nLab
topological string

Contents

Idea

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.

Classification

2d TQFT (“TCFT”)coefficientsalgebra structure on space of quantum states
open topological stringVect k{}_kFrobenius algebra AAfolklore+(Abrams 96)
open topological string with closed string bulk theoryVect k{}_kFrobenius algebra AA with trace map BZ(A)B \to Z(A) and Cardy condition(Lazaroiu 00, Moore-Segal 02)
non-compact open topological stringCh(Vect)Calabi-Yau A-∞ algebra(Kontsevich 95, Costello 04)
non-compact open topological string with various D-branesCh(Vect)Calabi-Yau A-∞ category
non-compact open topological string with various D-branes and with closed string bulk sectorCh(Vect)Calabi-Yau A-∞ category with Hochschild cohomology
local closed topological string2Mod(Vect k{}_k) over field kkseparable symmetric Frobenius algebras(SchommerPries 11)
non-compact local closed topological string2Mod(Ch(Vect))Calabi-Yau A-∞ algebra(Lurie 09, section 4.2)
non-compact local closed topological string2Mod(S)(\mathbf{S}) for a symmetric monoidal (∞,1)-category S\mathbf{S}Calabi-Yau object in S\mathbf{S}(Lurie 09, section 4.2)

References

The relation to topological M-theory/the topological membrane is discussed for instance in

Revised on November 17, 2014 15:36:23 by Urs Schreiber (82.113.98.106)