Questions tagged [conformal-field-theory]

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, statistical mechanics, and condensed matter physics.

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Questions on RG flows, CFTs, and UV and IR theories

In the space of field theories, Conformal field theories are fixed points in the RG flow. However, a lot of literature on CFT usually talks about a QFT being the RG flow between two CFTs: one UV and ...
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What is rational CFT?

The definition of rational CFT was rather confusing. There's were many references saying the different definitions: $c$ to be a rational number. the representation of CFT have finite primaries. (...
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Conformal flatness FLRW

The FLRW metric is known to be conformally flat, i.e., it is conformally related to the Minkowski metric. How I read this, is that it makes the FLRW metric expressible in static form. Is this correct? ...
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Find commutator $[P_\mu,K_\nu]$ in conformal group

We have conformal group with next element of this group: $$U=e^{i(P_\mu\epsilon^\mu-\frac{1}{2}M_{\mu\nu}\omega^{\mu\nu}+\rho D+\epsilon_\mu K^\mu)},$$ where $D$ is dilatation operator $$x^\mu\...
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Wick rotation of real-space trajectories

In CFT 2D, we often like to wick rotate the minkowski theory, to end up with an euclidean theory. Often, we then use complex coordinates to parametrize the plane, to make things easier. For instance, ...
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How to map a complex plane Mobius transformation to 1+1D Minkowski real plane?

So consider the $(x,t)$ plane endowed with the minkowski metric, namely: $$ds^2 = dx^2-dt^2.$$ It is well known that we can Wick rotate the time coordinate to get to the Euclidean metric. This can be ...
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What's minimal model?

Minimal model In Francesco conformal field theory page 210 only $$ c=1-\frac{6}{p(p+1)} $$ was the representation without the nonunitary representation. However, for the theory to be unitary, it only ...
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What is minimal model, unitary, rational, and irrational CFT? [closed]

Minimal model In Francesco conformal field theory page 210 only $$ c=1-\frac{6}{p(p+1)} $$ was the representation without the nonunitary representation. However, for the theory to be unitary, it only ...
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$R$ projection operator in entanglement entropy

I was reading a paper about entanglement entropy. The author introduced a real space cutoff operator $R$ which he claimed to project onto a real space subregion. Then he used $\bar P:=RPR$ as an ...
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How was the minimal model with a boundary related to the D brane?

Quote my advisor: The D brane was the boundary of the CFT However, in the development of the rational CFT, such as the minimal model, the D brane was not realized. Thus, when the boundary CFT was ...
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Is a running coupling constant a natural consequence in QFT, or is it a consequence of the "dressing-up" of particles?

The running coupling constant ("hold that constant!) is a well known phenomenon in quantum field theory. The constant varies with the energy of the interacting particles. I think this is rather ...
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How did the two copies of the Witt algebra become two copies of the Virasoro algebra in the CFT?

The Virasoro algebra \begin{equation} [L_m,L_n]=(m-n) L_{m+n} +\frac{c}{12} (m^3-m) \delta_{m+n,0} \end{equation} of the stress energy tensor $T$ was said to follow from the witt algebra of the local ...
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What does the non-commuting nature of the translation and dilation generators mean for the scaling dimension of a field?

I am reading about CFTs from the book by Di Francesco, Mathieu and Senechal and in page 98 was introduced to the conformal group and the algebra of the generators. In particular, we have the dilation/...
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Why is gravitational Chern-Simons action invariant under conformal transformation?

We know the action of topologically massive gravity in 3-dimentional spacetime is \begin{equation} \label{eq:EH} S=S_{\mathrm{EH}}+S_{\mathrm{CS}}=\frac{1}{\kappa^2}\int d^3x\sqrt{-g}R+\frac{1}{\mu}\...
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Extracting the central charge from correlation functions and normalization of operators

I have an elementary question about computing the central charge in a conformal field theory in dimension greater than 2. In principle, this is determined by the correlation functions involving the ...
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Symmetry restoration at high temperature

Weinberg in his famous article https://doi.org/10.1103/PhysRevD.9.3357 discusses how finite-temperature effects in a renormalizable quantum field theory can restore a symmetry which is broken at zero ...
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What is a "massive phase" in the string theory or CFT?

