Questions tagged [vertex-algebras]
The vertex-algebras tag has no usage guidance.
58
questions
12
votes
0answers
242 views
+100
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)...
2
votes
0answers
43 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, ...
3
votes
1answer
120 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 ...
2
votes
1answer
103 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
votes
1answer
393 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_{*...
2
votes
0answers
65 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
votes
0answers
190 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
votes
1answer
352 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
votes
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
votes
1answer
190 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
votes
0answers
228 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
votes
0answers
148 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
votes
1answer
175 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 ...
7
votes
0answers
143 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
votes
1answer
158 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 ...
12
votes
0answers
283 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
votes
2answers
290 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
votes
0answers
283 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
votes
1answer
191 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
votes
1answer
296 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
votes
1answer
140 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
votes
1answer
190 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
0answers
292 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
votes
2answers
733 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
votes
1answer
133 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
votes
2answers
268 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
votes
1answer
103 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
votes
1answer
711 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 ...
1
vote
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 $\...
5
votes
1answer
215 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, ...
28
votes
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
votes
1answer
274 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
votes
0answers
145 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
votes
1answer
275 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
votes
1answer
350 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
votes
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 $\Delta_a\ge 0$ ...
1
vote
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
votes
1answer
180 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
votes
2answers
435 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
votes
1answer
401 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
votes
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
$$
g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{...
2
votes
1answer
168 views
Commutators of Schur polynomials of Lie algebra elements
Question:
Is there a well-known formula for computing the commutators of Schur polynomials when the variables are Lie algebra elements? If the algebra has a particularly simple commutation relation, ...
12
votes
0answers
339 views
Does the de Rham version of Cohen's theorem hold in the $\infty$-setting?
One of the first results that one needs to prove in the theory of chiral algebras is a de Rham version of Cohen's theorem on the homology of $C_n$ spaces. This is achieved in Beilinson-Drinfeld's book ...
9
votes
3answers
570 views
properties of formal delta functions
The formal delta function is
$\,\,\displaystyle\delta(x):=\sum_{n\in\mathbb Z}x^n.
$
If we agree that expressions $(x+y)^n$ for $n\in\mathbb Z$ are always expanded in non-negative powers of the second ...
4
votes
0answers
167 views
Arithmetic analogs of vertex algebras?
Has anyone successfully defined and studied analogs of vertex algebras where the grading of the fields is by $(\log \mathbb Q)$ rather than $\mathbb Z$? What I mean is that the usual fields
$$
a(z) = ...
3
votes
0answers
224 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 $X$ ...
17
votes
2answers
708 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 ...
26
votes
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
votes
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
votes
2answers
771 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 ...
