Questions tagged [vertex-algebras]
The vertex-algebras tag has no usage guidance.
75
questions
4
votes
0answers
108 views
Computing theta functions of lattices in practice
I am motivated by a problem in 2d CFT to compute "generalized theta functions," expressions of the form
\begin{equation}
\vartheta_{L,u}(\tau) := \sum_{\alpha \in L} u(\alpha) q^{{\langle\...
3
votes
1answer
283 views
Why “holomorphic” vertex algebra?
I have a background quite far from vertex algebras, and it seems like a vertex algebra is holomorphic if basically there is only one irreducible module, namely itself. Why is it called holomorphic?
4
votes
1answer
210 views
Zhu's algebra for the Virasoro VOA
I am trying to understand the proof in the appendix of the following paper:
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.110.8757&rep=rep1&type=pdf
The paper discusses Zhu's ...
6
votes
0answers
126 views
What is known about “dimension two” vertex algebras?
In the paper Chiral Koszul duality, Gaitsgory and Francis develop a notion of a chiral algebra living on an arbitrary variety $X$. When $X=\mathbf{A}^1$ and the chiral algebra is translation invariant,...
3
votes
0answers
79 views
Logarithmic vector-valued modular functions and quasimodular forms with misleading modular weights
I have a somewhat imprecise question about functions with reasonably nice modular transformations that don't seem to fit nicely into what I understand of the plain vanilla theory of modular and ...
5
votes
0answers
126 views
BRST cohomology and vanishing cycles
Consider the $\mathbb{C}$-variety $\mathbb{A}^{1}$, equipped with the potential (ie global function) $P:=\frac{z^{n+1}}{n+1}$. We can form the twisted de Rham complex $H_{dR}(\mathbb{A}^{1},P)$ which ...
11
votes
1answer
197 views
Chiral homology for the Virasoro algebra and/or affine Lie algebra
I want to understand what concrete analytical objects are found in chiral homology of higher degree of a vertex algera (-module) $M$. More precisely: I can obtain conformal blocks on a surface $\Sigma$...
1
vote
0answers
51 views
Quantum-airy operators to specialised ones
Let $V$ be a vector space of finite or countable dimension over $\mathbb{C}$. Let $n$ denote the dimension of $V$ and the index set $I=\{1,2,\ldots,n\}$($n$ tend to $\infty$). Let us choose an order ...
1
vote
0answers
32 views
A reformulation of commutativity for intertwinning operators?
$\DeclareMathOperator{\Id}{\mathrm{Id}}\DeclareMathOperator{\Rep}{\operatorname{Rep}}$Let $V$ be a nice vertex algebra, and $M_1, M_2, M_3, M_4, M_5, M_6$ be modules over $V$. Assume that I have ...
2
votes
0answers
42 views
Understanding the intuition behind the $Q(z)$-tensor product
Let $z$ be a fixed non-zero complex number. Let $V$ be a vertex algebra, $W_1$, $W_2$, and $W_3$ be $V$-modules. Huang defines a $Q(z)$-intertwining map between these modules to be a linear map $F:W_1\...
3
votes
1answer
95 views
Intuition behind contragredient module of a VOA
Let $(V,Y)$ be a vertex operator algebra, and $V'$ be the graded dual of its underlying vector space. The contragredient module structure on $V'$ is given by $Y'$ defined by the formula:
$$\langle Y'(...
6
votes
0answers
78 views
Reference request: superconformal algebras and representations
I am looking for a book/monograph which deals with superconformal (vertex operator) algebras and their representation theory. What are some good books to understand to begin with the definition of a ...
3
votes
1answer
109 views
Zhu's $V/C_2(V)$ algebra
As an example, take the Virasoro algebra, i.e. $V$ is spanned by elements of the form $L_{-2}^{k_1} \cdots L_{-n}^{k_{n-1}} \Omega$ where $\Omega$ is the vacuum and $n \geq 2$. As I understand, we ...
3
votes
0answers
108 views
Twisted screening operators and twisted free-field realizations of $\mathcal{W}_n$ algebras
Let $\mathfrak{g}=\mathfrak{sl}_{n+1}$ and I am interested in the principal $\mathcal{W}$-algebra of $\mathcal{g}$ at self-dual level i.e. $k=- h ^{\vee} +1$, usually denoted by $\mathcal{W}_n$. Now ...
4
votes
2answers
200 views
Spectral Flow Invariance for Calabi-Yau Sigma Models
I am a mathematician who has become interested in some of the mathematics of string theory, of which I am largely ignorant, so please excuse any idiocies in what follows.
If $X$ is a Calabi-Yau $d$-...
8
votes
1answer
318 views
What is the Zhu algebra of a lattice vertex algebra?
Associated to a vertex algebra $V$ is an associative algebra $A(V)$, the Zhu algebra. Its defining property is approximately that the representations of $V$ and of $A(V)$ are the same.
