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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