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:

• 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)

On the relation to integrable systems:

• Mina Aganagic, Robbert Dijkgraaf, Albrecht Klemm, Marcos Marino, Cumrun Vafa, Topological Strings and Integrable Hierarchies, Commun. Math. Phys. 261 (2006) 451-516 [arXiv:hep-th/0312085, doi:10.1007/s00220-005-1448-9]

• Mina Aganagic, Miranda Cheng, Robbert Dijkgraaf, Daniel Krefl, Cumrun Vafa, Quantum Geometry of Refined Topological Strings, J. High Energ. Phys. 2012 19 (2012) [arXiv:1105.0630, doi:10.1007/JHEP11(2012)019]

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

• 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)

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

Relation to M2-branes:

• Jarod Hattab, Eran Palti: Non-perturbative topological string theory on compact Calabi-Yau manifolds from M-theory [arXiv:2408.09255]

• Jarod Hattab, Eran Palti: Emergent potentials and non-perturbative open topological strings [arXiv:2408.12302]

• Jarod Hattab, Eran Palti: On Calabi-Yau manifolds at strong topological string coupling [arXiv:2409.01721]

• Jarod Hattab, Eran Palti: Notes on integrating out M2 branes [arXiv:2410.15809]

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)

Knot invariants via topological strings and 5-branes

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

• Edward Witten, Chern-Simons gauge theory as a string theory, in: The Floer memorial volume, Progr. Math. 133, Birkhäuser (1995) 637-678 [doi:10.1007/978-3-0348-9217-9, arXiv/hep-th/9207094, MR97j:57052]

• Hirosi Ooguri, Cumrun Vafa: Knot Invariants and Topological Strings, Nucl. Phys. B 577 (2000) 419-438 [doi:10.1016/S0550-3213(00)00118-8, arXiv:hep-th/9912123]

• Sergei Gukov, Albert Schwarz, Cumrun Vafa: Khovanov-Rozansky Homology and Topological Strings, Lett. Math. Phys. 74 (2005) 53-74 [doi:10.1007/s11005-005-0008-8arXiv:hep-th/0412243]

• Sergei Gukov: Surface Operators and Knot Homologies, Fortschritte der Physik 55 5-7 (2007) 473-490 [doi:10.1002/prop.200610385, arXiv:0706.2369]

• Mina Aganagic, Cumrun Vafa, Large $N$ duality, mirror symmetry, and a Q-deformed A-polynomial for knots [arXiv:1204.4709]

Understanding this via NS5-branes/M5-branes:

• Edward Witten, Fivebranes and Knots, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 [arXiv:1101.3216, doi:10.4171/QT/26]

• Davide Gaiotto, Edward Witten, Knot Invariants from Four-Dimensional Gauge Theory, Advances in Theoretical and Mathematical Physics 16 3 (2012) [doi:10.4310/ATMP.2012.v16.n3.a5, arxiv:1106.4789]

• Edward Witten: Khovanov Homology And Gauge Theory, Geometry & Topology Monographs 18 (2012) 291-308 [pdf, arXiv:1108.3103]

• Sergei Gukov, Marko Stošić: Homological algebra of knots and BPS states, Geometry & Topology Monographs 18 (2012) 309-367 [doi:10.2140/gtm.2012.18.309, arXiv:1112.0030]

Review:

• Edward Witten, Khovanov Homology And Gauge Theory, Clay Conference, Oxford (October 2013) $[$pdf]

• Ross Elliot, Sergei Gukov: Section 1 of: Exceptional knot homology, Journal of Knot Theory and Its Ramifications 25 03 (2016) 1640003 [doi:10.1142/S0218216516400034, arXiv:1505.01635]

• Satoshi Nawata, Alexei Oblomkov: Lectures on knot homology, in: Physics and Mathematics of Link Homology, Contemp. Math. 680 (2016) 137 [doi:10.1090/conm/680, arXiv:1510.01795]

An alternative approach:

• Márk Mezei, Silviu S. Pufu, Yifan Wang: Chern-Simons theory from M5-branes and calibrated M2-branes, J. High Energ. Phys. 2019 165 (2019) [doi:10.1007/JHEP08(2019)165, arXiv:1812.07572]