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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How to derive Kac-Moody and Virasoro algebras from their descriptions as central extensions?

I am following the notes (https://arxiv.org/abs/hep-th/9904145) to learn conformal field theory, and want to know how to derive the contributions to the Virasoro and Kac-Moody algebras from the ...
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Conformal weight of a coset model, and a specific case

Given a coset model $(G\times SO(2d))/H$, what is the expression for its conformal weight (in terms of its central charge or, alternatively, in terms of the highest weights of irreducible ...
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41 views

Simple calculation on coordinate transformation of Lagrangian (Qaulls' CFT lecture note)

I have a question while reading "Lectures on conformal field theory" by Qaulls (https://arxiv.org/abs/1511.04074). $^1$ Question. I cannot find that the transformation (1.12) makes the action ...
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Weight $h$ as a function of the central charge $c$ in a Unitary Highest Weight Irrep of the Virasoro algebra

A highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by $(c,h)$, with $c$ the central charge and $h$ the "Witt weight" (the weight of the Witt algebra part of ...
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Is there a way to make this simple “derivation” of the Trace Anomaly correct?

I think I came up with a simple yet sketchy almost-proof of the trace anomaly (A.K.A. Weyl anomaly) in 2D CFT, but it has the wrong prefactor. I was wondering if anyone could assess whether this "...
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Conformal Invariance of Maxwell's Equations

I am currently doing some conformal field theory (in four dimensions) and want to show the invariance of Maxwell's equations under conformal transformations, in particular \begin{align} \partial_\...
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Field strength renormalization and the energy-momentum tensor

This question is about the connection between the energy-momentum tensor, dilation transformations, and field renormalization. From a Wilsonian perspective on renormalization we start out with a ...
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88 views

How to properly make sense of the $\mathcal{S}$-matrix as a correlator on a sphere?

In the book "Lectures on the Infrared Structure of Gravity and Gauge Theories" by Andrew Strominger, the author discusses in Chapter 3 the idea of "The $\mathcal{S}$-matrix as a Celestial Correlator". ...
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Dirac bracket and Poisson bracket, asymptotic symmetry

I am reading the paper arXiv:9906126. https://arxiv.org/abs/gr-qc/9906126 on the symmetry algebra at horizon (see also well known work done by Brown and Henneaux about the asymptotic algebra of AdS$...
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25 views

Relationship between boundary states and primary states of a Kazama-Suzuki model

In [1] and [2] the authors claim that the boundary states (not just the Ishibashi states) of a Kazama-Suzuki model are labelled in the same way as the primary states of the model, so that the boundary ...
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Irrational Conformal Field Theory v.s. Non-Unitary Conformal Field Theory?

Unitary conformal field theories (CFTs) with irrational (or including the special case of rational) central charge is called irrational conformal field theory (ICFT). Irrational conformal field ...
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Relation between scaling dimension and critical exponents for harmonic peturbations in $O(N)$ Wilson-Fisher (WF) in an old paper

I am reading the paper "Harmonic perturbations of generalized Heisenberg spin systems" (D J Wallace and R K P Zia, 1975) - https://iopscience.iop.org/article/10.1088/0022-3719/8/6/014/meta . The ...
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Are special conformal transformations continuous?

My understanding of special conformal transformations (SCTs) is fairly limited, but I believe that they are composed of an inversion, a translation and another inversion. Since inversions are discrete ...
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BMS Superrotations and supertranslations as conformal symmetries [duplicate]

Can both superrotations and supertranslations be understood as certain locally conformal transformations at the infinite on the sphere? Point 1: Recent research is highligthening the importance of ...
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In 2D CFTs what are the possible forms of correlation functions

I am following Sylvain Ribault's lectures on 2D CFT (https://arxiv.org/abs/1609.09523) in which he lays out 2D CFTs in an axiomatic format. In a CFT we assert (as an axiom) that there is a ...
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$SU(2)$ and $SO(3)$ WZW models

It seems that the $SU(2)_1$ and $SO(3)_1$ Wess-Zumino-Witten models are quite different despite the Lie algebras being identical. The $SO(3)_1$ model has central charge 3/2 and is equivalent to 3 ...
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39 views

OPE coefficients of primaries in a 2D CFT

I am trying to compute the OPE coefficients in a 2D CFT, and I am convincing myself of something that I know is not true but cant find my mistake. Given primaries $V_{\Delta_1}$ and $V_{\Delta_2}$ I ...
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Neumann Boundary Conditions for the Open string and the energy momentum tensor

I read in Polchinski's book, "String Theory", page 56, that for the open string the energy momentum tensor satisfies equation (2.6.26) at a boundary $$ T_{ab}n^a t^b=0 \,, $$ where $n^a$ and $t^a$ are ...
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The central charge and normal ordering

This question is about how the normal ordering in the energy momentum tensor for a free field is consistent with a non-vanishing vacuum expectation value implied by the transformation rules for a CFT. ...
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243 views

Do we care about CFTs in particle physics?

