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2
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1answer
92 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
141 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
172 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
194 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
372 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
577 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
14 views

World-sheet energy-momentum tensor and OPE

On p43 of Polchinski's book, it says that under the world-sheet translation $\sigma^a\rightarrow\sigma^a+\epsilon v^a$, $X^\mu\rightarrow X^\mu-\epsilon v^a\partial_a X^\mu$. And $$j^a=iv^b T_{ab},$$ ...
2
votes
1answer
91 views

Correlation functions and connection to ward identities

I have the following definition of a general correlation function $$ \langle \Phi(x_1)\dots \Phi(x_n)\rangle = \frac{1}{Z} \int [d\Phi] \Phi(x_1)\dots\Phi(x_n)e^{-S[\Phi]} $$ I have only just ...
2
votes
1answer
49 views

Conformal compatification of Minkowski and AdS

How do I show that the compactification of Minkowski is given by the quadric $$uv-\eta_{ij}x^{i}x^{j}=0$$ with an overall scale equivalence in the coordinates.I get that for $v \neq 0$, the surface ...
2
votes
1answer
83 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
114 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
103 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
129 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
72 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
216 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
181 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
141 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
144 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
205 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
33 views

Non translation invariant correlator in CFT

I'm doing an exercise on vertex operators in the CFT book by Di Francesco & al.; exercise 9.2 p.329 : Using mode expansion show that: $$\langle\tilde{\phi(z)}\tilde{\phi(w)}\rangle= - \text{ln} ...
2
votes
0answers
48 views

Possible Error in deriving conformal generator

My professor gave me the following derivation for the full generator of the Lorentz transformations. The starting point is to consider a subgroup of the conformal group that leaves the origin fixed ...
2
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0answers
52 views

Anomalies from a Renormaization Group Equation (RGE)

This is an approach to anomalies which seems unfamiliar to me.. Firstly what is this function $W$ which seems to satisfy the equation, $\frac{\partial W }{\partial g^{\mu \nu} } = \langle T_{\mu ...
2
votes
0answers
48 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
88 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
votes
0answers
85 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
votes
0answers
210 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
votes
0answers
176 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
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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
345 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
416 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 ...
2
votes
0answers
275 views

Trace of energy momentum tensor in $CFT_1$

There is a standard procedure to show that for CFT's in dimension $d\geq 2$, the trace of stress tensor vanishes. I think I can't apply those steps when I only have one dimension, say time, because if ...
2
votes
0answers
133 views

Argument for quantum theoretic conformality of $\cal{N}=2$ super-Chern-Simon's theory in $2+1$ dimensions -Part 2

This is in continuation to what I was asking here earlier - Argument for quantum theoretic conformality of $\cal{N}=2$ super-Chern-Simon's theory in $2+1$ dimensions Or one can look at this ...
1
vote
1answer
111 views

Large-N factorization of single-trace operators

Does anyone know where I can find a pedagogical explanation of large-N factorization in SU(N) gauge theories or nonlinear O(N) sigma models (in the latter case the trace corresponds to a dot product). ...
1
vote
1answer
63 views

Ghosts on Torus worldsheet

Why after the expansion, only 0-mode of bc-ghost contributes to the 4-points ghost function on a torus worldsheet? $$<c(z_1)b(z_2)\tilde{c}(\bar{z}_3)\tilde{b}(\bar{z}_4)>_{T^2} ...
1
vote
1answer
177 views

Compute the central charge of $bc$ conformal field theory

I have a s****d question, how to calculate the central charge of $bc$ conformal-field theory in Polchinski's string theory, Eq. (2.5.12)? For a $bc$ CFT given by $$S=\frac{1}{2\pi } \int d^2 z \,\,b ...
1
vote
1answer
193 views

Interaction potential analysis from $\phi^4$ model

In this paper, the authors consider a real scalar field theory in $d$-dimensional flat Minkowski space-time, with the action given by $$S=\int d^d\! x ...
1
vote
1answer
31 views

