nLab topological string

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Contents

Context

String theory

Ingredients

Critical string models

Extended objects

Topological strings

Backgrounds

Phenomenology

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 B→Z(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

General

Review:

On the relation to integrable systems:

On the relation to topological M-theory/the topological membrane:

On non-perturbative effects and Stokes phenomena in topological string amplitudes:

  • Murad Alim, Arpan Saha, Jörg Teschner, Iván Tulli, Mathematical structures of non-perturbative topological string theory: from GW to DT invariants (arXiv:2109.06878)

See also:

Relation to M2-branes:

On non-perturbative effects and resurgence in topological string theory:

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:

Computation via topological recursion

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

Knot invariants via topological strings and 5-branes

On realization of knot invariants/knot homology via topological string theory and BPS states:

Understanding this via NS5-branes/M5-branes:

Review:

An alternative approach:

Last revised on February 11, 2025 at 04:32:47. See the history of this page for a list of all contributions to it.

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