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Are umbral moonshine and umbral calculus connected?

In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
Daigaku no Baku's user avatar
4 votes
1 answer
427 views

WZW primary correlations in terms of current algebra?

Given the $\mathfrak{u}(N)$ algebra with generators $L^a$ and commutation relations $ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ , the WZW currents of $U(N)_k$ $$ J(z) = \sum_{n \in \mathbb{Z}} J^a_n z^{-n-...
Joe's user avatar
  • 535
5 votes
0 answers
127 views

Tensor product - Vertex / Chiral algebras

Two questions regarding tensor product of modules over vertex / chiral algebras: First question: For (rational?) vertex operator algebras there is a notion of fusion product of modules inducing a ...
E. KOW's user avatar
  • 742
4 votes
0 answers
106 views

BRST construction of coset VOAs

Most recent papers define cosets of $V_k(g)$ by $V_k(h)$, where $h\subset g$ - some affine (super-)Lie algebras, as a cohomology of a complex $$V_k(g)\otimes V_{-k}(h)\otimes ghosts$$ but I'm failing ...
Nikita Grygoryev's user avatar
3 votes
0 answers
58 views

Locally finite positive energy modules generated by singular vectors at positive levels?

This is question is about whether or not certain modules for an affine Lie algebra are generated by their singular vectors. I begin with some background. Backround on affine Lie algebras. Let $\...
user avatar
0 votes
0 answers
95 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 ...
Estwald's user avatar
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3 votes
2 answers
212 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 $\...
Sebastien Palcoux's user avatar
5 votes
0 answers
131 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 ...
Estwald's user avatar
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5 votes
1 answer
425 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^...
Pulcinella's user avatar
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13 votes
1 answer
588 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 ...
Pulcinella's user avatar
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10 votes
2 answers
423 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. ...
Estwald's user avatar
  • 1,331
9 votes
2 answers
560 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 ...
Estwald's user avatar
  • 1,331
2 votes
0 answers
81 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 ...
Ying's user avatar
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8 votes
1 answer
229 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 ...
Pulcinella's user avatar
  • 5,585
4 votes
1 answer
157 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 ...
espacodual's user avatar
7 votes
0 answers
137 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 ...
Pulcinella's user avatar
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3 votes
0 answers
65 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) \...
liouville's user avatar
1 vote
0 answers
60 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 ...
Arkadij's user avatar
  • 966
4 votes
0 answers
231 views

Vertex operator algebras and modular fusion categories

Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C} = \text{Rep}(\mathcal{V})$ be the tensor category of $\mathcal{V}$-modules. It is a conjecture by Vaughan Jones whether every ...
Sebastien Palcoux's user avatar
4 votes
0 answers
114 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\...
Pulcinella's user avatar
  • 5,585
2 votes
1 answer
177 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 ...
Pulcinella's user avatar
  • 5,585
2 votes
0 answers
114 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. ...
Pulcinella's user avatar
  • 5,585
5 votes
1 answer
201 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\ :=\ \...
Pulcinella's user avatar
  • 5,585
5 votes
1 answer
578 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 ...
Pulcinella's user avatar
  • 5,585
8 votes
1 answer
729 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 ...
Pulcinella's user avatar
  • 5,585
4 votes
0 answers
242 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\...
Justin Kulp's user avatar
3 votes
1 answer
565 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?
GTA's user avatar
  • 1,014
4 votes
1 answer
441 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 ...
clvolkov's user avatar
  • 323
6 votes
0 answers
204 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
153 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 ...
Christopher Beem's user avatar
6 votes
0 answers
212 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 ...
user avatar
13 votes
1 answer
386 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 $\Sigma$...
Simon Lentner's user avatar
1 vote
0 answers
61 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 ...
GGT's user avatar
  • 685
1 vote
0 answers
44 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 ...
Thibault Décoppet's user avatar
3 votes
0 answers
116 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\...
JeCl's user avatar
  • 1,001
4 votes
1 answer
222 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'(...
JeCl's user avatar
  • 1,001
7 votes
0 answers
107 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 ...
winawer's user avatar
  • 171
3 votes
1 answer
209 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 ...
JamalS's user avatar
  • 201
3 votes
0 answers
138 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 ...
user avatar
4 votes
2 answers
257 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 $d$-...
user avatar
10 votes
1 answer
834 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 ...
Pulcinella's user avatar
  • 5,585
4 votes
1 answer
138 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 $\...
Lelouch's user avatar
  • 857
15 votes
1 answer
515 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)...
Reimundo Heluani's user avatar
3 votes
0 answers
81 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, ...
user avatar
4 votes
1 answer
393 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 ...
user avatar
2 votes
1 answer
188 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 ...
JamalS's user avatar
  • 201
14 votes
1 answer
1k 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_{*...
user avatar
2 votes
0 answers
93 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 ...
user avatar
7 votes
0 answers
237 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 $...
Pulcinella's user avatar
  • 5,585
6 votes
1 answer
576 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" ...
user avatar