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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33 views

Renormalisation scheme dependence of beta function and scale/conformal invariance

This question arose when trying to reconcile the way in which the beta function is usually first introduced in the context of perturbative QFT with the statement that a vanishing beta function implies ...
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Conformal dimension of conserved current of WZW model

The holomorphic part of the conserved current of a Wess-Zumino-Witten model is given by $ J = - k \partial g g ^ { - 1 } $, where $ g $ is a map from $ S^ 2 $ to some Lie group. It is claimed that ...
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Conservation of current induced by two-dimensional scale-invariance

I'm writing my bachelor thesis in CFT (without ever having taken any courses in any field theory) and I'm trying to figure out why in two-dimensional scale-invariant theory the conservation of a ...
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1+1D conformal field theory for the critical point of non-Abelian symmetry breaking

In 1+1D, a spontaneous symmetry breaking of a finite group $G$ gives rise to critical point. What is the CFT for such a critical point.
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Continuous limit in Wilson renormalization group

I am trying to understand renormalization in Wilson approach. There's cool picture, which demonstrates flow of theories in IR: So, if one interested in UV limit, one need reverse flow and flow in ...
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Where do conformal symmetry breaking and the gravitational constant come from in conformal theories of gravity?

Conformal gravity theories are alternatives to GR which are conformally invariant. That is, if $g_{\mu\nu}$ is a metric solving the field equations of the theory, then so is $\Omega^2 g_{\mu\nu}$ for ...
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About variational methods, renormalization and $a$, $c$-theorems

Variational approximation Variational methods are an important technique, frequently applied for the approximation of complicated probability distributions, with the applications in statistical ...
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Degrees of freedom in quantum mechanics and $c$-theorem

In 2D there is notion of degrees of freedom. D.o.f. defined from correlation functions of stress-enegry tensor: $$ F(|z|^2) = z^4 \langle T_{zz}(z,\bar{z}) T_{zz}(0,0) \rangle \\ G (|z|^2) = \frac{z^...
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How to reconcile RG with CFT?

Textbooks often mention that if we perturb a UV theory in the "direction" of a relevant operator, the fixed point theory one eventually ends up with, under RG flow, is/can be a CFT. Now, ...
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1answer
68 views

Equivalents definitions of string field

One can define the Tachyon field in SFT via State-Operator correspondence by $\Phi(0)|0\rangle = c(0)e^{ikX(0)}|0\rangle = c_1|0;k\rangle$ with $|0;k\rangle = e^{ikX(0)}|0\rangle$. I'm trying to ...
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Percolation universality class

There's very good table of different universality classes: Ising model lies in the same universality class with $\phi^4$ theory. Ising in $d≥4$ have critical exponents for free scalar field. But I ...
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Geometric representations of affine Lie algebras

I have recently been reading literature on geometric constructions of representations of affine Lie algebras by Nakajima and others. In particular, the representations arise as cohomologies of moduli ...
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Vacuum Character in Compactified Boson Partition Function

For a generic $c \ge 1$ 2D CFT, I (wrongly?) expect to be able to write its torus partition function as $$Z(\tau, \bar\tau) = \chi_0(\tau) \bar \chi_0(\bar \tau) + \sum_{(h,\bar h) \ne (0,0)}n_{h,\bar ...
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Relationship between scaling dimension and mass in AdS/CFT

I've been reading Horatiu Nastase's notes on AdS/CFT, but I was confused about a certain relationship he claimed. If we compactify supergravity on $AdS_5\times S^5$, we may expand the fields in Kaluza-...
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Overall constant for the scalar propagator in AdS background

I am trying to solve Exercise 3.3 in TASI Lectures on AdS/CFT by João Penedones. It is solving for the scalar propagator $\Pi(X,Y)$ in AdS, and states as follows: $$ \begin{align} \frac{1}{2} J_{AB}J^...
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Conformal field theory in 2 dimensions and Riemann sphere

In many introductory textbooks on conformal field theories in two dimensions, the flat Euclidean manifold $\mathbb{R}^2$ is considered. Later, when the global conformal transformation is derived, $\...
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What does it mean for an extended operator to possess “local excitations”?

