The conformal-field-theory tag has no wiki summary.
2
votes
2answers
173 views
In what condition is a conformal flat manifold flat?
$g^{\mu\nu}(x)=\Omega^{2}(x)g'^{\mu\nu}(x)$ is a conformal transformation.
If $g'^{\mu\nu}$ is flat, what kind of $\Omega(x)$ is choosed can make $g^{\mu\nu}$ flat.
We can think about any dimension ...
16
votes
3answers
253 views
Interpreting the CS/WZW correspondence
It is understood that there is a correspondence between the 3d Chern-Simons topological quantum field theory (TQFT) and the 2d Wess-Zumino-Witten conformal quantum field theory (CQFT). A good summary ...
9
votes
1answer
231 views
$\text{Rep}(D(G))$ as representation category of a vertex operator algebra
The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as ...
2
votes
0answers
77 views
Subbundles of conformal blocks bundles over $\overline{\mathcal{M}}_{g,n}$
Let $\mathfrak{g}$ be a semisimple split Lie algebra over a field $k$ of characteristic zero. Let $l$ be a non-negative integer and let $P_{l}^{n}$ be the set of $n$-tuples of dominant integral ...
2
votes
0answers
55 views
Vertex Operators with nonlinear terms in Conformal Field Theory
Suppose that we have a simple theory described by the following Hamiltonian
$$
H=\sum_{k>0}k a_k^{\dagger}a_k,
$$
where $[a_k,a_{k'}^{\dagger}]=\delta_{k,k'}$. We can usually define the Vertex ...
7
votes
1answer
183 views
Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)
A quick Question:
Is there some duality known between the quasi Hopf algebra
$D^\omega(H)$ of a finite group $H$ to an orbifold model (such as
SU(2)/$G$ or SO(3)/$G$ orbifold of some ...
20
votes
5answers
955 views
Verlinde's formula
"Verlinde's formula" predicts the dimension of the space of conformal blocks of a chiral CFT.
Depending on...
• which chiral CFT one considers (does one restrict to WZW models, or not?)
...
1
vote
0answers
101 views
What are Euler density and Weyl invariants?
I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold)
Do many (which?) of them vanish when ...
3
votes
0answers
241 views
A gentle introduction to CFT [closed]
1) Which is the definition of a conformal field theory?
2) Which are the physical prerequisites one would need to start studying conformal field theories?
(i.e Does one need to know supersymmetry? ...
7
votes
0answers
182 views
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:
$$
...
5
votes
0answers
163 views
Are there exactly solvable CFTs?
I am wondering if there are CFTs such that n-point correlation functions in them of the fields (may be the primaries or of some notion of twist fields) is exactly known.
Are there such?
Aren't ...
11
votes
1answer
770 views
Is this error in this paper of Langlands fixable?
The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker (1), but the mathematicians found their proof insufficient, so that, FQS (2) and Langlands (3), published in the ...
4
votes
0answers
79 views
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, ...
18
votes
1answer
679 views
Has the conformally field theoretic metric on a Calabi-Yau variety been proved to exist?
Let $(X,g)$ be a compact Kähler manifold. Physics allows us to consider a supersymmetric
sigma model with target $(X,g)$, which is a N=2 two-dimensional field theory.
From the two-dimensional point ...
3
votes
0answers
127 views
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 ...
2
votes
0answers
208 views
fake Calabi-Yau threefold
(1) What is a "fake Calabi-Yau threefold"? Can I describe as a complex or symplectic manifold with trivial canonical bundle, but no compatible Kahler structure? Some mathematicians actually seem to ...
7
votes
1answer
504 views
wrapping M5-branes on a Riemann surface
AdS/CFT gives us a way to use geometry to study field theory! I am trying to wrap M5-branes on a Riemann surface $\Sigma_{g}$. In my problem, for a Riemann surface in 11d, the normal bundle is max ...
