The conformal-field-theory tag has no wiki summary.
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1answer
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Explanation of Cardy's “a theorem”
There seems to have been some discussion of Cardy's "a-theorem" recently:
“It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, Sd, when suitably ...
5
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1answer
634 views
Some questions on Conformal Field Theory, Current algebras and the Sugawara construction
Since I don't know how to add another question to an already existing topic,
I'm opening a new thread. However I'm referring to:
Beginners questions concerning Conformal Field Theory
As noted, a ...
4
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1answer
163 views
Defining a CFT using beta-functions
Won't it be correct to define a CFT as a QFT such that the beta-function of all the couplings vanish?
But couldn't it be possible that the beta-function of a dimensionful coupling vanishes but it ...
4
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1answer
89 views
Massive excitations in Conformal Quantum Field Theory
Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of ...
4
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1answer
186 views
Conformal transformation equation
I am currently reading Kiritsis's string theory book, and something bugs in the CFT (fourth) chapter. He derives the equation that should satisfy an infinitesimal conformal transformation $x^{\mu} ...
4
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1answer
117 views
Do a 1-dimensional conformal theory exist?
can we have in physic or can we speak about 1-d conformal theory in physics ??
for example in this one dimensional theory what would be the generators $ x \partial _{x} $ or $ \partial _{x} $ ??
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2answers
244 views
Witten Index and letter partition function
I haven't seen any reference which explains these things and I am not sure of all the steps of the argument or the equations. I am trying to reproduce here a sequence of arguments that I have mostly ...
4
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1answer
164 views
Argument for quantum theoretic conformality of $\cal{N}=2$ super-Chern-Simon's theory in $2+1$ dimensions
I am using the standard symbols of $V_\mu$ for the gauge field, $\lambda$ for its fermionic superpartner and $F$ and $D$ be scalar fields which make the whole thing a $\cal{N}=2$ vector/gauge ...
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2answers
473 views
Superconformal algebra
I had earlier also asked a question about super conformal theories and I am continuing with that, now with more specific examples. I am quite puzzled with it given that I see no book explaining even ...
4
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1answer
116 views
Name of fermionic CFT theory
I'm looking for a name or references to theories that include a stress energy tensor of the form
$$T(z)=A:\phi^\alpha\partial\phi_\alpha:(z)+B:\prod_{i=1}^{D}\phi^i:(z)$$
$\alpha=1,...,D$.
Where ...
4
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1answer
70 views
How can we have massive states of strings and CFT on the string worldsheet at the same time?
Ok, so we can have conformal invariance on a string world sheet. However, it is well known that to preserve conformal symmetry we require states to be massless. So how is it that string theories ...
4
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1answer
84 views
Virasoro TT OPE in Polchinski's book
I'm trying to understand eq. 2.2.11 in Polchinski's first book.
He's computing
$$:\partial X^\mu(z)\partial X_\mu(z): :\partial' X^\nu(z')\partial' X_\nu(z'):$$
Now, I understand why this ...
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3answers
351 views
Special conformal transformations and locality
In the conformal symmetry, used in some QFT theories, the infinitesimal generators, applying to space-time, are all linear (translations, rotations, boosts, dilatation), except the special conformal ...
4
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1answer
117 views
QM with complex eigenvalues
What class of theories/physical systems own finite/infinite complex eigenvalues? I do know that e.g., quasinormal modes of BH do have complex eigenvalues, but are they finite or infinite in number? ...
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0answers
49 views
How to understand Modular transformation in topological order?
Topological order in (2+1)D is described by its ground state degeneracy and the braiding statistics and topological spins of excitations. People believe that these information is all encoded in ground ...
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Trace of stress tensor vanishes ==> Weyl invariant
You often see in textbooks the statement that ${T^\mu}_\mu = 0$ implies Weyl invariance or conformal invariance. The proof goes like
$\delta S \sim \int \sqrt{g} T^{\mu\nu} \delta g_{\mu\nu} \sim ...
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0answers
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Defects in 3+1 TFTs/2+1 CFTs
I would like to know of good pedagogic references to learn about the notion of "defects" in TFTs and CFTs. I am specially interested in 3+1 TFTs (.and probably about their relation to 2+1 CFTs..)
