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1answer
35 views

What are marginal fields in CFT?

In this article they call weight $(h,\bar{h})=(1,1)$ fields marginal. Why are these fields called marginal? Why are they to be distinguished.
3
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1answer
38 views

Reading list and book recommendation on Conformal Field Theory

I have a background in QFT, GR and differential geometry at the level of a master student in theoretical physics. I would like to touch the area of CFT. I know the textbook of Philippe Di Francesco. ...
1
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1answer
29 views

The spin and weight of a primary field in CFT

A primary field in Conformal Field Theory transforms as $$\phi (z,\bar{z}) =\left(\frac{dz}{dz'} \right)^h \left(\frac{d\bar{z}}{d\bar{z}'} \right)^\bar{h}\phi (z',\bar{z}') $$ under a conformal ...
3
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1answer
42 views

The contribution to the one loop beta function for the WZW model

When the Wess-Zumino-Witten model $$S_{WZW}=\frac{k}{4\pi}\int d^2 z \, \, \mathrm{Tr}[\partial u \bar{\partial}u^{-1} ]+ \frac{k}{12\pi}\int d^3 \sigma \epsilon^{ijk}\, ...
4
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2answers
119 views

Path integral derivation of the state-operator correspondence in a CFT

Below, I paraphrase the path integral derivation of the state-operator correspondence in David Tong's notes on CFT (see pdf here). This is my interpretation of the text in that pdf, so please correct ...
3
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0answers
61 views

Non-abelian bosonization

Reading this review about non-abelian bosonization, Non-abelian bosonization by I.Karmazin, I stumbled about two questions Below equation 6, I don't get the final point in the statement about the ...
5
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0answers
70 views

Zamolodchikov's c-theorem paper

I am reading the 1986 paper [1] where Zamolodchikov proves the c-theorem and I would like to understand how equations (7a), (7b) and (8) are derived from the Callan-Symanzik equation. For ...
3
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1answer
118 views

What is conformal gauge?

I often see in physics articles on gravity such notion as conformal gauge and Weyl transformation. They use Conformal gauge to change coordinates to transform metrics from arbitrary $$ds^2=g_{\mu ...
2
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1answer
76 views

Temperature in CFT

Non-vanishing Temperature can break conformal symmetry(Can anyone show this point explicitly), my question is that in AdS/CFT the temperature of boundary field theory is non-zero, why the boundary ...
2
votes
1answer
30 views

A paradoxical equation in RNS string fermionic part

It is well known for RNS string, $i.e.$, worldsheet supersymmetry formalism, the open string NS sector has worldsheet fermion expansion: \begin{equation} \psi^{\mu}_{\pm} = \frac{1}{\sqrt 2} \sum_{r ...
1
vote
1answer
70 views

OPE of fermionic field bosonization in string theory, in Polchinski 10.3.12

In Polchinski's String Theory Vol. 2, equations 10.3.12 are $$e^{iH(z)}e^{-iH(-z)}~=~\frac{1}{2z}+i\partial H(0)+2zT_B^H(0)+O(z^2)\tag{10.3.12a}$$ ...
3
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2answers
78 views

Relation between conformal and topological field theories

The Chern-Simons (CS) theory is a topological quantum field theory (TQFT). The question is, is a conformal field theory (CFT) a topological quantum theory? Or the reverse, topological quantum field ...
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0answers
46 views

What is the central charge about? [closed]

I have two very basic questions. What is meant by the term conformal field theory? What is the central charge in a conformal field theory?
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0answers
38 views

CFT calculation of the Partitionfunction of $2+1$ dimensional gravity

I want to reproduce formula (4.29) in http://arxiv.org/abs/0804.1773v1 given by: $$ Z=Tr(q^{L_{0}}\bar q^{\bar L_{0}})=|q|^{-2k} \prod^{\infty}_{n=2}\frac{1}{|1-q^{n}|^{2}} $$ Where the trace is ...
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0answers
32 views

Identitity operator-state mapping in CFT

In a general CFT, how can the identity operator be mapped into a state? Can you give examples to demonstrate the systematic way to do the mapping? Take 2D CFT (in String Theory) as examples. For ...
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0answers
25 views

What is Hilbert series?

What is the Hilbert series in the context of conformal field theory? I often see this terminology in many papers related with super conformal field theories which i don't know well, and after heard ...
5
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2answers
97 views

What is IR CFT and UV CFT?

