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,...

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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 C)/\...
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495 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): $$\...
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215 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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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 ...
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198 views

Conceptual question about field transformation

(c.f Conformal Field Theory by Di Francesco et al, p39) From another source, I understand the mathematical derivation that leads to eqn (2.126) in Di Francesco et al, however conceptually I do not ...
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290 views

Hermitian conjugation in Radial Quantization

I'm a little confused about Hermitian conjugation in a radially quantized CFT. Now, in the Minkowski theory, Hermitian conjugation leaves the coordinates invariant, i.e. $t^\dagger = t$ and $x^\dagger ...
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158 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 ...
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326 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 ...
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92 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 ...
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193 views

Why can we not choose the stress tensor in a CFT to be identically symmetric?

The stress tensor for a conformal field theory (or any quantum field theory) can be derived from the action $S$ by the functional derivative $$T^{\mu \nu} ~=~ -\frac{2}{\sqrt{|g|}}\frac{\delta S}{\...
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452 views

Two and three point function of primary fields of arbitrary fields

I was looking at this paper hep-th/0011040 and I found the following equation: $$ \langle C_{\mu_1 \dots \mu_l} \mathcal{O}^{\mu_1 \dots \mu_l}(x_1) D_{\nu_1 \dots \nu_l} \mathcal{O}^{\nu_1 \dots \...
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738 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 ...
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169 views

A question about vertex operator

(skip disclaimer) I have a question about writing raising and lowering operators in the Schroedinger basis in the section of vertex operator in Polchinski's string theory vol 1 p.68. It is given ...
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1answer
268 views

Time-ordered Derivative and Equal-time Commutator

In Green, Schwarz & Witten Superstring theory, Vol. I, page 141, I don't understand how pulling the derivative inside the Time-ordered product can give an Equal-time Commutator: $$\tag{3.2.44} \...
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202 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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657 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 ...
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276 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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How can I show that inversion is continuously connected to a reflection?

From Ex 3.1 in the TASI lectures on the conformal bootstrap: http://arxiv.org/abs/1602.07982 the problem is the inversion map (with Euclidean signature) $$ I\colon x^\mu \mapsto \frac{x^\mu}{x^2} $$ ...
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114 views

(Euclideanized) QFT on $S^d$ vs $S^{d-1}\times S^1$

Broadly I would like to understand what is the difference in the physical interpretation of a (Euclideanized) QFT which is on space-time $S^d$ and which is on a space-time $S^{d-1}\times S^1$. In ...
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380 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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Are second-order phase transitions always scale/Lorentz invariant?

I know that both scale invariance and Lorentz invariance typically emerge at second-order phase transitions, but is there a proof or a counterexample? (I know that it's believed that any theory that ...
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Connection between the M5 brane and (2, 0) superconformal field theory

I have read that the worldvolume theory of the M5 brane is a $(2, 0)$ superconformal field theory (SCFT). But I have also learnt from talks that the $(2, 0)$ theory lacks a Lagrangian description. ...
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Target Space Lorentz Invariance vs. World Sheet Weyl Invariance

The Polyakov action, $S\sim \int d^2\sigma\sqrt{\gamma}\, \gamma_{ab}\partial^a X^\mu \partial ^b X_\mu$, has the well known classical symmetries of world sheet diffeomorphism invariance, world ...
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Is there a maximum number of fixed points that a QFT can have?

I was wondering: is there a maximum number of (trivial and non-trivial) fixed points that a QFT can have (as a function of the space-time dimension and field content in the QFT)?
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Mode operators in the Virasoro algebra

This questions concerns Exercise 2.11 in Polchinski. We are asked to compute the commutator $$L_{m}(L_{-m}|0;0\rangle) - L_{-m}(L_{m} |0;0\rangle)$$ By plugging the mode expansions, we use the ...
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153 views

Chiral Scale and Conformal Invariance in 2D QFT

I am reading a paper by Hofman and Strominger. In the appendix A, I have reproduced the equations (A10). Now they made a statement that "The Jacobi identity can be used to show that $O_h$ and $O_p$ ...
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Why the bosonic part of the superconformal group $SU(2,2|1)$ is $SO(4,1) \times U(1)_R$?

