Timeline for CFT as an axiomatic field theory
Current License: CC BY-SA 4.0
10 events
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| Oct 9, 2022 at 15:00 | comment | added | André Henriques | @AndiBauer The situation is not the same for full CFT. For full CFT, the Weyl anomaly well-understood: there is a central extension of the 2-dim complex cobordism category by $\mathbb R$, given by the Liouville action functional, and a full CFT is a symmetric monoidal functor out of this centrally extended 2-dim complex cobordism category. For chiral CFTs, all the things that you mention in your remark are things I've thought about for many years (and talked about to all the experts I could think of), but I still don't know how to fit them together. | |
| Oct 8, 2022 at 21:53 | comment | added | Andi Bauer | Is the situation the same for full CFT? Or do there exist precise worked-out examples? | |
| Oct 8, 2022 at 21:51 | comment | added | Andi Bauer | @AndréHenriques Why would it not just be a topological 3-manifold with conformal structure only at its boundary? I mean apart from the fact that for a UMTC with non-zero chiral central charge, one needs either a $p_1$-structure on the 3-manifold, or a 4-dimensional "anomaly bulk" which only matters up to cobordism. In the latter case the 2-dimensional conformal boundary would need a 3-dimensional anomaly bulk as well and I'm unsure how the functorial TQFT depends on this anomaly bulk. Is this vaguely related to what the problem is, or has just nobody managed yet to work out a full example? | |
| Oct 7, 2022 at 9:27 | comment | added | André Henriques | @AndiBauer: ("AQFT" usually refers to the Haag-Kastler approach – I prefer "Functorial QFT".) The statement that chiral CFT is a Functorial QFT with a 3D bulk RT theory, and a ∂ with conformal structure is well-accepted in the community. But note that I do not know how to make it precise, and neither do any of the experts I've talked to (and I've talked to many!). Specifically, the question I don't know the answer to is <With what (local) geometric structure must a 3-manifold with boundary be equipped in order to support 3d Chern-Simons theory in its bulk, and chiral WZW on its boundary?> | |
| Oct 7, 2022 at 3:27 | comment | added | Andi Bauer | Thanks for the reference, I'll look into it! | |
| Oct 7, 2022 at 3:27 | comment | added | Andi Bauer | So which of the terms I mentioned are more related to chiral CFT? Vertex operator algebra probably? I thought the OPE is for the full CFT, but maybe not? I thought the Virasoro algebra was for the full CFT since it combines the holomorphic and anti-holomorphic part, is that wrong? I really don't know too much about "conventional" CFT. | |
| Oct 7, 2022 at 3:22 | comment | added | Andi Bauer | @AndréHenriques I am aware of the difference between full and chiral CFT. In my understanding, also chiral CFT can be formulated as an AQFT that is a combination of a 3-dimensional bulk Reshetikhin-Turaev 3-2-1-extended TQFT and a boundary with conformal structure, where the bulk ribbons can terminate at the boundary. A full CFT arises from a chiral one by pulling back the cartesian product with the interval, i.e., considering a thin layer of bulk between two boundaries. If you can describe how the many structures of "conventional" (chiral) CFT fit into this pictures, I'd also be very happy. | |
| Oct 6, 2022 at 22:00 | comment | added | André Henriques | I'm not sure that you're aware that "Full CFT" and "Chiral CFT" are two distinct notions. The stuff you that describe in your first paragraph sounds like full CFT, but the stuff that you discuss later sounds like chiral CFT. For more details, you may check out the beginning of my course notes: andreghenriques.com/Teaching/CFT-2020.pdf | |
| Oct 6, 2022 at 18:09 | comment | added | Abdelmalek Abdesselam | I don't think there is an abundance of worked out examples. For Liouville theory see arxiv.org/abs/2112.14859 although this is as far from the TQFT, finite-dimensional $V$ situation as possible. | |
| Oct 6, 2022 at 17:34 | history | asked | Andi Bauer | CC BY-SA 4.0 |