When reading through some articles, one encountered a vocabulary termed as a "massive phase" in the string theory or the CFT, in the case where a theory followed an RG flow between UV and IR....
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Defining normal-ordering in an OPE calculation

Suppose $\phi_1, \phi_2, \psi$ and $\bar{\psi}$ are free fields of a two-dimensional CFT with propagators on the plane given by $$\phi_1(y)\phi_1(z) \sim -\log(y-z),\quad\phi_2(y)\phi_2(z) \sim -\log(...
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Conformal Cyclic Cosmology

In this conformal cyclic cosmology, where did the very first eon come from? How did the very first eon start with possible lowest entropy?
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Conformal cyclic cosmology, black hole and Weyl curvature

Roger penrose in his theory "conformal cyclic cosmology" states that after a google there would be a giant black hole. Does black hole have Weyl curvature? If it does, then we know that ...
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What does the exponentiated generator of scale transformation do when it acts on a function? [duplicate]

We know that $d/dx$ is the generator of translation in the sense that $$e^{ad/dx}f(x)=f(x+a)\tag{1}$$ which can be easily be proved from the Taylor series of $f(x+a)$. Studying the very basics of ...
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Associativity of the Kac-Moody OPE

The Kac-Moody OPE reads \begin{equation} J^a(z_1)J^b(z_2)\sim \frac{k\delta^{ab}}{2(z_{12})^2}+\frac{f^{abc}J^c(z_1)}{z_{12}}+\mathcal{O}(z_{12}^0)\text{ descendants}\qquad (1) \end{equation} However, ...
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Representations of minimal model primary fields in the Coulomb-gas Formalism

This question is a cross-post from MO (link). Is it known how to construct the primary field operators of the unitary minimal models $\mathcal{M}(m+1,m)$ in the Coulomb gas formalism? As far as I can ...
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Holograhpic Dual of Replica Manifold

Following https://arxiv.org/abs/1501.05315, the conformal map $w=z^\alpha$ can be interpreted as a conical defect geometry on the bulk side. In particular, we can obtain the replica manifold by the ...
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Are there massless excitations when a system is at a $U(1)$ critical point?

I am trying to understand the nature of the energy spectrum of a system at various critical point types. For instance, in the following paper the authors shows that for the transverse field Ising ...
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Why for 6d SCFT we consider (1,0) and (2,0) only?

It is known to have a stress-energy tensor we must have the supercharges $\mathcal{N}<2$. My confusion is why in most cases we consider chiral supercharges, and what is the problem with $\mathcal{N}...
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What does "conformally coupled scalar" mean?

"Conformally coupled scalar $\phi$" - I encounter it a lot, but I can't find what it means.
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Programs/Scripts for plotting penrose diagrams

What programs do people generally use to plot penrose diagrams? I need to plot some simple ones for the dS and AdS and Schwarzschild metric and my hand drawn ones dont look very good.
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2 votes
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Prove that the fusion/crossing matrix was a function of $c,h$ only and was invariant between different theories

The fusion matrix or crossing matrix $$ F_{nm}\begin{bmatrix}i&j\\k&l\end{bmatrix} $$ relates the 4 point correlation function in the different channels. How to show that it was was a function ...
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Why was the A-D-E type minimal model distinguished?

One was reading the paper where one encountered a word called "A Type minimal model", which seemed to indicate some historical identification. Latter one found it in the Wikipedia that in ...
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General derivation of Supervirasoro algebra

I'm looking for a derivation derivation of the ($\mathcal{N} = 1$) Supervirasoro algebra (NS sector) that does't just apply to specific examples. Most books/papers either just cite the result, or ...
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6 votes
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String theory: Conformal invariance and Conformal Killing Vectors

I am confused by the relation between the invariance of the Polyakov action under conformal transformations and the Conformal Killing Vectors (CKVs) appearing during the process of quantization. Let ...
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What is boundary cosmological constant in boundary Liouville field theory?

In this paper, Liouville field theory with conformally invariant boundary is studied. The action is: $$ \int_{\Gamma}\left(\frac{1}{4 \pi}\left(\partial_{a} \phi\right)^{2}+\mu e^{2 b \phi}\right) d^{...
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How to calculate the OPE of the $X_L(z_1)X_L(z_2)$ in the free boson theory from the mode expansion?

