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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What do we mean by radial quantisation in CFT?

When we quantise QFT we do that in equal time slices. In CFT it is useful to use equal radius slices. Why is that the case? And what does it mean?
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Hermitian Conjugation in 2D CFT

When we take the hermitian conjugate of an operator in D dimensions we have: $$ \mathcal{O}_{flat}(r,\vec{n})^\dagger=\frac{1}{r^{2\Delta}}\mathcal{O}_{flat}\left(\frac{1}{r},\vec{n}\right) $$ where $\...
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Ising model operators

Ising model formulated as lattice theory with local degrees of freedom described by $s_i$ $i\in 1, \dots, N$ and energy: $$ E[\sigma_i] = -J\sum_{<ij>} s_i s_j $$ From $s_i$ I can construct a ...
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Special conformal transformation maps line and circle to line and circle [migrated]

I want to show that a special conformal transformation maps line and circle to line and circle and find an example of it maps line to circle. The statement seems very true from the picture of the wiki ...
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Polchinski massless vertex operator in the Polyakov approach p105 Eq. (3.6.16)

I am trying to check the Weyl transformation of the massless vertex operator in Polchinski closed bosonic string in the Polyakov approach (p105, Eq 3.6.16).To do that one needs to calculate something ...
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Confusion with central charge in CFT and improvement of energy-momentum tensor

In theory of free scalar field $$ S = \int d^2x \;\partial_\mu \phi \partial^\mu \phi $$ $$ \langle \phi(z) \phi(w)\rangle \propto \ln(z-w) $$ Exist family of energy-momentum tensors (new term ...
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Is there a definitive prediction of mass of new particle by String/CFT, based on which we could approve or discard that theories?

Does String/CFT predict experimentally measurable mass of a new particle which could be experimentally discovered by building new accelerator or using existing accelerators, but seeking for specific ...
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Could String theory / CFT be classified as pseudoscience in its current form? [duplicate]

Richard Feynman was saying that those theories were dishonest. Those were selling sweet story of unique compactification, supersymmetry which never breaks, that they can predict parameters of Standart ...
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Three-point function in CFT

Reason for deriving the 3-point function Searching for a derivation of the three-point function constraints in CFT online I have realised that there is no derivation of the 3-point function. Most ...
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Is a function of conformal ratios invariant under conformal transformations?

If I have a function $f:=f(r,s)$ a function of the conformal ratios $r$ and $s$ only, i.e. for example: $$r := \frac{(x1-x2)^2(x3-x4)^2}{(x1-x3)^2(x2-x4)^2} \qquad, \qquad s := \frac{(x1-x4)^2(x2-x3)^...
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Constraining the 2-point correlation function

Consider the two-point function $$ \langle\mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\rangle=f(x_1,x_2) $$ If the operators are in a CFT, we can constrain this function using the symmetries of the theory. ...
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Polchinski Weyl Anomaly from perturbing the flat background. Eq (3.4.22)

In deriving the Weyl anomaly for the bosonic string using a perturbation around a flat background, Polchinksi uses Eq. (3.4.22), i.e. $$ \ln \frac{ Z[\delta+h] }{Z[\delta]} \approx\, \frac{1}{8\pi^2}\...
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Constructing W-algebras

I am following the algorithm in W-algebras with two and three generators, in order to construct consistent (anti-)commutator relations for a particular W-algebra. I am considering $W(2,4,4)$ where ...
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Particles and states in string theory

In QFT, put simply, we have some quantized field which lives on our spacetime and the excitations of this field correspond to particles. So some particular excitation would correspond to a particle at ...
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Conformal weight of one dimensional scalar fields

In string theory a conformal transformation changes the metric, $g_{\mu\nu} \rightarrow \Omega(\tau, \sigma)g_{\mu\nu}$ with $g_{\mu\nu}$ the two-dimensional metric on the (Polyakov) string worldsheet....
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How is $ \phi_1\partial_\mu\phi_2-\phi_2\partial_\mu\phi_1 $ a primary field?

