Questions tagged [vertex-algebras]

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$H^4$ of the Monster

The Monster group $M$ acts on the moonshine vertex algebra $V^\natural$. Because $V^\natural$ is a holomorphic vertex algebra (i.e., it has a unique irreducible module), there is a corresponding ...
André Henriques's user avatar
31 votes
3 answers
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 ...
John Baez's user avatar
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30 votes
5 answers
6k 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?) &...
André Henriques's user avatar
22 votes
1 answer
876 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 ...
Ramesh Chandra's user avatar
17 votes
2 answers
885 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 ...
Jamie Vicary's user avatar
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17 votes
1 answer
686 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 ...
André Henriques's user avatar
15 votes
2 answers
899 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 ...
André Henriques's user avatar
15 votes
1 answer
506 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
15 votes
0 answers
403 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 ...
Gro-Tsen's user avatar
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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_{*...
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13 votes
2 answers
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 ...
André Henriques's user avatar
13 votes
2 answers
1k 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 ...
Syu Gau's user avatar
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13 votes
1 answer
525 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 ...
Edwin Beggs's user avatar
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13 votes
0 answers
406 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 ...
Reimundo Heluani's user avatar
12 votes
1 answer
363 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
12 votes
1 answer
540 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
  • 5,506
12 votes
1 answer
580 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 "...
Bipolar Minds's user avatar
11 votes
1 answer
390 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 ...
André Henriques's user avatar
11 votes
1 answer
824 views

different N=2 SUSY structures on the chiral de Rham complex of a Calabi-Yau manifold?

The context In a beautiful paper, Malikov-Schechtman-Vaintrob defined a canonical sheaf of vertex algebras equipped with a differential on any manifold $X$ (either in the $C^\infty$, complex analytic ...
DamienC's user avatar
  • 8,103
11 votes
1 answer
764 views

Is there a canonical map from the cohomology of orbifold chiral de Rham on an orbifold to the cohomology of chiral de Rham on a crepant resolution?

The two-variable elliptic genus is a topological invariant of almost-complex manifolds that takes values in power series. These power series turn out to describe weak Jacobi forms when the manifold ...
S. Carnahan's user avatar
  • 45k
10 votes
1 answer
807 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
  • 5,506
9 votes
2 answers
491 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
  • 1,187
9 votes
2 answers
341 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(...
André Henriques's user avatar
9 votes
2 answers
538 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 ...
Gytis's user avatar
  • 383
9 votes
3 answers
834 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 ...
André Henriques's user avatar
9 votes
2 answers
381 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
  • 1,187
8 votes
1 answer
685 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
  • 5,506
8 votes
1 answer
318 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 $\Delta_a\ge 0$ ...
Gytis's user avatar
  • 383
8 votes
1 answer
216 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
  • 5,506
7 votes
1 answer
280 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, ...
Theo Johnson-Freyd's user avatar
7 votes
1 answer
374 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 $...
user avatar
7 votes
0 answers
132 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
  • 5,506
7 votes
0 answers
106 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
7 votes
0 answers
235 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
  • 5,506
7 votes
0 answers
180 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. ...
მამუკა ჯიბლაძე's user avatar
6 votes
2 answers
2k views

About state-field correspondence

In the definition of vertex algebra, we call the vertex operator state-field correspondence, does that mean that it is an injective map?? Are there some physical interpretations about state-field ...
Xuexing Lu's user avatar
6 votes
2 answers
582 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 ...
user avatar
6 votes
1 answer
391 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 ...
Bin Gui's user avatar
  • 555
6 votes
1 answer
380 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 ...
Gytis's user avatar
  • 383
6 votes
1 answer
548 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-...
Marcel Bischoff's user avatar
6 votes
1 answer
564 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
6 votes
0 answers
200 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,...
6 votes
0 answers
206 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
6 votes
0 answers
226 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 ...
André Henriques's user avatar
6 votes
0 answers
136 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^{...
Slava Rychkov's user avatar
5 votes
1 answer
372 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
  • 5,506
5 votes
1 answer
303 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 ...
Bin Gui's user avatar
  • 555
5 votes
1 answer
277 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 ...
Yuji Tachikawa's user avatar
5 votes
1 answer
197 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
  • 5,506
5 votes
1 answer
545 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
  • 5,506