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 ...

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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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Homotopy group of the conformal group [migrated]

I would like to know which are the first three homotopy groups of the conformal group SO(4,2): $$ \pi_n(SO(4,2))=? \quad n=1,2,3 $$
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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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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 = ...
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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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Algebra of conformal transformation in 2D

In case of 2 dimensional conformal field theory the infinitesimal change in the coordinate satisfy the Cauchy Riemann conditions, so we can say that 2d conformal transformation is nothing but the ...
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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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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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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, ...
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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 ...
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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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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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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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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 ...
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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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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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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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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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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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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 ...
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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: ...
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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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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 ...
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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 ...
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What is the global Virasoro symmetry generators in BTZ spacetime?

This is the case of AdS3, how about BTZ? The picture is from arxiv:1506.01353.
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Generally speaking, given two CFTs, how can we check that they are equivalent?

We can calculate their central charges. And beyond this, are there some other general approaches? Or we must find an isomorphism of operators between the two theories?
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Reference request: 2D conformal field theory and functions on the triangular lattice

I don't have much of a physics background and was wondering if anyone knows what is meant by "conformally invariant" functions defined on the plaquettes of the honeycomb lattice (ie functions defined ...
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Existence of lagrangians at strong coupling

It is well known that some QFT do not admit a lagrangian formulation (like the $(2,0)$ SCFT in $d=6$). Up to my understanding, all the examples that I know of non lagrangian theories are always ...
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Can a scalar field transform nontrivially under a local special conformal transformation?

Is there any way to have a scalar field that transforms non-trivially under local special conformal transformations? Just by the index structure, I can see that the possibilities are $$\begin{align} ...
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CFT and temperature

I have tried to think about this for some time but could not really go anywhere. Sorry for the sloppy question and thanks for any pointer. My question is about CFT at finite temperature and ...
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“Irreversibility” of the RG flow

In his remarkable work, Zamolodchikov proved a theorem regarding two dimensional QFT Renormalization Group (RG) flow, describing a monotonically decreasing function in the flow parameter which is ...
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Operator product expansion in 2D cft

My question is about an orbifold CFT theory in 2 dimension. Forgetting about the details, the problem is If I want to find the OPE of two operators, let's say, $O^{++}(z)$ and $O^{--}(z)$, i can ...
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Can the terms in the microscopic model with nonzero conformal spin generate some new term(s) under RG (renormalization group) flow?

As in the book Bosonization and Strongly Correlated Systems at page 66, it says that "We see that the original perturbation with nonzero conformal spin generates the perturbation with zero conformal ...
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Orbifold actions and twist operators

A twist operator $\sigma$ is the operator that acts on the untwisted vacuum $|0\rangle$ to create a twisted vacuum $\sigma|0\rangle$. States belonging to the twisted sector of an orbifold are built on ...
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Simple conceptual question conformal field theory

I come up with this conclusion after reading some books and review articles on conformal field theory (CFT). CFT is a subset of FT such that the action is invariant under conformal transformation ...