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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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Stress-Energy Tensor and Conformal Invariance in String Theory

Since the Euler-Lagrange Equations corresponding to the Polyakov Action implies no dependance on the auxillary metric we arrive at the constraint $T_{ab}=0$. We then change to lightcone coordinates $++...
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Do Two Operators Need A Term In Their OPE Whose Weight Is Their Combined Weight?

My logic is as follows. Suppose I have two operators, $O_1$ and $O_2$. I place a copy of each operator near the origin, and a copy of each operator distance $d$ away, for some large $d$. I can first ...
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Derivation of Holomorphic Ward Identities in Franceso's CFT

In equation 5.37 of francesco's CFT he writes the Ward Identities for traslation symmetry in the language of holomorphic functions. He goes from \begin{equation} \frac{\partial}{\partial x^\mu} \...
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Mismatch between conformal generators and conformal transformations as changes of variables

Introduction It is known that under changes of coordinates different fields transform according to their tensorial nature (scalar, vector, etc.) like$^{[1]}$ \begin{equation} \phi(x)\rightarrow\phi'...
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What do we mean by scale invariance in a classical field?

First of all, I read many questions but they don't seem to answer my specific question. So, here it goes According to Francesco's Conformal Field Theory and many other books, a scale transformation \...
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What is $Z$ useful for in a CFT?

As an example, the partition function of a free boson on a torus with modular parameter $\tau$ is, $$Z(\tau,\bar{\tau}) = \frac{1}{|\eta(\tau)|^2}.$$ In quantum field theory, the partition function ...
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rigorous definition of coherence length at mean field theory

so as far as I know, when we are doing mean field theory, in qft, we expand a action of a theory around a classical solution. so we find a classical solution, than we add quantum mechanical ...
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Where does the string worldsheet live?

Suppose you define a Euclidean two-dimensional CFT for a scalar field $X$ of dimension length, $$ S = \int d^2x h{}_{\mu\nu} \partial^\mu X \partial{}^\nu X \, , $$ and $h{}_{\mu\nu}$ is the metric of ...
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Conformal transformation vs diffeomorphisms

I am reading Di Francesco's "Conformal Field Theory" and in page 95 he defines a conformal transformation as a mapping $x \mapsto x'$ such that the metric is invariant up to scale: $$g'_{\mu \nu}(x'...
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Derivation for the commutation relation of $\alpha_m^\mu$ and $\alpha_n^\nu$

I have been trying to derive the commutation relation of $\alpha_m^\mu$ and $\alpha_n^\nu$ in a closed-string mode expansion, but I found an extra factor of $2$ that ruins things out: Given $\dot X = ...
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$\phi R$ term for scalar field in a curved background

Condider the following action for a free scalar field $\phi$ in a curved background $$S=\int dx\Big( \frac12g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+\gamma \phi R\Big)$$ Here $g_{\mu\nu}$ is a ...
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Inversion Formula in Conformal Field Theory

I was working on my thesis on Conformal Bootstraps. For that, I now have to use the inversion formula to get anomalous dimensions in the $\phi^3$ theory. Can anyone suggest any good reference(s) to ...
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Question about a formula in Wakimoto representations

In conformal field theory book by Francesco, Philippe, Mathieu, Pierre, Sénéchal, David, page 662 equation (15.281) I don't quite understand why the second line equation develop a $\partial^2 \varphi$...
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Is every classical field theory with dimensionless couplings conformally invariant?

I'm trying to learn conformal field theory and getting rather frustrated, because I can't find any source that gives decent examples or straightforward logic. In most sources I have found, conformal ...
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Connected components of conformal group $ {\rm Conf}(p,q)$ containing $P$, $T$ and conformal inversion are same or different?

As we known (see this post), the global conformal group for $\mathbb{R}^{p,q}$ is $$ {\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}$$ The global conformal group ${\rm Conf}(p,q)$ has 4 ...
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Interacting conformal field theories in spacetime dimensions higher 6?

