The vertex-algebras tag has no usage guidance.

**21**

votes

**3**answers

865 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 ...

**4**

votes

**1**answer

70 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

**0**answers

79 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
$$
...

**5**

votes

**1**answer

172 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 ...

**3**

votes

**1**answer

110 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 ...

**7**

votes

**1**answer

135 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

**0**answers

95 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

**1**answer

114 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 ...

**7**

votes

**1**answer

231 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 ...

**11**

votes

**1**answer

238 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 ...

**5**

votes

**0**answers

93 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 ...

**2**

votes

**1**answer

129 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, ...

**7**

votes

**0**answers

205 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 ...

**8**

votes

**3**answers

399 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

**0**answers

128 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

**0**answers

174 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 $X$ ...

**13**

votes

**1**answer

448 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 ...

**21**

votes

**5**answers

2k 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?)
...

**3**

votes

**0**answers

191 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 ...

**14**

votes

**2**answers

632 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

473 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 ...

**14**

votes

**2**answers

898 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 ...

**11**

votes

**1**answer

613 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 ...

**35**

votes

**1**answer

2k views

### 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 ...

**11**

votes

**1**answer

518 views

### Is there a canonical map between the cohomology of orbifold Chiral de Rham on an orbifold and 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 ...