Questions tagged [vertex-algebras]

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1answer
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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 ...
1
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1answer
75 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 ...
11
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1answer
358 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_{*...
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0answers
61 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 ...
6
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0answers
182 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
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1answer
288 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" ...
2
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0answers
58 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 $\...
5
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1answer
181 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
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0answers
223 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
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0answers
139 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 ...
4
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1answer
170 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 ...
6
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0answers
135 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
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1answer
150 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 ...
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0answers
273 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
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2answers
272 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
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0answers
279 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
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1answer
185 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
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1answer
294 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
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1answer
135 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
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1answer
187 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
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0answers
288 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
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2answers
702 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
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1answer
132 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
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2answers
263 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
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1answer
102 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
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1answer
695 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 ...
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0answers
161 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 $\...
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1answer
211 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, ...
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3answers
1k 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
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1answer
271 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
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0answers
143 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 $$ ...
6
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1answer
272 views

When two vertex (operator) algebras can be patched-up to a full CFT on a genus 0 surface?

Theorem 3 of the nLab article "Full field algebra" states that Theorem 3. Two vertex operator algebras $V$ may appear as the left and right chiral halfs of a full conformal field theory precisely ...
5
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1answer
335 views

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

Let $(G,+)$ be a finite Abelian group. We say $q\colon G\to \mathbb{T}$ is a non-degenerated quadratic form, if $q(-a)=q(a)$ and the symmetric function $$ b(g,h) =q(g+h)q(g)^{-1}q(h)^{-1} $$ is a non-...
8
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1answer
188 views

Fourier series of a Wightman field

From a proof that 2D Wightman CFT leads to a vertex algebra [1]: Let $$ Y(a,z):=\frac{1}{(1+z)^{2\Delta_a}}\Phi_a\left(i\frac{1-z}{1+z}\right),\quad\text{with}\quad |z|<1. $$ Here $\Delta_a\ge 0$ ...
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0answers
174 views

Understanding the Segal-Sugawara construction

I am trying to understand the Segal-Sugawara construction from the book "vertex algebras and algebraic curves" by Frenkel, Ben-Zvi in 3.4.8. As an absolute layman in the area of mathematical physics ...
4
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1answer
178 views

Closest point to a dual lattice point (in particular for root lattices!)

Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...
8
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2answers
431 views

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

On p. 13 of "Vertex Algebras for Beginners", 2nd edition, Kac writes: "Under certain assumptions and with certain additional data one may reconstruct the whole QFT from these chiral algebras, but we ...
12
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1answer
395 views

q-Virasoro and q-Heisenberg algebras

The literature has definitions (seemingly plural, though they might be linked) of a $q$-deformed Virasoro algebra. But is there any link of these to a $q$-deformed Heisenberg algebra? (Classically ...
6
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0answers
121 views

Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form $$ g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{...
2
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1answer
163 views

Commutators of Schur polynomials of Lie algebra elements

Question: Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
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0answers
334 views

Does the de Rham version of Cohen's theorem hold in the $\infty$-setting?

One of the first results that one needs to prove in the theory of chiral algebras is a de Rham version of Cohen's theorem on the homology of $C_n$ spaces. This is achieved in Beilinson-Drinfeld's book ...
9
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3answers
555 views

properties of formal delta functions

The formal delta function is $\,\,\displaystyle\delta(x):=\sum_{n\in\mathbb Z}x^n. $ If we agree that expressions $(x+y)^n$ for $n\in\mathbb Z$ are always expanded in non-negative powers of the second ...
4
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0answers
167 views

Arithmetic analogs of vertex algebras?

Has anyone successfully defined and studied analogs of vertex algebras where the grading of the fields is by $(\log \mathbb Q)$ rather than $\mathbb Z$? What I mean is that the usual fields $$ a(z) = ...
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0answers
223 views

Does the following first order approximation of the Kapranov-Vasserot infinitesimal loops still do any job?

Let $X$ be a scheme over, say, a field $k$. Let us denote $\mathrm{Spec}(k[\varepsilon])$ by $T$ and its (unique) $k$-point by $0\in T$. Call the first order infinitesimal cone $C_{T,0}(X)$ over $X$ ...
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2answers
703 views

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

The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as ...
25
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5answers
4k views

Verlinde's formula

"Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT. Depending on... • which chiral CFT one considers (does one restrict to WZW models, or not?) &...
4
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0answers
253 views

$\mathbb Z/2$-orbifolds in Virasoro representations, CFTs, VOAs

Suppose that ${\rm Vir}_c$ is a rational Virasoro algebra with central charge $c$. Then ${\rm Vir}_c$ has finitely many irreducible modules $M_h$, parametrised by the highest weights $h$. Furthermore ...
15
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2answers
763 views

Is this a vertex algebroid?… What is vertex algebroid?

A couple of day ago, I was lamenting to a friend about the fact that I have no idea what vertex algebroids are. During our discussion, I came up with a guess of what a vertex algebroid might be. I'm ...
16
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1answer
587 views

Character of parity-twisted supersymmetric VOA module — question inspired by the Stolz-Teichner program

I'll begin with some background that is unnecessary for the actual question, but that might be interesting to the reader: Topological modular forms ($TMF$) is a generalized cohomology theory whose ...
13
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2answers
1k views

I'm looking for a Virasoro-module whose character is 1+ 240q+ 2160q^2+ 6720q^3…

Let $E_4(q)=1+ 240q+ 2160q^2+ 6720q^3+\ldots $ be the Eisenstein series of weight 4, also known as the theta-series of the $E_8$-lattice. I'm looking for a $\mathbb N$-graded vector space $V$ of ...