In vertex ...
4
votes
1answer
102 views
coset of affine Lie algebra
In many books about conformal field theory, when we talk about a coset $\mathfrak{g}_k/\mathfrak{h}_{k'}$, we would talk about how the modules of $\mathfrak{g}_k$ are decomposed into those of $\...
15
votes
1answer
422 views
q-series identity related to Jackson-Slater, proof required
The question:
I have been trying to prove the following $q$-series identity for quite some time now:
$$
\sum_{k \geq 0} \frac{q^{2k^2}}{(q)_{2k}} = \sum_{m,k \geq 0} \frac{q^{m^2 + 3k m + 4k^2}}{(q)...
3
votes
0answers
58 views
Composition of operators in $w_{1+\infty}$ and $W_{1+\infty}$
The algebra $W_{1+\infty}$ can be defined as a central extension of the lie algebra $w_{1+\infty}$ (defined as being spanned by $\left(-\partial_z \right)^m z^{-k}$ ). See for example: Alexandrov, ...
4
votes
1answer
214 views
Examples of simple vertex operator algebras (VOAs)
A vertex operator algebra $V$ is called simple if $V$ is a simple $V$-module. What are some examples of simple VOAs? Are there lots of examples or this is a very strong condition? Is there a ...
2
votes
1answer
140 views
Vertex operator algebras and isomorphism of graded vector spaces
I have two vertex operator algebras and I would like to show that as graded vector spaces, they are isomorphic, rather than as algebras.
The issue is I have not found anything in the literature that ...
12
votes
1answer
624 views
Factorization and vertex algebra cohomology
A chiral algebra on a smooth curve $X$, in the sense of Beilinson-Drinfeld, is a right $D_{X}$-module with a chiral bracket, which is a map $\mathcal{V}^{\boxtimes 2}(\infty\Delta)\rightarrow \Delta_{*...
2
votes
0answers
81 views
Representation theoretic definition of wavefunctions of an integrable hierarchy?
I am reading Kac's book on infinite dimensional lie algebras. In the last chapter, he starts with a highest weight module of an affine lie algebra $\mathfrak{g}(A)$, and uses it to define tau ...
7
votes
0answers
217 views
$X$ with $H^*(X)=$affine Verma module?
Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over $\mathbf{C}$, and $\widehat{\mathfrak{g}}_\kappa$ the associated affine Lie algebra. It is the central extension of the loop algebra $...
6
votes
1answer
463 views
$GL(\infty)$ group action through the boson-fermion correspondence
Every point of the Sato Grassmannian can be used to generate a tau function of the KP hierarchy. In addition, the Sato Grassmannian can be seen as a subset of the "second quantized fermion Fock space" ...
3
votes
0answers
64 views
Free almost commutative vertex algebras
Given a commutative $k$-algebra $A$, we can freely generate a commutative vertex algebra by formally adjoining a derivation. We obtain a functor $CAlg_{k} \rightarrow CVAlg_{k} $, which I'll denote $\...
7
votes
1answer
248 views
Deformations of Vertex Algebras
As the title suggests, I'm interested in deformation theory of vertex algebras and their representations.
In the paper https://arxiv.org/abs/1806.08754, the authors construct, for a vertex algebra $...
4
votes
0answers
241 views
Are vertex operator algebras ever conspiratorial?
I have a vertex operator algebra (VOA) $V$ with all niceness properties (unitary, rational, CFT type, etc). Its Lie algebra $\mathfrak{g} = V_1$ of spin-$1$ fields is large, and I understand how the ...
6
votes
0answers
185 views
Orthogonality relations for characters of VOAs?
If $G$ is a finite group, the characters of its irreps satisfy
$$
\langle \chi_1,\chi_2\rangle := \frac{1}{|G|}\sum_{g\in G} \chi_1(g)\; \overline{\chi_2(g)} = \delta_{\chi_1,\chi_2}.
$$
Alexei ...
5
votes
1answer
217 views
Classification of quasi-lisse vertex algebras
Quasi-lisse vertex algebras were introduced by Arakawa and Kawasetsu in Quasi-lisse vertex algebras and modular linear differential equations . They satisfy the property that the normalized character ...
7
votes
0answers
162 views
A single vertex operator - bare bones explanation?
There is an ocean of literature (and a sea of popular texts inside) on vertex algebras, including quite a lot of Q & A here on MO, and I am trying to read some random selections from time to time. ...
5
votes
1answer
183 views
Are extensions of regular vertex operator algebras also regular?
Let $U$ be a simple VOA which is self-dual and of CFT type (i.e., $U\simeq U'$, and $U$ has grading $U=\bigoplus_{n\in\mathbb N}U(n)$ with $U(0)$ spanned by the vacuum vector $\Omega$). Let $V$ be a ...