In ...
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1answer
707 views
Why/How is this Wick's theorem?
Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$.
$$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$
...
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2answers
214 views
How are functional determinants of Laplace-type operators used in physics?
Many mathematical papers concerning the $\zeta$-regularized Determinant of Laplace-type operators refer for motivation to the broad use of such determinants in mathematical physics, especially in ...
3
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1answer
94 views
Conformal Quantum Mechanics
I heard the term Conformal Quantum Mechanics used today.
What exactly does this mean?
Why would one want to study this?
3
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1answer
126 views
What are the conserved charges related to the Virasoro generators?
I have just learned from reconsidering my demystified book, that when conformally maping the worldsheet of a closed string to the complex plain by using the transformation $z = e^{\tau + i\sigma}$ ...
3
votes
1answer
286 views
A certain regularization and renormalization scheme
In a certain lecture of Witten's about some QFT in $1+1$ dimensions, I came across these two statements of regularization and renormalization, which I could not prove,
(1) $\int ^\Lambda \frac{d^2 ...
3
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1answer
158 views
Wick Order and Radial Ordering in CFT
I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case):
...
3
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1answer
64 views
Spectral properties of CFT
What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac ...
3
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1answer
74 views
How does the conformal Ward identity guarantee a vanishing 3-point function in this case?
I was looking through some conformal Ward identity related things when I noticed that this paper (arXiv:1212.3788) writes in their equation (33), a 3-point function between a conserved current and two ...
3
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1answer
118 views
Question on Section 9.1.3 in “Conformal Field Theory” by Philippe Di Francesco et. al
Question on Section 9.1.3 in "Conformal Field Theory" by Philippe Di Francesco et. al.
The basic idea of the Coulomb-gas formalism is to place a background charge in the system, making the $U(1)$ ...
3
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1answer
200 views
Superpartner for the stress-energy tensor
I would like to understand what is meant when one introduces a generator $G(z)$ as the superpartner of the energy-momentum tensor $T(z)$.
How does one decide that this $G(z)$ should have a ...
3
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1answer
141 views
commutation of operator product expansion
In CFT, when we have an OPE:
$$O_1(z)O_2(w)=\frac{O_2(w)}{(z-w)^2}+\frac{\partial O_2(w)}{(z-w)}+...$$
this holds inside a time-ordered correlation function, so $O_1(z)O_2(w)=O_2(w)O_1(z)$. Does it ...
3
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1answer
225 views
String matrix models with c>1
Question 1: What is the status of string random matrix models (~ triangulated random surfaces) with c>1?
In my limited search, I have just come across a few papers by Frank Ferrari (in 2000-2002) on ...
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0answers
48 views
Questions about classical and quantum scale invariance
This is kind of a continuation of this and this previous questions.
Say one has a free "classical" field theory which is scale invariant and one develops a perturbative classical solution for an ...
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0answers
124 views
Inclusion of information about external particles to calculate scattering amplitudes
In this (schematic) equation to calculate the scattering amplitude A by integrating over all possible world sheets and lifetimes of the bound states
$$ A = \int\limits_{\rm{life time}} d\tau ...
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Wilson lines, boundary conditions, surface defects of TQFTs
I asked the following question in mathematics stack exchange but I'd like to have answers from physicists too;
I have been studying (extended) topological quantum field theories (in short TQFTs) from ...
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60 views
Relating the deformation of Calabi-Yau metrics and the conformal quantum field theories
(v2)
As I read e.g. in this question, the nice holonomy group features of Calabi-Yau manifolds are valuable regarding supersymmetry (I suspect because it's a symmetry involving the target manifold, ...
3
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0answers
130 views
Central charge at the fixed point of the ${\cal N}=2$ Landau-Ginzburg theory in $1+1$ dimensions
Let me first believe that the ${\cal N}=2$ Landau-Ginzburg theory does in the IR flow to a non-trivial fixed point and that if the potential is of the form $\Phi ^k$ then the central charge of the CFT ...
