A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In 2D, the infinite-dimensional algebra of local conformal transformations normally permits exact solution or classification of such theories. Further use for CFT applications to string theory,...

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Wick Theorem, ordering & CFT

I'm having a little trouble with correlation functions wick theorem and ordering in the context of OPE and CFT, for string theory. (1) My first question, the propagator is: $$<X(z) X(w)> = \...
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3answers
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Quantum field theory variants

Wikipedia describes many variants of quantum field theory: conformal quantum field theory topological quantum field theory axiomatic/constructive quantum field theory algebraic quantum field theory ...
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Why/How is this Wick's theorem?

Let $\phi$ be a scalar field and then I see the following expression (1) for the square of the normal ordered version of $\phi^2(x)$. $$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 ...
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64 views

Dual of the Identity operator (AdS/CFT)

We know that in a CFT the spectrum of gauge invariant operators must contain an Identity operator (for the operator algebra to close). For those CFTs that admit a holographic dual what does the ...
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1answer
65 views

AdS/CFT: why is the fifth coordinate in AdS space inversely proportional to an energy scale?

In several different articles about the AdS/CFT correspondence, it is stated that one can show that the fifth coordinate $z$ on the AdS side, in coordinates such that the AdS metric becomes: $$ds^2 = ...
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1answer
104 views

Total quantum dimension of excitations in the Toric code

In the Toric code, the excitations are e, m, fermion $\epsilon$ and vacuum. Thus, the total quantum dimension is $D= \sqrt{\sum{d_{a}^{2}}} = 2$. It seems one takes into account all sorts of possible ...
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32 views

confused about the definition of conformal transformation [duplicate]

Recently, I read some books and articles about conformal field theory and I find there exists two completely different views about conformal transformation... The first is that: Conformal ...
4
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2answers
470 views

2D Ghost CFT and two-point functions

For some reason I am suddenly confused over something which should be quit elementary. In two-dimensional CFT's the two-point functions of quasi-primary fields are fixed by global $SL(2,\mathbb C)/\...
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0answers
55 views

How to derive the effective action or Vertex generating functional of complex scalar field?

I am studying-self QFT. Recently, I am studying and trying to follow the calculation of the Ginzburg-Landau free energy functional of superconductor in this paper:http://arxiv.org/abs/hep-ph/0108256. ...
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29 views

What is the meaning of conformal mass on a branched manifold?

Consider the following QFT on the "branched" manifold, $\mathbb{H}^2 \times S^1_q$ where $S^1_q$ is an unit radius circle whose angular coordinate goes from $0$ to $2\pi q$, and hence the "branching". ...
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1answer
57 views

Why does the Weyl transformation preserve angles in string theory?

The Weyl invariance symmetry of the Polyakov action is said to be considered as the invariance of the theory under a local change of scale which preserves the angles between all lines. However, why ...
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1answer
122 views

What is the conformal mode of a metric?

I have a problem in terminology. This article talks about the conformal mode of a physical metric. I know what a conformal transformation is. But what is the conformal mode of a metric?
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29 views

Dilations in momentum space

I don't quite understand what's going on here. Let's suppose I have a dilation in real space. The generator is $D=x^j \partial_j$, so an infinitesimal dilation is $\delta x^i = Dx^i = x^j \partial_j ...
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1answer
100 views

Classical conformal invariance

So I am trying to understand classical conformal invariance. So we move gently from general coordinate invariance to Weyl invariance to conformal invariance, and now they start out with this thing ...
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45 views

CFTs and hyperbolic spaces

Is there a good way to explain why its natural to consider CFTs to be on hyperbolic spaces? Even if I forget about CFTs is there a good way to explain if there is a way that the hyperbolic space can ...
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0answers
60 views

How can the stress tensor components of a worldsheet CFT in general background be (anti)-holomorphic?