Why in $d=4$ $\mathcal{N}=1$ SCFT the bosonic part of the superconformal group $SU(2,2|1)$ is $SO(4,1) \times U(1)_R$? More generally how can I determine the such a thing in other theories? Is there ...
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297 views

Virasoro operators commutation relations [closed]

For the commutation relation in quantising the bosonic string $$\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$$ we can then calculate this for $m=-n$ in between the vacuum ...
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141 views

Link between anomalous dimensions and fractal dimensions

I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a ...
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516 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 ...
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Expressions of action and energy momentum tensor in bc conformal field with central charge equals one

I have a question with conformal field theory in Polchinski's string theory vol 1 p. 51. For $bc$ conformal field theory $$ S=\frac{1}{2\pi} \int d^2 z b \bar{\partial} c $$ $$ T(z)= :(\partial b) ...
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185 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 ...
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Analogy for the AdS/CFT Correspondence

Some time ago, I heard about a simple analogy for the AdS/CFT correspondence to something in everyday life. Consider a room filled with furniture, with the walls of the room covered in mirrors. The 2D ...
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792 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 particular I'm referring to ...
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158 views

Relation between conformal and topological field theories

The Chern-Simons (CS) theory is a topological quantum field theory (TQFT). The question is, is a conformal field theory (CFT) a topological quantum theory? Or the reverse, topological quantum field ...
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772 views

What is conformal gauge?

I often see in physics articles on gravity such notion as conformal gauge and Weyl transformation. They use Conformal gauge to change coordinates to transform metrics from arbitrary $$ds^2=g_{\mu \...
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416 views

Are diffeomorphisms a proper subgroup of conformal transformations?

The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not ...
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170 views

The spin and weight of a primary field in CFT

A primary field in Conformal Field Theory transforms as $$\phi (z,\bar{z}) =\left(\frac{dz}{dz'} \right)^h \left(\frac{d\bar{z}}{d\bar{z}'} \right)^\bar{h}\phi (z',\bar{z}') $$ under a conformal ...
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1answer
157 views

Even-branes in IIA and odd-branes in IIB

The R-R sector of IIA and IIB are respectively given as, $8_s \otimes 8_c = [1]\oplus [3] = 8_v \oplus 56_t$ $8_s \otimes 8_s = [0]\oplus [2] \oplus [4]_+ = 1 \oplus 28 \oplus 35_+$ Now looking at ...
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467 views

Some questions about the free Fermionic partition function on a circle (Ginsparg's CFT lectures)

The following questions are based on these lectures, http://arxiv.org/abs/hep-th/9108028 I would like to know what is the relationship between the last equation on page 82 ($(L_0)_{cyl} = L_0 - \...
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2answers
519 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 ...
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1answer
350 views

What is a su(2) level k algebra

What is meant by su(2) level k algebra ? Is it a lie algebra of some lie group ? What is the relation with SU(2) group. I see it in the context of quantum hall edges. Googling and google-booking for a ...
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186 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 ...
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1answer
355 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 $z = e^{\tau + i\sigma}$ ...
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420 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?
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670 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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560 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 k}...
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191 views

Idea behind Compactified Boson

On p. 167 of his Conformal Field Theory, Di Francesco introduces "Compactified Boson". He says: The invariance of the free-boson Lagrangian [...] with respect to translations $\varphi(x) \...
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374 views

Massless fields in curved spacetimes

I read the following statement in one of Penrose's paper zero rest-mass field equations can, with suitable interpretations, be regarded as being conformally invariant. I take this to imply that ...
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Correlator of energy-momentum tensor and OPE

In http://arxiv.org/abs/hep-th/9108028 Equation (2.22), the correlation function of then energy-momentum tensor with some primary fields is We can view this as sum over the OPE of the energy-...