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3
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0answers
225 views

A gentle introduction to CFT [closed]

Which is the definition of a conformal field theory? Which are the physical prerequisites one would need to start studying conformal field theories? (i.e Does one need to know supersymmetry? Does one ...
3
votes
0answers
193 views

Traceless of stress-energy tensor in $d=2$

This is a question regarding Francesco, section 4.3.3. In this section, he considers the two-point function $$ S_{\mu\nu\rho\sigma}(x) = \left< T_{\mu\nu}(x) T_{\rho\sigma}(0)\right> $$ He then ...
3
votes
0answers
122 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 ...
3
votes
0answers
141 views

Conformal symmetry of Navier-Stokes?

This question is in reference to the paper arXiv:0810.1545 Can someone help understand this scaling argument and the proof(?) that there is a conformal symmetry in Navier-Stoke's equation? (..am I ...
3
votes
0answers
141 views

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 ...
3
votes
0answers
125 views

Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius

Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained ...
3
votes
0answers
81 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
votes
0answers
175 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
votes
0answers
85 views

How to construct a 2D conformal field on a mirrored annulus with a looping pattern?

I would like to construct a 2D conformal field on an annulus in which the inner and outer boundaries are like mirrors, and can be approximated by regular polygons (with the same number n of mirror ...
2
votes
3answers
559 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 ...
2
votes
2answers
117 views

A question about Virasoro algebra

(skip disclaimer) I have a question in Polchinski's string theory book volume 1 p54, related to the Virasoro algebra. Introducing complex coordinates $$w=\sigma^1 + i \sigma^2 $$ $$z=\exp (-i ...
2
votes
2answers
103 views

Identity of Operator Product Expansion (OPE)

I have one more s****d question in Polchinski's string theory book, Eqs. (2.3.14a) $$ j^{\mu}(z) :e^{ik \cdot X(0,0)}:~ \sim~ \frac{k^{\mu}}{2 z} :e^{ik \cdot X(0,0)}:,$$ where $j^{\mu}_a ...
2
votes
1answer
113 views

representation of conformal group in d>2

In P. Di Francesco, P. Mathieu, D. Snchal they fix the generators of the conformal group acting on a scalar field by somewhat arbitrarily defining $$\Phi'(x)=\Phi(x)-i\omega_a G_a\Phi(x)$$ and by ...
2
votes
2answers
88 views

Factor of two differences for free field Green's functions in conformal field theory

I have a question about the expressions for free field Green's functions in conformal field theory. It comes from three origins 1) In Polchinski's string theory volume I p36, it is given $$ ...
2
votes
2answers
308 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 ...
2
votes
1answer
55 views

Normal ordering of the identity operator

I'm puzzled about what should be the normal ordering of the identity operator (or any proportional operator): looking at it from the "Fock space operators POV",the prescription is to move all the ...
2
votes
1answer
58 views

A question related to “old covariant quantization” of string theory

I have a question about "old covariant quantization" in Polchinski's string theory p. 123. It is said The only nontrivial condition at this level is $(L_0^{\rm m} + A) | \psi \rangle =0 $, ...
2
votes
1answer
249 views

Energy-Momentum Tensor in Conformal Field Theory

Basically, I would really like it if somebody just explained to me what is going on here. Please use any physics lingo you feel is necessary, but explain what you mean. I am just having trouble ...
2
votes
1answer
82 views

Anticommuting relation in $bc$ CFT

(skip disclaimer) I have a question about conformal field theory in Polchinski's string theory vol 1 p. 61. Given anticommuting fields $b$ and $c$ and the Laurent expansions $$ b(z) = ...
2
votes
3answers
210 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
votes
1answer
271 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
votes
1answer
790 views

conformal, Weyl transformations, apparent discrepancies and confusions

Because of the apparent discrepancy of how some CFT and GR books define conformal transformation unlike in string theory area, I wanted to get rid of all the confusion from McGreevy's lecture notes: ...
2
votes
1answer
68 views

A question about the Bosonization of the Thirring model

Is there a way or sense in which one can Bosonize this kind of a Lagrangian, $L = \bar{\psi}\gamma^\mu \partial _\mu \psi + f(x) \bar{\psi}\psi$ for $f(x)$ being some function on space-time. ...
2
votes
1answer
66 views

Does $c = 0$ implies that the theory is “empty”?

