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 ...

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Why do three dimensional gauge theories flow to conformal theories in the infrared?

What is meant with the fact that Super Yang-Mills flows to a conformal field theory in the infrared? Also, is this a general fact or does this depend on the fact of considering a certain class of ...
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Derivation of Eq. 7.12 in the review paper of Kraus

I'm reading "Lectures on black holes and the $AdS_3/CFT_2$ correspondence" by Kraus. http://arxiv.org/abs/hep-th/0609074 I don't know how one can obtain Eq.7.12. My stupid question is how to obtain ...
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Vertex operator - state mapping in Polchinski's book

In Polchinski's textbook String Theory, section 2.8, the author argues that the unit operator $1$ corresponds to the vacuum state, and $\partial X^\mu$ is holomorphic inside couture $Q$ in figure ...
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Do primary fields (in a CFT) satisfy the wave equation?

In a 2-dimensional CFT, do primary fields generally satisfy the wave equation? I know that if they have either purely holomorphic or purely anti-holomorphic, they do. But what about a general primary ...
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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 ...
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Orbifold actions and twist operators

A twist operator $\sigma$ is the operator that acts on the untwisted vacuum $|0\rangle$ to create a twisted vacuum $\sigma|0\rangle$. States belonging to the twisted sector of an orbifold are built on ...
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185 views

Charged CFT observables and AdS/CFT

I have a simple question regarding the holographic dictionary when mapping operators on the CFT side to those in AdS. One piece of the dictionary is that a global symmetry maps onto a gauge symmetry ...
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What are the quantum dimensions of the primary fields for SU(N) level k Kac-Moody current algebras?

The CFT of the $\mathrm{SU}(N)$ level $k$ Kac-Moody current algebra has many Kac-Moody primary fields. I wonder if any one has calculated the quantum dimensions of those Kac-Moody primary fields. I ...
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269 views

Wick's theorem for calculating OPE

I am trying to understand a calculation using Wick's theorem. Let $T(z)$ be the analytic part of a stress-energy tensor, and $\phi(z)$ a free boson field. Now, ...
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Dual of the Identity operator (AdS/CFT)

We know that in a CFT the spectrum of gauge invariant operators must contain an Identity operator (for the operator algebra to close). For those CFTs that admit a holographic dual what does the ...
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Do “typical” QFT's lack a lagrangian description?

Sometimes as a result of learning new things you realize that you are incredibly confused about something you thought you understood very well, and that perhaps your intuition needs to be revised. ...
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BTZ Black Hole Central Charge and Conformal Weight

I have been trying to reproduce a calculation (equation 4.12) in this paper http://arxiv.org/pdf/1107.2678v1.pdf by Carlip reviewing the derivation of the effective central charge of the BTZ Black ...
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When does the correlator of a string of fields and the current vanish “sufficiently fast” at infinity and Ward's identity?

One consequence of the Ward identity (cf. Di Francesco et al) is that it means variation of correlators under infinitesimal transformation is zero. This can be seen by integrating the ward identity, ...
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Meaning of conformal field theory [closed]

Can anyone summarize what conformal field theory is actually about,i.e. 1) what are its goals? (for example, to study such-and-such fields/functions/maps/etc. to see whether they have such-and such ...
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“Light” states in critical $O(N)$ model in $2+1$ (and holography)

Let me split the question in a few parts, Can someone give me a reference which explains the CFT properties of the critical $O(N)$ model in $2+1$? Like how are the CFT correlators (in a $1/N$ ...
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Some questions about calculation central charge in a CFT in $d$ spacetime dimensions

This is based on this paper, http://arxiv.org/abs/hep-th/0212138 For a CFT on a $S^d$ spacetime (of radius R) it seems to be claimed that the central charge is given by, $ c = \langle \int_{S^d_R} ...
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RG flow from a UV scale invariant field theory to a gapped phase in the IR

On the section 3 of http://arxiv.org/abs/1309.2921 the authors consider the RG flow from a scale invariant field theory in the UV to a gapped theory in the IR. The theory is couple to a background ...
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Identity in CFT

I heard and read couple of times reference to a certain identity in conformal field theory (maybe specific to two dimensions). The identity relates the trace of stress-energy tensor to the beta ...
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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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Is string theory over a time varying background a conformal field theory to all orders in perturbation theory?

