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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Construction of Primaries of WZWN CFT

Is it possible to construct primaries of $SU(2)_{k+1}$ by using primaries of lower levels?. E.g. If I have a primary of $SU(2)_2$, let's say $\Phi^{(1/2)}$, the field with spin $1/2$ and another ...
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26 views

How rigorous is the description of the edge of the Kitaev Honeycomb as a CFT?

My understanding of the Kitaev honeycomb model is that high-level abstract properties (anyons and their braid statistics) can be seen to emerge the microscopics of the model (fermions and vortices). ...
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How to define the distance between two points in a conformal transformed space?

Consider a particular conformal transformation $x^\mu\rightarrow x'^\mu$, and the metric of a flat space transforms in the following way, $$\eta_{\mu\nu}\rightarrow g'_{\mu\nu}=\Lambda^2(x)\eta_{\mu\...
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Conformal invariance of Maxwell equation in presence of external current

It is known that pure electrodynamics in curved space-time is invariant under Weyl transformations $$ \tag{1} g_{\mu\nu} \to \Omega(x)g_{\mu\nu}, \quad F_{\mu \nu} \to \Omega^{-1}(x)F_{\mu\nu}. $$ ...
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How can I show that inversion is continuously connected to a reflection?

From Ex 3.1 in the TASI lectures on the conformal bootstrap: http://arxiv.org/abs/1602.07982 the problem is the inversion map (with Euclidean signature) $$ I\colon x^\mu \mapsto \frac{x^\mu}{x^2} $$ ...
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OPE coefficents and commutation relations, and OPE with stress tensor

Basic question about conformal field theory: In a conformal field theory in $d\geq 3$ dimensions, what is the relation between commutation relations and OPE coefficients? In particular, because ...
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51 views

Transformation of quasi-primary field in CFT [on hold]

As it is well known that in conformal field theory the energy-momentum tensor is a quasi-primary field, and its transformation law under conformal transformation $z\rightarrow w(z)$ is \begin{...
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25 views

How is the two-point function of an operator dual to a scalar ADS field obtained in ADS/CFT?

The two point function of an operator dual to a scalar field in ADS/CFT can obtained directly from computation of the on-shell action in momentum space and then taking it back to position space. The ...
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40 views

Correlator of energy-momentum tensor and OPE

In http://arxiv.org/abs/hep-th/9108028 Equation (2.22), the correlation function of then energy-momentum tensor with some primary fields is We can view this as sum over the OPE of the energy-...
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1answer
57 views

Central charges in 2D CFT and Virasoro algebra

Suppose we quantize some classical CFT algebra given by generators which satisfy $$[l_n,l_m]=(n-m)l_{n+m},$$ $$[\overline{l}_n,\overline{l}_m]=(n-m)\overline{l}_{n+m},$$ $$[l_n,\overline{l}_m]=0.$$ ...
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35 views

Where to look up calculations in CFT? [duplicate]

I am a mathematician, but occasionally I need the answers to questions that might already have been calculated by physicists. Recently I've found myself asking questions like "what is the space of 0-...
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124 views

conformal symmetry vs time ordering

In Quantum field theories we generally calculate time ordered correlation functions. But it seems that in a conformal field theory I can use conformal symmetry to destroy time ordering. Let me look ...
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27 views

Transformation law for the eigenfunction representation of dilations (scale transformations) in CFT

On page 32 section 2.1.2. of http://arxiv.org/abs/hep-th/9905111 a representation $\phi(x)$ for the conformal group is chosen such that $D\phi(x) = -i \Delta\phi(x) $. After that it is stated that ...
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79 views

Conformal Gravity

Lubos, in his comment to a question, says that (http://physics.stackexchange.com/q/61281) First of all, one can't gauge a symmetry without modifying (enriching) the field contents. Gauging a ...
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1answer
40 views

OPE coefficients identity operator

when I have two canonically normalized operators $\phi_{1}$ and $\phi_{2}$ and I want to compute their OPE in terms of the identity operator, is there any way to actually calculate the first levels of ...
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1answer
91 views

Virasoro Algebra vs Witt Algebra

I'm reading some notes on CFT, and there's a strange topic that I find quite confusing. We define the Witt algebra to be the generators of conformal transformations on the complex plane. $l_n = -z^{...
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What is the origin of the five-form field in bubbling Ads geometries?

