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I suspect the expected structure of $Rep(V)$ common to all vertex algebras $V$ is something like "abelian pseudomonoidal category" and I don't think you can say much else. The abelian structure follo …

The terminology is explained earlier on that page and the previous page in the paper. On the same page, we see that they set $\mathfrak{g} = \mathfrak{su}(2) \times \mathfrak{su}(2)$, and let the suba …

I expect there will never be a classification of simple VOAs, unless perhaps one is only sorting according to very rough criteria. This is because there are too many of them - even the rational case …

Let $\mathfrak{g}$ be a complex Lie algebra with a distinguished nonzero central element $x$, and let $V$ be an irreducible representation of $\mathfrak{g}$. The usual proof of Schur's lemma can be a …

I can't address all uses by all physicists, but in many contexts, they consider only representations at a fixed level that admit a well-behaved energy grading. That is, sometimes an energy grading is …

Section 5 of Borcherds, Barnard, Lectures on Quantum Field Theory is a discussion of the 0-dimensional spacetime case, which gives finite dimensional integrals.

The Feigin-Frenkel isomorphism is globalized by the global quantum geometric Langlands conjecture, proposed by Stoyanovsky, and refined by Gaitsgory and his collaborators. See Gaitsgory's 2016 collec …

The answer to your question is Yes. Consider your covering group $C$ as a central extension: $$1 \to N \to C \to G \to 1$$ and suppose it is given by a 2-cocycle $\alpha \in H^2(G, N)$. Then for any …

I don't know what paper you are reading, but you can find examples in most textbooks. For example, Frenkel and Ben-Zvi's book "Vertex algebras and algebraic curves" has a treatment of modules in chap …

In general, we expect field theories to be described by some higher categorical structures, where bulk models are assigned objects (also called 0-morphisms), domain walls are assigned morphisms (also …

For the case of vector spaces graded by an abelian group (with braiding determined by an abelian 3-cocycle following Joyal-Street), this was done by Dong and Lepowsky in their 1993 book "Generalized V …

If your VOA $V$ is not rational, then it is quite unlikely that its category of representations is a modular tensor category. That is, you can safely conclude that Theorem 3 contains an unstated assu …

I'm not an expert on black holes, but I can give you a couple pointers. From work of Bekenstein and Hawking in the 1970s, we are pretty sure that macroscopic black holes in our 3+1 dimensional univer …

I think the suggested example is not a good fit for illustrating a tensor product decomposition, because $L^2$ functions on an interval are most naturally identified with states of a single particle i …

The half-edges incident to a 4-valent vertex can be labeled 1,2,3,4. There are three ways to split them into pairs: 12-34, 13-24, and 14-23. Another way to think of the three pairings is by consider …