Questions tagged [conformal-field-theory]
The conformal-field-theory tag has no usage guidance.
68
questions with no upvoted or accepted answers
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local equivalence of loop group representations
Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group
$$
L_IG ...
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15
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216
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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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346
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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 ...
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conformal blocks for beginners
I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...
- 3,717
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380
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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 ...
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526
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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)....
- 607
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319
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The space-time dimension of the N-superstring theory?
Let $\mathfrak{W}$ be the Lie algebra generated by $d_{n} = ie^{in\theta}\frac{d}{d\theta}$ and $\mathfrak{Vir} = \mathfrak{W} \oplus C \mathbb{C}$ its central extension:
$$
[L_m,L_n]=(m-n)L_{m+n}+\...
- 25.9k
9
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0
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605
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Quantum Drinfeld-Sokolov reduction for a module
There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the ...
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330
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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 ...
- 42.5k
8
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2k
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Neveu-Schwarz and Ramond sector in the free fermion CFT
My question is about the Neveu-Schwarz and the Ramond sector in the free fermion CFT.
The setup is as follows.
We consider two dimensional Minkowski space with a point removed $M = \mathbb{R}^{1, 1}-\...
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7
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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 ...
- 2,901
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466
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Extreme unitary minimal models of conformal field theory
Some of the best understood conformal field theories are the 2D unitary minimal models $\mathcal{M}(m+1,m)$ indexed by the integer $m\ge 2$ and with central charge
$$
c=1-\frac{6}{m(m+1)}\ .
$$
I ...
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7
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552
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Is there an E8 symmetry in the zero-field Ising model?
In the paper On classification of modular tensor categories by Rowell, Stong and Wang, they list the Ising modular category $I$ as having 3 objects $1$, $\sigma$ and $\psi$, with fusion rules $\sigma^...
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Choice of framing in Gravitational Chern Simons
I was trying to understand formula(2.21) in Witten's paper "Quantum Field Theory and Jones Polynomial"(link: https://projecteuclid.org/euclid.cmp/1104178138) (Page 360).
There, it was mentioned, the ...
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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 ...
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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 ...
- 42.5k
6
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0
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888
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Understanding Segal's definition of conformal field theory
I have a fundamental problem in understanding Segal's definition of a conformal field theory:
On the one hand his monoidal CFT-functor is a formalization of the fact that, physically, the integrand ...
- 1,661
5
votes
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answers
170
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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 "...
- 19.6k
5
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137
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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 ...
- 10.3k
5
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120
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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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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 ...
5
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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 ...
- 223
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111
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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 $...
- 969
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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 ...
- 732
4
votes
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answers
97
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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 ...
4
votes
0
answers
276
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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 ...
- 2,901
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111
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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 $\mathbb{C}[L_{-1},L_{-2},\dotsc]...
- 1,746
4
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233
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Computing theta functions of lattices in practice
I am motivated by a problem in 2d CFT to compute "generalized theta functions," expressions of the form
\begin{equation}
\vartheta_{L,u}(\tau) := \sum_{\alpha \in L} u(\alpha) q^{{\langle\...
- 41
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305
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Intuition for conformal nets
I was planning on reading the work of Arthur Bartels, Christopher L. Douglas and André Henriques on the 3-category of conformal nets as discussed in these papers: Coordinate-free nets, Conformal ...
- 430
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answers
224
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What is the value of the partition function of CFT on a compact conformal manifold?
Is the value of the partition function of a 2d CFT on a compact conformal manifold well defined? Or is there some kind of "anomaly" that makes it dependent on a bulk or some other kind of further ...
- 2,901
4
votes
0
answers
195
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3d Chern-Simons TQFT of gauge group (E8)$_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 ...
- 10.3k
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answers
107
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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}...
- 343
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0
answers
108
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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 ...
- 546
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293
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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 ...
- 616
4
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0
answers
300
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Proving conformal invariance of a field theory by property of its stress energy tensor.
I have a question about proving conformal invariance of a field theory by property of its stress energy tensor.
In physics there is argument that when the stress-energy tensor is traceless, symmetry,...
- 49
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596
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Correlation functions of complex operators
One defines the "scaling dimension" (as opposed to "engineering dimension") of an operator $\cal{O}$ as $[\cal{O}]$ such that if $\cal{O}(t^{-1}x) = t^{[\cal{O}]}\cal{O}(x)$ then the Lagrangian in ...
- 3,453
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votes
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answers
165
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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:
$$\int \mathrm{d}x^...
3
votes
0
answers
135
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Summing over roots of a simple Lie algebra and Deligne series
For a simple Lie algebra $\mathfrak{g}$ we can define a Killing form $K(X,Y) \equiv \frac{1}{2 h^\vee}\operatorname{tr}(\mathfrak{ad}_X \mathfrak{ad}_Y)$, where $\mathfrak{ad}_X Y \equiv [X, Y]$ as ...
- 857
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answers
64
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Hecke operators for modular form with respect to $\Gamma_{\theta}(2)$ subgroup
The congruence subgroup $\Gamma_{\theta}(2)$ is defined as:
$$\Gamma_{\theta}(2)=\left\{\gamma\in SL(2,\mathbb{Z})|\gamma\equiv\left(\begin{array}{cc}1 & 0\\ 0 & 1\end{array}\right) \...
- 31
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204
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Vertex operator algebras and modular tensor 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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3
votes
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199
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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 ...
- 191
3
votes
0
answers
39
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 ...
- 378
3
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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). ...
- 437
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132
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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{...
- 10.3k
3
votes
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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 ...
user35360
3
votes
0
answers
100
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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 $\pi$...
- 555
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votes
0
answers
74
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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 ...
- 31
3
votes
0
answers
170
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Self-dual vertex algebras
Let $(V,Y)$ be a self-dual conformal vertex algebra. For instance, it could be the vertex algebra associated to a positive definite, even, unimodular quadratic form. I look for a formula to compute
$$
...
- 2,324
3
votes
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572
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relation between AGT- conjecture and CNV-correspondence
Is there any relation between AGT conjecture 0906.3219 and the 4d-2d correspondence of 1006.3435 ?
For pure SYM of $\mathcal{N}=2$ , SU(2) guage group thoery, we know the explicit instanton ...
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2
votes
0
answers
316
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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 ...
