# Questions tagged [conformal-field-theory]

The conformal-field-theory tag has no usage guidance.

151
questions

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### Representations of minimal model primary fields in the Coulomb-gas Formalism

This question is in some sense a follow-up to [1]: is it known how to construct the primary field operators of the unitary minimal models $\mathcal{M}(m+1,m)$ in the Coulomb gas formalism? (This would ...

2
votes

1
answer

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### Conformal groupoid

I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is ...

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3
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### Elegant proofs of $\bar{\partial}z^{-1} = 2\pi \delta_0$

For a function $f(x,y)$ on $\mathbb{R}^2,$ defined possibly outside the origin, write
$$\int_\epsilon ' f \,dx\,dy : = \int_{\mathbb{R}^2\setminus D_\epsilon}f \, dx\,dy,$$
(the integral on the ...

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### Number of solutions of an infinite linear system

Let $F_1(z), F_2(z), F_3(z), \cdots$ be an infinite sequence of functions of a continuous variable $z\in \Omega$ with $\Omega$ an open subset of $\mathbb{C}$. The functions $F_n(z)$ are holomorphic on ...

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### Why do quantum observables form an associative algebra in some contexts?

In elementary quantum mechanics, we learn that quantum observables are self-adjoint operators that act on the Hilbert space of states.
However, in more advanced context, we talk of local operators, ...

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### Hypergeometric embedding of conformal blocks into twisted cohomology of configurations

In brief terms, the identification of
$\mathfrak{sl}_2$ lowering operators "$f$" applied "in a conformal block" at the $i$th puncture $z_i$ in the Riemann sphere with
the "...

16
votes

1
answer

679
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### From a physicist: How do I show certain superelliptic curves are also hyperelliptic?

As the title suggests, I am a physicist and have a question about how to show certain superelliptic curves are also hyperelliptic. The superelliptic Riemann surfaces in question has the form $$w^n = \...

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### What are the generators and relations of the conformal cobordism category?

According to a definition by Segal, a $2$-dimensional CFT is a symmetric monoidal functor from the category of oriented conformal cobordisms to the cateogry of projective complex vectorspaces. Coming ...

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### Reference on the Chern-Simons theory and WZW models for mathematicians

I would like to ask if there are any beginner friendly references for learning CS theory and WZW models. It seems that most mathematical texts on the subjects begin with convenient definitions that ...

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### Is there a treatment to relate the Multiple scale analysis or scale separation to the usage of the CFT especially in the perturbation?

In the recent years one started to think weather the abstract group treatment to the Multiple-scale analysis and the scale separation could be uniformly obtained. Especially, one attempted to search ...

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### Is the timelike free boson CFT a valid CFT as per Segal's functorial CFT prescription?

Is the timelike free boson CFT a valid CFT as per Segal's functorial CFT prescription?
I am aware that the Euclidean free boson theory is a well-defined CFT, but I was wondering whether one might run ...

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### Wightman QFTs corresponding to minimal models

Is it known (rigorously) whether or not there exist (1+1)D Wightman QFTs which can (in some reasonable sense) be said to correspond to physicists' unitary minimal models $\mathcal{M}(m+1,m)$, $m\in\...

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### How do we give mathematical meaning to 'physical dimensions'?

In so-called 'natural unit', it is said that physical quantities are measured in the dimension of 'mass'. For example, $\text{[length]=[mass]}^{-1}$ and so on.
In quantum field theory, the dimension ...

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### The semiclassical limit of Virasoro reps $\varphi_{n,1}$ produces certain $\mathfrak{sl}_2$ reps — what is the connection to KdV?

The semiclassical ("light") limit $c\to \infty$ of the irreducible Virasoro representation $\varphi_{n,1}$ with highest weight $h_{n,1}\to -\frac{n-1}{2}$ is $\mathbb{C}[L_{-1},L_{-2},\dotsc]...

