Questions tagged [conformal-field-theory]

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8
votes
0answers
302 views

Differential version of $G\mapsto H^3(G,\mathbb Z)$?

Let $\mathit{cLieGrp}^{\mathrm{inj}}$ be the category of compact connected Lie groups, and injective continuous group homomorphisms. Is there a reasonable functor (some kind of degree $3$ differential ...
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 ...
3
votes
1answer
75 views

Non-stationary Lamé equation and WZW/Virasoro conformal blocks

The non-stationary Lamé equation $\left(2i \pi \kappa \partial_\tau -\partial_x^2+g(g-1)\wp(x)\right)\psi(x,\tau)=E \psi(x,\tau)$ appears as a BPZ type equation (for the 2 points Virasoro conformal ...
4
votes
0answers
135 views

What is the value of the partition function of CFT on a compact conformal manifold?

Is the value of the partition function of a 2d CFT on a compact conformal manifold well defined? Or is there some kind of "anomaly" that makes it dependent on a bulk or some other kind of further ...
2
votes
0answers
83 views

Monoidal category of irreducible highest weight modules of the Virasoro algebra

I'm trying to see if I can construct a monoidal category $\mathbf{C}$ whose objects are the irreducible unitary highest weight representations of the Virasoro algebra. I am thinking on doing the ...
5
votes
1answer
505 views

The use of Schur's lemma for Lie algebras in physics (CFT)

Anytime a one-dimensional central extension appears in the physics literature, immediately they assume that in any irreducible representation the central charge will be a multiple of the identity, ...
1
vote
1answer
468 views

Holomorphic maps from upper half plane to itself (or equivalently Poincare disc to itself)

Suppose I parametrize complex plane by coordinates,$$z = x+i y,\ \bar z=x-i y$$ then the upper half plane, $\mathbb H_+$ is given by $y>0$. I am looking for chiral coordinate transformations, $f(z)$...
3
votes
0answers
90 views

GKO (or coset) construction - all possible highest weights $h$

I am reading the famous paper "Unitary Representations of the Virasoro and Super-Virasoro Algebras" by Goddard, Kent, Olive. From a compact simple Lie algebra $\mathfrak{g}$ and a Lie subalgebra $\...
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
0answers
100 views

Conformal group and cobordism

In this post, I am exploring my thoughts on the implementation of conformal symmetry group structure and cobordism relations. Namely, I like to know what has been done and explored in the past? on ...
4
votes
0answers
114 views

3d Chern-Simons TQFT of gauge group (E8)$_1$ = SO(16)$_1 \otimes$ a trivial spin TQFT = Cartan E$_8$ matrix

In this post, we like to relate the following 3 bosonic TQFTs that can be defined on generic non-spin manifold $M^3$. Given a non-abelian Chern-Simons (CS) TQFT of a gauge group $G$ and the $k$ named ...
5
votes
0answers
162 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-...
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 ...
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 ...
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 ...
11
votes
0answers
143 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
76 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-...
5
votes
1answer
173 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 ...
7
votes
1answer
329 views

Vector bundle structure of conformal block bundle

My question is about the conformal block bundle, which (following Kohno's "Conformal Field Theory and Topology") is constructed as follows: Consider the projection map onto the first $n$ coordinates \...
17
votes
2answers
612 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
547 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 $...
10
votes
3answers
473 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 ...
5
votes
0answers
99 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 ...
2
votes
0answers
98 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\...
12
votes
1answer
374 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 ...
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
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 ...
10
votes
1answer
187 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 ...
4
votes
0answers
85 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}...
5
votes
0answers
74 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
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(...
4
votes
0answers
60 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 ...
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. ...
2
votes
1answer
83 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 $$ ...
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 ...
7
votes
2answers
377 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
84 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 $\pi$...
9
votes
0answers
363 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)....
33
votes
7answers
4k 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
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 $\...
3
votes
0answers
49 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
337 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 ...
9
votes
0answers
227 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
117 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
318 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 ...
6
votes
1answer
476 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
157 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)...
3
votes
0answers
195 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 ...
8
votes
1answer
337 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 ...