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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Picture Number Operator in String Theory

My question concerns the ghost picture charge/picture number operator in the RNS formalism of Superstring theory. In particular I refer to page 403 of "Basic Concepts of String Theory" by R. ...
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What is my mistake in solving the commuation relation $[P_\mu , K_\nu]$

The aim is to obtain the commutation relation: $$[P_\mu , K_\nu]= -2\eta_{\mu\nu}D + 2L_{\mu\nu}$$ I have been trying to solve this for a while now and I get different answers, but not the one I am ...
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Checking $\xi$ solves the conformal killing equation

Problem: I am trying to prove that, using: $$\xi^\mu (x) = a^\mu + \omega ^\mu_\nu x^\nu + \sigma x^\mu + b^\mu x^2 -2b_\nu x^\nu x^\mu \tag{1}$$ and $$\kappa = \sigma -2b_\nu x^\nu \tag{2}$$ ...
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Argyres-Douglas CFT

Adding of mass in supersymmetric gauge theories will affect structure of moduli space by creating new singular point (picture and some statements from Matteo Bertolini: Lectures on Supersymmetry): ...
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How do I get from the conformal transformation equation to the conformal killing equation?

I am unable to obtain the conformal killing equation: $$2\kappa(x) \eta_{\mu\nu}= \partial_\mu \xi _\nu + \partial_\nu \xi_\mu\tag{1}$$ Theory: I understand that the conformal transformation is: $$...
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What is the meaning of a flat metric invariant up to a Weyl factor?

When studying CFT I was told that: A conformal transformation is a spacetime transformation that leaves the flat metric invariant up to a Weyl factor. What is the meaning of leaving a metric ...
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Finding conformal dimension from given Lagrangian

How can I find conformal dimension $(h,\bar{h})$ of field $\Phi$, if Lagrangian density is $$\mathcal{L}=\frac{-\dot\iota}{2}\Big(\Phi^\dagger\frac{\partial\Phi}{\partial x^0}+\Phi\frac{\partial\Phi^\...
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How to check conformal invariance of a Lagrangian?

The Lagrangian is $$\mathcal{L}=\frac{-\dot\iota}{2}\Big(\Phi^\dagger\frac{\partial\Phi}{\partial x^0}+\Phi\frac{\partial\Phi^\dagger}{\partial x^0}+\Phi^\dagger\frac{\partial\Phi^\dagger}{\partial x^...
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Special conformal transformations as isometries of $\rm AdS_2$

It is claimed in these lecture notes (page 87) that a continuous isometry of AdS in Poincare coordinates is the special conformal transformation, $\delta x_\mu = 2 c \cdot x x_\mu - x^2 c_\mu$ for $c_\...
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Finite conformal transformations of fields

I want to work out the finite change of a Field in Conformal Field Theory. In Di Francesco's Conformal Field Theory he states "In principle we can derive [it] from the [local generators at x=0]" but ...
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Torus two-point blocks and a-monodromy for the 2D Ising CFT

I was trying to use some concrete example to understand the a-monodromy and b-monodromy in the proof of the Verlinde formula. On the Yellow Book, I found the following results for the torus two-point ...
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Representations of Conformal Group

I want to work out the Representations of the Conformal Group. I work with Francesco's Conformal Field Theory. He stats in equation 4.30 that $$e^{i x^\rho P_\rho}K_\mu e^{-i x^\rho P_\rho}= K_\mu +...
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Commutator of two “CFT charges”

In class it was shown that $$ i[Q_\epsilon,T^{\mu\nu}] = -(\epsilon\cdot\partial)T^{\mu\nu} - \partial_\rho (\epsilon^\mu T^{\rho\nu}) + \partial^\nu(\epsilon_\rho T^{\rho\mu}) $$ with $$ Q_\...
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Where is this Virasoro null from?

Let's consider the Virasoro algebra with a generic $c$. Take a primary $|h\rangle$ and I try to look for its level-9 nulls: Mathematica spits out 3 solutions $$ h = \frac{1-c}{3}, \quad \frac{1}{3}(53-...
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Path integral for free fermion on torus

If one will consider free fermion on torus,one will face with different spin structures. There are four spin structures, usually labeled ±±. The ++ spin structure has a single positive chirality zero-...
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Conformal Invariance of the Scalar Field

Consider a scalar field with action $$S(\phi)=\int_Md^Dx\partial_\mu\phi\partial^\mu\phi.$$ Following the book on Conformal Field Theory of Di Francesco, Mathieu and Sénéchanl, they claim that under a ...
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Kac-Moody primary OPE

I am reading a paper and on page 13-14 (PDF page 15-16), they say that, The fermionic generators [$G^\pm$ and $\tilde{G}^\pm$] are Virasoro and affine Kac-Moody primaries with weights $h= 3/2 $ ...
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Causality, branch cut choice and analytic continuation of Euclidean 2-pt. correlator in 2D CFT

In 2D CFT, the Euclidean two point correlator of a primary operator $\mathcal{O}$ with conformal weights $h$, $\bar{h}$ is given by $$ \begin{align} \langle\mathcal{O}(z,\bar{z})\mathcal{O}(0,0)\...
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Non-zero Euclidean commutator in 2D CFT?

In a Euclidean QFT, commutators of operators vanish for any spacetime separation. This can be argued very simply by using the path integral representation of the correlator, wherein operators become ...
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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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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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