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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Why is the limiting operator in the CFT state-operator correspondence well-defined, and why is conformal symmetry necessary?
Consider a Euclidean CFT in radial quantisation, and let $S$ be the unit sphere centred on the origin. The state-operator correspondence says that any state $\Psi_S$ living on $S$ can be prepared by a ...
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OPEs, Path integrals some basic questions
I have these basic questions about the OPEs and path integrals. In this question, I am considering string theory on worlds sheet with conformal gauge.
Q1. When we write OPE between two operators say $...
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Ising model rescaling
Consider the 2D classical Ising model. It's understood that there is a critical temperature $T_c$, and that the correlation length $\xi(T)$ defined by:
$$\langle \sigma_i \sigma_j \rangle_\mathrm{...
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How to expand conformal normal ordering / operator product expansion?
I'm working with Polchinski and my question is related to chapter 2 of his book String Theory, Volume 1: Introduction to the Bosonic String.
What I believe to understand
An operator product expansion (...
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How to define volume of Weyl transformation and diffeomorphism groups?
I have a trouble in defining Weyl transformation and diffeomorphism group volumes in the formal expression of string partition function on some manifold $M$:
\begin{eqnarray}
Z_M=\frac{\int Dg [DX]_g}{...
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$\mathbb{Z}_2$ symmetry after orbifold
In Ginsparg's lecture note, section 8.5, the author mentions a new $\mathbb{Z}_2$ symmetry of the orbifold theory: its generator is called $\tilde g$, and it maps any state $\psi \in \mathcal{H}_\text{...
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Contour integral for commutator of fermionic fields
Suppose we have primary fields $A$ and $B$ which have the OPE,
$$A(z) B(w) = \frac{1}{z-w} = -B(z)A(w), \quad |z| > |w|,\tag{1}$$
so they have fermionic statistics. Now I was curious how this would ...
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Free boson twisted boundary condition and $T^2$ partition function
Many CFT textbooks discuss free boson theory and free fermion theories on the torus.
The partition function for the boson theory (without compactification and orbifold) is obtained by summing over the ...
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2-point spin correlator in CFT
I'm currently studying CFT and trying to write correlation functions, mas having trouble when it comes to operators with spin.
The problem that I have is to write the 2 point correlation function of ...
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Advanced string theory book recommendations [duplicate]
Let's say I fully understand Polchinski books on string theory
Are there any recommended and more advanced book on strings?
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CFT description of polynomially degenerate, critical spin-chain
For length $L$ spin chains described by conformal field theories, there's a nice a way to extract the central charge via fitting the following ansatz for the entanglement entropy of the ground state:
$...
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What is $v$ in conformal field theory?
In reading about conformal field theory applied to spin chains of length $N$, I've seen the following expression several times, describing how the central charge $c$ can be extracted from the ground ...
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Spin and scale dimension of canonical spin-1/2 fields in (1+1)d
I am reading the book "Non-perturbative methods in 2 dimensional quantum field theory" by Abdalla, Abdalla and Rothe and have some questions about the Chapter 2.4 "Bosonization of ...
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Do conserved currents have to be primary?
In many texts about CFT it is proven that spin-1 conserved currents have the dimension $d-1$. In the proof it is used that, sometimes only implicitly, the current $J^\mu$ is a primary operator. ...
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Given a finite group, how to figure out which chiral algebra can realise this symmetry?
The classification of the "minimal models" of chiral algebra gives us rational conformal field theories in two dimensions. For example, the classification of unitary representations of ...
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How does one know which operator appears in an OPE?
Let us consider a primary operator $\mathcal{O}$ in a free CFT. How would one find the OPE of:
\begin{equation}
:\mathcal{O}^2(x): :\mathcal{O}^2(y):
\end{equation}
(where :_: denotes normal-ordering)...
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Open-close string amplitude
My question is regarding Polchinski question number 6.12 in which we have to find the amplitude for two open and one close tachyon strings.
I found this solution by Matthew Headrick ("A solution ...
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Di Francesco et al.'s CFT - additional corrections to free-energy for strip geometries on a lattice?
In classical spin systems, there's a nice way to extract the central charge of the model by looking at finite-size corrections to the free energy of strips of length $L$ and width $W$ in the limit of ...
