Timeline for How does the scalar TV invariant of a 3-manifold with boundary fit into the TQFT picture?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 3, 2020 at 12:21 | vote | accept | Calvin McPhail-Snyder | ||
| Aug 13, 2020 at 15:01 | comment | added | Calvin McPhail-Snyder | @TianYang Thanks! That also looks relevant. | |
| Aug 3, 2020 at 21:37 | comment | added | Tian Yang | I think arxiv.org/pdf/1807.03327.pdf Proposition 5.3. may answer your question. | |
| Jun 29, 2020 at 19:34 | answer | added | Calvin McPhail-Snyder | timeline score: 1 | |
| Jun 24, 2020 at 15:32 | comment | added | Calvin McPhail-Snyder | That might be the right direction: it suggests how you obtain the formulas in Theorem 1.1 of arxiv.org/pdf/1701.07818.pdf. Maybe I should read that paper more carefully: I think Section 3.2 answers at least part of my question. | |
| Jun 24, 2020 at 15:12 | comment | added | Ian Agol | Maybe for a manifold with boundary, it’s the inner product of the appropriate RT invariant with itself (in the RT vector space associated to the boundary, which may have an indefinite inner Hermitian inner product)? | |
| Jun 24, 2020 at 15:10 | comment | added | Ian Agol | Actually, what I suggested probably doesn’t work, given that TV is the square of RT. sciencedirect.com/science/article/pii/0040938394000530 | |
| Jun 24, 2020 at 13:48 | comment | added | Calvin McPhail-Snyder | Applying your comment, a manifold with boundary would be a linear functional on the vector space associated to its boundary. I suppose this is also what happens in the usual formalism, but here the vector space has a much more topological interpretation than the usual combinatorial state space of TV theory. I'm still not sure how to get a number instead of a covector in a natural, topological way. Maybe you close up the manifold in some canonical fashion? | |
| Jun 23, 2020 at 22:58 | comment | added | Ian Agol | Given a surface, one may consider the formal linear span of all the manifolds bounding it. arxiv.org/abs/math/0503054 One may pair two such manifolds together by gluing along the surface, then evaluate $TV_r$ on the closed manifold. Then $TV_r$ gives a pairing on this space, by bilinear extension. One can quotient by the null-space of this pairing, maybe this gives a finite-dimensional vector space associated to the surface? However, I think the pairing is indefinite in the Chen-Yang case. | |
| Jun 23, 2020 at 17:44 | history | edited | Calvin McPhail-Snyder | CC BY-SA 4.0 |
fix citation
|
| Jun 23, 2020 at 16:13 | history | edited | YCor |
edited tags
|
|
| Jun 23, 2020 at 15:24 | history | asked | Calvin McPhail-Snyder | CC BY-SA 4.0 |