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12
votes
2answers
384 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
170 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
66 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
105 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
63 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
132 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
280 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
203 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
82 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
72 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
239 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
90 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
356 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
118 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
626 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
187 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
248 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
130 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
318 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
146 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
120 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
107 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
401 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
75 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
139 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
120 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
198 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
89 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
97 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
373 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?
1
vote
0answers
87 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
86 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 ...
2
votes
1answer
175 views

Stress-energy Trace of Massless Klein Gordon Field

I've calculated the trace of the stress-energy for a massless KG field and I keep getting $T = - (\partial \phi)^2$ in 3+1 dimensions. I'm using $$T_{\mu\nu} = \partial_\mu \phi \partial_\nu \phi - ...
2
votes
1answer
98 views

Transferring CFT correlations from $\mathbb{R}^3$ to $S^3$

There seems to be a simple method to transfer a CFT's correlations from $\mathbb{R}^3$ to $S^3$ but I am not understanding why it is supposed to work. The idea is that somehow because, $ds^2_{S^3} = ...
1
vote
0answers
68 views

Action of conformal generators on fields

I am calculating the action of the conformal generators on fields, to be more precise on wavefunctions. For now, I'm classical. I will just paste the part of my report on this to show what I am ...
3
votes
0answers
105 views

Moduli Space of $\mathcal{N}=4$ SYM on $\mathbb{R} \times S^3$

When we define $\mathcal{N}=4$ SYM on flat Minkowski space, the supersymmetric vacua are parametrized by scalars living in the cartan subalgebra of the gauge group. A generic point in the moduli space ...