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7
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2answers
248 views

Operator Product Expansion (OPE) in Conformal Field Theory

We denote local operators of a conformal field theory (CFT) as $\mathcal{O}_i$ where $i$ runs over the set of all operators. Formally, the operator product expansion (OPE) is given by, ...
5
votes
0answers
175 views

Is string theory over a time varying background a conformal field theory to all orders in perturbation theory?

When computing the first order perturbative corrections to string theory over a curved background, we find the background has to be Ricci-flat if the dilaton is constant and we have no fluxes. Such is ...
2
votes
1answer
76 views

Number of zero-modes on the sphere

Is it true that a field of conformal dimension $h$ (integer or half integer) has $1-2h$ zero-modes on the sphere, if $1-2h \geq0$. This seems to be right for different ghost fields : $c$ has ...
5
votes
2answers
122 views

Vertex operator - state mapping in Polchinski's book

In Polchinski's textbook String Theory, section 2.8, the author argues that the unit operator $1$ corresponds to the vacuum state, and $\partial X^\mu$ is holomorphic inside couture $Q$ in figure ...
12
votes
2answers
389 views

Why are there no particles in conformal theories?

In Matt Strassler's recent post (here) he makes the statement that scale invariant (I assume he means conformally invariant, more generally) theories have no particles in them. What's the reason for ...
5
votes
0answers
178 views

What is the connection between Conformal Field Theory and Renormalization group in QFT?

As I know, the fundamental concept of QFT is Renormalization Group and RG flow. It is defined by making 2 steps: We introduce cutting-off and then integrating over "fast" fields $\widetilde{\phi}$, ...
6
votes
0answers
173 views

Toda equations and surface operator

I would like to know the reason why the equation (14) in the paper by Yamada is called the Toda equation. \begin{equation} \left[\frac12\sum_{i=1}^N\left(y_i\frac{\partial}{\partial ...
3
votes
1answer
68 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
3
votes
1answer
106 views

Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is ...
3
votes
1answer
66 views

Stress tensor in product of 2D CFTs

I was struggling with a question, hoping someone could point me in the right direction. I'm interested in 2D CFTs on a cylinder. I want to take the tensor product of two CFTs. My questions are these: ...
8
votes
3answers
120 views

“tmf$(n)$ is the space of supersymmetric conformal field theories of central charge $-n$”

I read this intriguing statement in John Baez' week 197 the other day, and I've been giving it some thought. The post in question is from 2003, so I was wondering if there has been any progress in ...
5
votes
1answer
135 views

Klein factors and Conformal Field Theory

Consider the mode expansion of a (chiral) scalar field confined to a disc with circumference L: $$ \phi(x) = \phi_{0} + p_{\phi} \frac{2\pi}{L} x + \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} ...
8
votes
1answer
281 views

About the general expression of trace anomaly and CFT partition functions

I have put up a question here, http://mathoverflow.net/questions/139685/proof-of-the-general-expression-for-anomaly-in-a-cft-and-its-partition-function Here I am putting up a slightly different ...
5
votes
1answer
350 views

Time-ordering vs normal-ordering and the two-point function/propagator

I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between $::$) is ...
2
votes
1answer
83 views

Steepest descent for Mellin-type integration

Here I would like to see the behavior of a function as an integral when its argument (which is a parameter in the integral) goes to zero. If I try to evaluate an integral $$I(\lambda) = ...
3
votes
1answer
210 views

CFT and the conformal group

Equations 2-7 on page 21 of these notes, http://www.math.ias.edu/QFT/fall/NewGaw.ps seems to give a fairly compact definition of what a CFT is. But I have two questions, This definition is ...
2
votes
1answer
83 views

Normal ordering of the identity operator

I'm puzzled about what should be the normal ordering of the identity operator (or any proportional operator): looking at it from the "Fock space operators POV",the prescription is to move all the ...
3
votes
0answers
73 views

Virasoro Operators commutation relations

For the commutation relation in quantising the bosonic string $\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$ we can then calculate this for $m=-n$ in between the vacuum ...
4
votes
1answer
116 views

Why should a holomorphic function be expanded in Laurent series rather than Taylor series?