I am trying to understand why $$ \phi_1\partial_\mu\phi_2-\phi_2\partial_\mu\phi_1 $$ is a primary field that we can consider in a CFT, provided that $\phi_1,\phi_2$ are primary. Since I see ...
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‘t Hooft loops, yang mills, and “free energy”

When reading through the N=4 SYM s-duality literature, one will encounter some interesting properties i.e. twists (on boundary conditions), non-trivial electric and magnetic fluxes, the so called “...
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Correspondance between free fermion CFT partition function and Virasoro character

In these lectures on conformal field theory, the author calculates the partition function for the free fermion on a torus. By calculating the contributions from different boundary conditions, they ...
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Why allow only one singularity at $z =0$ when defining the local conformal algebra in 2D?

The reference I'm using for this question is the review "Applied CFT" by Paul Ginsparg. In section 1.2 (Conformal algebra in 2 dimensions) he argues that if the metric is the Euclidean one $g_{\mu\nu}=...
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Green's function on torus

I have a question about the Green's function $G(z,w)$ on torus which takes the form (for example the first equation in the paper https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n2-p03-...
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Operator-state correspondence in quantum mechanics

Operator-state correspondence is usual in $d\geq 2$. See for example Operator-state correspondence in QFT. Is some kind of Operator-state correspondence in 1d CFT or more generally in quantum ...
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Conformal symmetry extension for super p-branes?

Conformal symmetry plays a vital role in string and superstring theories. Why can not we generalize conformal symmetry to objects of greater dimensions as branes? Where are we with this puzzle in 2020?...
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Vertex algebra confusion

In Blumenhagen's book on CFT, the authors have defined $\bar{v}(\bar{z})$ to be the antiholomorphic part of the vertex operator for a free bosonic CFT, $V(z,\bar{z})=:\exp{(\alpha X(z,\bar{z})}):$ ...
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Verifying infinitesimal form of 2.4.26 from Polchinski

According to equation 2.4.26 of Polchinski's String Theory Volume 1, the finite transformation of the energy momentum tensor is $$ (\partial z')^2 T'(z') = T(z) - \frac{c}{12} \{z',z\}$$ Where $$\{...
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How do fields transform under special conformal transformations?

A Question in Classical Field Theory $\underline{\text{Assumption 1}}$: The definition of a transformation specifies how both the coordinates and the fields transform: They are namely $(1$-$1)$ and $(...
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Witten's description of WZW conformal blocks

I am reading this paper by Witten - Geometric Langlands From Six Dimensions. In section 4.1, he gives a description of the vector space of conformal blocks of the current algebra associated to a ...
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Minimal models and Lattice models

How does one see that the minimal model M(4,3) is the Ising model ? And how can I argue out that the fields contained in M(6,5) but with the non-diagonal modular invariant partition function ...
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OPE of stress tensor in CFT

I come aross an OPE between stress tensor components in CFT which is \begin{equation} T(z)\bar{T}(\bar{w})\sim -\frac{\pi c}{12}\partial_{z}\partial_{\bar{w}}\delta^{(2)}(z-w)+... \end{equation} I am ...
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What are the necessary conditions for a CFT to have a holographic dual? [duplicate]

The number of degrees of freedom of a CFT is given by its central charge $c$. From the bootstrap point of view, any CFT is characterized by the knowledge of its "CFT data", i.e. the scaling dimensions ...
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$T$-duality symmetry of $SU(2)_1$ WZW model

For bosons at self-dual radius, the CFT has T-duality symmetry. My question is can we realize this symmetry on the lattice model? for example antiferromagnetic spin chain.
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Selfstudy Conformal Field Theory and Lie algebra from scratch roadmap [closed]

A couple of months ago, I stumbled upon Conformal Quantum Mechanics (CQM) which was really interesting yet confusing. I didn't have the prerequisites so I couldn't follow the paper. I want to first ...
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On the use of the Cauchy-product in deriving the stress-energy tensor constraint for a string in light-cone coordinates

I shall be following the notation used in David Tong's String Theory notes. In bosonic string theory, after fixing the gauge of the Polyakov action using Diff/Weyl invariance to make the two-...
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Is the Casimir energy in CFT an observable?