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Verlinde formula of correlation functions
The well-known Verlinde formula states that under $S$-transformation, the characters of a (rational) chiral algebra transform into each other,
$$
\chi_i (- 1/\tau) = \sum S_{ij} \chi_j(\tau) \ ,
$$
...
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How does $u$-$v$ symmetry hold for 2D conformal blocks?
The explicit solution for the conformal blocks that appear in the $\phi$ 4-point function is
$$
g_{\Delta,\ell}(z,\bar z)\sim k_{\Delta+\ell}(z)k_{\Delta-\ell}(\bar z) + k_{\Delta+\ell}(\bar z)k_{\...
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Is the spin of a primary field in 2D necessarily integer or half-integer?
The primary field in a 2D CFT is defined by the transformation property
\begin{align}
\phi^{'}(w) = \left(\frac{dw}{dz}\right)^{-h} \left(\frac{d\bar{w}}{d\bar{z}}\right)^{-\bar{h}} \phi(z)
\end{...
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Infinitesimal conformal transformation on the complex plane as Laurent series [duplicate]
In the book "Conformal field theory" by Di Francesco et al, or the BPZ paper and many other places, the infinitesimal conformal transformation in the (compactified) complex plane/Riemann ...
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Pedagogical reference to learn how to work with CFTs?
I know very little about AdS/CFT, but I am considering looking more into quantum gravity for future research topics. From those of you who have already "conquered" AdS/CFT, I was wondering ...
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Visualizing the conformal compactification diagram of $G$
I asked a question a year and 3 months ago on mathstackexchange but after 3 bounties and still no answer I've decided to try here. Here's the link: conformal compactification.
Construct a conformal ...
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Anyon and state spaces
I start learning about anyons, but I'm confused by a few Hilbert spaces.
First of all, it is said that anyons are "excitations" with anyonic statistics. By that I would imagine they are ...
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Noether current for ``local" conformal transformation?
If the fields $b$ and $c$ have conformal weight $\lambda$ and $1-\lambda$ and action is:
$$S = \frac{1}{2\pi} \int d^2z \, b \bar \partial c,$$
under conformal transformations $z \rightarrow z+\...
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Relation between WZW model and gauge transformation
I came across this question while reading Chapter 15 of Conformal Field Theory by Di Francesco.
So the action of the Weiss-Zumino-Witten(WZW) model is as follows:
$$S = \frac{1}{4a^2}\int d^2x {\rm Tr}...
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Will $L_n$ commute with $f(z)I$? and Ward identity for 0 conformal weight
This might be a stupid question.
In some situation I have to calculate a form like this.
$$
L_n z^N |\psi\rangle \overset{?}{=} z^N L_n|\psi\rangle
$$
In the appendix $B$ of BPZ https://www....
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Conformal invariance and tracelessness of the energy-momentum tensor: contradictory statements
Before starting my question, let me define a couple terms to avoid the confusion that usually accompanies this topic:
I define the $c$-number valued energy-momentum tensor as $T^{\mu\nu} = \frac{2}{\...
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Four-point function and OPE in 2d minimal models
In Di Francesco's book, the four point function in minimal model is studied in eq(9.88),
$$
\langle
\phi_{(r_1, s_1)}(0)
\phi_{(2,1)}(z)
\phi_{(r_3, s_3)}(1)
\phi_{(r_4, s_4)}(\infty)
\rangle\\
\sim |...
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In AdS/CFT, do we have Weyl-invariance or only conformal invariance?
The asymptotic symmetry group of $AdS_{d+1}$ is $SO(d,2)$, which just so happens to be the conformal group of $d$-dimensional Minkowski spacetime. Therefore the boundary dual, if it exists, has ...
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Number (dimension of the space) of conformal blocks
In 2d CFTs, we can study correlation functions of holomorphic primaries. For example, in section 8.3.3 of Di Francesco's book, the BPZ differential equation satisfied by
$$
\langle \phi_{2,1}(z_0) \...
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Modular invariance of WZW CFT and weight lattice/coroot lattice
I am reading the big yellow Book "conformal field theory" by Francesco et al (see equation 14.312-14.315). I am confused with the modular transformation of the Theta function under modular $...
