Questions tagged [vertex-algebras]
The vertex-algebras tag has no usage guidance.
101
questions
45
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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 ...
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 ...
30
votes
5
answers
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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?)
&...
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 ...
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 ...
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 ...
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 ...
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)...
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 ...
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_{*...
13
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2
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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 ...
13
votes
2
answers
1k
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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 ...
13
votes
1
answer
525
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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 ...
13
votes
0
answers
406
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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 ...
12
votes
1
answer
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$...
12
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1
answer
540
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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 ...
12
votes
1
answer
580
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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 "...
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 ...
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 ...
11
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1
answer
764
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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 ...
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 ...
9
votes
2
answers
491
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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 ...
9
votes
2
answers
341
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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(...
9
votes
2
answers
538
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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 ...
9
votes
3
answers
834
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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 ...
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. ...
8
votes
1
answer
685
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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 ...
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$ ...
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 ...
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, ...
7
votes
1
answer
374
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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 $...
7
votes
0
answers
132
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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 ...
7
votes
0
answers
106
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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 ...
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 $...
7
votes
0
answers
180
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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. ...
6
votes
2
answers
2k
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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 ...
6
votes
2
answers
582
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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 ...
6
votes
1
answer
391
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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 ...
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 ...
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}
$$
is a non-...
6
votes
1
answer
564
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$GL(\infty)$ group action through the boson-fermion correspondence
Every point of the Sato Grassmannian can be used to generate a tau function of the KP hierarchy. In addition, the Sato Grassmannian can be seen as a subset of the "second quantized fermion Fock space" ...
6
votes
0
answers
200
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What is known about "dimension two" vertex algebras?
In the paper Chiral Koszul duality, Gaitsgory and Francis develop a notion of a chiral algebra living on an arbitrary variety $X$. When $X=\mathbf{A}^1$ and the chiral algebra is translation invariant,...
6
votes
0
answers
206
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BRST cohomology and vanishing cycles
Consider the $\mathbb{C}$-variety $\mathbb{A}^{1}$, equipped with the potential (ie global function) $P:=\frac{z^{n+1}}{n+1}$. We can form the twisted de Rham complex $H_{dR}(\mathbb{A}^{1},P)$ which ...
6
votes
0
answers
226
views
Orthogonality relations for characters of VOAs?
If $G$ is a finite group, the characters of its irreps satisfy
$$
\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}.
$$
Alexei ...
6
votes
0
answers
136
views
Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x
Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form
$$
g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{...
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^...
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 ...
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 ...
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\ :=\ \...
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 ...