In the context of defect conformal field theory, we consider in operator product expansions "local excitations" of the defect (see e.g. text between eq. $(1.1)$ and $(1.2)$ in the paper ...
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Non-supersymmetric CFT in $d=4, 5, 6$

There's no known interacting CFT in $d>6$, see Interacting CFT in $d>6$ Also we know a lot CFT in $d=2$ (minimal models for example) and in $d=3$ (WF fixed points in $4-\epsilon$ approach to ...
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Weyl Anomaly Derivation in Polchinski Eq (3.4.21)

In Polchinski's longer derivation of the Weyl anomaly, he arrives at the result (equation 3.4.19): $$ \ln{\frac{Z[g]}{Z[\delta]}} = \frac{a_1}{8\pi} \int d^2\sigma \int d^2\sigma' g^{1/2} R(\sigma) G(\...
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Interacting CFT in $d>6$

There's an expectation that there aren't interacting CFT in $d>6$. As I understand, main reason for this is due to the scaling dimension of ordinary scalar fields and Dirac fields. This lead to ...
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Recombination phenomena in CFT

Now I study very interesting lectures Superconformal symmetry and representations and I face some statements, which are unclear to me. In unitary CFT there are unitary bounds for dimensions of ...
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A specific coset model

When studying certain 4D $\mathcal{N} = 2$ SCFT, we are led to considering the following coset (having in mind $su(4) \oplus u(1) \subset so(8)$) $$ \frac{\widehat {so(8)}_{-2} \oplus \widehat{su(4)}...
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Question on Eq. 7.40 of “Conformal Field Theory” by Di Francesco et. al

I am trying to understand the second line of (7.40), which I've written below. $$ \begin{align}\langle \alpha|\alpha\rangle &= c_\alpha h^{n(\alpha)}[1 + O(1/h)]\\ \langle \alpha | \beta \rangle &...
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A Question about Central Extension of Lie Groups in Physics

I've been studying central extensions of Lie groups in quantum physics. I found an interesting fact which is that physicists seem only interested in central extensions of symmetry groups by $U(1)$, ...
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Can a CFT have multiple primary operators with same scaling dimension and/or spin?

In CFT ($d>2$), 2-point functions (between two scalar primary operators, for example) vanish unless the operators have same scaling dimension. This leads me to wonder whether a CFT can have two ...
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Two dimensional conformal field theories with changing central charge

For two-dimensional conformal field theories it is usually assumed that the cental charge is fixed (for simplicity let's assume that $c=\bar c$). Is there a generalization or a concept that uses the ...
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Connections between conformal field theory and particle physics

I'm wondering if one could use conformal field theory to predict any facts in (experimental) particle physics? I think CFT shares some similarities with QCD, but I'm not sure if one could use CFT to ...
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Existence of bound state in (higher dimensional) conformal field theory

I'm wondering why correlation function is a more fundamental object in conformal field theory than the notion of particle. By conformal symmetry, I think we can still discuss about massless particle. ...
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$F$-symbols for compact Lie groups

Moore and Seiberg (1989) prove that rational CFTs are classified by the braiding matrices $$ B\begin{bmatrix}j_1&j_2\\i&k \end{bmatrix}\colon \bigoplus_p V_{j_1p}^i\otimes V_{j_2k}^p\to V_{...
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Worldsheet CFT away from criticality

Can we obtain the worldsheet CFT describing string theory as a fixed point of some renormalization group flow (although I assume it leads breaking of diffeomorphism)? In other words, any irrelevant ...
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Rearrangement lemma in normal ordering

I try to calculate energy-momentum tensor from sugawara construction of wakimoto representation for SU(2)K current in 2d CFT. but at first I have to understand rearrangement lemma. so can anybody ...
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Physical system that is described by the conformal group's irreductible representations

I am currently studying the conformal group, i.e the 15-dimensional group that is associated with the conformal symmetry of spacetime. Before working on that group, I've work on the Poincaré Lie group....
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2answers
66 views

1- Loop partition function (Ramond-sector)

When calculating a part of the trace for the partition function of the fermionic Ramond-sector in light-cone coordinates, I'd like to understand how we get to the result $\left(\frac{\theta \left[1/2;...
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Relevant operators in Ising model

Why in 3d Ising near critical point there are only two relevant deformations? I am interested in experimental arguments and also in theoretical explanation. For example, in 3D Ising Model and ...
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Why CFT operators are eigenvalues of Dilatations?

