Questions tagged [vertex-algebras]
The vertex-algebras tag has no usage guidance.
98
questions
0
votes
0
answers
44
views
Some version of non-commutative Wick formula
Let $V$ be a vertex algebra. The traditional non-commutative Wick formula is a tool to calculate term like $[a_\lambda:bc:]$. However, I need to calculate terms of the form $[:ab:_\lambda c]$. I found ...
2
votes
2
answers
141
views
Simple modular tensor category and zero entries in its S-matrix
Question 1: Is there a simple modular fusion category with a zero entry in its S-matrix?
(or equivalently, with a fusion matrix of zero determinant?)
Yes, by this answer below providing the example $\...
4
votes
0
answers
83
views
Functoriality of Feigin–Frenkel duality
For a simple Lie algebra $\mathfrak{g}$, we have the W-algebra of level $k$, denoted by $\mathcal{W}^k(\mathfrak{g})$. Using Wakimoto free field realization and screening operators, Feigin and Frenkel ...
5
votes
1
answer
197
views
Defining extended TQFTs *with point, line, surface, … operators*
$\newcommand\Cob{\mathrm{Cob}}\newcommand\Vect{\mathrm{Vect}}\DeclareMathOperator\Rep{Rep}$The ordinary definition of a TQFT is:
Defnition: A $d$-dimensional TQFT is a symmetric monoidal functor $\Cob^...
11
votes
1
answer
296
views
What is the Zhu algebra of a vertex algebra "really"?
Given any vertex algebra $V$, you can give a particular quotient $\DeclareMathOperator\Zhu{Zhu}\Zhu V=V/\cdots$ an algebra structure using (a small amount of) the vertex algebra structure. As far as ...
3
votes
0
answers
63
views
Zhu's induction for affine vertex algebras
For a vertex operator algebra $V$, Yongchang Zhu defined the so-called Zhu's algebra $\operatorname{Zhu}(V)$ of $V$. It's an associative algebra with underlying vector space being some quotient of $V$....
2
votes
0
answers
65
views
Homological properties of representations of vertex operator algebras
For a vertex operator algebra $\mathbb{V}$, one cares about representations of $\mathbb{V}$ (sometimes with some additional requirements, like the full subcategory $\mathcal{O}$ of $\mathfrak{g}$-$\...
9
votes
2
answers
272
views
What is the meaning of chiral in the context of vertex algebras?
There are many objects in mathematics that have the term "chiral" in their name, for instance, chiral algebra by Beilinson and Drinfeld, chiral de Rham complex, chiral Koszul duality etc. ...
8
votes
2
answers
372
views
Proofs of the Frobenius characteristic map
Let $\mathfrak{S}_n$ be the symmetric group on $n$ letters, $\mathsf{Rep}(\mathfrak{S}_n)$ be the abelian category of finite dimensional complex representations of $\mathfrak{S}_n$. A classical result ...
2
votes
0
answers
74
views
DHR superselection and DR reconstruction in low spacetime dimensions
Given a completely rational net on $\mathbb{R}$, the Doplicher-Haag-Roberts (DHR) category is a modular fusion category (MFC) identical to that associated with the corresponding vertex operator ...
7
votes
1
answer
187
views
What structure does Rep(vertex algebra) have?
Let $V$ be a vertex algebra. If $V$ is particularly nice, it is known that its category $\text{Rep} V$ of modules is a modular tensor category, see e.g. [1] [2].
However, this has always seemed to me ...
2
votes
0
answers
94
views
Is there a non-pointed simple integral modular fusion category?
The weakly group-theoretical conjecture (supporting a negative answer to [ENO11, Question 2]) states as follows:
Statement 1: Every integral fusion category is weakly group-theoretical.
We wonder ...
4
votes
1
answer
129
views
Coordinate principal bundle over a curve
I am trying to understand more about geometric interpretation of vertex algebras following "Vertex Algebras and Algebraic Curves" by Ben-Zvi and Frenkel, but I am in trouble with the ...
7
votes
0
answers
117
views
Learning roadmap for admissible representations of $\widehat{\mathfrak{g}}$ (affine Lie algebras)
Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over $\mathbf{C}$. A priori one might expect the representation theory of the affine Lie algebra $\widehat{\mathfrak{g}}$ (the Lie ...
3
votes
0
answers
53
views
Hecke operators for modular form with respect to $\Gamma_{\theta}(2)$ subgroup
The congruence subgroup $\Gamma_{\theta}(2)$ is defined as:
$$\Gamma_{\theta}(2)=\left\{\gamma\in SL(2,\mathbb{Z})|\gamma\equiv\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right) \...
1
vote
0
answers
58
views
Alternative definition of physical states
Suppose that we have a vertex operator algebra $V$ with a conformal element $\omega$ and the associated conformal field
$$
Y(\omega,z) = \sum_{k\in \mathbb{Z}} L_kz^{-k-2}\,,
$$
where $L_k$ satisfy ...
