4
$\begingroup$

I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every conformal manifold with $n$ boundary circles. This map is subject to the constraint that gluing two boundary circles of a manifold amounts to taking the trace of the two according $V$ components. By "conformal manifold", I mean an equivalence class of 2-manifolds with conformal structure, which is labelled by a finite set of real parameters for a fixed topology. Also, to make gluing unambiguous we need a base point on each boundary component.

What do all the conventional notions of CFT mean in this picture? It seems to me that the "vertex operator algebra", and the "operator product expansion" are very closely related to the "tensor" which the axiomatic CFT assigns to the pair of pants with different conformal parameters. I also understand that the conformal annuli form a $\mathbb R^+\times U(1)$ group under stacking ($\mathbb R^+$ for the with/radius ratio of the annulus, and $U(1)$ for the shift of the base point), and the two according generators are the Hamiltonian $H$ and momentum $p$ of the CFT. Is there any way to see that if $V$ is finite, the axiomatic CFT is actually an axiomatic TQFT? Also is there any way to derive that the spectrum of $H$ consists of these infinite towers? In other words, how do I arrive at the Virasoro algebra starting from the axiomatic CFT? Is there any example for which the axiomatic-CFT tensors assigned to different conformal manifolds have been worked out explicitly? Is there an explicit generators-and-relations presentation of the conformal cobordism category? Is it even true that all the properties of conventional CFT follow from only the axiomatic-CFT axioms? If this is not rigorously known yet, what do people believe is the case?

$\endgroup$
9
  • $\begingroup$ 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. $\endgroup$ Oct 6, 2022 at 18:09
  • $\begingroup$ 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 $\endgroup$ Oct 6, 2022 at 22:00
  • $\begingroup$ @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. $\endgroup$
    – Andi Bauer
    Oct 7, 2022 at 3:22
  • 3
    $\begingroup$ @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?> $\endgroup$ Oct 7, 2022 at 9:27
  • 1
    $\begingroup$ @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. $\endgroup$ Oct 9, 2022 at 15:00

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.