From the polchinski page 238, given \begin{equation} [x_L,p_L] =[x_R,p_R]=i\tag{8.2.14} \end{equation} and the mode expansion $$\begin{equation} \begin{split} X_L(z) = x_L -i\frac{\alpha'}{2}p_L \ln z ...
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Lorentz boost as a conformal transformation

The conformal group is the set of transformation that preserve angles. With this idea, then a conformal transformation is such that $x\rightarrow x^\prime$ and $$ g^\prime_{\mu\nu}(x^\prime) = \Omega(...
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Dilation operator acting on $x$-dependent field

I've been studying conformal field theory (CFT) and got the following "apparent" inconsistency. Let's take dilation ($D$) and translation ($P_\mu$, 4-momentum) generators that according to ...
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The confusion over the invariance of the correlation function and the mutually local field in the CFT

Consider the correlation function $$\langle \Pi_{i=1}^n V_i(z_i,\bar z_i) \rangle$$ such as $$\langle V_1(z_1) V_2(z_2) \rangle,(z_1>z_2)$$ by position the $z_i$ correctly, the exchange of the ...
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Anti-holomorphic contribution to 2d conformal algebra

I am reading Ginsparg's notes on 2D-CFT, and I am deeply confused about why Ginsparg states after (1.8) that the conformal algebra for 2d Euclidean space consists of two copies of the Witt algebra. My ...
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Computing a conformal block in 2D CFT

The following comes from Di Francesco et al., section 6.6.4. The exercise is number 6.5. A conformal block is given by $$\mathcal{F}^{21}_{34}(p\mid x) = x^{h_p-h_3-h_4}\sum_{\{k\}}\beta^{p\{k\}}_{34}...
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Critical exponents and scaling dimension

It is often stated that the scaling exponents, e.g. $\alpha$ and $\beta$, of the critical 2D Ising model are related to the scaling dimensions $\Delta_{\sigma}$ and $\Delta_{\epsilon}$ of the ...
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Why are CFTs not usually studied in momentum space?

Conformal symmetry in QFT has been extremely useful for physics. However, while most of QFT is usually done in momentum space, CFTs are usually studied in position space or in terms of Mellin ...
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How to show that in 2D CFT the marginal operator must have $(h,\bar h)=(1,1)$?

A related post might be What are marginal fields in CFT? where Qmechanic♦ pointed to Ginsparg secion 8.6. However, I heard about two argument. Claim 1:In a $D$ dimension CFT, the marginal operator ...
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What does the Verma module in the reducible Virasoro algebra represent?

In the conformal field theory book by Francesco, Mathieu, Senechal, the Verma module is built from a primary field $|\phi\rangle$, and if one of the descendants is a singular vector $|\chi\rangle$, ...
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Finite transformations in linear dilaton conformal theory

In the linear dilaton conformal theory, we have a stress-energy tensor $T(z)=-\frac{1}{α′}:∂X(z)∂X(z):+V∂^2X(z)$. Via the OPE with $T(z)$, the infinitesimal transformations of operators can be easily ...
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Poincaré Symmetry becoming Mobius Symmetry for Euclidean Theory on Riemann Sphere

I've just started reading some introductory notes by Goddard and Gaberdiel on CFTs. The authors start by considering a Euclidean signature meromorphic field theory on the Riemann sphere. They state ...
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Symmetry of Scalar Action Associated with (Conformal) Killing Tensor

Short version: Consider the action for a scalar field coupled to the Ricci scalar in $d$ spacetime dimensions: $$S = -\frac{1}{2}\int d^dx \, \left(\nabla_\mu \phi \nabla^\mu \phi + \xi R \phi^2\right)...
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Ground state in radial quantization -- Why isn't $\phi(0) |0\rangle = |0 \rangle$?

I am trying to reconcile two perspectives on the ground state defined through the path integral. In Tom Hartman's gravity lectures (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf) he says ...
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Could 2D critical quantum system be described by 3D conformal field theory?

It is well known that 1D quantum critial systems are described by 2D cft. Could 2D critical quantum system be described by 3D conformal field theory?
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Is there a character ring for quantum groups?

It is a well known fact that for any (reasonable) group $G$, the character ring and the representation ring are isomorphic, $$ \chi_{R_1}(g)\chi_{R_2}(g)=\chi_{R_1\otimes R_2}(g),\qquad g\in G $$ Is ...
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Is central charge related to symmetry?

I am currently reading the paper "Theory of finite-entanglement scaling at one-dimensional quantum critical points" by Pollmann et. al. and I am trying to understand the central charge in ...
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