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Integrating infinitesimal transformations in CFT
Consider a conformal transformation $z\mapsto f(z)$ of the complex plane described infinitesimally by the vector field $X\partial+\bar{X}\bar{\partial}$. I was wondering how one can, starting from the ...
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General OPE for a primary field with itself [closed]
I am working through the book Statistical Field Theory [Giuseppe Mussardo 2nd Ed] for CFT.
In the problem session, there is
[10.6. Operator product expansion in the channel of the identity operator]
...
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How to interpret multivalued fields in 2D CFT?
In the notes I've seen on 2D Conformal Field Theory, we derive the Witt Algebra by considering infinitesimal transformations of the form
\begin{align}
z' &= z + \epsilon z^{n+1}
\end{align}
which ...
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Does this DEFINE a CFT?
I have a discrete correlated system defined on a squared grid in $d=2$, all euclidean. I have a random field at each point given by a local function of Grassmann variables (I wouldn't say fermions ...
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Does a CFT need a UV regulator?
I have a very basic question about conformal field theory: Does the partition function need a UV regulator, or is it finite even without? That is, does $\int D\phi \exp(-S)$ converge, or do we need to ...
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Conformal invariance in 2d and correlation functions
It is well-known that 2d global conformal invariance constrains the 2, 3-point functions to some very simple form, and 4-point function must be
$$
f(\eta, \bar \eta) \prod_{i < j}z_{ij}^{...} \bar ...
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CFT Description of the 2D Ising Model as both a free fermion theory and a $\varphi^4$ Landau theory
There are numerous Stack Exchange answers that explain how to construct a free fermion CFT ($c = 1/2$) which describes the critical point of a 2D Ising model.
However there are also sources that ...
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Is there any way to write a non-local operator nicely?
Consider the boundary(the real axis) of the 2d CFT or the D-brane. They were non local object but with well known description.
Consider the following expression,
$$\langle 0| \phi_1(z_1) \phi_2(z_2)|B\...
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Conformal Transformations [duplicate]
I am getting confused if the conformal transformations are transformations of charts on the manifold, or something else. Basically, I can imagine a Euclidean plane, and two observers one with chart $x$...
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What Conformal Field theories are currently known to exist?
Conformal field theories (CFTs) pop up all over physics, especially in condensed matter and string theory. Their existence puts strong constraints on what quantum field theories can exist, since every ...
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How to write the Hamiltonian for the free fermion on the plane?
On a cylinder, the Hamiltonian and mode expansion for the (periodic) free fermion are
$$H=\sum_{n\ge \frac12} \omega n\,\, c_n^\dagger c_n, \quad \psi(x)=\sum_{n\ge\frac12}i\left(c_n e^{-iknx}-c_n^\...
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Conformal Ward identity minus sign error
I'm trying to track down what seems like a fairly crucial minus-sign error in Di Francesco et al's conformal field theory book. The minus sign has to do with the Ward identity for Lorentz ...
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Does Liouville theory have modular invarinace?
The liouville theory was very similar to the free theory but with the introduction of interaction. However, the method to approach the question seemed to be some what different from that of the free ...
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Independent parameters of a conformally invariant object depending on four points
I was considering a conformally invariant object that depends on four points in space: $\Phi(x_1, x_2, x_3, x_4)$. This object apparently depends on 16 different quantities (the four coordines of each ...
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Vectors in AdS/CFT: scaling dimension and near boundary behaviour
I'm trying to understand how the near boundary expansion of a field in AdS$_{d+1}$ is related to the conformal dimension of the corresponding operator in the dual CFT$_d$.
I use coordinates in which ...
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Can D-brane vibrate and generate more allowed boundary conditions?
Previously the D-branes were said to be fixed hyperplane or a boundary where a string could end. The commonly known boundary conditions were Dirichlet Brane and the Neumann Brane and the superposition ...
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Regularisation of supersymmetric two-point function
In four-dimensional Minkowski space, one of the building blocks for two-point functions in CFTs is the squared modulus of the separation between the two points $x_1$ and $x_2$,
$$x_{12}^2 = x_{12}^ax_{...
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