Timeline for What are "branes", and why do they form a category?
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12 events
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| Aug 24, 2022 at 2:15 | comment | added | David Ben-Zvi | @JakeMcNamara yes that's right, the $A_\infty$-structure only uses disks. The full Gromov-Witten potential / topological string partition function is of course extremely interesting but I don't know of a purely "algebraic/categorical" role for it | |
| Aug 23, 2022 at 16:41 | comment | added | Jake McNamara | Maybe this is my confusion: does the Floer differential/composition include contributions from surfaces of any topology, or only strips/disks? I would expect the TQFT category of boundary conditions to only include strips/disks, while the string amplitudes would include manifolds of any topology. It's been a while since I thought about the math of this :) | |
| Aug 23, 2022 at 16:40 | comment | added | Jake McNamara | @DavidBen-Zvi Ah, I see your point! There are indeed two distinct choices of BRST charge, according to whether or not you couple to 2d gravity. I guess your point is that for a TQFT, the composition of boundary conditions is actually (homotopy) associative, while for a non-topological QFT I would expect something like an OPE composition law (for the identity boundary condition, this is just the normal OPE of codimension-2 operators). | |
| Aug 23, 2022 at 15:42 | vote | accept | Anton Hilado | ||
| Aug 23, 2022 at 15:07 | comment | added | David Ben-Zvi | The $A_\infty$ or dg structure of the category of branes is again a general feature of twisted SUSY theories (and part of the "derived" form of TQFT as in Costello, Lurie et al). But we're not coupling to 2d gravity / integrating over moduli spaces of curves -- or more concretely, we're not taking "string states" (BRST cohomology for the Virasoro algebra / killing the loop rotation symmetry / passing from Hochschild to cyclic homology/ ..) which would take us beyond the realm of QFT. | |
| Aug 23, 2022 at 15:04 | comment | added | David Ben-Zvi | @JakeMcNamara I don't think string theory is relevant for the structure of the category of branes or the KW story (though of course it's wonderful to have outside tools+historical perspective). The BRST differential, as for other twisted SUSY theories, enforces the homotopy trivialization of the stress tensor, which is why we get a TQFT. | |
| Aug 23, 2022 at 12:13 | comment | added | Jake McNamara | @DavidBen-Zvi I certainly agree that the Atiyah-Floer story is happening in 4d/2d SUSY or topological QFT, but isn't there a small subtlety with the category of branes? In particular, aren't the $n$-ary composition maps defined by string amplitudes, not correlation functions? I.e., for the SUSY QFT, one must integrate at least over moduli, while for the TQFT, I would expect to need to sum over topologies (perhaps the higher genus contributions vanish?). Relatedly, the BRST differential is encoding the enforcement of worldsheet diffeomorphism as a gauge symmetry. | |
| Aug 23, 2022 at 9:11 | comment | added | Severin Bunk | @მამუკაჯიბლაძე I think a standard reference is the book "Dirichlet branes and mirror symmetry" edited by Douglas and Gross. For something shorter, there are also Paul Aspinwall's notes "D-Branes on Calabi-Yau manifolds". | |
| Aug 23, 2022 at 6:51 | comment | added | მამუკა ჯიბლაძე | What would be a standard reference for this description of the category of branes? | |
| Aug 23, 2022 at 2:39 | comment | added | David Ben-Zvi | Thanks for the great explanations and welcome to MO! I would argue that Kapustin-Witten is happening purely in 4d QFT (supersymmetric or topological), and the story being discussed is happening purely in 2d (SUSY or T)QFT. "Branes" here (perhaps a misnomer, but a standard one) just means (supersymmetric or topological) boundary conditions for 2d field theory rather than the (closely related) fancier and wilder beasts that live in string theory.. though again your answer explains nicely why such objects should for categories and the relevant ones. | |
| S Aug 22, 2022 at 23:49 | review | First answers | |||
| Aug 23, 2022 at 0:35 | |||||
| S Aug 22, 2022 at 23:49 | history | answered | Jake McNamara | CC BY-SA 4.0 |