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1
answer

103
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### Modular S-matrix of (p,q) minimal model

What is the expression for the modular S-matrix of (p,q) minimal model? The Wiki https://en.wikipedia.org/wiki/Minimal_model_(physics) does not provide S-matrix

6
votes

1
answer

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### Tensor representations of the quantum algebra $U_q(\mathfrak{sl}(2))$ at the roots of unity

I'm trying to understand how the representation theory of $U_q(\mathfrak{sl}(2))$
works and I had a look to some books and lecture notes available on the internet. The case of $q^m\neq1$ is discussed ...

1
vote

2
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376
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### What is a simplified intuitive explanation of conformal invariance? [closed]

Can the concept of conformal map and conformal Invariance be explained in very general terms, preferably in high school/undergrad-level Mathematics? Abstracting away from the applications in physics (...

4
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### Computing theta functions of lattices in practice

I am motivated by a problem in 2d CFT to compute "generalized theta functions," expressions of the form
\begin{equation}
\vartheta_{L,u}(\tau) := \sum_{\alpha \in L} u(\alpha) q^{{\langle\...

5
votes

1
answer

283
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### Uses for (Framed) E2 algebras twisted by braided monoidal structure

$\newcommand{\C}{\mathcal{C}}$ $\newcommand{\g}{\mathfrak{g}}$
If $\C$ is a monoidal category (not necessarily a symmetric monoidal category), it's possible to define the notion of an algebra object $...

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4
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### Mathematical predictions of AdS/CFT

What sorts of mathematical statements are predicted by the AdS/CFT correspondence?
My "understanding" (term used very loosely) is that this correspondence isn't a mathematically rigorous ...

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### Reference for NIM-rep theory for non-commutative fusion rings?

The literature on nonnegative integer matrix representations (NIM-reps) seems to be focused on commutative fusion rings, since a primary motivation there is for rational conformal field theory (RCFT). ...

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### $e^{2\pi ic_{-}/8}$ and $e^{2\pi ic_{-}/24}$ in unitary modular category (UMC)

Background
Unitary modular categories (UMC) do not capture the central charge $c_{-}$ of the topological quantum field theory (TQFT). However, there is a relation that fixes,
$c_{-}\bmod
8$:
\begin{...

3
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1
answer

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### Intuition behind contragredient module of a VOA

Let $(V,Y)$ be a vertex operator algebra, and $V'$ be the graded dual of its underlying vector space. The contragredient module structure on $V'$ is given by $Y'$ defined by the formula:
$$\langle Y'(...

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### Intuition for conformal nets

I was planning on reading the work of Arthur Bartels, Christopher L. Douglas and André Henriques on the 3-category of conformal nets as discussed in these papers: Coordinate-free nets, Conformal ...

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### Tsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free

I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to ...

3
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1
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### Zhu's $V/C_2(V)$ algebra

As an example, take the Virasoro algebra, i.e. $V$ is spanned by elements of the form $L_{-2}^{k_1} \cdots L_{-n}^{k_{n-1}} \Omega$ where $\Omega$ is the vacuum and $n \geq 2$. As I understand, we ...

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### Twisted screening operators and twisted free-field realizations of $\mathcal{W}_n$ algebras

Let $\mathfrak{g}=\mathfrak{sl}_{n+1}$ and I am interested in the principal $\mathcal{W}$-algebra of $\mathcal{g}$ at self-dual level i.e. $k=- h ^{\vee} +1$, usually denoted by $\mathcal{W}_n$. Now ...

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### coset of affine Lie algebra

In many books about conformal field theory, when we talk about a coset $\mathfrak{g}_k/\mathfrak{h}_{k'}$, we would talk about how the modules of $\mathfrak{g}_k$ are decomposed into those of $\...

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### 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

1
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153
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### 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

1
answer

123
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### 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
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### 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 ...

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### 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 ...

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votes

1
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### 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, ...

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1
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### 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, $f(z)$...

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1
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### 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 $\...

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### 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 ...

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### 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
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0
answers

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### 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 ...

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### $T\bar{T}$ 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-...

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1
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### 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 ...

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1
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### 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 ...

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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 ...

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### 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 ...

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### 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 (...

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### 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 ...

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1
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### 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
\...

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### 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 $...

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1
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### 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 $...

12
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### 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 ...