In all textbooks/lecture notes on string theory (e.g. Polchinski, page 43 at the bottom) it is proven that, as the stress tensor is traceless and conserved, $T^a_a=\partial^a T_{ab}=0$, we have $\...
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0answers
65 views

Generalized gravitational entropy and entanglement entropy

What are the differences (if any) between Generalized gravitational entropy (Lewkowycz-Maldacena) and holographic entanglement entropy (Ryu-Takayanagi)? More specifically, I was wondering following ...
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1answer
94 views

For what values of $\lambda$ is the distribution $(x-i\varepsilon)^\lambda$ positive?

I've been reading the famous unpublished paper by Luescher and Mack "The energy momentum tensor of critical quantum field theories in 1+1 dimensions". In the proof of their main theorem, page 7 of the ...
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76 views

Anomaly polynomial of Hitchin system $\mathcal{N}=2$ 4d SQFT

I would like to ask about mathematical background of this object. So, I am trying to puzzle out with 4d $\mathcal{N}=2$ SQFT. As far as I can gather this theory can be described in terms of ...
2
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1answer
328 views

What is a ghost number?

I am currently studying CFT chapter of Becker,Becker,Schwarz and am trying to understand what the ghost number is in BRST Quantization. From what I gather BRST Quantization is used to add an extra ...
2
votes
1answer
61 views

In 1d criticality, what is the relation between the universality class and central charge?

I want to know how to obtain the universality class of the phase transition from the central charge "c" in one dimensional model. If c is less than 1, there is a one-to-one correspondence. But if c is ...
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0answers
88 views

Problem with OPE (from Polchinski) [closed]

I was reading Polchinski, Vol. 2 pag 12, while I found (10.3.12a): $$ e^{iH(z)}e^{-iH(z)}=\frac{1}{2z} + i\partial H(0) + 2zT^H_B(0) + O(z^2).\tag{10.3.12a} $$ Now I tried to do the OPE, what I ...
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0answers
37 views

Correlator $bc$ system [closed]

I have da doubt with bc system. Polchinski says (2.5.10) $$ b(z)b(0)~=~O(z). \tag{2.5.10} $$ I tried to compute the correlation function With eom, using eq (2.5.6b) by Polchinki, removing the ...
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0answers
83 views

Connection between the M5 brane and (2, 0) superconformal field theory

I have read that the worldvolume theory of the M5 brane is a $(2, 0)$ superconformal field theory (SCFT). But I have also learnt from talks that the $(2, 0)$ theory lacks a Lagrangian description. ...
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1answer
237 views

What are the quantum dimensions of the primary fields for SU(N) level k Kac-Moody current algebras?

The CFT of the $\mathrm{SU}(N)$ level $k$ Kac-Moody current algebra has many Kac-Moody primary fields. I wonder if any one has calculated the quantum dimensions of those Kac-Moody primary fields. I ...
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1answer
171 views

Conformal blocks in 2D CFTs

I have studied conformal field theories in two dimensions and I understand the basic idea behind conformal blocks too. But I never completely realized what they are when it comes to computing them. ...
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84 views

Target Space Lorentz Invariance vs. World Sheet Weyl Invariance

The Polyakov action, $S\sim \int d^2\sigma\sqrt{\gamma}\, \gamma_{ab}\partial^a X^\mu \partial ^b X_\mu$, has the well known classical symmetries of world sheet diffeomorphism invariance, world ...
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2answers
94 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 ...
2
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0answers
46 views

Large Spin operators and radial quantization

I've been reading the paper "Comments on operators with large Spin" (here) and I am having some trouble understanding the following: In section 2, they begin by studying, in a conformal field theory ...
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0answers
36 views

Decoupling of Weyl factor in critical dimension

In his paper called "Quantum geometry of bosonic strings", A.M.Polyakov quantizes a bosonic string using path integrals over the space of all metrics on the worldsheet. A critical dimension renders $D ...
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51 views

OPE in a general $d$-dimensional CFT

I am looking for a good reference which explains how to express an OPE in a general $d$-dimensional CFT for bosonic and fermionic fields. Precisely, I don't understand the reason of the appearance of ...
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77 views

Is there a maximum number of fixed points that a QFT can have?

