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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Conformal symmetry and group in arbitrary dimensions [duplicate]

As far as i understand, the full symmetry of relativity is conformal symmetry. This is represented by the conformal group $ \operatorname{Conf}(1, 3) $ Of Minkowski spacetime which is $ \mathbb{R}^{1, ...
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Double Discontinuity In CFT

In the paper Analyticity in Spin in Conformal Theories Simon defines the double discontinuity as the commutator squared in (2.15): $$\text{dDisc}\mathcal{G}\left(\rho,\overline{\rho}\right)=\left\...
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Radial ordering in CFT

Consider the following quantum two-point function (without assuming radial time ordering), $$\begin{align} \langle 0 | \hat{T}(y)\hat{T}(z) |0 \rangle & = \sum_{n,m}y^{-(m+2)}z^{-(n+2)}\...
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Relation between the Casimir energy and the central charge in CFT in general

In 2d CFT we know that the Casimir energy of the vacuum is proportional to the conformal central charge $c$. $$ F_L=f_0 L-\frac{\pi c}{6 L} \tag{1} $$ where $F$ is the free energy and L is the ...
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Why radial quantization gives different spectrum?

For example we work with 1+1D massless free boson, in canonical quantization we allow creation operators at any momentum so the Hamiltonian has continuous spectrum. But if we conformally map to a ...
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Planar spin in two-dimensional CFT

I have several questions regarding the definition of planar spin. I was reading the big yellow book (by Di Francesco et. al.) Section 5.1.5 looks a little mysterious. Look at 5.25, which is the two-...
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Correlation function of excited states in the Ramond sector of 2 d free fermion?

$\newcommand{\ket}[1]{|#1\rangle}$ I'm hoping to find an algorithm to do three point function calculation for generic excited states in the Ramond sector of 2d free fermion. In the NS sector, it's ...
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Field transformation under conformal transformation

In 1 (see references below), I'm trying to derive how a spinless field transforms under a conformal transformation, specifically eq. (2.41). CFT references/lectures are the most confusing I've seen ...
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Is there an exact correspondence between Seiberg-Witten theory and mirror symmetry?

Seiberg-Witten solution gives an algebraic geometrical description of the quantum moduli of 4d $\mathcal{N}=2 $ SUSY gauge theory. However, the solution seems purely constructive and does not enjoy ...
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Can Toric Code have a gapless boundary?

The toric code model is known to have two types of "gapped" boundaries, namely, the rough boundary and the smooth boundary. See, for example, Chap. 4.1 of this beautiful review https://arxiv....
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Is 2d CFT partition function invariant under $SL(2,\mathbb{Z})$ or $SL(2,\mathbb{C})$?

In Applied Conformal Field Theory by Paul Ginsparg page 8, the globally defined infinitesimal generators $\{l_{-1},l_0,l_1\} \cup \{\bar l_{-1},\bar l_0,\bar l_1\}$ resulted the finite form of the ...
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How the partition be invariant if the correlator not invariant under global conformal transformation?

Under the global conformal transformation $$\tau \rightarrow \frac{a\cdot \tau +b }{c\cdot \tau + d}, ad-bc=1, a,b,c,d\in\mathbb{Z} $$ the partition function is invariant $$Z(\tau,\bar \tau)= Z( \frac{...
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How was holomorphic function (local) restricted to special conformal group (global) in 2d conformal transformation? [closed]

An example could be found on this pdf file and the discussion was the 2d conformal transformation. Usually, the conformal transformation was derived locally such that the local conformal ...
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Why are tidal forces conformally invariant?

Tidal forces are encoded in the Weyl-tensor $C^\mu_{\nu\lambda\sigma}$. It is well-known that the Weyl-tensor is invariant under conformal transformations: $g'_{\mu\nu}(x) = \Omega(x)g_{\mu\nu}(x)$. ...
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Gravity dual of the string world-sheet CFT?

The AdS/CFT correspondence conjectures a duality between a $(D+1)$ dimensional gravity theory in asymptotic AdS spacetime with a $D$ dimensional conformal field theory. Is there any sense in asking ...
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What are the fusion rules of E-series minimal models?

Virasoro minimal models are supposed to be solved, which is certainly true in the A-series and D-series, but the E-series models are more exotic. Do we at least know the (non-chiral) fusion rules? The ...
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A question on BRST current of the Bosonic string, why we choose $c(z)$ as generator?