Are there any papers which directly tackle the question of whether or not there exists interacting CFTs in spacetime dimensions higher than 6? It has been proven that there do not exist any ...
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I want to know the conformal weights of spinors in 2D

I want to know the conformal weights(or dimensions) of left/right-moving fermions in 2D, ${\cal N}=(2,2)$ superconformal theory. More specifically, what is the left/right-moving conformal dimension ($...
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Resource on free conformal field theory [duplicate]

In QFT, it is easy to find resource on a free scalar quantum field theory. When I tried to search for a free conformal field theory, I could not find a good resource. Can anyone either give some ...
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Confusion on Carlip's approach to Cardy Formula

I'm reading Carlip's approach to Cardy Formula (https://arxiv.org/abs/hep-th/9806026). He considers the partition function on the torus of modulus $\tau$ to be $$\mathcal{Z}(\tau, \bar{\tau})= \text{...
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Correlator of a single vertex operator

In any textbook on CFT vertex operators $V_\alpha(z,\bar{z})=:e^{i\alpha\phi(z,\bar{z})}:$ are introduced for the free boson field $\phi(z,\bar{z})$ and their correlation function is computed $\...
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Simultaneous shifted diagonalization of bunch of operators

I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule $$ \Big[ L_i ,L_j \Big]= (i-j)(L_{i+j}-\frac14 L_{i+j-1}).....*$$ My question is ...
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Luttinger model, conform field theory and RG

I'm studying the Luttinger model and I'm having a hard time understanding its relation with conformal field theory. I'd like to know something about this. Is the Luttinger model conformal only at ...
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Half Witt algebra

I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule $$ \Big[ L_i ,L_j \Big]=\frac18 \frac{(2i+2j-1)(2j-2i)}{(2j+1)(2i+1)}L_{i+j-1}-\...
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How does the Weyl anomaly imply $\langle T^{\mu}_{\mu} \rangle \neq 0$

I want to consider the case of euclidean field theory in 2 dimensions with the action $$S[\phi]=\int \! d^2\!x \sqrt{\det(g)}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi$$ which leads to a partition ...
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How do we understand the results of $1/N$ or $\epsilon$ expansion beyond leading orders?

When we do $1/N$ expansions in, say, 2+1$D$ $O(N)$ models and try to extract all kinds of critical exponents from it, we get the following results for the scaling dimensions of various operators up to ...
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Aharony-Bernman-Jafferis-Maldacena (ABJM) and k=1 Chern Simons matter

I have read recently that the partition function / half-BPS wilson vev (w/ NG probe) of a Chern-Simons matter theory with N=6 U(N)k x U(N)-k super-conformal symmetry (ABJM) on S3 is proportional to ...
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Resources on Gubeser-Klebanov-Polyakov (GKP) strings and N=4 Super-Yang Mills dual description

*I have learned recently that the Gubeser-Klebanov-Polyakov string / folded string in AdS3 (if I recall correctly, and I assume with some additional virasoro constraints, etc) is dual to large-spin ...
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Understanding Moonshine via Heterotic E8xE8 (resubmitted, originally asked on mathematics stack exchange)

Recently I have become familiar with the conjectured relationship of monstrous moonshine and pure (2+1)-dimensional quantum gravity in AdS with maximally negative cosmological constant and, it’s being ...
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Scalar-Scalar-Spin(l+1/2) correlator in CFT

Is the $\langle \phi^{(0)} \phi^{(0)} \psi^{(l)} \rangle$ in a CFT zero ? Where $\phi$ and $\psi$ are spin-0 and spin-$(l+1/2)$ fields respectively and $l$ is an integer. If so, please, explain why ? ...
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But how exactly do you calculate the Joukowsky Airfoil, within a minimal margin of error?

After reading a fair bit of theory around the uses of conformal mapping to solve for the forces of lift acting on a wing, or a 2D cross section of the wing, in relation to the angle of attack. However ...
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Free Field Realization of Current Algebras and its Hilbert space

I have some conceptual confusion regarding the interplay between current algebras, their free field representation and the Hilbert space generated from it. Let's sketch a simple example, $\mathfrak{...
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Double-trace operators in CFT?