I'm wondering if there is more than the empty theory (no local fields, identically vanishing stress energy tensor) that can have central charge $c$ equals to $0$? My intuition tells me no, the stress ...
2
votes
1answer
94 views

Transferring CFT correlations from $\mathbb{R}^3$ to $S^3$

There seems to be a simple method to transfer a CFT's correlations from $\mathbb{R}^3$ to $S^3$ but I am not understanding why it is supposed to work. The idea is that somehow because, $ds^2_{S^3} = ...
2
votes
1answer
80 views

Conformal dimensions of the energy-momentum tensor

Currently I am reading the Di Francesco-Mathieu-Sénéchal textbook on conformal field theory. Above the equation 5.52, the author argues that the EM tensor should have scaling dimension 2 and spin 2. ...
2
votes
1answer
75 views

Virasoro operator in “old covariant quantization”

I met some problem about the Virasoro operator in "old covariant quantization" in Polchinski's string theory vol I p 123. It is given $$L_0^{\rm m}=\alpha' p^2 + \alpha_{-1} \cdot \alpha_1 + \cdots ...
2
votes
1answer
123 views

About the conserved charge for the ghost number current in $bc$ conformal field theory

(skip disclaimer) I have a question about the conserved charge for the ghost number current in $bc$ conformal field theory in Polchinski's string theory p62. It is said For the ghost number ...
2
votes
1answer
160 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
votes
1answer
180 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 ...
2
votes
2answers
357 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 ...
2
votes
1answer
544 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 = ...
2
votes
1answer
79 views

Steepest descent for Mellin-type integration

Here I would like to see the behavior of a function as an integral when its argument (which is a parameter in the integral) goes to zero. If I try to evaluate an integral $$I(\lambda) = ...
2
votes
1answer
82 views

OPE and 4-point correlation function in CFT_d

I'm reading this paper where to determine the coefficient $C^{\phi\phi O}(x_{12},\partial_2)$ of the OPE (p.10) $$\phi^\alpha (x_1)\phi^\beta (x_2)=C_\phi ...
2
votes
1answer
102 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
votes
1answer
126 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
votes
1answer
69 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 ...
2
votes
1answer
197 views

Tachyon vertex operator (Polchinski's book)

I would like to know how does Polchinski in his book "derive" what is the "tachyon vertex operator" (..as say stated in equation 3.6.25, 6.2.11..) I can't locate a "derivation" of the fact that ...
2
votes
1answer
167 views

What is the operator for the edge current of a fracional quantum Hall state?

The edge of a fractional quantum Hall state is a chiral conformal field theory. In the Laughlin case it corresponds to the chiral boson, $$ S = \frac{1}{4\pi} \int dt dx ...
2
votes
1answer
122 views

constraint on scaling dimension

How can we show that for any scalar operator $\Delta\geq1$ (where $\Delta$ is the scaling dimension)? Where can I find a reference for reading where it comes from?
2
votes
1answer
139 views

Constraining two-point functions of boundary operators on the disk

I'm trying to understand the constraints on the disk CFT correlation function $\langle O_1(y_1)O_2(y_2)\rangle$, where the $O_i$'s are boundary operators that are not necessarily primary. It's a ...
2
votes
1answer
176 views

Electromagnetic current-current correlators

Let the free electromagnetic current $J_\mu(x)$ be = $:\bar{\psi}(x)\gamma_\mu Q \psi(x):$ where $::$ is the normal ordering. In this expression why is $Q$ thought of as a "charge operator" instead ...
2
votes
0answers
46 views

stress tensor in Kazama-Suzuki construction

This is a technical question about equation (2.42) of the original paper [KS] of the Kazama-Suzuki construction. I think the authors did a simple substitution ...
2
votes
0answers
70 views

Regulating an infinite sum in the $bc$ CFT

The EM tensor of the $bc$-CFT is $$ T(z) = \colon \partial b c \colon - \lambda \partial \colon b c \colon $$ After expanding in a mode expansion, we find $$ T(z) = \sum_{m} \frac{1}{z^{m+2}} \sum_n ...
2
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0answers
74 views

Half-integer Spin and “natural conformal dimension”

If we consider a classical field theory for a massless particle of integer spin $s$, in a curved space-time, one finds that it is "naturally" conformal in a space-time of dimension $2+2s$ For ...
2
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0answers
165 views

Stokes' theorem in complex coordinates (CFT)

I am studying CFT, where I encounter Stokes' theorem in complex coordinates: $$ \int_R (\partial_zv^z + \partial_{\bar{z}}v^{\bar{z}})dzd\bar{z} = i \int_{\partial R}(v^{z}d\bar{z} - v^{\bar{z}}dz). ...
2
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0answers
154 views

Conformal group in two dimensions [closed]

How can one show in a group-theoretical way that each of SO(d,2) and SO(d+1,1) is isomorphic to two-copies of Virasoro algebra for d=2?
2
votes
0answers
61 views

Is Wick rotation invariant under proper conformal transformations?

Is Wick rotation invariant under proper conformal transformations? Why or why not? Does Wick rotation apply to conformal field theories? $(1-i\epsilon )T$ is not invariant under proper conformal ...
2
votes
0answers
288 views

Definitions of the Normal Ordering Operator in CFTs and QFTs

Recall the normal ordering of bosonic operators in QFT is defined by a re-arrangement of operators to put creation operators to the left of annihilation operators in the product. This is designed to ...
2
votes
1answer
382 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 ...