This question is related to these others: mostly this one, but also this one and this one. Do we care about CFTs in particle physics? Let me explain. Suppose we don’t know anything about string ...
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How does it make sense to talk about the size of a string if the string action is conformally invariant?

From what I understand the Polyakov action in string theory is essentially something like $$S(\xi, g, G)=\kappa \int_{\Sigma} d \mu_{g} \operatorname{Tr}_{g} \xi^{*} G$$ where $\Sigma$ is a given ...
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Conformal transformation

I am reading some lecture notes on Conformal Field Theory by Joshua D. Qualls (https://arxiv.org/abs/1511.04074). At the end of page 5 of these notes, it is stated that the four momentum transforms ...
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2-dimensional QFTs invariant under area-preserving diffeomorphisms

In introductory textbooks & lecture notes on conformal field theory, it is usually stated that solving the highly nontrivial dimensional quantum field theory in 2 dimensions is possible due to the ...
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Generators of 2D global conformal group in terms of differential operators?

I'm looking for a reference that lists generators of two dimensional global conformal group on a complex sphere in terms of differential operators that may act on quasi primary fields $\phi(z,\bar z)$....
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Holographic duals of (super)gravity sigma models

Consider a (super)gravity theory on asymptotically AdS spacetime $N$ with fixed conformal boundary $\partial N$ coupled to scalars $\phi_i$ taking values in a manifold $M$, possibly in addition to ...
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Why any expectation value can be computed by this path integral, and not just the time-ordered ones?

This is quite a basic question about the path integral. In Polchinki's String Theory book, Chapter 2, he says: Expectation values are defined by the path integral $$\langle \mathscr{F}[X]\...
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Level truncation in String Field Theory

In string field theory, to study tachyon condensation at level-2, the string field is truncated to $$\Psi=t c_{1}|0\rangle+ u c_{-1}|0\rangle+ v L_{-2}^{(m)} c_{1}|0\rangle,$$ where $t,u, v$ are ...
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Is the leading order contribution to the double-trace operator anomalous dimension always $O(1/N^2)$?

Is the leading order contribution to the double-trace operator anomalous dimension always $O(1/N^2)$ ? I noticed that the double-trace contribution in Polchinski's paper hep-th/0907.0151 gets an ...
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CFT correlators and effective string picture

I have been reading https://arxiv.org/abs/hep-th/9702015 by Maldacena and Strominger. Authors derive emission rate of Kerr-Newmann black hole via standard asymptotic matching first. Then rederive ...
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Large $c$ limit and connected correlation functions in $2d$ QFT

EDIT: This question has been edited thanks to a comment. One of my definitions was wrong, so I have rewritten the whole question. I was reading this paper about $T \bar{T}$ deformations of $2d$-QFTs ...
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Conformal dimension of fields over groups

Group Field Theory (GFT) deals with field theory over group space for instance $$S_2 = \int dg_1dg_2 \mathcal{K}(g_1, g_2)\phi(g_1)\phi(g_2) +\int dg_1dg_2dg_3\mathcal{V}(g_1, g_2, g_3)\phi(g_1)\phi(...
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Matching AdS and CFT symmetries

The isometries of AdS in $D+1$ dimensions and the conformal symmetries in $D$ are isomorphic as Lie algebras. However, the generators on each side have a physical interpretation. In the bulk we have ...
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Why has the free boson a charge $c=1$ in 2D CFT?