I was wondering: is there a maximum number of (trivial and non-trivial) fixed points that a QFT can have (as a function of the space-time dimension and field content in the QFT)?
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49 views

Calculating OPE of Graviton Vertex Operator [duplicate]

Consider Exercise 2.8 in Polchinski's String Theory book. We are asked to compute the weight of $$f_{\mu \nu}:\partial X^{\mu} \bar{\partial}X^{\nu}e^{ik\cdot X}:$$ I have carried out the usual ...
4
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0answers
122 views

Mode operators in the Virasoro algebra

This questions concerns Exercise 2.11 in Polchinski. We are asked to compute the commutator $$L_{m}(L_{-m}|0;0\rangle) - L_{-m}(L_{m} |0;0\rangle)$$ By plugging the mode expansions, we use the ...
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0answers
148 views

Calculation of OPE in Polchinski

Consider Exercise 2.8 in Polchinski's String Theory book. We are asked to compute the weight of $$f_{\mu \nu}:\partial X^{\mu} \bar{\partial}X^{\nu}e^{ik\cdot X}:$$ I have carried out the usual ...
2
votes
1answer
150 views

Operator product expansion energy momentum tensor

We have the following equation from Polchinski (2.4.6) $$ T(z)X^{\mu}(0) \sim \frac{1}{z}\partial X^{\mu}(0) , \tag{2.4.6} $$ where $T(z)$ is defined as $T(z) = -\frac{1}{\alpha'} :\partial X^{\mu} \...
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1answer
90 views

Special OPE in $\beta\gamma$ system

I would like to find the OPE $$\beta(z)\gamma(w)^{-1}\tag{1}$$ given $$\beta(z)\gamma(w)~\sim~\frac{1}{z-w}\tag{2}$$ from the $\beta\gamma$-system in CFT. Is it possible?
3
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0answers
110 views

Anyonic Braiding and Conformal Field Theory

I am looking for resources (both pedagogical and newer research articles) on the connection between topological quantum computation and conformal field theory. In particular, a CFT description of ...
4
votes
2answers
295 views

Analogy for the AdS/CFT Correspondence

Some time ago, I heard about a simple analogy for the AdS/CFT correspondence to something in everyday life. Consider a room filled with furniture, with the walls of the room covered in mirrors. The 2D ...
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0answers
226 views

Ishibashi states and Cardy states in CFT

What are the Ishibashi states and Cardy states in CFTs? I am familiar with conformal field theory language. It would be great if someone can discuss about the basic idea of these states and their ...
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194 views

Monstrous Moonshine outside of String Theory

My question concerns applications of monstrous moonshine, which is the connection between the $j$-function and the monster group. Recently, physicists have applied it to string theory and, ultimately, ...
10
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1answer
344 views

Can you take the cutoff to infinity at a conformal fixed point?

A conformal fixed point is defined by $$\beta(g)=0$$ We hence know that couplings, masses and dimensions of operators do not flow in the effective Lagrangian when we change the renormalization ...
6
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1answer
734 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 $\widetilde{\phi}$, ...
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2answers
748 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 \...
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1answer
188 views

Polchinski Equation (7.2.4)

On page 209 of Polchinski's string theory book he writes down the expectation value of a product of vertex operators on the torus; equation $(7.2.4)$. The derivation is analogous to an earlier ...
2
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1answer
97 views

Standard derivation of Witt algebra

I have been studying Conformal field theory for the past one week from the books by Blumenhagen and Di Francesco etal. If I understand correctly, whenever one talks of 'local (infinitesimal) ...
6
votes
2answers
499 views

Conformal transformation equation

I am currently reading Kiritsis's string theory book, and something bugs in the CFT (fourth) chapter. He derives the equation that should satisfy an infinitesimal conformal transformation $$x^{\mu} \...