Why all operators which we consider in CFT have fixed Dilatation value? As I know in general QFT we haven't such requirement. What if one will consider questions about operators, which are not ...
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What does it mean by conformal group act projectively and unitarily? [closed]

What does it mean by conformal group act projectively, unitarily, and projective unitarily?
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How does operator product expansion work and how does it connect to Ward Identity in CFT?

In 2D CFT From Paul Ginsparg's paper, the operator product expansion took nice form for primary fields, i.e. (Eq. 2.10) $$T(z)\Phi(\omega,\bar{\omega}) = \frac{h}{(z-\omega)^2} \Phi(\omega,\bar{\...
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Conformal primaries in momentum space

Consider the Fourier transform of a conformal primary $O$ $$\tilde{O}(k) = \int d^dx e^{ik\cdot x} O(x)$$ Now consider the transformation of the momenta $k \to \lambda k$, so that the above reads $$\...
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Young tableaux for conformal group representation

This is taken from arXiv:1910.14051, pg 32: Decomposing this $SO(d+ 1, 1)$ representation into $SO(1, 1)× SO(d)$ representations as in (A.4), we find $$\square \underset{\operatorname{SO}(1,1) \times ...
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Is there any difference between signature $(1,1)$ and $(2,0)$ in 2D CFT?

Is there any difference between signature $(1,1)$ and $(2,0)$ in 2D CFT? The only thing I could thought of was that the previous one had Lorentz symmetry and the later one was Euclidean (rotation), ...
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Scattering amplitude of open + closed strings and D-Branes

When calculating the scattering amplitude of $n$ open string tachyons and $m$ closed string tachyons on the disk, I'd like to understand why choosing the open string tachyons to be attached to $D_{25}$...
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OPEs of vertex operators

Suppose we have two chiral bosons $a(z)$ and $b(z)$ with operator product expansions (OPEs), $$a(z)a(w) = \log(z-w), \quad b(z)b(w) = -\log(z-w)$$ as well as $a(z)b(w)=0$, including only singular ...
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+50

One-point function in CFT on an infinite strip through scaling analysis

In Philippe Di Francesco's book on Conformal Field Theory in section 11.2.3 on the Infinite Strip, the one point function of a primary operator (with scaling dimension $\Delta$) is calculated by ...
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Are there applications for the cancellation of one-loop divengeces in $SO(8192)$ bosonic string theory?

I recently read in Polchinski's textbook on string theory (Volume one, page 229) that an orientifold of the 26-dimensional bosonic string can be considered. Investigating further, I found that after ...
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37 views

Question about solutions to Killing equation in Simmons-Duffin CFT notes

In David Simmons-Duffin's TASI lectures on conformal bootstrap, in section 2.4, the author derives the Killing equation (eq. 20) corresponding to spacetime translation symmetry (spacetime dimensions $...
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Overall constant for the three-point function of a spin 2 primary operator and two scalars

I am trying to solve Exercise 2.5 in the lecture notes on AdS/CFT by João Penedones (link: https://arxiv.org/pdf/1608.04948.pdf). The problem is as follows: Conformal symmetry fixes the three-point ...
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80 views

Representations of the Conformal Group in d Dimensions

In the first edition of Conformal Field Theory by P. Di Francesco, P. Mathieu and D. Sénéchal, to obtain the full generators of the conformal symmetry, they used a trick: First, study the action of ...
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Free theory in AdS/CFT

In $AdS_5$/$CFT_4$, scalar fields of mass $m$ are mapped to conformal operators of dimension $\Delta$ via the dictionary: $$\frac{m^2}{L^2} = \Delta(\Delta - 4)\tag{$*$}$$ From the string theory ...
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Traceless energy momentum tensor and energy spectrum

We have a $D$ dimensional flat minkowskian spacetime, and a field theory with $T_{\mu \nu}$ symmetric, traceless ($T^{\mu}_{\mu} = 0 $) and conserved ($\partial^{\mu} T_{\mu \nu} = 0$). We also assume ...
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Coupling constant dependence of the central charge

In an interacting CFT, how is the central charge related to the coupling constants of the interaction?

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