I have been reading the paper: Bubbling Ads space and the 1/2 BPS geometries[hep-th/0409174]. In the paper they look at 1/2 BPS states in the field theory which are dual to D3 branes IIB supegravity. ...
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CFT conformal weight vs. scaling dimension

I was wondering if anybody could clarify what the difference between the conformal scaling dimension $\Delta$ and the conformal weight $h$ is? Is it correctly understood that $\Delta$ is related to ...
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121 views

Partition functions for a (3+1)-d TQFT

It is well known that for a Chern-Simons theory defined on an arbitrary (2+1)-d oriented manifold, its partition function can be evaluated based on Witten's surgery method. My question is: is there a ...
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45 views

Entanglement in Quantum field theory [duplicate]

How is entanglement represented in a field theory? For instance how can I represent a maximally entangled state such as a Bell state? Would such an approach also apply in a Conformal field theory ...
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How to calculate anti analytic conformal weights

In books like Mussardo's "Statistical Field Theory" and "Conformal Field Theory" of Di Francesco et al, there is no clear explanation on how to calculate the anti analytic conformal weights $\bar \...
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Conformal Transformation: Minkowski sheet to cylinder

What conformal transformation can I make to 2d Minkowski with metric $ds^2=-dt^2+dx^2$ to show that it is conformal to a cylinder?
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Conformal Connections in Physics [closed]

For my diploma thesis in Mathematics I investigate conformal connections (as an example of Cartan connections). All in all the thesis should deal with geometric aspects (associated bundles, (pseudo)-...
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Why are CFT descriptions of String Theory inherently perturbative and how can it be circumvented?

Field theories like QED/QCD are a priori non-perturbative theories. Perturbatively you can describe them by Feynman diagrams which essentially sum over all topologies of virtual particle creation and ...
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CFT: from States to Operators

I'm having trouble finding the general algorithm for moving from states to operators under the state-operator correspondence in a CFT. Does anyone have any hints as to how one might go about ...
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Laughlin wave function and CFT

I have a question regarding Eq. (3.5) in Moore & Read's paper. They said \begin{equation} \Psi_{\text{Laughlin}}=\left\langle\prod_{i=1}^{N}e^{i\sqrt{q}\phi(z_i)}\exp\left[-i\int \mathrm d^2z^{\...
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What is a zero temperature horizon?

While reading the paper "Disorder horizons: Holography of randomly disordered fixed points" by Hartnoll and Santos, I came across this: We are interested in solutions with a zero temperature ...
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1answer
51 views

Does the two-point function of free field reveal conformal anomaly?

Consider free scalar field in two dimensions with the standard action written in complex coordinates $S=\int d^2z\, \partial \phi\bar{\partial}\phi$. The two-point correlation function is known to be $...
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156 views

Why is conformal field theory so important?

I just started escaping the world of quantum mechanics and looking to study quantum field theory. I heard of AdS/CFT and also heard that CFT is of much importance. Now I do not get why having ...
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88 views

Does Operator Product Expansion form an algebra?

The operator product algebra in CFT is defined as $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega}).$$ ...
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$\mathcal{N}=4$ SYM on a rotating three-sphere

In the following paper: http://arxiv.org/abs/hep-th/9911124, the authors claim that the dual field theory to the five dimensional Kerr-AdS black hole is a $\mathcal{N}=4$ SYM on a rotating three-...
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76 views

Why do people care about Mathieu groups and related things? (Something about monstrous moonshine)

Before I begin, let me say I don't know anything about what I am asking. This morning for somewhat random reasons I decided to google moonshine and related things. As it were I discovered my ignorance ...
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How to show that $L_n^\dagger=L_{-n}$ for the Virasoro generators in CFT?

It seems to be a common knowledge that for the Virasoro generators in CFT the rule of hermitian conjugation reads $$L_n^\dagger=L_{-n}$$ There is probably more then one way to show this. I ask for a ...
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1answer
61 views

A conformal mapping between a Rindler Wedge and a causal diamond. What is the right map and how do I see it does what is expected?