What is IR CFT and UV CFT? In many physics related materials, they often mention IR, and UV. I think it is related with regularization (I remember in QFT, there is UV cutoff in some regularization ...
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0answers
69 views

Beta function of the non-linear sigma model

In chapter 7.1.1. inTong's notes about String Theory could someone sketch how can I show the statements that he nmakes around eq. 7.5 That the addition of the counterterm can be absorbed by ...
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0answers
71 views

How to derive the scale factor for special conformal transformation? [closed]

By definition a conformal transformation of the coordinates is an invertible mapping $x\rightarrow x'$ which leaves the metric invariant upto a scale factor: \begin{equation} g_{\mu\nu}'(x') = ...
3
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0answers
102 views

Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two ...
2
votes
1answer
47 views

Ward identities and operator product expansions

Polchinski's (2.3.11) gives the Ward Identity $$i\epsilon[Res_{z\rightarrow z_0}j(z)\mathcal A(z_0,\bar z_0)+\bar {Res}_{\bar z\rightarrow \bar z_0}\tilde j(\bar z)\mathcal A(z_0,\bar ...
1
vote
2answers
64 views

World-sheet energy-momentum tensor and OPE

On p43 of Polchinski's book, it says that under the world-sheet translation $\sigma^a\rightarrow\sigma^a+\epsilon v^a$, $X^\mu\rightarrow X^\mu-\epsilon v^a\partial_a X^\mu$. And $$j^a=iv^b T_{ab},$$ ...
1
vote
1answer
57 views

Ward Identity in CFT

This is about Polchinski's eq(2.3.11). It says that $$Res_{z\rightarrow z_0}j(z)\mathcal A(z_0,\bar z_0)+\bar{Res}_{z\rightarrow z_0}\tilde j(\bar z)\mathcal A(z_0,\bar z_0)=\frac1{i\epsilon}\delta ...
2
votes
1answer
93 views

Operator product expansion in CFT

I'm on Polchinski's p39. Can someone please tell me the steps in the equivalence below? $$\exp\left[\frac{\alpha'}4\int d^2z_4 d^2z_5\ln|z_5-z_4|^2\frac{\delta}{\delta X^\mu(z_4,\bar ...
2
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0answers
38 views

Non translation invariant correlator in CFT

I'm doing an exercise on vertex operators in the CFT book by Di Francesco & al.; exercise 9.2 p.329 : Using mode expansion show that: $$\langle\tilde{\phi(z)}\tilde{\phi(w)}\rangle= - \text{ln} ...
2
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0answers
52 views

Possible Error in deriving conformal generator

My professor gave me the following derivation for the full generator of the Lorentz transformations. The starting point is to consider a subgroup of the conformal group that leaves the origin fixed ...
2
votes
2answers
152 views

How do you measure numerically the central charge of a system?

Let's say that you are doing some Monte-Carlo simulations of a statistical system on a lattice and you observe scale invariance, meaning that you are at a conformal point. Can you get a numerical ...
5
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3answers
180 views

What makes General Relativity conformal variant?

I have a question regarding the well known fact that General Relativity is not a conformal invariant theory or to put it in other words about the fact that it is conformal variant: What are the ...
1
vote
1answer
51 views

How can one calculate the central term of the conformal field theory algebra (and show it's really the virasoro algebra)?

So I'm following Szabo's book "An Introduction to String Theory and D-brane Dynamics (2nd ed, 2011); still on the canonical treatment in chapter 3. After doing a mode expansion, we get (up to a ...
5
votes
0answers
64 views

BTZ Black Hole Central Charge and Conformal Weight

I have been trying to reproduce a calculation (equation 4.12) in this paper http://arxiv.org/pdf/1107.2678v1.pdf by Carlip reviewing the derivation of the effective central charge of the BTZ Black ...
1
vote
1answer
58 views

CFT Entanglement Entropy - relation between translations and the stress-energy tensor

In a recent paper on CFT entanglement entropy, I want to understand the defintion of a certain partition function. They consider a metric space $S^1 \times \mathbb{H}^{d-1}_q$ with metric: $$ ...
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2answers
148 views

Dilation operator in CFT viewed as 'hamiltonian'?