I'm new to the forum, I will try to make my asking as clear as possible. I'm currently writing a 40-minutes talk on the BRST quantization of the Bosonic string, mostly following Polchinski's Book. The ...
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What is special about conformal field theory in 2d? [duplicate]

In the most of textbooks about CFT, the special case of 2d is noticed in which complex coordinates play important role and it reads some results like the conformal transformation of energy-momentum ...
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CFT In Embedding Space

I am trying to figure out how a translation or a conformal transformation explicitly look like in embedded space. Given a CFT in Euclidian (or Minkowski) coordinates $x^\mu$ we can embedded them in $d+...
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Why the existence of an associative OPE for a CFT$_2$ is presented as an extra axiom in this presentation?

In the book "Mathematical Introduction to Conformal Field Theory" by Schottenloher, the author introduces in Chapter 9 one axiomatic definition of what a CFT in two dimensions is. The first ...
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Confusion about the derivation of stress tensor OPE from Ward Identity

I apologize for any difficulty in expressing my review. Allow me to briefly summarize the material and then pose my question. Review In David Tong's string lecture note, he derives the OPE between ...
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How to derive the dimension of conformal Killing vector fields on the Riemann sphere? Is it metric independent?

Context In all string textbooks and lecture notes, they derive the CKV on the sphere by considering the flat plane first, i.e. $(\mathbb{C},\delta_{\mu\nu})$. Then, write it in complex variables $$z = ...
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What's the exact definition of fields in conformal field theory?

For example we work with a 2d scalar field $\phi$. I guess $\phi$, $\partial_z\phi$, $\partial_{\bar z}\phi$ are fields, are there more? Is it true that all fields are in the form of $\partial_z^i\...
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Confusion regarding simplifying normal ordered products in CFT

I am studying CFT on my own and have some confusion regarding applications of Wick's Theorem to simplify normal ordered products to time ordered products. Wick's theorem is fairly straightforward, ...
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State-Operator Correspondence and symmetry in CFT in general dimension

Let us assume to have a QFT ($\mathcal{L}$) with translational, Lorentz, scale and conformal invariance. I ask because we can, for example the free scalar free theory, canonically quantize the system ...
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CFT Radial Quantization Raising and Lowering Operators Sign Question [closed]

Following Slava Rychkov (Page 41, Or here on arxiv page 39), I am trying to show that the momentum operator raises the scaling dimension. I've seen the other related questions on this matter by Y. ...
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Can a relevant operator's OPE with itself only include the identity and irrelevant operators?

I am interested in correlation functions in critical spin chains, and I'm trying to understand the consequences of conformal field theory for these correlation functions. I should warn that I do not ...
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Special conformal transformation convention

This is a question not about the Physics (or Math) but about the convention. One usually defines (as in the yellow book Di Francesco) the special conformal transformation (SCT) as \begin{equation} x^{...
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From $H_3^+$-WZW to Liouville gravity: Why should I use auxiliary fields?

Context: auxiliary fields I am new to the field of CFT (conformal field theory), and I am reading two articles, $H^+_3$ WZNW model from Liouville field theory, by Y. Hikida and V. Schomerus; and Free ...
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$(\mathcal{S}\mathcal{T})^3=\mathcal{S}^2=+1$ mistake in CFT big yellow book?

In Conformal Field Theory Philippe by Di Francesco, Pierre Mathieu David Sénéchal Sec 10.l. Conformal Field Theory on the Torus eq.10.9 says the modular transformation $\mathcal{T}$ and $\mathcal{S}$ ...
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Conformal mapping of Euclidean Schwarzschild AdS black hole

I have been trying to understand this for some time. given the Euclidean S-AdS black hole metric, $$ ds^2 = f(r)d\tau^2 + \frac{1}{f(r)} dr^2 + r^2d\Omega^{2}_{D-2} $$ From what I understand this has ...
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Quantum corrections in Holography

AdS/CFT stablish that there is some kind of correspondence between the ${\cal N}=4$ SYM theory and strings in $AdS_5\times S^5$ space-time. I know, for instance that 1/2 BPS operators like Tr$(\phi_1^...
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Coupling two Ising chains via an energy-energy interaction

Consider the transverse-field Ising model on a chain with periodic boundary conditions: $$ H = -\sum_{i=1}^{L} \sigma_{i}^z \sigma_{i+1}^z + h \sigma_{i}^x$$ There's a phase transition at $h=1$, which ...
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Derivation of the Conformal Ward Identity in Di Francesco et al

