Questions tagged [conformal-field-theory]
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176 questions
5 votes
1 answer
246 views
How to calculate S-matrix of critical 3-state Pott's model without considering W-algebra?
Let's consider the critical 3-state Potts model. According to conformal field theory, it corresponds to a CFT with a central charge $c=\frac{4}{5}$. However, there are 10 characters for $c=\frac{4}{5}$...
1 vote
0 answers
125 views
Contragradient Module for the Kac-Moody Vertex Operator Algebra
For a vertex operator algebra $(V,Y,\left|0\right>)$ and a $\mathbb{Z}$-graded module $(M,Y_M)$, one can form the contragradient module $(M^{\lor},Y_{M^{\lor}})$ with underlying $\mathbb{Z}$-graded ...
- 1,146
2 votes
0 answers
138 views
Quillen bundles and 2D CFTs
Roughly speaking, a (mathematical) genus-$0$ conformal field theory (CFT) is a projective symmetric monoidal functor $Z$ from $C$ to $GrVec$ [1], where $GrVec$ is the category of graded complex vector ...
- 5,748
0 votes
0 answers
115 views
modular properties of macmahon function?
How does the MacMahon function for counting plane partitions $M(q) = \frac{1}{(1-q^n)^n}$ behave under modular transformations? For instance for $q= e^{2 \pi i \tau}$ where $\tau \rightarrow -1/\tau$.
- 11
1 vote
0 answers
110 views
Boundary-condition-changing Operators for Free Boson BCFT with Dirichlet Boundary Conditions (or more general BCFTs)?
(NOTE: This is a crosspost from this Physics.SE post) Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. ...
- 545
3 votes
1 answer
246 views
Gaussian free field from Liouville quantum gravity?
If $\Sigma$ is a Riemann surface, there are two measures on $\text{H}^s(\Sigma)$: the Gaussian free field $h(z)$ and the Gaussian multiplicative chaos $\mu(z)= \lim_{\epsilon\to0} e^{\gamma h_\...
- 6,181
2 votes
0 answers
157 views
Knot invariants in WZW CFT via Holographic Principle
In the physics literature the Holographic Principle relates theories in the bulk and the theories in the asymptotic boundary. While the bulk theory is the 3D Chern-Simons theory, the corresponding ...
- 5,748
4 votes
1 answer
543 views
WZW primary correlations in terms of current algebra?
Given the $\mathfrak{u}(N)$ algebra with generators $L^a$ and commutation relations $ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ , the WZW currents of $U(N)_k$ $$ J(z) = \sum_{n \in \mathbb{Z}} J^a_n z^{-n-...
- 545
5 votes
0 answers
174 views
Tensor product - Vertex / Chiral algebras
Two questions regarding tensor product of modules over vertex / chiral algebras: First question: For (rational?) vertex operator algebras there is a notion of fusion product of modules inducing a ...
- 1,146
4 votes
1 answer
341 views
Properties of the stress energy tensor in Wightman formulation of CFT
In various papers that I have been reading about applying the Wightman axioms to conformal field theory, the authors write things like the following about the stress-energy tensor: $$\int \mathrm{d}x^...
5 votes
0 answers
186 views
BRST construction of coset VOAs
Most recent papers define cosets of $V_k(g)$ by $V_k(h)$, where $h\subset g$ - some affine (super-)Lie algebras, as a cohomology of a complex $$V_k(g)\otimes V_{-k}(h)\otimes ghosts$$ but I'm failing ...
1 vote
0 answers
75 views
Splitting of the conformal group into $PSL(2,\mathbb{R})$ and other factorizations
In 1+1 dimensions of Minkowski spacetime, the conformal group can be split into two copies of $PSL(2,\mathbb{R})$ acting on null lines. I'm curious to know if a similar split exists for the conformal ...
- 424
0 votes
0 answers
169 views
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 ...
- 1,593
2 votes
1 answer
280 views
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. ...
- 1,593
2 votes
0 answers
607 views
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 ...
12 votes
2 answers
739 views
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. ...
- 1,593
3 votes
0 answers
170 views
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 ...
- 867
2 votes
0 answers
131 views
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 ...
- 437
1 vote
0 answers
136 views
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 ...
- 867
1 vote
0 answers
366 views
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 ...
- 424
3 votes
1 answer
341 views
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 $$\theta_{2}(q)=\sum_{n \in \mathbb{Z}}q^{(n+\frac{1}{2})^{2}}$...
- 31
5 votes
0 answers
457 views
CFT as an axiomatic field theory
I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every ...
- 3,095
3 votes
0 answers
109 views
Hecke operators for modular form with respect to $\Gamma_{\theta}(2)$ subgroup
The congruence subgroup $\Gamma_{\theta}(2)$ is defined as: $$\Gamma_{\theta}(2)=\left\{\gamma\in SL(2,\mathbb{Z})|\gamma\equiv\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right) \...
- 31
5 votes
1 answer
301 views
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 ...
- 1,465
2 votes
0 answers
192 views
Arithmetic analogues in Liouville quantum gravity
I recently discovered about Minhyong Kim's work on what can be coined "Arithmetic Gauge Theory/Arithmetic Chern-Simons Theory". Since Liouville quantum gravity is fully understood, I was ...
- 133
4 votes
0 answers
275 views
Vertex operator algebras and modular fusion categories
Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C} = \text{Rep}(\mathcal{V})$ be the tensor category of $\mathcal{V}$-modules. It is a conjecture by Vaughan Jones whether every ...
- 27.6k
3 votes
0 answers
272 views
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 ...
- 301
4 votes
1 answer
252 views
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 ...
- 439
8 votes
3 answers
1k views
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 ...
- 8,062
4 votes
0 answers
66 views
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 ...
- 418
2 votes
0 answers
261 views
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, ...
- 71
5 votes
0 answers
199 views
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 "...
- 21.1k
16 votes
1 answer
816 views
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 = \...
- 163
6 votes
0 answers
229 views
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 ...
- 3,095
10 votes
2 answers
1k views
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 ...
- 71
2 votes
0 answers
156 views
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 ...
- 131
11 votes
1 answer
584 views
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\...
- 301
29 votes
2 answers
4k views
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 ...
- 3,745
4 votes
0 answers
148 views
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]...
- 1,914
1 vote
1 answer
905 views
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
- 4,864
7 votes
1 answer
352 views
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 ...
2 votes
2 answers
2k views
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 (...
- 157
4 votes
0 answers
337 views
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\...
- 41
5 votes
1 answer
674 views
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 $...
- 8,062
22 votes
4 answers
3k views
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 ...
- 1,499
3 votes
0 answers
112 views
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). ...
- 437
5 votes
0 answers
196 views
$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{...
- 10.8k
4 votes
1 answer
302 views
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'(...
- 1,031
3 votes
0 answers
405 views
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 ...
- 438
7 votes
2 answers
592 views
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 ...
- 585