26 votes

$H^4$ of the Monster

In arXiv:1707.08388, I calculate that the cohomology class you described has order 24 and that it is not a characteristic class in the ordinary sense.
18 votes

What's the supersymmetric analogue of the Monster group?

There is a super analog constructed just as you describe with the Conway group $Co_0$ replacing the Monster and commuting with the superconformal algebra. The construction is described in detail in: ...
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16 votes
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71, the Monster, and c = 24 CFTs

Schellekens' enumeration is exhaustive in the following sense: the degree 1 subspace of the meromorphic CFT/vertex algebra is naturally a Lie algebra, and it is known that this Lie algebra must be one ...
  • 176
10 votes

Why "holomorphic" vertex algebra?

Those in a hurry can skip to the bottom. Those with some leisure time can read the details. To explain the answer, it is helpful to remember what vertex algebras are. Suppose you have a quantum field ...
10 votes
Accepted

Why are VOA characters modular forms (geometrically)?

I'm sure someone here can handwave the intuition behind these statements much better than me. This handwaving version goes like this, one is interested in computing vacuum 1-point functions on a torus,...
9 votes

What are advantages of chiral algebras over vertex algebras?

Some comments: It is not necessarily true that chiral algebras are essentially conformal vertex algebras, as chiral algebras are allowed to vary over the curve in a way that vertex algebras are not. ...
  • 5,586
9 votes

Chiral homology for the Virasoro algebra and/or affine Lie algebra

There isn't much written up on how to compute the higher chiral homology of vertex algebras. Besides the original work of Beilinson and Drinfeld that covers in particular the universal cases of ...
8 votes
Accepted

What are braided vertex algebras?

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 ...
  • 43.4k
8 votes
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Factorization and vertex algebra cohomology

If $X$ is taken to a be a formal disc $D$, then we obtain a vertex algebra. I believe in this case the above construction produces the vertex algebra cohomology (with coefficients in the adjoint ...
7 votes
Accepted

Do we have a braided tensor category for vertex algebra modules by using conformal blocks on an arbitary compact Riemann Surface?

In general, you won't get a vertex tensor category, because you don't get well-defined unit behavior when you use conformal blocks on higher genus surfaces. Huang-Lepowsky assume the vertex operator ...
  • 43.4k
7 votes

The proof that a vertex algebra can lead to a Wightman QFT

You might find "An Introduction to Conformal Field Theory" by M. Gaberdiel (arXiv:hep-th/9910156v2) useful. He has a brief discussion of how in some cases chiral algebras can be assembled into ...
  • 5,161
6 votes

$\text{Rep}(D(G))$ as representation category of a vertex operator algebra

A lot has happened in the last four years, and we now have lots of positive results. The current state of knowledge is given in Evans-Gannon, "Reconstruction and Local Extensions for Twisted Group ...
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6 votes
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q-Virasoro and q-Heisenberg algebras

The main sources are Awata et al or Frenkel-Reshetikhin. In http://arxiv.org/pdf/q-alg/9507034v5.pdf section 4, you can see the q,t case. You can also look at http://arxiv.org/pdf/q-alg/9505025v1.pdf ...
  • 938
6 votes

Annihilation operators in a vertex algebra

Suppose there exists $v,w \in V(d)$ such that $v_{(d)}w \neq 0$. And now consider $(Tv)_{(d)}w = -2d \,v_{(d)}w \neq 0$. Notice also that by skew-symmetry your condition being true for $d>n$ ...
  • 161
6 votes
Accepted

Linear independence of genus-one correlation functions

A proof of this property can be found in Proposition 2.2 of Huang's paper Vertex operator algebras and the Verlinde conjecture. This proof actually uses Zhu's algebras to simplify the discussions (...
  • 555
6 votes

q-series identity related to Jackson-Slater, proof required

The following proof is due to George E. Andrews (who is now a coauthor on the mentioned paper). Consider the following families of polynomials indexed by $n$ $$ S_n = \sum_{k \geq 0} q^{2k^2} \binom{...
5 votes
Accepted

Simple current extensions in VOA theory and CFTs

I'm not a physicist, so I can't say anything authoritative on your first question, but I can say something about questions 2. and 3. You are correct that in VOA theory, simple currents are invertible ...
5 votes

What are advantages of chiral algebras over vertex algebras?

The main advantage of chiral algebras over vertex algebras is that they admit "very functorial" definitions, and this helps more general concepts and constructions appear naturally. The usual ...
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5 votes
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When two vertex (operator) algebras can be patched-up to a full CFT on a genus 0 surface?

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 ...
  • 43.4k
5 votes
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Deformations of Vertex Algebras

The complex considered on that article is a linear algebraic version of a complex constructed by Tamarkin in his ICM address https://arxiv.org/abs/math/0304211 As he points out, in the case of ...
4 votes
Accepted

Do all non-degenerate quadratic forms come from positive even lattices?

Edited: I have missed your "positive". The signature of the lattice modulo 8 depends on the form only (some people call this Brown invariant and van der Blij theorem; Nikulin below calls this just the ...
4 votes
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Annihilation operators in a vertex algebra

The answer seems to be yes for quasi-primary $v$ if $V$ has a suitable invariant bilinear form. Then one can identify $v_{(n)} w$ with its pairing with the vacuum, and obtains it as the appropriate ...
4 votes
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Modular tensor category associated to an even integral lattice and the lattice automorphism

Edit: I've thought about this question again, and I think the answer is more positive than what I said in an earlier version. I will assume $L$ is positive-definite, since we need that to make $V_L$ ...
  • 43.4k
4 votes
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Are extensions of regular vertex operator algebras also regular?

Update: I think it will be useful to have a more coherently written answer, since I have learned more since my original response. Under the assumptions of the question, the answer is yes, the vertex ...
4 votes
Accepted

coset of affine Lie algebra

This is typically given by the commutant, or coset construction. You take the vector subspace of $\mathcal{R}_\text{vac}[\mathfrak{g}_k]$ spanned by vectors $v$ satisfying $Y(u,z)v \in \mathcal{R}_\...
  • 43.4k
4 votes

What is the Zhu algebra of a lattice vertex algebra?

I'm expanding Reimundo Heluani's link, which gives the answer when $L$ is an even positive definite lattice. Write $\mathfrak{h}=L\otimes_{\mathbf{Z}}\mathbf{C}$. Every $\alpha\in L$ gives two ...
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3 votes

Simple current extensions in VOA theory and CFTs

Regarding your first question, physicists are interested in classifying modular invariant partition functions for two-dimensional rational conformal field theories. Simple currents are a useful tool ...
  • 5,161
3 votes

Poisson vertex algebra

This is an exercise problem and it is more proper to ask it on stack exchange. Your problem is that you do not know how to evaluate $Y_{-}(a\cdot b,z)$ for arbitrary $a,b$ in the Poisson vertex ...
  • 31
3 votes
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Some examples of vertex algebra modules

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 ...
  • 43.4k
3 votes
Accepted

Commutators of Schur polynomials of Lie algebra elements

No, there will be no nice formula for $[V_{\mu}^{(n)},V_{\lambda}^{(m)}]$ in general. Compute first $$ V_{\mu}(z_1) V_{\lambda}(z_2)=p(z_1,z_2) : V_{\mu}(z_1)V_{\lambda}(z_2) :$$ $$ V_{\lambda}(z_2)...
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