Questions tagged [conformal-field-theory]
The conformal-field-theory tag has no usage guidance.
4
votes
0answers
68 views
$T\bar{T}$ deformation: Stress-energy momentum tensor deformed in CFT and in QFT for various $d$-dimensions
The $T\bar{T}$ deformation is based on the original work of Zamolochikov [1] explored deformations of two-dimensional conformal field
theories (CFT) by an operator that is quadratic in the stress-...
3
votes
1answer
87 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 ...
4
votes
1answer
94 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
2answers
123 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 ...
10
votes
0answers
104 views
Geometry of Affine Kac-Moody Algebras
I recently asked this question on phys.SE and it was suggested to me to ask it here.
One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric ...
4
votes
0answers
57 views
GSO projection and $H^d(M, \mathbb{Z}_2)$
This follows up the comment which suggests that asking the later 2nd part of subquestion in "GSO (Gliozzi-Scherk-Olive) projection and its Mathematics" as a new different question
GSO (Gliozzi-Scherk-...
4
votes
1answer
130 views
Nonlinear sigma models with non-compact groups / target spaces
A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T.
The target manifold T is equipped with a Riemannian metric g. Σ is ...
1
vote
0answers
67 views
Vector Bundle Structure of Conformal Block Bundle
Apologies in advance if this question is too elementary but I wasn't expecting any luck asking on math.stackexchange. Anyway, my question is about the conformal block bundle, which (following Kohno's "...
16
votes
2answers
426 views
What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?
Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$
superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $...
24
votes
1answer
525 views
Has the $E_8$-based generating function for squares numbers been proven?
In his 2004 paper Conformal Field Theory and Torsion Elements of the Bloch Group, Nahm explains a physical argument due to Kadem, Klassen, McCoy, and Melzer for the following remarkable identity. Let $...
8
votes
3answers
411 views
Is there a discrete lattice analogue of conformal transformations?
There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under ...
4
votes
0answers
80 views
Modular $S$-matrix for an extended affine Lie algebra
This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.
In this paper, the authors ...
1
vote
0answers
75 views
Modular transformation of affine characters of non-simply connected groups$.$
Consider an (untwisted) affine algebra corresponding to a compact and simply-connected Lie group $G$. Under a modular transformation, its characters transform as (cf. 9612078)
$$
\chi_\mu\to\sum_{\nu\...
9
votes
1answer
285 views
Compactification of 6d (2, 0) SCFT on 4-manifolds
This question is about the 6d (2, 0) superconformal field theory (also called 'theory X' by some people). This SCFT, which can be considered as a relative quantum field theory (see here for a ...
10
votes
1answer
209 views
Linear independence of genus-one correlation functions
Let $V$ be a vertex operator algebra with all the good finiteness properties that people usually assume (positively graded, $C_2$-cofinite, $V\cong V'$, etc.)
Let $W$ be a module for $V$, not ...
4
votes
0answers
201 views
Reference Request: Vertex Algebras
I am currently a graduate student in mathematics with an interest in vertex algebras. I am comfortable with the algebraic aspects and would like to learn more about the geometric aspects. The issue is ...
10
votes
1answer
172 views
Is the “Ramond sector” invariant of a 3-framed lattice always divisible by 24?
For the purposes of this question, a rank-$r$ (integral) lattice is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It ...
3
votes
0answers
77 views
Moduli spaces for the TCFT map $HH(L) \to GW(X)$
Let $L$ be a Lagrangian submanifold of a closed symplectic manifold $X$. What I gather from Costello (see specifically $\S$2.5 there), is that one expects to have a morphism of closed TCFT's
$\tag{1}...
4
votes
0answers
55 views
Conformal Dimension and Highest Weight States of Coset CFT
I am trying to understand the vertex operator algebras of the following form:
$$\frac{U(M|N)_{k_1}}{U(L)_{k_2}}$$
Where $U(M|N)$ is the unitary supergroup, $U(L)$ is the usual unitary group, and $...
9
votes
2answers
241 views
Annihilation operators in a vertex algebra
Let $V=\bigoplus_{d\in\mathbb N}V(d)$ be a Möbius-covariant vertex algebra with $V(0)=\mathbb C$.
Recall that a vector $v\in V$ is called quasi-primary if $L_1v=0$.
For $v\in V(d)$, we write $Y(...
3
votes
0answers
54 views
Section of the spinor bundle over $S^{1}$ that extend to sections of the spinor bundle over $D^{2}$
Let $\mathbb{S} \rightarrow S^{1}$ be the spinor bundle associated to the connected double cover $\text{Spin}(S^{1}) \rightarrow S^{1}$. Let $\mathbb{D} \rightarrow D^{2}$ be the spinor bundle ...
1
vote
1answer
85 views
Two definitions of conformal inclusion
Assume that $V$ is a vertex operator algebra, and the VOA $V'$ is a vertex subalgebra of $V$. The notion that $V'\subset V$ is a conformal inclusion has different meanings in different literatures. ...
2
votes
1answer
78 views
Limits of a quasiperiodic function with two pseudoperiods
Let $\beta$ be a real number such that $\beta^2\notin\mathbb{Q}$. For any smooth function $f$ on $\mathbb{R}$ that decreases sufficiently at infinity, for example a Gaussian function, let us define
$$
...
20
votes
1answer
632 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 ...
7
votes
2answers
305 views
What is the strongest known RSW result in planar percolation?