12
votes
0answers
340 views
Vertex algebras and factorization algebras
It is often said that vertex algebras are a special case of factorization algebras. In particular, in their book "FAs in QFT" Costello/Gwilliam construct a functor from a certain class of 2d "...
6
votes
2answers
356 views
Simple current extensions in VOA theory and CFTs
I apologize in advance if this is too broad and off-topic here. I have seen some papers in the field of vertex operator algebras (VOA) theory about simple current extensions. As far as I understand ...
14
votes
0answers
319 views
The Monster Moonshine Module from the engineering or algorithmic point of view
From what I understand (see, e.g., this question), the Monster Moonshine Module is a kind if "third generation" (or "second quantization"?) after the Golay code (with automorphism group $M_{24}$) and ...
3
votes
1answer
240 views
Poisson vertex algebra
Suppose $vir_{c}= \operatorname{span}\langle L_{-2}v_{c},L_{-3}v_{c},....\rangle$ is a vector space spanned by Virasoro algebra. Then we have a symmetric algebra $Sym(vir_{c})$. For this symmetric ...
11
votes
1answer
329 views
Linear independence of genus-one correlation functions
Let $V$ be a vertex operator algebra with all the good finiteness properties that people usually assume (positively graded, $C_2$-cofinite, $V\cong V'$, etc.)
Let $W$ be a module for $V$, not ...
5
votes
1answer
172 views
Some examples of vertex algebra modules
Recently I'm learning the vertex modules. In the paper, there are a lot of abstract theory about the module theory,for instance the $C_{2}-$cofinite conditions and associated variety. I hope to find ...
5
votes
1answer
221 views
Modular tensor category associated to an even integral lattice and the lattice automorphism
Let $(L,\langle -,-\rangle)$ be an even integral lattice, and let $(A,q)$ be the associated discriminant form: $$
A=L^*/L, \quad q(a)=e^{\pi i \langle a,a\rangle}.
$$
We let $\hat L$ to be the ...
5
votes
0answers
335 views
Reference Request: Vertex Algebras
I am currently a graduate student in mathematics with an interest in vertex algebras. I am comfortable with the algebraic aspects and would like to learn more about the geometric aspects. The issue is ...
12
votes
2answers
916 views
What are advantages of chiral algebras over vertex algebras?
In Chiral Algebras, Beilinson and Drinfeld introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-modules. On the other hand, There is already a notion of vertex ...
3
votes
1answer
138 views
Does the Leclerc-Thibon involution exchange vertex operators of the first and second type?
This question is about $U_q ( \hat{\mathfrak{sl}}_2 )$ representation theory. There is a notion of vertex operators $\Phi_{\pm }(z)$ of first and $\Psi_{\pm}(z)$ of the second type. They are defined ...
9
votes
2answers
293 views
Annihilation operators in a vertex algebra
Let $V=\bigoplus_{d\in\mathbb N}V(d)$ be a Möbius-covariant vertex algebra with $V(0)=\mathbb C$.
Recall that a vector $v\in V$ is called quasi-primary if $L_1v=0$.
For $v\in V(d)$, we write $Y(...
2
votes
1answer
110 views
Two definitions of conformal inclusion
Assume that $V$ is a vertex operator algebra, and the VOA $V'$ is a vertex subalgebra of $V$. The notion that $V'\subset V$ is a conformal inclusion has different meanings in different literatures. ...
22
votes
1answer
754 views
71, the Monster, and c = 24 CFTs
The largest prime in the order of the Monster group is $71$. This number $71$ shows up at various places:
The minimal faithful representation has dimension $196883 = 47.59.71$
The Monster group can ...
1
vote
0answers
167 views
Wightman axioms to Vertex algebra, the inspiration for the infinitesimal translation operator T?
In section 1.1, 1.2 of Kac's book Vertex Algebras for Beginners, he deduces the axioms of vertex algebras (or more precisely, right chiral algebras) from the Wightman axioms for $2$d CFT.
Denote $\...
5
votes
1answer
224 views
What are braided vertex algebras?
The notion of vertex algebra, like any reasonable algebraic notion, makes sense inside any (sufficiently linear) symmetric monoidal category. The standard pictures of the operator product, however, ...
31
votes
3answers
2k views
What's the supersymmetric analogue of the Monster group?
Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...
6
votes
1answer
321 views
Do we have a braided tensor category for vertex algebra modules by using conformal blocks on an arbitary compact Riemann Surface?
In Huang & Lepowsky's series of papers A theory of tensor products for module categories for a vertex operator algebra, they defined for a rational vertex algebra $V$ the $P(z)$ tensor product of ...
3
votes
0answers
156 views
Self-dual vertex algebras
Let $(V,Y)$ be a self-dual conformal vertex algebra. For instance, it could be the vertex algebra associated to a positive definite, even, unimodular quadratic form. I look for a formula to compute
$$
...