2
votes
2answers
250 views
pseutotensor category
In the paper B. Bakalov, A.D'Andrea and V.G.Kac, Theory of finite pseudoalgebras, section 3, one finds the following definition of pseudotensor category:
A pseudotensor category is a class of ...
22
votes
6answers
4k views
Explanations for mathematicians, about the falsifiability (or not) of string theory [closed]
Like many other mathematicians, I think string theory very attractive. This theory has wonderfully influenced many new topics in mathematics (I myself have worked on one of them), but it's not the ...
5
votes
5answers
387 views
Example for non equivalent rational full CFTs with same modular invariant (partition function)
I am looking for a counter example which shows, that a full rational 2D CFT (with respect to a given chiral subtheory) is not characterized by its modular invariant partition function. People tell me ...
11
votes
2answers
776 views
What happens to Virasoro at c=25?
The Virasoro algebras $Vir_c$ are a family of infinite dimensional Lie *-algebras parametrized by a real number $c$, called the central charge.¹
I hear that there exist two critical values of the ...
12
votes
2answers
474 views
Is this a vertex algebroid?… What is vertex algebroid?
A couple of day ago, I was lamenting to a friend about the fact that I have no idea what vertex algebroids are.
During our discussion, I came up with a guess of what a vertex algebroid might be.
I'm ...
1
vote
1answer
168 views
Analytic curve on Riemann surface
Suppose there is a closed analytic curve $C$ on a Riemann surface $S$, that is the image of a map $\gamma$ from the equator $E$ of the Riemann sphere to the surface $S$ which is a restriction of a ...
11
votes
2answers
507 views
What do correlation functions compute in CFT?
I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power ...
10
votes
0answers
524 views
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 ...
1
vote
0answers
222 views
definitions of primary fields
I have come across two similar definitions of primary fields in conformal field theory. Depending on what I am doing each definition has its own usefulness. I expect both definitions to be compatible ...
1
vote
0answers
196 views
Algebra out of a set of modules of a Lie algebra? Fusion
The problem I faced is how to organize a set of finite-dimensional irreducible representations $U_\alpha$ of some simple Lie algebra $g$ into an Lie algebra $A$ that contains $g$ as a Lie subalgebra ...
4
votes
1answer
161 views
Why is there a discrepancy between the normalizations of the central terms for the commutation relations of the Virasoro versus Neveu-Schwarz Lie algebras?
Following the standard conventions in the literature, the commutation relations of the Virasoro Lie algebra are given by
$$[L_m,L_n]=(m-n)L_{m+n}+\delta_{m,-n}\frac1{12}(m^3-m)c,$$
$$[c,L_n]=0.$$
...
14
votes
1answer
379 views
Character of parity-twisted supersymmetric VOA module — question inspired by the Stolz-Teichner program
I'll begin with some background that is unnecessary for the actual question, but that might be interesting to the reader:
Topological modular forms ($TMF$) is a generalized cohomology theory whose ...
13
votes
5answers
2k views
The Chern-Simons/Wess-Zumino-Witten correspondence
I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship.
I guess in the condensed matter ...
5
votes
1answer
324 views
Is there a fusion rule in positive characteristic?
Verlinde's fusion gives a certain "tensor product" of representations of loop groups. The category of representations of loop groups has (essentially equivalent) two incarnations. One is analytic, ...
3
votes
1answer
326 views
eigenspinors of Dirac operator
$M$ compact manifold. Let $\lambda$ be an eigenvalue for the Dirac operator of multiplicity greater than 2. I'm interested in showing the existence of two linearly independant eigenspinors $u$ and $v$ ...
4
votes
0answers
312 views
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 ...
12
votes
1answer
450 views
What is known about the connection of positive energy representations of loop groups and modular forms
At the end of Section 14.1 in Pressley, Segal "Loop Groups" there is the remark that
the partition function is a modular function in the sense that the Dedekind $\eta$ function is a modular form. I ...
42
votes
3answers
2k views
What exactly is the relation between string theory and conformal field theory?
