Timeline for Seiberg-Witten theory in 4d is categorification of Seiberg-Witten in 3d
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 25, 2017 at 16:46 | comment | added | Chris Gerig | Gukov is probably using "categorification" in a vague way. | |
| Mar 25, 2017 at 16:38 | comment | added | Chris Gerig | The "SW invariant" I refer to is an integer, and it is obtained from a closed 4-manifold (cobordism with empty ends) by stretching out two balls, so you get a cobordism from 3-sphere to 3-sphere and hence a map on homologies which (roughly speaking) is $\mathbb{Z}\to\mathbb{Z}$, where the image of 1 is the SW invariant. | |
| Mar 25, 2017 at 9:40 | comment | added | Gorbz | Hi, thanks. I know that book. My confusion is in what sense the "cobordism" is equal to the SW invariants in 4d. These are polynomial invariants. On the other hand homologies are vector spaces. How are polynomial invariants categorification of a vector space, I thought that that should be a 2 category! | |
| Mar 25, 2017 at 3:26 | comment | added | Chris Gerig | Cobordism = 4-dimensional manifold with 3-manifold boundaries. Associated to each 3-manifold is the Seiberg-Witten-Floer homology, and a cobordism induces maps between those homologies. Definitive reference: "Monopoles and 3-manifolds" by Kronheimer-Mrowka. | |
| Mar 25, 2017 at 3:11 | comment | added | Gorbz | @ChrisGerig Firstly homology should be categorified to a 2-category if I am correct. Secondly physically what is this cobordism in 4d? Any references? | |
| Mar 25, 2017 at 2:44 | comment | added | Chris Gerig | It's just the "TQFT framework" where in 3 dimensions you have homologies (generated by solutions to 3-dimensional SW equations), and in 4-dimensions you have cobordism maps between homologies (by counting solutions to 4-dimensional SW equations that are asymptotic to the 3-dimensional generators), and when you take the cobordism to have empty boundary (i.e. a closed 4-manifold) you get the numerical SW-invariant (roughly speaking). | |
| Mar 24, 2017 at 23:15 | history | asked | Gorbz | CC BY-SA 3.0 |