory. Notably the toplogical analogs of the D-branes of the physical string – the A-branes and B-branes – have been studied in great (mathematical) detail, giving rise to homological mirror symmetry and, eventually, the notion of TCFT itself.
Also the perspective of string theory as the dimensional reduction of a conjectured UV-completion of 11-dimensional supergravity – “M-theory” – has its analog for topological strings, going, accordingly, by the term topological M-theory.
|2d TQFT (“TCFT”)||coefficients||algebra structure on space of quantum states|
|open topological string||Vect||Frobenius algebra||folklore+(Abrams 96)|
|open topological string with closed string bulk theory||Vect||Frobenius algebra with trace map and Cardy condition||(Lazaroiu 00, Moore-Segal 02)|
|non-compact open topological string||Ch(Vect)||Calabi-Yau A-∞ algebra||(Kontsevich 95, Costello 04)|
|non-compact open topological string with various D-branes||Ch(Vect)||Calabi-Yau A-∞ category||“|
|non-compact open topological string with various D-branes and with closed string bulk sector||Ch(Vect)||Calabi-Yau A-∞ category with Hochschild cohomology||“|
|local closed topological string||2Mod(Vect) over field||separable symmetric Frobenius algebras||(SchommerPries 11)|
|non-compact local closed topological string||2Mod(Ch(Vect))||Calabi-Yau A-∞ algebra||(Lurie 09, section 4.2)|
|non-compact local closed topological string||2Mod for a symmetric monoidal (∞,1)-category||Calabi-Yau object in||(Lurie 09, section 4.2)|
wikipedia topological string theory