# Questions tagged [conformal-field-theory]

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

164
questions

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### Some version of non-commutative Wick formula

Let $V$ be a vertex algebra. The traditional non-commutative Wick formula is a tool to calculate term like $[a_\lambda:bc:]$. However, I need to calculate terms of the form $[:ab:_\lambda c]$. I found ...

2
votes

1
answer

109
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### Difference between two definitions of affine Lie algebras

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, we have the notion of affinization of $\mathfrak{g}$, which is the central extension of the corresponding loop algebra.
...

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131
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### Segal's axioms for CFT

In Segal's papers about Conformal Field theory, https://www2.math.upenn.edu/~blockj/scfts/segal.pdf, in section $1$, he describes the evolution of a system (a string moving about in a manifold $M$) by ...

9
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### What is the meaning of chiral in the context of vertex algebras?

There are many objects in mathematics that have the term "chiral" in their name, for instance, chiral algebra by Beilinson and Drinfeld, chiral de Rham complex, chiral Koszul duality etc. ...

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### Summing over roots of a simple Lie algebra and Deligne series

For a simple Lie algebra $\mathfrak{g}$ we can define a Killing form $K(X,Y) \equiv \frac{1}{2 h^\vee}\operatorname{tr}(\mathfrak{ad}_X \mathfrak{ad}_Y)$, where $\mathfrak{ad}_X Y \equiv [X, Y]$ as ...

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### DHR superselection and DR reconstruction in low spacetime dimensions

Given a completely rational net on $\mathbb{R}$, the Doplicher-Haag-Roberts (DHR) category is a modular fusion category (MFC) identical to that associated with the corresponding vertex operator ...

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### Singular vectors/null states in algebra $\mathfrak{su}(2)_{-4/3}$

I encounter recently admissible affine Lie algebras when visiting some physics problems. I am reading Adamovic's A construction of admissible $A_1^{(1)}$-modules of level $−4/3$.
In section 3, it is ...

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78
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### Relation between projective representation and the representation of the universal cover of a Lie Group

I am reading this paper, in what says exactly:
"Weare dealing with a ray representation os the conformal group AND THEREFORE with a representation of the universal covering group of the conformal ...

3
votes

1
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140
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### Equation about Jacobi Theta Functions

Reading some Conformal Field Theory, I came across the following equation
about the Jacobi Theta functions without any justification:
Let $$\theta_{2}(q)=\sum_{n \in \mathbb{Z}}q^{(n+\frac{1}{2})^{2}}$...

4
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185
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### CFT as an axiomatic field theory

I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every ...

3
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### Hecke operators for modular form with respect to $\Gamma_{\theta}(2)$ subgroup

The congruence subgroup $\Gamma_{\theta}(2)$ is defined as:
$$\Gamma_{\theta}(2)=\left\{\gamma\in SL(2,\mathbb{Z})|\gamma\equiv\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right) \...

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256
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### The role of estimates in field theories

I have been taking a look at some papers in constructive quantum field theory and I got the impression that there is a systematic of estimating things like e.g the effective action or the free energy ...

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76
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### Arithmetic analogues in Liouville quantum gravity

I recently discovered about Minhyong Kim's work on what can be coined "Arithmetic Gauge Theory/Arithmetic Chern Simmons Theory". Since Liouville quantum gravity is fully understood, I was ...

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144
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### Vertex operator algebras and modular fusion categories

Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C}=Rep(\mathcal{V})$ be the tensor category of (ususal) $\mathcal{V}$-modules. It is a well-known open-problem whether every ...

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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 ...

3
votes

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162
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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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### 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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106
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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
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1
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706
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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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132
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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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603
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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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109
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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 ...

11
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1
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411
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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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106
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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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183
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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

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133
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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 ...

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2
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668
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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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193
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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
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346
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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
votes

1
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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
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1
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166
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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
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1
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148
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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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answers

182
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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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102
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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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970
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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, ...

4
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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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### 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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219
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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 ...