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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 …

I can answer your literal question. Not everyone studies exotic $\mathbb{R}^4$, because the universe of mathematical and theoretical physics is a big one with many interesting ideas, and there's no r …

I'm not an expert in this area, but I've heard that algebraic topologists run into congruences quite often. For example, the stable homotopy groups of spheres are almost always finite abelian groups, …

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 …

I disagree (though not particularly strongly) with the comments claiming that you should learn classical mechanics first. You don't need much physics background to learn to do basic calculations with …

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 …