In 2d free conformal field theory, there is an operator equation: $$ \partial\bar\partial\hat{X}^\mu\left(z,\bar z\right)=0 $$ Why can it have Laurent expansion like this below rather than Taylor ...
5
votes
2answers
241 views

A calculation problem on Conformal Field Theory

This problem is on Di Francesco's book I. It's exercise 7.1: Calculate the norm of the following vector, where $\lvert h\rangle$ is the state of highest weight. $$L_{-1}^n\lvert h\rangle$$ I have ...
0
votes
0answers
35 views

Shrinking and Expanding objects in a CFT

In a conformal field theory, is it possible to construct a machine that shrinks or expands objects?
2
votes
1answer
92 views

A question about the Bosonization of the Thirring model

Is there a way or sense in which one can Bosonize this kind of a Lagrangian, $L = \bar{\psi}\gamma^\mu \partial _\mu \psi + f(x) \bar{\psi}\psi$ for $f(x)$ being some function on space-time. ...
2
votes
1answer
72 views

Does $c = 0$ implies that the theory is “empty”?

I'm wondering if there is more than the empty theory (no local fields, identically vanishing stress energy tensor) that can have central charge $c$ equals to $0$? My intuition tells me no, the stress ...
9
votes
4answers
357 views

Symmetries of a Free Massless Scalar in Two Dimensions

On p. 49 of Polchinski's book, he says: "Incidentally, the free massless scalar in two dimensions has a remarkably large amount of symmetry -- much more than we will have occasion to mention." Does ...
4
votes
1answer
119 views

Wick's theorem for calculating OPE

I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field. Now, ...
6
votes
0answers
95 views

d=2 O(3) sigma model becomes “conformal antiferromagnet”

In Advanced topic in quantum field theory / M. Shifman on page 251 the author discusses the fact that the theta term is topological and does not affect the equations of motion. Then he said: "In ...
25
votes
1answer
627 views

Sigma Models on Riemann Surfaces

I'm interested in knowing whether sigma models with an $n$-sheeted Riemann surface as the target space have been considered in the literature. To be explicit, these would have the action ...
9
votes
1answer
189 views

Why do we assume local conformal transformations are symmetries in 2D CFT

The global conformal group in 2D is $SL(2,\mathbb{C})$. It consists of the fractional linear transforms that map the Riemann sphere into itself bijectively and is finite dimensional. However, when ...
1
vote
0answers
162 views

Morphisms between chiral CFTs

This is a question about terminology. Given two vertex algebras $V_1$ and $V_2$ (= chiral CFTs), there are two kinds of maps $V_1\to V_2$ that one might want to consider. 1) Morphisms of VOAs that ...
3
votes
1answer
254 views

A key relation in Di Francesco's book on Conformal Field Theory

Recently I am reading Di Francesco's book Volume I on Conformal Field Theory. In order to reduce the number of fields in a correlator, the calculation of Operator Algebra is extremely important. To do ...
9
votes
0answers
131 views

Mathematical motivation of OPE?

In Peskin & Schroeder (and also Cheng which I have skimmed through) they motivate the Operator Product Expansion with a lot of words. Is there any way to motivate it mathematically, e.g. Taylor ...
4
votes
0answers
68 views

(Euclideanized) QFT on $S^d$ vs $S^{d-1}\times S^1$

Broadly I would like to understand what is the difference in the physical interpretation of a (Euclideanized) QFT which is on space-time $S^d$ and which is on a space-time $S^{d-1}\times S^1$. In ...
8
votes
2answers
325 views

Central charge in a $d=2$ CFT

I've always been confused by this very VERY basic and important fact about two-dimensional CFTs. I hope I can get a satisfactory explanation here. In a classical CFT, the generators of the conformal ...
4
votes
1answer
153 views

Hermitian conjugation in Radial Quantization

I'm a little confused about Hermitian conjugation in a radially quantized CFT. Now, in the Minkowski theory, Hermitian conjugation leaves the coordinates invariant, i.e. $t^\dagger = t$ and $x^\dagger ...
3
votes
1answer
121 views