3
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0answers
135 views
Toda equations and surface operator
I would like to know the reason why the equation (14) in the paper by Yamada is called the Toda equation.
\begin{equation}
\left[\frac12\sum_{i=1}^N\left(y_i\frac{\partial}{\partial ...
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1answer
248 views
What is the meaning of the concepts of “operator mixing” (and anomalous dimensions) [closed]
I am looking for an explanation about the idea of "operator mixing" and its associated concept about when anomalous dimension has to be thought of as a matrix.
For example this idea is slightly ...
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1answer
144 views
Conformal fields on compactified manifolds? An apparent paradox!
I would appreciate it if someone tells me how a cft on a compactified manifold (e.g. by means of periodic boundary conditions) can be meaningful? The global conformal invariance is broken due to the ...
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2answers
442 views
Question on Conformal Field Theory
Since every question has to be asked in a seperate topic,
I'm asking a question refering to the following topic:
Beginners questions concerning Conformal Field Theory
In particula I'm refering to the ...
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2answers
191 views
Invariance of Maxwell's Equations under inverting variables - Reference and use
Some months ago, an ArXiv paper mentioned in passing that Maxwell's Equations were invariant under reciprocating the variables, or at least this results in a dual set of Maxwell Equations. (Actually I ...
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3answers
87 views
Spherical inversion in terms of special conformal transformation
I want to consider conformal maps on suitable compactifications of $\mathbb{R}^{n}$. I know that a special conformal transformation: $$x_i\mapsto\frac{x_i-x^{2}b_i}{1-2b\cdot x+b^{2}x^{2}}$$ can be ...
2
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1answer
252 views
Generalized propagator
I don't understand how to calculate this generalized two-point function or propagator, used in some advanced topics in quantum field theory, a normal ordered product (denoted between ::) is subtracted ...
2
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1answer
141 views
2D Ghost CFT and two-point functions
For some reason I am suddenly confused over something which should be quit elementary.
In two-dimensional CFT's the two-point functions of quasi-primary fields are fixed by global $SL(2,\mathbb ...
2
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1answer
123 views
Conformal Invariance and the OPE
I am reading Polchinski's String Theory now and am having trouble with a couple of things mentioned on Page 45 under "Conformal Invariance and the OPE". Here is the paragraph I am reading. The ...
2
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1answer
190 views
Equivalent definitions of primary fields in CFT
I have come across two similar definitions of primary fields in conformal field theory. Depending on what I am doing each definition has its own usefulness. I expect both definitions to be compatible ...
2
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1answer
151 views
A certain $\cal{N}=2$ superconformal theory (or is it?)
I want to look at the following theory in $1+1$ dimensions with $\Phi$ being the chiral superfield,
$L = \int d^2x d^4\theta \bar{\Phi}\Phi - \int d^2x d^2\theta \frac{\Phi^{k+2}}{k+2} - \int d^2x ...
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2answers
323 views
Why is a critical quantum system described by a conformal theory in one higher dimension of space?
These questions are linked, so I've asked them in a single post:
Why is a critical one-dimensional many-body system a two-dimensional conformal field theory?- Why the switch from 1D to 2D?
What does ...
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1answer
382 views
How do you obtain the commutation relations at non-equal times (for the edge of a fractional quantum Hall state)?
The edge of a fractional quantum Hall state is an example of a chiral Luttinger liquid. Take, for the sake of simplicity, the edge of the Laughlin state. The Hamiltonian is:
$$H = ...
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1answer
94 views
Symmetries in Wilsonian RG (2)
This question is related to the paper http://arxiv.org/abs/1204.5221 and is a continuation of the previous question Symmetries in Wilsonian RG
In the liked paper why do the equalities in equation ...
2
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1answer
101 views
Current operators for compactified CFTs
Intuitively I feel that if you compactified open bosonic strings on a product of $n$ circles such that each radius is fine-tuned to the self-dual point then the CFT of these $n$ world-sheet fields ...
2
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1answer
54 views
Application of Ward identities for OPE under scaling and rotations
I think this is a very straightforward question but I don't see it right now. In Tong's notes on String theory (http://www.damtp.cam.ac.uk/user/tong/string/four.pdf) section 4.2.3 he defines the ...