What is the conceptual difference between so called "single-trace" and "double-trace" (or "multi-trace") operators e.g., in a Conformal Field Theory?
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Existence of solutions to crossing with the Gliozzi's method

The Gliozzi's method in the Conformal Bootstrap consists in finding approximate solutions to the crossing equation $$ \sum_{\Delta,L} \mathsf{p}_{\Delta,L} \, \frac{v^{\Delta_\varphi}G_{\Delta,L}(u,v)-...
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Conformal Ward identities for spinor operators

How do you derive conformal Ward identities for operators with spin? You can see in Penedones's notes (page 6) ( https://arxiv.org/abs/1608.04948 ) a brief derivation of Ward identities for general ...
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Lattice model realization of $SU(2)$ WZW model at level $k$?

Is there any lattice model realization of the following model: $c=1$ boson at the self-dual radius, or the $SU(2)$ WZW model at $k=1$. This is a question inspired by: Orbifolds of the $c =1$ ...
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Braiding matrix from CFT first principles

Various CFT models are known to produce representations of braid groups. A famous example is the $SU(2)$ WZW model at level $k$, for which the braiding matrix for the case of two fundamental irreps ...
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Conformal symmetry, Weyl symmetry, and a traceless energy-momentum tensor

I'm trying to drill down the exact relation between conformal symmetry, Weyl symmetry, and tracelessness of the energy-momentum tensor. However, I'm getting quite confused because every book I can ...
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What is the actual definition of conformal invariance?

I've seen a large variety of slightly different definitions of conformal invariance. For simplicity I'll only consider scale invariance, which is already confusing enough. Some of the definitions are: ...
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2D global conformal transformations and the $z= \frac1w$ argument

For instance in Blumenhagen's CFT, there is a standard argument which determines that globally defined conformal transformations on the Riemann sphere where $$l_n = -z^{n+1} \partial_z$$ is an ...
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How to define an Operator Product Expansion (OPE) on arbitrary Riemann surface for a CFT?

Whenever we define the OPE of a 2D CFT, we do so (at least in the literature that I have come across) on the complex plane. Similarly, the commutation relations between conformal generators $L_n$ and ...
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What is radial ordering?

In my String theory lecture radial ordering was introduced and I don't understand what it is. My first guess was $$R(A(z)B(w)) = A(z)B(w)\Theta(|z|-|w|) + B(w)A(z)\Theta(|w|-|z|).$$ But then we have ...
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Known Conformal Field Theories? [closed]

What is a comprehensive list of all known Conformal Field Theories e.g. in 2 dimensions? CFT references usually talk about just minimal models and maybe Wess-Zumino-Witten model. I'm curious to learn ...
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Why can we not define asymptotic states in CFTs?

I have known that we can't define asymptotic states in CFTs, because we can't use Fock spaces to describe CFTs. But is that right and why? I want to know some details about it.
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Does the vanishing of the one-loop beta-function imply no running to all orders?

This question sounds ridiculous, but bear with me. I am having a hard time reconciling the following two facts: Classical global symmetries can become anomalous upon quantization, and the anomalous ...
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Primary field in CFT and path integral

I should feel ashamed to ask such a naive question, but anyway let me start with the $\phi^4$ theory in the Minkowski spacetime, which has a Lagrangian of the form $$\frac{1}{2}(\partial\phi)^2-\frac{...
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Constraints on correlation functions of Quasi Primary Fields

I have problems understanding constraints on correlation functions of quasi primary fields (QPF) following DiFrancesco's Conformal field theory book. In chapter 4, section 4.2.1, a QFP is defined as a ...
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Weyl anomaly in 2d CFT (string theory lectures by D.Tong)

In his lectures on String Theory (http://www.damtp.cam.ac.uk/user/tong/string.html), Tong gives a proof of the Weyl anomaly, using equation $(4.36)$. It seems wrong to me. Here he uses the OPE ...
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Geometry of Affine Kac-Moody Algebras

One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric quantization, using the Kähler structure of various $G/H$ spaces. Can one perform a ...
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2d CFT and first intersection of vanishing curves

In the search for 2d Unitary CFT's, we use the Kac determinant to find the null curves, and remove regions in parameter space(c and h space) where the determinant is negative. Now, for $h>0$ and $ ...
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Scalar Product and Adjoint Operator in CFT

$\newcommand{ip}[2]{\left< #1, #2\right>} \newcommand{\d}{\, \mathrm d \lambda^n}$For a Hilbert space $(H,\ip \cdot \cdot)$ and an operator$^1$ $A$ the adjoint of $A$ is defined via the ...