Even-branes in IIA and odd-branes in IIB

The R-R sector of IIA and IIB are respectively given as, $8_s \otimes 8_c = [1]\oplus [3] = 8_v \oplus 56_t$ $8_s \otimes 8_s = [0]\oplus [2] \oplus [4]_+ = 1 \oplus 28 \oplus 35_+$ Now looking at ...
3
votes
0answers
108 views

Questions on entanglement entropy

If the spatial entangling surface is $M$ then it seems that one way to get the entanglement entropy is to think of the QFT on the manifold $S \times M$ where $S$ is a 2-manifold with the metric, ...
4
votes
1answer
404 views

Classical theories and AdS/CFT

When I was editing the Physics.SE tag wiki for ads-cft, I initially wrote something on the lines of : The AdS/CFT correspondence is a special case of the holographic principle. It states that ...
5
votes
2answers
103 views

Do primary fields (in a CFT) satisfy the wave equation?

In a 2-dimensional CFT, do primary fields generally satisfy the wave equation? I know that if they have either purely holomorphic or purely anti-holomorphic, they do. But what about a general primary ...
1
vote
1answer
38 views

Why does a null state correspond to a field that any correlator containing it vanishes?

I am reading the 7th chapter of Di Francesco's CFT book. It builds, for example in section 7.3, a null state |x> which is orthogonal to the whole Verma Module. The author asserts that the field x ...
3
votes
1answer
76 views

Explicit definition of the energy operator in the Ising model

I've simulated a few 2d Ising models at critical temperature on triangular lattice and I'm now trying to check that the correlation functions are right. I alraedy did it for the spin operator ...
2
votes
0answers
48 views

stress tensor in Kazama-Suzuki construction

This is a technical question about equation (2.42) of the original paper [KS] of the Kazama-Suzuki construction. I think the authors did a simple substitution ...
4
votes
1answer
140 views

Decomposition of representations of the Virasoro algebra under $sl(2)$

The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight ...
1
vote
1answer
125 views

Large-N factorization of single-trace operators

Does anyone know where I can find a pedagogical explanation of large-N factorization in SU(N) gauge theories or nonlinear O(N) sigma models (in the latter case the trace corresponds to a dot product). ...
10
votes
0answers
203 views

Orbifold CFT of SU(2)/G and SO(3)/G

In this paper by Dijkgraaf, Vafa, Verlinde, Verlinde, orbifold CFT is discussed. In that paper, it outlined that orbifold CFT provides a way to generate the new theories from the old known ones. ...
2
votes
0answers
90 views

Regulating an infinite sum in the $bc$ CFT

The EM tensor of the $bc$-CFT is $$ T(z) = \colon \partial b c \colon - \lambda \partial \colon b c \colon $$ After expanding in a mode expansion, we find $$ T(z) = \sum_{m} \frac{1}{z^{m+2}} \sum_n ...
2
votes
2answers
98 views

Factor of two differences for free field Green's functions in conformal field theory

I have a question about the expressions for free field Green's functions in conformal field theory. It comes from three origins 1) In Polchinski's string theory volume I p36, it is given $$ ...
8
votes
1answer
379 views

Is Conformal Symmetry Local or Global?

I'm just brushing up on a bit of CFT, and I'm trying to understand whether conformal symmetry is local or global in the physics sense. Obviously when the metric is viewed as dynamical then the ...
1
vote
0answers
110 views

Why Liouville theory is interesting? [closed]

What makes Liouville theory subject to relatively intense research field?
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vote
0answers
88 views

Central charges c and topological ground state degeneracy GSD

A 2+1D topological field theory (topologically ordered states), implies that the topological ground state degeneracy (GSD) on $T^2$ torus (2D manifold without boundary). For example a level k U(1) ...
2
votes
0answers
88 views

Half-integer Spin and “natural conformal dimension”

If we consider a classical field theory for a massless particle of integer spin $s$, in a curved space-time, one finds that it is "naturally" conformal in a space-time of dimension $2+2s$ For ...