# Relation between Topological String Theory and Physical String Theory?

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I'm familiar with topological string theory from the mathematical perspective.  In my narrow world, the topological string partition function is given by the Gromov-Witten partition function, which is equivalent to the Donaldson-Thomas partition function via the MNOP change of variables.  Finally, these two are really equivalent to the Gopakumar-Vafa invariants under another change of variables.  These partition functions are enumerating worldsheet instantons, ideal sheaves, and D-branes, respectively.

To my understanding, topological string theory is not physically realistic for the following reason: modeling a string as a Riemann surface, means you're of course considering it moving in time.  However, in a Calabi-Yau compactification, you have a copy of the Calabi-Yau fibered at every point of spacetime. So you have a string "moving in time" yet simultaneously sitting at one point in time.  I guess the resolution is that the topological sector doesn't see time, and is physically unrealistic in this sense.

So the "physical" string theories are the Type II A & B, heterotic theories, etc.  How does topological string theory relate to these physical theories?  There's the famous "string theory moduli space diagram with the six string theories; where does the topological theory fit here?  Maybe the correct way to think of this is that each of these theories have their "topological sector" where you can compactify the theory on some Calabi-Yau threefold, and the Gromov-Witten partition function actually computes some physical observables.  Is this at all on the right track?

I'm interested in this question for its own right, but there's also a more specific reason.  I recently computed that there's a generating function of Gopakumar-Vafa invariants which precisely agrees with the generating function of a certain black hole degeneracies.  Of course, the Gopakumar-Vafa side is in topological string theory.  The black hole story is apparently, by considering "Type IIA compactified on the six-torus $T^{6}$."  Does anyone have any instincts as to why these agree?  Does it seem to people like it's most likely accidental?  (For example, due to the fact that there are only so many low weight modular and Jacobi forms)

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The topological string amplitudes agree with certain of the type II superstring amplitudes ("physical string" amplitudes). Review of this relation is in:

• Andrew Neitzke, Cumrun Vafa, section 6 of "Topological strings and their physical applications", hep-th/0410178

and lecture notes include:

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

The original articles on this are:

• 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)

and more recent developments are in

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

Regarding the relation of topological string to black hole microstates, I suppose you are aware of

Regarding the question of how this fits into the "M-theory" picture:

The answer to this is meant to be "topological M-theory", see section 7 of the above review by Neitzke-Vafa, and see

• Robbert Dijkgraaf, Sergei Gukov, Andrew Neitzke, Cumrun Vafa, "Topological M-theory as Unification of Form Theories of Gravity" (arXiv:hep-th/0411073)

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

More references on the nLab at topological string

.

answered Jun 14, 2017 by (6,015 points)

@UrsSchreiber Thanks a lot for the literature navigation; I'll try to digest some of these today!  Do you mind commenting on how the Gopakumar-Vafa papers relate to blackhole microstates?  Is there some relation between the Gopakumar-Vafa invariants and Strominger-Vafa work on black hole microstates?

@scpietromonaco You can find the references in the short review on the subject, which Urs recommended first.

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DT invariants are directly related to the physical string theory: there are indices counting supersymmetric states of some D-branes wrapped on a Calabi-Yau 3-fold, where the D-branes are dynamical objects of the physical string theory (the relation with the mathematical definition comes from the fact that the low energy effective theory on a D-brane is an ordinary super-Yang-Mills theory. Ideal sheaves are instantons in a U(1)-gauge theory over a spacetime effectively noncommutative because of a non-zero B-field).

Gopakumar-Vafa invariants are directly related to the physical string theory: they are indices in M-theory counting supersymmetric states of M2-branes wrapped on 2-cycles of a Calabi-Yau 3-fold. M-theory compactified on a Calabi-Yau 3-fold has five noncompact dimensions and M2-branes wrapped on 2-cycles looked like for large charges as spinning black holes in five dimensions (see https://arxiv.org/abs/hep-th/9910181 ) and so Gopakumar-Vafa invariants are directly related these black holes degeneracies. Remark that there is not yet a complete direct mathematical definition of Gopakumar-Vafa invariants (which is related to the physical fact that multi-M2 branes form a strongly coupled system difficult to understand).

Topological string on a Calabi-Yau 3-fold is related to physical string theory because, as recalled in Urs Schreiber answer, they compute supersymmetric term is the low energy effective action of type IIA string theory compactified on the Calabi-Yau 3-fold.

From a physics perspective, the mathematical relation between these three objects follows from relations between physical string theories. Topological string theory is related to Gopakumar-Vafa invariants because M-theory is the strong coupling of type IIA string theory. Topological string is related to DT invariants (MNOP) because of the S-duality property of type IIB string theory.

A more complicated relation between topological string and four dimensional black holes degeneracies is given by the OSV conjecture https://arxiv.org/abs/hep-th/0405146

answered Jun 16, 2017 by (5,120 points)

Thanks a lot, very helpful.  So before applying any duality, I should think of DT invariants as computing physical indices in physical Type IIB string theory?  (BTW, I think we do finally have a complete mathematical definition of Gopakumar-Vafa invariants from Maulik and Toda recently (https://arxiv.org/abs/1610.07303).  They're defined as Euler characteristics of a very exotic cohomology theory)

I agree that the paper of Maulik and Toda is the state of the art on the question of defining Gopakumar-Vafa invariants but I would not say that they give a complete answer: in general, they conjecture the existence of orientation data with nice properties and it is not known in complete generality that such orientation data exist.

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