From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are ...
2
votes
1answer
119 views

Correlation functions and connection to ward identities

I have the following definition of a general correlation function $$ \langle \Phi(x_1)\dots \Phi(x_n)\rangle = \frac{1}{Z} \int [d\Phi] \Phi(x_1)\dots\Phi(x_n)e^{-S[\Phi]} $$ I have only just ...
4
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1answer
118 views

Conceptual question about field transformation

(c.f Conformal Field Theory by Di Francesco et al, p39) From another source, I understand the mathematical derivation that leads to eqn (2.126) in Di Francesco et al, however conceptually I do not ...
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0answers
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0answers
107 views

Derivation of the Noether current

(c.f Di Francesco et al, Conformal Field Theory, pp40-41) I am trying to derive eqn (2.142) or $\delta S = \int d^d x \partial_{\mu}j^{\mu}_a \omega_a$ in the book CFT by Di Francesco et al. I have ...
1
vote
1answer
90 views

Generator of the Special Conformal Transformation

In this thread Integrating the generator of the infinitesimal special conformal transformation, the generator of the 'flow' of the transformation is written as $$G_b = 2(b \cdot x)x - x^2 b,$$ where ...
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1answer
74 views

Conformal compatification of Minkowski and AdS

How do I show that the compactification of Minkowski is given by the quadric $$uv-\eta_{ij}x^{i}x^{j}=0$$ with an overall scale equivalence in the coordinates.I get that for $v \neq 0$, the surface ...
6
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3answers
282 views

Integrating the generator of the infinitesimal special conformal transformation

(c.f Di Francesco, Conformal Field Theory chapters 2 and 4). The expression for the full generator, $G_a$, of a transformation is $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta \omega_{a}} \partial_{\mu} ...
4
votes
1answer
134 views

Why is string theory a two dimensional quantum (conformal) field theory on its worldsheet?

In string theory, we quantize the two dimensional field theory on the string's worldsheet. I have a question about this kind of quantization of string theory: did we have similar theory for point-like ...
3
votes
1answer
132 views

Anyons: Effect of braiding on fusion multiplicities

In the theory of non-abelian anyons, essential information is stored in the fusion multiplicities or Verlinde coefficients $N_{ab}^c$. Having the Pants Decomposition in mind, it is possible to use ...
5
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2answers
177 views

What uniquely defines a CFT?

So, I am quite new to CFT (and a as descriptive answer as possible would be appreciated). I want to know what uniquely defines a CFT in 2D and otherwise. Firstly in 2D, What defines a CFT? So I ...
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0answers
34 views

When does the correlator of a string of fields and the current vanish “sufficiently fast” at infinity and Ward's identity?

One consequence of the Ward identity (cf. Di Francesco et al) is that it means variation of correlators under infinitesimal transformation is zero. This can be seen by integrating the ward identity, ...
1
vote
1answer
67 views

Null State Level 2 in CFT

I'm reading Cardy's notes on CFT. He states the following in section 4.3: $$\hat L_n\left(\hat L_{-2}|\phi_j\rangle-(1/g)\hat{L^2}_{-1}|\phi_j\rangle\right)=0.$$ I tried to work this out explicitly ...
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0answers
55 views

Anomalies from a Renormaization Group Equation (RGE)

This is an approach to anomalies which seems unfamiliar to me.. Firstly what is this function $W$ which seems to satisfy the equation, $\frac{\partial W }{\partial g^{\mu \nu} } = \langle T_{\mu ...
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2answers
194 views

Commutation relations of the generators of the conformal group

My question is from P.98 of the book by Di Francesco on Conformal Field theory. He gives the six non-vanishing commutation relations between the elements $P_{\mu}, D, L_{\mu \nu}$ and $K_{\mu}$ ...
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2answers
200 views

Decoupling of Holomorphic and Anti-holomorphic parts in 2D CFT

This maybe a very naive question. I have just started studying CFT, and I am confused by why we have two separate parts of everything in CFT (operator algebras and hilbert space), the holomorphic ...
2
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1answer
121 views

Are diffeomorphisms a proper subgroup of conformal transformations?

The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not ...
1
vote
1answer
104 views

Translation and Dilation transformations within the conformal group

I am using Di Francesco's book P.39. The equation that the generators of the transformations satisfy is given by: $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta w_a} \partial_{\mu} \Phi - \frac{\delta ...
6
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0answers
129 views

2d Ising model in CFT and statistical mechanics

When I recently started to read about conformal field theory, one of the basic examples there is the so called Ising model. It is characterized by certain specific collection of fields on the plane ...