# Questions tagged [conformal-field-theory]

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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 ...
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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. ...
131 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 ...
273 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. ...
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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 ...
1 vote
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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 ...
1 vote
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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 ...
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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 ...
1 vote
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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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### 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 ...
175 views

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

This question is in some sense a follow-up to : 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 ...
162 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 ...
508 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 ...
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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 "...
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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 ...
256 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 ...
162 views

### 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 ...
125 views

### 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 ... 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 $\... 8 votes 0 answers 327 views ### Differential version of$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 1 answer 166 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 1 answer 148 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 0 answers 182 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 0 answers 102 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 ... 7 votes 1 answer 970 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, ... 4 votes 1 answer 1k 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 1 answer 176 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$\...
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 ...