When computing the first order perturbative corrections to string theory over a curved background, we find the background has to be Ricci-flat if the dilaton is constant and we have no fluxes. Such is ...
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A calculation problem on Conformal Field Theory

This problem is on Di Francesco's book I. It's exercise 7.1: Calculate the norm of the following vector, where $\lvert h\rangle$ is the state of highest weight. $$L_{-1}^n\lvert h\rangle$$ I have ...
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Identify the coefficients of Operator Product Expansion (OPE)

Sorry I have a stupid question in Polchinski's string theory book vol 1, p46. For a holomorphic function $T(z)$ with a general operator $\mathcal{A}$, there is a Laurent expansion $$T(z) A(0,0) \sim ...
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294 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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Is there a general relationship between the conformal weight of a field and its (classical) scaling dimension?

A field $\phi(z)$ has the conformal weight $h$, if it transforms under $z\rightarrow z_1(z)$ as $$ \phi(z) = \tilde{\phi}(z_1)\left(\frac{dz_1}{dz}\right)^h $$ The (classical) scaling dimension can ...
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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 ...
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Why is string theory a two dimensional quantum (conformal) field theory on its worldsheet?

In string theory, we quantize the two dimensional field theory on the string's worldsheet. I have a question about this kind of quantization of string theory: did we have similar theory for point-like ...
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Commutation relations of the generators of the conformal group

My question is from P.98 of the book by Di Francesco on Conformal Field theory. He gives the six non-vanishing commutation relations between the elements $P_{\mu}, D, L_{\mu \nu}$ and $K_{\mu}$ ...
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256 views

Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is ...
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Representations of the Conformal Group in terms of the Poincare Group Reps

The Conformal group contains the Poincare group. Typically, if you take a representation of a group and then look at it as a representation of a subgroup, the representation will be reducible. I often ...
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Some questions about the paper, “AdS description of induced higher spin gauge theory”

I am referring to this paper. I guess that in this paper one is trying to relate the massless spin $s$ gauge fields in $AdS_4$ to conformal spin $s$ theory on $S^3$. So am I right that the ...
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417 views

What is on the AdS side in AdS/CFT supergravity or string theory?

What really is on the AdS side in AdS/CFT, does it always have to be string theory or is sometimes supergravity "enough" or better suited to do calculations? From the answers to my earlier question, ...
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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 ...
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Why should a holomorphic function be expanded in Laurent series rather than Taylor series?

In 2d free conformal field theory, there is an operator equation: $$ \partial\bar\partial\hat{X}^\mu\left(z,\bar z\right)=0 $$ Why can it have Laurent expansion like this below rather than Taylor ...
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451 views

Expectation value of the stress-energy tensor in 2-D CFT

Due to a previous question, I am confused with the expectation value of the stress-energy tensor in a 2-D conformal field theory. Let's take the example of string theory, to sketch the problem. ...
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Decomposition of representations of the Virasoro algebra under $sl(2)$

The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight ...
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558 views

Classical theories and AdS/CFT

When I was editing the Physics.SE tag wiki for ads-cft, I initially wrote something on the lines of : The AdS/CFT correspondence is a special case of the holographic principle. It states that ...
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422 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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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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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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730 views

CFT and the conformal group

Equations 2-7 on page 21 of these notes, http://www.math.ias.edu/QFT/fall/NewGaw.ps seems to give a fairly compact definition of what a CFT is. But I have two questions, This definition is ...
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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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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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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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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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Path integral derivation of the state-operator correspondence in a CFT

Below, I paraphrase the path integral derivation of the state-operator correspondence in David Tong's notes on CFT (see pdf here). This is my interpretation of the text in that pdf, so please correct ...
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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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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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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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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 ...