In the free scalar field theory in 2D conformal field theory, we consider the correlation functions of the derivatives of the fields, i.e. $$\langle \partial \phi(z) \partial \phi(w) \rangle, \tag{1}$...
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Implications of null-fields for operator algebra in CFT

In the context of Conformal Field Theory (CFT), I have a primary field $\phi_{(r,s)}$ with a level 1 null-descendant, i.e. $(r,s)=(1,1)$ and $h_{(1,1)}=0$. My goal is to understand how this condition ...
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Can all CFT state be prepared through scale invariant MERA

It is known that in numeric computation, scale invariant MERA is useful for representing a CFT vacuum state. Is the converse true? i.e. all CFT vacuum state (the quantum state with translation and ...
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39 views

Factor of $1/2$ in $TT$-OPE [closed]

I'm trying to calculate the TT OPE in a bosonic theory. I'm missing a factor of 1/2 in the least-singular term. We have (following Di Francesco) $$\langle \partial \phi(z) \partial \phi(0) \rangle = \...
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Noether's theorem for arbitrary conformal coordinate transformations

I have been reading Introduction to Conformal Field Theory by Blumenhagen and Plauschinn. Equation (2.19) on page 19 states that if our theory is invariant under a general conformal transformation $x^\...
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Weyl transformation of Ricci tensor

We define the Weyl transform as, $$ \tilde{g}_{\mu\nu}=\Omega^2g_{\mu\nu}, $$ wherein $\Omega^2$ is a scalar function of space-time $x$. The Weyl transformed Christoffel symbol can be obtained by ...
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Importance of an extra total derivative term in Liouville theory

In this paper on boundary Liouville theory, the authors have introduced an extra term, $-\partial_{\sigma}^2\phi$, (the last term in the equation below) in defining the stress tensor of the Liouville ...
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71 views

Is the gauge/gravity (or AdS/CFT) duality believed to be exact?

I was wondering about the implications of the gauge/gravity (or AdS/CFT in a more restrictive sense) duality for the way we deal with physical theories, and I was wondering if the duality was believed ...
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Completeness relation of spin matrices

I was reading Hugh Osborne's notes on Conformal Field theory and came across a completeness relation which seems easy to prove but I am unable to do it. ${(s_{\mu\nu})}_{\alpha}^{\beta}{(s^{\mu\nu})}...
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Is there an explicit mapping between N free bosonic fields and the $SU(N)_1$ WZW model + free boson?

Witten's nonabelian bosonization tells us that $N$ free Dirac fields can by written in terms of an $SU(N)_1$ WZW model and one free boson. But bosonization also tells us that we could just as well ...
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Supercurrent of the $bc$-$\beta\gamma$ SCFT

In Polchinksi's Sec. 10.1, the $bc$-$\beta\gamma$ SCFT is introduced with action $$S_{BC} = \frac{1}{2\pi} \int d^2z (b \bar \partial c + \beta \bar \partial \gamma)$$ and supercurrent $$T_F = -\...
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1answer
51 views

Anomalous dimension of double-trace operators

Is it true that if a single-trace operator, say, $O$ acquires an anomalous dimension $\gamma_o$, then the anomalous dimension of the double-trace operator $O^2$ is $2\gamma_o$? If no, can anyone ...
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Why the full conformal symmetry is $Vir\otimes \overline{Vir}$ instead of $Vir\oplus \overline{Vir}$

In 2D CFT, we have the Virasoro generators $L_m$ and the generators $\bar L_m$, which are such that $[L_m,\bar L_n]=0$. Hence I thought that the full conformal algebra was $Vir\oplus \overline{Vir}$. ...
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85 views

Normal ordering by contour integral in CFT

In chapter 6 of Di Francesco, they introduce the normal ordering $$ (AB)(w) = \oint_w \frac{ dz }{ 2\pi i (z-w) }A(z) B(w)\ .\tag{6.130}$$ So far so good. But then starting eq (6.139) $$ \oint_w \...
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Tachyon mode in Brane-Antibrane system (Type II)

Lets First start with coincident $D3$ brane- $D3$ Brane system. Superstring stretch between these two contain a Tachyon mode But GSO projection removes this. For open string, $D$ branes serve just as ...
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Can conformal transformations in $\mathbb{R}^{1,1}$ be analytically continued to $\mathbb{R}^{2,0}$?

In 1+1 dimensions, 2D Minkowski space, a conformal transformation is given by two real functions (of one variable). After Wick rotating the time dimension, giving us 2 dimensional Euclidean space, ...
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Which AdS/CFT correspondences have been found so far?

When I read about AdS/CFT correspondence, there always comes the most famous example of conjectured correspondence, which is the one between type IIB string theory (AdS side) and $\mathcal{N}=4$ ...
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Finite conformal transformations of fields from infinitesimal

I know that in conformal field theories conformal group acts not by pushforwards but (e.g. for scalar field $\phi$ with conformal dimension $\Delta$) $$ \phi(x) \mapsto \phi'(x') = \left| \frac{\...