Ward Identity in CFT

This is about Polchinski's eq(2.3.11). It says that $$Res_{z\rightarrow z_0}j(z)\mathcal A(z_0,\bar z_0)+\bar{Res}_{z\rightarrow z_0}\tilde j(\bar z)\mathcal A(z_0,\bar z_0)=\frac1{i\epsilon}\delta ...
1
vote
1answer
100 views

Lightcone singularity of a 3 point function in CFT

I had a quick question regarding the title of the question. In e.g. 2D CFT (for simplicity), the three point function of three operators with conformal dimension $a$, $b$ and $c$ are given as $$ ...
1
vote
1answer
144 views

How to derive Eq. (2.4.23) in Polchinski's string theory book

Given the Operator Product Expansion (OPE) of a product of the energy momentum tensors $$T(z)T(0) = \frac{ \eta^{\mu}_{\mu} }{2z^4} - \frac{2}{\alpha' z^2} :\partial X^{\mu} \partial X_{\mu}(0): + ...
1
vote
1answer
167 views

Definition of CFT

A standard QFT cannot be defined as a set of Poincare-invariant correlation functions because this does not take into account the possibility of non-perturbative effects (e.g. instantons) Can we ...
1
vote
1answer
167 views

Exercise QFT and CFT

Consider the action functional $S[z;t_1,t_2]=\int_{t_1}^{t^2}[g(z,\bar{z})\dot{z}\dot{\bar{z}}]^{\frac{1}{2}}dt$ with $z(t)$ a complex path with end points $z_i=z(t_i),\; i=1,2$. $g(z,\bar{z})$ is a ...
1
vote
1answer
44 views

How can one calculate the central term of the conformal field theory algebra (and show it's really the virasoro algebra)?

So I'm following Szabo's book "An Introduction to String Theory and D-brane Dynamics (2nd ed, 2011); still on the canonical treatment in chapter 3. After doing a mode expansion, we get (up to a ...
1
vote
1answer
26 views

CFT Entanglement Entropy - relation between translations and the stress-energy tensor

In a recent paper on CFT entanglement entropy, I want to understand the defintion of a certain partition function. They consider a metric space $S^1 \times \mathbb{H}^{d-1}_q$ with metric: $$ ...
1
vote
2answers
66 views

Generator of the Special Conformal Transformation

In this thread Integrating the generator of the infinitesimal special conformal transformation, the generator of the 'flow' of the transformation is written as $$G_b = 2(b \cdot x)x - x^2 b,$$ where ...
1
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1answer
59 views

Null State Level 2 in CFT

I'm reading Cardy's notes on CFT. He states the following in section 4.3: $$\hat L_n\left(\hat L_{-2}|\phi_j\rangle-(1/g)\hat{L^2}_{-1}|\phi_j\rangle\right)=0.$$ I tried to work this out explicitly ...
1
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1answer
53 views

Translation and Dilation transformations within the conformal group

I am using Di Francesco's book P.39. The equation that the generators of the transformations satisfy is given by: $$iG_a \Phi = \frac{\delta x^{\mu}}{\delta w_a} \partial_{\mu} \Phi - \frac{\delta ...
1
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1answer
59 views

Class of scalar field actions invariant under conformal transformations

From the actions in $d$ dimensions given by $$S = \int d^dx \,\, \partial_{\mu}\phi \partial^{\mu} \phi + g \phi^k$$. What is the condition that needs to be $k$ so that the theory is invariant ...
1
vote
1answer
37 views

Why does a null state correspond to a field that any correlator containing it vanishes?

I am reading the 7th chapter of Di Francesco's CFT book. It builds, for example in section 7.3, a null state |x> which is orthogonal to the whole Verma Module. The author asserts that the field x ...
1
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1answer
75 views

Identify the weight of operator under conformal transformation

I have a stupid (homework tag may be suitable =_=) question about the problem 2.7 in Polchinski's string theory volume 1. Why the weight of operator $:e^{ik\cdot X}:$ is $(\frac{\alpha'k^2}{4}, ...
1
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1answer
189 views

A question about defining a classical CFT

This is kind of related to this, Defining a CFT using beta-functions So what would be the right definition of a CFT even classically? Is it true that classically one will call a theory scale ...