One of the weakest estimates conjectured to hold for critical planar percolation models (and proved in many cases) is the so-called RSW estimate. RSW estimate is the statement that the probability of ...
3
votes
0answers
77 views
Does von Neumann density imply strong additivity of a conformal net?
Let $\mathcal A$ be a conformal net, and let $\mathcal J$ be the set of all proper open sub-intervals of $S^1$.
We say that $\mathcal A$ satisfies von Neumann density, if for any representation $\pi$...
8
votes
0answers
260 views
Mysterious relationship between central charges of conformal field theories and the Beraha numbers
Background:
Conformal field theories (CFTs) in two dimensions are partially characterized by a so-called central charge (characterizing the central extension of the Virasoro algebra which defines it)....
30
votes
7answers
3k views
Why is conformal invariance only possible for massless theories?
I'm conscious that this isn't necessarily a research level question, but I've asked this question on mathstackexchange, and received no answer. So I'm trying it here.
A usual mantra in field theories ...
1
vote
0answers
130 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 $\...
3
votes
0answers
41 views
VOA affine algebras
What is the difference between decomposition of tensor product
of affine algebra VOA modules and their fusion product decomposition?
I heard that for the former central extension parameter is ...
7
votes
1answer
232 views
Variation of the Green function with respect to the metric
Consider a (closed) Riemann surface and let $G(x,y)$ be the Green function of the Laplace-Beltrami operator. We can informally identify $G$ with the two-point correlation function for the Gaussian ...
8
votes
0answers
162 views
Equivalence classes of Wilson lines in $SU(2)_k$ Chern-Simons theory
One basic aspect of the 3D TQFT/2D CFT correspondence that I'd like to understand better is the following. It is often said that the ground states of Chern-Simons theory on a (spatial) torus are in ...
4
votes
1answer
110 views
Strong additivity of the conformal net of an affine simple Lie algebra
Let $\mathfrak g$ be a complex simple Lie algebra, $l$ be a natural number, and $V=V^l(\hat g)$ be the vertex operator algebra of the affine Lie algebra $\hat{\mathfrak g}$ at level $l$. We know that $...
14
votes
0answers
299 views
What is the mathematical name for the anomaly for an action of a group on a lattice conformal field theory?
Suppose $V$ is a (bosonic) chiral conformal field theory which is "holomorphic" in the sense that its category of vertex modules is trivial. (The definition of "chiral conformal field theory" might be ...
5
votes
0answers
316 views
Lagrangian formulation of the Ising model as a conformal field theory
An important example of conformal field theory is the 2d Ising model, more precisely its scaling limit when the size of the lattice goes to zero. I am not an expert in the field, but this is the only ...
1
vote
1answer
142 views
Large spin expansion of affine $\mathfrak{su}(2)_k$ characters
There is a problem I am trying to solve for some time now which in a few words boils down to computing the coset characters for
$$
\frac{\mathfrak{su}(2)_k\oplus\mathfrak{su}(2)_\ell}{\mathfrak{su}(2)...
2
votes
0answers
131 views
The Hitchin connection as a twisted D-module?
The famous Hitchin connection is a flat projective connection on the (projectivization of) the vector bundle of non-abelian theta functions, over the moduli space of curves $\mathcal{M}_g$. There are ...
7
votes
1answer
301 views
Conformal blocks in genus zero
In section 10.4 of "Vertex Algebras and Algebraic Curves", Ben-Zvi & Frenkel (second edition), the authors claim that for any vertex algebra V, the space of one-pointed conformal blocks with ...
28
votes
3answers
1k views
What's the supersymmetric analogue of the Monster group?
Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...
6
votes
1answer
215 views
Do we have a braided tensor category for vertex algebra modules by using conformal blocks on an arbitary compact Riemann Surface?
In Huang & Lepowsky's series of papers A theory of tensor products for module categories for a vertex operator algebra, they defined for a rational vertex algebra $V$ the $P(z)$ tensor product of ...
3
votes
0answers
131 views
Self-dual vertex algebras
Let $(V,Y)$ be a self-dual conformal vertex algebra. For instance, it could be the vertex algebra associated to a positive definite, even, unimodular quadratic form. I look for a formula to compute
$$
...
6
votes
1answer
248 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
268 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
0answers
276 views
Extreme unitary minimal models of conformal field theory
Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge
$$
c=1-\frac{6}{m(m+1)}\ .
$$
I ...
8
votes
1answer
166 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 $\Delta_a\ge 0$ ...
12
votes
1answer
360 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
556 views
Understanding Segal's definition of conformal field theory
I have a fundamental problem in understanding Segal's definition of a conformal field theory:
On the one hand his monoidal CFT-functor is a formalization of the fact that, physically, the integrand ...
10
votes
2answers
968 views
How should a mathematician approach the physics literature concerning percolation?
I would like to read some of the physics literature on two-dimensional percolation, however in attempting this I have run into two problems. (1) Physics papers on percolation are (relatively) hard ...
6
votes
1answer
359 views
central charge and Calabi-Yau dimension
I would like to know if there is any setting where the two notions of
central charge of 2D conformal field theories,
Calabi-Yau dimension of fractionally Calabi-Yau categories
can be understood as "...
1
vote
0answers
157 views
Conformal welding of annuli
This question is similar to that stated in Conformal Welding Reference:
Let $\Sigma$ be a 1-dim compact connected complex manifold with boundary $\partial \Sigma= \partial\Sigma^+ \cup \partial\...