Maybe it would be helpful for me to summarize the little bit I
think know. A 2D CFT assigns a Hilbert space ${\cal H}$ to a circle and
an operator
$$A(X): {\cal H}^{\otimes n}\rightarrow {\cal ...
32
votes
1answer
1k views
H^4 of the Monster
The Monster group $M$ acts on the moonshine vertex algebra $V^\natural$.
Because $V^\natural$ is a holomorphic vertex algebra (i.e., it has a unique irreducible module), there is a corresponding ...
18
votes
0answers
636 views
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 ...
10
votes
1answer
425 views
Is there a canonical map between the cohomology of orbifold Chiral de Rham on an orbifold and the cohomology of Chiral de Rham on a crepant resolution?
The two-variable elliptic genus is a topological invariant of almost-complex manifolds that takes values in power series. These power series turn out to describe weak Jacobi forms when the manifold ...
14
votes
1answer
653 views
Codes, lattices, vertex operator algebras
At the end of "Notes on Chapter 1" in the Preface to the Third Edition of Sphere packings, lattices and groups, Conway and Sloane write the following:
Finally, we cannot resist calling attention ...
2
votes
0answers
456 views
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 ...
2
votes
1answer
464 views
Even lattices and binary codes
I have a maybe simple question about even positive-definite lattices and lattices coming from binary codes. They seemed to be used in framed vertex operator algebras.
What is known about even ...
15
votes
3answers
703 views
central extensions of Diff(S^1) and of the semigroup of annuli
$\mathit{Diff}(S^1)$ refers to the group of orientation preserving diffeomorphisms of the circle. The semigroup of annuli $\mathcal A$ is its "complexification": the elements of $\mathcal A$ are ...
2
votes
0answers
279 views
Collection of charged line segments in 2D - where do electric field lines meet it?
Suppose we have a collection of charged line segments in 2D. I'd like to be able to do two things :
from an arbitrary point in the plane, follow the electric field and find where it meets the ...
8
votes
2answers
811 views
Character of the Basic Representation for Affine E_8 in Terms of Jacobi Theta Functions
When $\mathfrak g$ is a complex, simple, simply laced Lie algebra of rank r then the (specialized) character of the basic representation for the corresponding affine Lie algebra $\hat {\mathfrak g}$ ...
19
votes
1answer
710 views
Conformal blocks vector bundles on $\overline{M}_{g}$ in terms of generalized theta functions?
Conformal field theory uses representation theory to produce various vector bundles on the Deligne-Mumford compactified moduli spaces $\overline{M}_{g}$ and $\overline{M}_{g,n}$, known as bundles of ...
3
votes
1answer
847 views
Witten Index, letter partition function and superconformal representations.
Except in a few papers I have seen so little written about this that I am not sure I can even frame this question properly.
I would like to know of expository references and explanations on the ...
5
votes
1answer
2k views
3D conformal mappings
Are there analogues to conformal mapping in 3 dimensions?
I have a specific example I am trying to solve.. Laplace's equation in 3D with slightly complicated rectilinear boundaries. (Think of ...
16
votes
5answers
3k views
What do mathematicians currently do in conformal field theory (or more general field theory)
I am wondering what currently our mathematicians do related to conformal field theory, (I know currently it is a central topic, but I have only a vague idea what mathematicians do in there), or more ...
6
votes
2answers
2k views
Normal Ordering with Vertex Operators in Conformal Field Theory
The "definition" of the normal ordering in CFT looks a bit vague to me.
I found the definition in terms of exponentiated functional derivative pretty opaque.
Also in this context it might help if ...
8
votes
1answer
415 views
How many embeddings are there of super-Virasoro into n Fermions?
What is the space of N=1 super-Virasoro vertex superalgebras inside the c=n/2 free fermion vertex superalgebra? [Said differently, how many Neveu-Schwartz vectors are there in n fermions?] Answers ...
