topological string



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”)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)



Review includes

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

See also

Relation to black hole microstate counting

Disucssion of black holes in string theory via the topological string’ Gopakumar-Vafa invariants:

Relation to physical string amplitudes

The following includes discussion of superstring string scattering amplitudes in terms of topological string scattering amplitudes (for review see NeitzkeVafa04, section 6 and Antoniadis-Hohenegger 07:

  • M. Bershadsky, S. Cecotti, Hirosi Ooguri, Cumrun Vafa, Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes, Commun.Math.Phys.165:311-428,1994 (arXiv:hep-th/9309140)

  • I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, Topological Amplitudes in String Theory, Nucl.Phys. B413 (1994) 162-184 (arXiv:hep-th/9307158)

  • K.S. Narain, N. Piazzalunga, A. Tanzini, Real topological string amplitudes, JHEP (2017) 2017:80 (arXiv:1612.07544)

Computation via topological recursion

Computation via topological recursion in matrix models and all-genus proofs of mirror symmetry is due to

Revised on October 20, 2017 18:23:20 by Urs Schreiber (