3
votes
0
answers
144
views
Vertex operator algebras and modular fusion categories
Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C}=Rep(\mathcal{V})$ be the tensor category of (ususal) $\mathcal{V}$-modules. It is a well-known open-problem whether every ...
4
votes
0
answers
102
views
Going between the abstract and the concrete notions of chiral homology
Let $X$ be a smooth algebraic curve over $\mathbf{C}$, and let $\mathcal{V}$ be a factorisation algebra over $X$, whose fibre above $x\in X$ is the vertex algebra $V$.
Note that $\mathcal{V}\in\...
2
votes
1
answer
148
views
Confusion about Wakimoto's chiral differential operators on $\mathbf{P}^1$
It is a classical result of Wakimoto that the sheaf of chiral differential operators $D_{ch}$ on $\mathbf{P}^1$ has global sections
$$D_{ch}(\mathbf{P}^1)\ \simeq\ L_{-2}(\mathfrak{sl}_2)$$
the simple ...
1
vote
0
answers
88
views
What is the factorization algebra/space of an affine W algebra?
The affine vertex algebra $V_k(\mathfrak{g})$ factorizes, i.e. comes from a factorisation space, the Beilinson Drinfeld Grassmannian. Similarly, lattice vertex algebras have a factorization analogue.
...
5
votes
1
answer
185
views
Existence of orbifold vertex algebras – current status?
Let the finite group $G$ act on a vertex algebra $V$. It is expected that there are certain vector spaces $V_g$ (with the structure of $g$ twisted $V$ modules), with $V_1=V^G$, and
$$V/G\ :=\ \...
4
votes
1
answer
440
views
Twisted differential operator, chiral differential operator, $???$ (continue the sequence)
Let $X$ be a smooth variety.
One can define the notion of a sheaf of twisted differential operators (TDO) on $X$. They "quantise" functions on $T^*X$. Examples include the usual sheaf of ...
8
votes
1
answer
580
views
Why are VOA characters modular forms (geometrically)?
In Zhu's seminal paper, he proves (5.3.2) that if $V$ is a vertex algebra the character of all of its modules are modular forms! (This is not literally true- there are conditions).
I have always found ...
4
votes
0
answers
191
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
1
answer
449
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
1
answer
387
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
0
answers
180
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,...
4
votes
0
answers
125
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 ...
6
votes
0
answers
192
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
1
answer
315
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
0
answers
58
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
0
answers
41
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 ...
3
votes
0
answers
90
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
1
answer
178
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:
$$\langle Y'(...
6
votes
0
answers
95
views
Reference request: superconformal algebras and representations
I am looking for a book/monograph which deals with superconformal (vertex operator) algebras and their representation theory. What are some good books to understand to begin with the definition of a ...
3
votes
1
answer
161
views
Zhu's $V/C_2(V)$ algebra
As an example, take the Virasoro algebra, i.e. $V$ is spanned by elements of the form $L_{-2}^{k_1} \cdots L_{-n}^{k_{n-1}} \Omega$ where $\Omega$ is the vacuum and $n \geq 2$. As I understand, we ...
3
votes
0
answers
124
views
Twisted screening operators and twisted free-field realizations of $\mathcal{W}_n$ algebras
Let $\mathfrak{g}=\mathfrak{sl}_{n+1}$ and I am interested in the principal $\mathcal{W}$-algebra of $\mathcal{g}$ at self-dual level i.e. $k=- h ^{\vee} +1$, usually denoted by $\mathcal{W}_n$. Now ...
4
votes
2
answers
234
views
Spectral Flow Invariance for Calabi-Yau Sigma Models
I am a mathematician who has become interested in some of the mathematics of string theory, of which I am largely ignorant, so please excuse any idiocies in what follows.
If $X$ is a Calabi-Yau $d$-...
10
votes
1
answer
680
views
What is the Zhu algebra of a lattice vertex algebra?
Associated to a vertex algebra $V$ is an associative algebra $A(V)$, the Zhu algebra. Its defining property is approximately that the representations of $V$ and of $A(V)$ are the same.
In vertex ...
4
votes
1
answer
122
views
coset of affine Lie algebra
In many books about conformal field theory, when we talk about a coset $\mathfrak{g}_k/\mathfrak{h}_{k'}$, we would talk about how the modules of $\mathfrak{g}_k$ are decomposed into those of $\...
15
votes
1
answer
490
views
q-series identity related to Jackson-Slater, proof required
The question:
I have been trying to prove the following $q$-series identity for quite some time now:
$$
\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
0
answers
74
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
1
answer
335
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
1
answer
166
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 ...
14
votes
1
answer
895
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
0
answers
88
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
0
answers
231
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
1
answer
531
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
0
answers
79
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
1
answer
336
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 $...