Questions tagged [vertex-algebras]
The vertex-algebras tag has no usage guidance.
101
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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 ...
4
votes
1
answer
402
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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 $U(N)_k$
$$ J(z) = \sum_{n \in \mathbb{Z}} J^a_n z^{-n-...
4
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0
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Tensor product - Vertex / Chiral algebras
Two questions regarding tensor product of modules over vertex / chiral algebras:
First question: For (rational?) vertex operator algebras there is a notion of fusion product of modules inducing a ...
4
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0
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95
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BRST construction of coset VOAs
Most recent papers define cosets of $V_k(g)$ by $V_k(h)$, where $h\subset g$ - some affine (super-)Lie algebras, as a cohomology of a complex
$$V_k(g)\otimes V_{-k}(h)\otimes ghosts$$
but I'm failing ...
3
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0
answers
57
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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 $\...
0
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0
answers
85
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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 ...
3
votes
2
answers
201
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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 $\...
5
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0
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123
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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
372
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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^...
12
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1
answer
542
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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 ...
9
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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. ...
9
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2
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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
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0
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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 ...
8
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1
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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
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0
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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
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1
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155
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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
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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
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0
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64
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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
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60
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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
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0
answers
204
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Vertex operator algebras and modular tensor 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
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113
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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
171
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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 ...
2
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0
answers
108
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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
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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\ :=\ \...
5
votes
1
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545
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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
686
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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
232
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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
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539
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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
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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
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0
answers
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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
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150
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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
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0
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206
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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 ...
12
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1
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363
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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
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59
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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
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0
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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
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0
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108
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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\...
4
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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'(...
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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
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1
answer
202
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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
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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
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2
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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
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1
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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
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1
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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
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1
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506
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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
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0
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80
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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
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1
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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
181
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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
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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
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92
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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
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0
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235
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$X$ 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 $...