We know that if we transform a 2d conformal field theory from a plane to a cylinder with perimeter $L$, the ground state energy will be shifted by $$E = -\frac{c}{24L}$$ due to the Schwarzian ...
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Holographic AdS/CFT intepretation of Cosmological Constant

I'm trying to understand how the value of AdS cosmological constant (or, equivalently, of AdS radius) affects the boundary CFT in AdS/CFT correspondence. I'd be glad of discovering something about the ...
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Most general Lagrangian in Conformal Quantum Mechanics

This question has already been asked and answered in Most general Lagrangian in CFT in 0+1D. However I am just partially convinced with the answer. The idea is to construct the most general ...
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Isomorphism between the conformal group and SO

In CFT, one constructs the generators of various confromal transformations, and re-expresses them in terms of $J_{ab}$'s that manifestly satisfy commutation relations of $so(d+1,1)$ (taking Euclidean ...
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4-point function in CFT

Im currently trying to understand the form of the 4 point function in CFT, i.e. how to derive equation 4.62 in Di Francesco et al. In particular, the coefficients of the $x_{ij}=|x_i-x_j|$. For four ...
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Bosonization left and right moving fields in 1D

In’s Senechal’s Bosonization review (https://arxiv.org/pdf/cond-mat/9908262.pdf)for the free boson he defines separate left and right moving parts for the field $\phi$ As $$\phi(x,t)=\phi(x-vt)+\bar{\...
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Ghost exorcisms of fields?

In Mack's paper "D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes", he makes the following statement ...
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Equivalence between OPE associativity and four-point crossing symmetry

I'm reading Simmons-Duffin CFT Lecture Notes, where it's stated that one can recover the OPE associativity from the four-point correlator crossing symmetry. It seems supposed to be a very trivial ...
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The mathematical structure of $\widehat{su(2)}_k$

Some of my colleagues work on CFT's and quantum groups and I hear them talk a lot about $\widehat{su(2)}_k$ algebras. According to them (and the general physics literature) these are what ...
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How tensor fields transform under special conformal transformation?he

I am trying to find special conformal charges for free maxwell field. We have diffferent transformation rules for lorentz transformation and scaling transformation. what will it be for SCT for vector ...
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Weyl SUSY vs Conformal SUSY

Is it possible to add the generators of dilatations to Poincare superalgebra in any dimensions with any number of supercharges without adding the full superconformal generators? I have only seen ...
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Definition of dangerously irrelevant operator

(Disclaimer: There is already a question about dangerous irrelevant operators which has not been very successful. However, the question there is quite broad, and here I aim to ask a more precise ...
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Electric field produced by a capacitor consisting of two parallel plates of different lengths: field lines and edge effect

It is known that the following integral equation describes the electrostatic field produced by a capacitor consisting of two parallel circular plates, derived in this paper (download for free) $$f(x)...
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Why do we need conformal compactification to define the global conformal group?

First I have the definition of a conformal map. Let $(M,g)$ and $(M',g')$ be two pseudo-Riemannian manifolds of same dimension. Let $U\subset M$ and $V\subset M'$, we say that a smooth map of maximal ...
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Weyl- Squared Lagrangians

I'm studying conformal gravity theories, in particular I read that if we take $L=\sqrt{g}C_{abcd}C^{abcd}$ where $C$ is the Weyl tensor the theory we get is not unitary. What does it means unitary at ...
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Appearence of higher spin algebra

I start from massles & free scalar field theory in $d$-dimenisonal space. It is clear that this theory has conformal symmetry. My question is devoted to derivation of conserved current $$J_{\mu_1\...
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Calculating CFT on 2D curved manifolds

In 2D we can always choose coordinates in a coordinate patch so that the metric is conformally flat $$g_{\mu\nu}(x)=\kappa(x)\delta_{\mu\nu}$$ A simple example is the sphere $S^2$ in stereographic ...
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What does it mean when we say the kinematic space of the time slice of Ads3 is ds2?

I have been going through this paper Integral Geometry and Holography the authors in page 19 demonstrate the idea of kinematic space using $Ads_3$, they start off with a hyperboloid model and show ...

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