nLab topological string

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”)	coefficients	algebra structure on space of quantum states
open topological string	Vect${}_k$	Frobenius algebra $A$	folklore+(Abrams 96)
open topological string with closed string bulk theory	Vect${}_k$	Frobenius algebra $A$ with trace map $B \to Z(A)$ 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${}_k$) over field $k$	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$(\mathbf{S})$ for a symmetric monoidal (∞,1)-category $\mathbf{S}$	Calabi-Yau object in $\mathbf{S}$	(Lurie 09, section 4.2)

Related concepts

• TCFT

  • A-model

  • B-model

• CP^N sigma-model

• quantum cohomology ring

• topological M-theory, topological membrane

References

General

Review includes

• Andrew Neitzke, Cumrun Vafa, Topological strings and their physical applications, talk at Simons Workshop in Mathematics and Physics 2004 (hep-th/0410178)

• I. Antoniadis, S. Hohenegger, Topological Amplitudes and Physical Couplings in String Theory, Nucl.Phys.Proc.Suppl.171:176-195,2007 (arXiv:hep-th/0701290)

• Marcel Vonk, A mini-course on topological strings (arXiv:hep-th/0504147)

• Andrew Neitzke, Nonperturbative topological strings, 2005 (pdf)

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

• Jan de Boer, Paul de Medeiros, Sheer El-Showk, Annamaria Sinkovics, Open $G_2$ Strings (arXiv:hep-th/0611080)

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:

• Wikipedia topological string theory

• Lotte Hollands, Topological Strings and Quantum Curves (arXiv:0911.3413)

• Mina Aganagić, Cumrun Vafa, Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots, arxiv/1204.4709

• Min-xin Huang, Recent Developments in Topological String Theory (arXiv:1812.03636)

Relation to black hole microstate counting

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

• Rajesh Gopakumar, Cumrun Vafa, M-Theory and Topological Strings–I (arXiv:hep-th/9809187)

• Rajesh Gopakumar, Cumrun Vafa, M-Theory and Topological Strings–II (arXiv:hep-th/9812127)

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:

• Michael Bershadsky, Sergio Cecotti, Hirosi Ooguri, Cumrun Vafa, Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes, Commun. Math. Phys. 165 (1994) 311-428 [arXiv:hep-th/9309140, doi:10.1007/BF02099774]

• 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

• Vincent Bouchard, Albrecht Klemm, Marcos Marino, Sara Pasquetti, Remodeling the B-model, Commun.Math.Phys.287:117-178, 2009 (arXiv:0709.1453)

• Bertrand Eynard, Amir-Kian Kashani-Poor, Olivier Marchal, A matrix model for the topological string I: Deriving the matrix model (arXiv:1003.1737)

• Bertrand Eynard, Amir-Kian Kashani-Poor, Olivier Marchal, A matrix model for the topological string II: The spectral curve and mirror geometry (arXiv:1007.2194)

• Bertrand Eynard, Nicolas Orantin, Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture (arXiv:1205.1103)

• Bohan Fang, Chiu-Chu Melissa Liu, Zhengyu Zong, All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds (arXiv:1310.4818)