I am going through the calculations in arXiv:1312.7856 [hep-th]. These involve a conformal map between the Minkowski Rindler Wedge ($\mathcal{R}$), given by $X^1 \geq 0,X^+\geq 0,X^-\geq 0 \quad$ (...
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$i\epsilon$ in CFT correlation functions

M. Luescher in his talk on p.6 writes that the 2-point correlation function of a Hermitian local field $O_k$ of scaling dimension $d=3-k$ looks like $$ \langle 0| O_k(x) O_k(y) |0\rangle = A_k (x-y-i ...
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1answer
77 views

Monodromy matrix and differential equations

What is the significance of monodromy matrix in the context of differential equations? I have seen some papers(1,2,3 etc) in CFT which use the monodromy method to compute conformal blocks at large ...
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1answer
55 views

Applicability of Cardy's “doubling trick” to the 2D Ising Model

In Section 11.2.2 of the book on Conformal Field Theory by di Francesco, Mathieu, and Senechal (page 417), the two point function on the Upper Half Plane is written as being equal to the four point ...
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On the c-theorem

I have been reading a few papers on CFT and AdS/CFT regarding the c-theorem and I have a few questions regarding c-theorems: a) Why is it that the c-theorem is usually considered for only unitary ...
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AdS3/CFT2 duality for free bulk scalar

I am looking at lecture notes by Kaplan. Through chapters 4, 5, and 6 he takes the free field in $\mathrm{AdS}_3$ to the boundary to create a CFT2 primary field. The result is equation (6.5): $$\...
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1answer
35 views

Special conformal transformation of stress-energy

Consider a 2d CFT, e.g. a single bosonic degree of freedom. The $TT$ OPE is $$ T(w) T(z) = \frac{c/2}{(z-w)^4} + \frac{2 T(w)}{(z-w)^2} + \frac{\partial T(w)}{z-w} + \text{regular terms}. $$ Does ...
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129 views

Relation of conformal symmetry and traceless energy momentum tensor

In usual string theory, or conformal field theory textbook, they states traceless energy momentum tensor $T_{a}^{\phantom{a}a}=0$ implies (Here energy momentum tensor is usual one which is symmetric ...
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Correlation function for ghosts in 2D CFT

In Di Fracenso, page 117, it is explained that the correlation function for two primary fields $\phi_1,\phi_2$ of weights $h_1,h_2$ is constrained to be of the form $\langle\phi_1(z)\phi_2(w)\rangle$=...
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In a 2D CFT of a free boson $X$, is $X$ a primary field?

A primary field $\mathcal{O}(w,\bar{w})$ with weight $(h,\bar{h})$ is defined by having the following OPEs with the stress tensor: $T(z)\mathcal{O}(w,\bar{w})=\frac{h\mathcal{O}}{(z-w)^2}+\frac{\...
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What is known about the physics of Planckbrane (another brane) in Randall–Sundrum model?

Randall–Sundrum model imagines that our universe is a five-dimensional anti-de Sitter space and the elementary particles except for the graviton are localized on a (3 + 1)-dimensional brane or branes. ...
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Cardy counting for time dependent theories

In certain limits of 2 dimensional CFT's, it is possible to compute the entropy of the theory in terms of the density of states which is given by the Cardy formula $$S = 2\pi \left(\sqrt{\frac{c_R \...
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Conformal field theories that are not conformally invariant

Is it possible to construct a field theory that is Weyl-invariant but not conformally invariant? Any references?
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Question about the superconformal index

According to arXiv:1507.08553v1, the superconformal index, defined by $$I(\beta_j) = \mbox{Tr}_{\mathcal{H}}(-1)^F e^{-\gamma\{Q,Q^\dagger\}}e^{-\sum_{j}\beta_j t_j}$$ is independent of the ...
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1answer
68 views

Fields in the action of the Non-linear Sigma Model (WZW)

I am trying to understand the action of the nonlinear sigma model in the context of understanding WZW-models. On Wikipedia, its action is given as $S_k\left(\gamma\right)=-\frac{k}{8\pi}\int_{S^2}\...
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1answer
109 views

Normal Ordering in String Theory: Polchinsky vs. all others

Polchinsky defines normal ordering in string theory as: $$:X^\mu(z,\bar z)X^\nu(w,\bar w): = X^\mu(z,\bar z) X^\nu(w, \bar w) + \frac{\alpha'}{2} \eta^{\mu\nu} \log |z-w|^2$$ and for more ...
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What is a “dynamically generated scale” physically?

A theory like QCD with massless quarks in four dimensions has no explicit mass parameters in its classical Lagrangian. At the quantum level however, instead a mass scale Λ is generated dynamically at ...