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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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    $\begingroup$ (minor nitpick to this interesting question) Six string theories? Have you discovered a new one beyond type I, IIa, IIb, heterotic-O(32),heterotic-$E_8\times E_8$? (note that M-theory is not strictly speaking a string theory, since it has only 2- and 5-branes, but no fundamental 1-branes/strings) $\endgroup$ – ACuriousMind Jun 13 '17 at 10:40
  • $\begingroup$ @ACuriousMind Ah I see, does 11d SUGRA not count? I had thought this was some limit of M-theory where when the one of the dimensions contracts, the M2-branes appear as string and you get a string theory? And the low energy effective action is some SUGRA. Maybe that is totally wrong. $\endgroup$ – Benighted Jun 13 '17 at 16:45
  • $\begingroup$ Yes, you get a String theory, that is IIA. $\endgroup$ – Rexcirus Jun 15 '17 at 10:13
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The crucial idea to understand why topological string theory cannot be in itself a fundamental description of nature is because, by construction, all the operators of the theory (including the energy momentum tensor) are BRST exact.

The immediate consequence of the latter fact is that the theory does not support any propagating degree of freedom in its spectrum. For example; gravitational waves are absent because the graviton vertex operator is BRST exact. That does not imply that your'e unable to change the target manifold or that gravity or gauge fields are unimportant in the theory, quite the opposite, the theory is dynamical, the background moduli can be varied by adding exactly marginal operators into the worldsheet path integral measure (mathematically speaking: modifying the vacuum to the extent of preserving derived equivalence of derived categories of coherent sheaves), gravity and gauge fields impose important constrains (like the Calabi-Yau one) and produce an impressively rich instanton dynamics without analogues in physical string theory. Examples are the "quantum spacetime foam" definition of the theory (DT-GW equivalence) or the reinterpretation of the full perturbative expansions in terms of D-instanton expansions.

Nevertheless, topological string is physically relevant. For an entire list of beautiful applications see https://www.youtube.com/watch?v=rilO7U6OyoM or read this.

Intuitively, you can think of the topological string as computing instanton degeneracies in local Calabi-Yau. Sometimes the latter is powerful enough to rewritte the entire BPS degeneracies of some 4d and 5d black holes, to exactly compute F/D terms in N=2 supergravities or one-loop corrections to gauge couplings on heterotic backgrounds (see pag. 131), the famous reinterpretation of the the self-dual part of N=4 SYM in 4d https://arxiv.org/abs/hep-th/0312171 and more recently, to compute indices of some objects(M/E-strings) relevant for the dynamics of six dimensional SCFTs (see https://www.youtube.com/watch?v=3uNxzZoCA-w or this overview.

What is the physical relationship between physical and topological string theory? the intuitive answer is that the latter is computing some local instanton degeneracies coming from physical string theories compactified on Calabi-Yau spaces.

Perhaps is worth to mention that the holy grail of the topological string theory is to find a description that computes the the entire instanton degeneracies of say M2/M5 branes (in the GV understanding of the theory) in the case of a compact Calabi-Yau. That's one of the biggest dreams of most topological string theorists :) For more information see https://www.youtube.com/watch?v=Fmri3ke8Q-g.

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    $\begingroup$ Thanks, this is a great answer to an old question I'm still interested in! I'm intrigued by your last paragraph: in the world of enumerative geometry, you might say the holy grail is to compute the entire GW/DT/GV partition function for a compact Calabi-Yau threefold in all of its Kahler classes. Right now, all known partition functions only include a subset of the Kahler classes, except for the non-compact local geometries. Is this more or less the mathematical counterpart to the physics holy grail you describe in the last paragraph? $\endgroup$ – Benighted May 20 '20 at 23:06
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    $\begingroup$ It was a pleasure. I'm happy that the answer was useful. And yes, the way you describe my last paragraph from the mathematical perspective is absolutely correct. I couldn't have described it more clearly. $\endgroup$ – Ramiro Hum-Sah May 20 '20 at 23:32
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    $\begingroup$ This is cool! Don't physicists have an ansatz for any compact CY3 $X$? Maybe you could say if my understanding is roughly correct: the genus $g$ Gromov-Witten potentials $F_{g}(X)$ are expected to be sections of a degree $2g-2$ line bundle over the stringy Kahler moduli space (or by the mirror map, the moduli space of complex structures on the mirror). In this sense, the $F_{g}(X)$ are behaving at least in spirit like "generalized automorphic forms." This is something I've seen in my work lately. Is this more or less the story? I think from BCOV? $\endgroup$ – Benighted May 21 '20 at 3:31
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    $\begingroup$ Do physicist have an ansatz? Yes, recently a physical ansatz has been raised. The relevant paper is this and you and you can watch this talk for an overview of the progress (the anatz is shown in 17:52). From the mathematical perspective: Your intuition is bright. From the mathematical perspective: Your intuition is bright. An analogy of why you are on the right track: Topological string theory can be reformulated as a maximally supersymmetric U(1) gauge theory $\endgroup$ – Ramiro Hum-Sah May 21 '20 at 20:55
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    $\begingroup$ ... with the usage of the GW/DT correspondence. The problem can be summarized as our lack understanding on how to integrate over the moduli space of ideal sheaves of compact Calabi-Yau threefolds. But, famously Vafa and Witten have computed the partition function of the maximally supersymmetric (N=2) gauge theory for CY twofolds; In solving the theory, S-duality was crucial. $\endgroup$ – Ramiro Hum-Sah May 21 '20 at 21:06

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