I am reading section 5.2.2. (titled The Conformal Ward Identity) from Conformal Field Theory by Di Francesco et al. The authors write \begin{align} \partial_\mu(\epsilon_\nu T^{\mu\nu}) &= \...
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How to derive WZW model’s energy-momentum tensor? The result is of course the Sugawara construction

I want to know how to derive WZW’s energy-momentum tensor. We know WZW action is $$ S_{WZW}[g]=\frac{k}{16\pi}\int d^2x Tr(\partial g^{-1}\partial g) - \frac{ik}{24\pi} \int_B d^3y \epsilon_{abc} Tr(h^...
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Wick rotation of CFT three-point function

Let $\langle O_1\cdots O_n\rangle$ be a Euclidean CFT$_d$ correlation function. I know that we can analytically continue to Lorentzian signature as follows. Let $x_i = (\tau_i,\mathbf{x}_i)\in\mathbb{...
Gold's user avatar
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CFT algebra calculation [closed]

Hi I'm reading https://arxiv.org/abs/2006.13280 and following its calculation but I'm stuck in page16, eqn 4.4a. I got different result than the one in the paper, but I can't find the point I missed. ...
Positron3873's user avatar
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Noether charge for dilatations in terms of creation and anihilation operators

I am trying to compute the conserved charge for a continuous diatation symmetry for the massless real scalar field in four dimensions terms of creation and annihilation operators. Then I have, $$\...
Gabriel Palau's user avatar
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How do Dedekind's eta function arise while computing the partition function of a compact scalar field over circle?

I am following the book String Theory in a nutshell (From Elias Kiritsis). In chapter 4.18, it takes a theory following the action: $$S=\frac{1}{4\pi l_s^2}\int X\square X\ d\sigma,\tag{4.18.1}$$ $$ \...
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Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model

According to P. Francesco et al. conformal field theory book the central charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which ...
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About Noether currents for conformal symmetries

I´m reading "Conformal algebra on Fock space and conjugate pairs of operators" of Klaus Sibold and Eden Burkhard, there the authors write all Noether currents in terms of the energy momentum ...
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Trace of stress tensor in 2D average null energy condition

I was looking through Zamolodchikov's derivation of the $c$-theorem and stumbled across an equation which says the following - $$\Theta = T^\mu_\mu = 4T_{z\bar{z}}.$$ As far as I understand, for two ...
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Ricci Scalar Curvature under conformal transformation

Consider the Klein-Gordon equation in curved spacetime with metric $g$ $$\square_g \phi - \xi R \phi = 0$$ and consider a conformal transformation $$g \longmapsto \tilde{g} = \Omega^{2} g \quad , \...
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Scale transformation of scalars in curved backgrounds

I am puzzled by the concept of scalar fields that arise in conformal field theory in curved backgrounds. In general relativity, so far as I understand it, a scalar field is basically a function ...
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WZW primary fields / correlations in terms of current algebra?

Cross-posted from a Mathoverflow thread! Answer there for a bounty ;) Given the $\mathfrak{u}_N$ algebra with generators $L^a$ and commutation relations $ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ , the ...
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Reverse AdS/CFT correspondence?

$\text{AdS}_{n}/\text{CFT}_{n-1}$ correspondence provides a dictionary for one-to-one mapping observables in bulk gravity to boundary conformal field theories. However, does the reverse correspondence ...
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Phase diagram of the Ashkin-Teller model for unequal intraplane couplings

The 2d Ashkin-Teller model is a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric model consisting of two planes of Ising models on the square lattice with an interplane four-spin coupling: $$H = -\sum_{&...
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Is the celestial sphere we actually see the Riemann Sphere?

I've been watching a few lectures by R. Penrose where he seems to say that what we see around us is the Riemann sphere. He usually gives the example of an observer floating in deep space or if the ...
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Strange Wick rotation in the computation of string partition function

In order to compute the one-loop vacuum-to-vacuum amplitude for the bosonic string, one runs into \begin{equation} Z(\tau) = V_D (q \bar{q})^{-D/24} \int \frac{d^Dk}{(2 \pi)^D} \exp({- \pi \alpha^\...
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OPE Coefficients in Holography

I am having trouble reproducing a calculation from the paper "Holography from Conformal Field Theory". In a 2d CFT, consider an operator $\mathcal{O}$ in mean field theory (MFT) with ...
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