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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Dilations in momentum space

I don't quite understand what's going on here. Let's suppose I have a dilation in real space. The generator is $D=x^j \partial_j$, so an infinitesimal dilation is $\delta x^i = Dx^i = x^j \partial_j ...
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47 views

CFTs and hyperbolic spaces

Is there a good way to explain why its natural to consider CFTs to be on hyperbolic spaces? Even if I forget about CFTs is there a good way to explain if there is a way that the hyperbolic space can ...
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How can the stress tensor components of a worldsheet CFT in general background be (anti)-holomorphic?

In all textbooks/lecture notes on string theory (e.g. Polchinski, page 43 at the bottom) it is proven that, as the stress tensor is traceless and conserved, $T^a_a=\partial^a T_{ab}=0$, we have $\...
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102 views

Classical conformal invariance

So I am trying to understand classical conformal invariance. So we move gently from general coordinate invariance to Weyl invariance to conformal invariance, and now they start out with this thing ...
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Generalized gravitational entropy and entanglement entropy

What are the differences (if any) between Generalized gravitational entropy (Lewkowycz-Maldacena) and holographic entanglement entropy (Ryu-Takayanagi)? More specifically, I was wondering following ...
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79 views

Anomaly polynomial of Hitchin system $\mathcal{N}=2$ 4d SQFT

I would like to ask about mathematical background of this object. So, I am trying to puzzle out with 4d $\mathcal{N}=2$ SQFT. As far as I can gather this theory can be described in terms of ...
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1answer
123 views

What is the conformal mode of a metric?

I have a problem in terminology. This article talks about the conformal mode of a physical metric. I know what a conformal transformation is. But what is the conformal mode of a metric?
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1answer
337 views

What is a ghost number?

I am currently studying CFT chapter of Becker,Becker,Schwarz and am trying to understand what the ghost number is in BRST Quantization. From what I gather BRST Quantization is used to add an extra ...
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1answer
62 views

In 1d criticality, what is the relation between the universality class and central charge?

I want to know how to obtain the universality class of the phase transition from the central charge "c" in one dimensional model. If c is less than 1, there is a one-to-one correspondence. But if c is ...
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37 views

Correlator $bc$ system [closed]

I have da doubt with bc system. Polchinski says (2.5.10) $$ b(z)b(0)~=~O(z). \tag{2.5.10} $$ I tried to compute the correlation function With eom, using eq (2.5.6b) by Polchinki, removing the ...
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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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Problem with OPE (from Polchinski) [closed]

I was reading Polchinski, Vol. 2 pag 12, while I found (10.3.12a): $$ e^{iH(z)}e^{-iH(z)}=\frac{1}{2z} + i\partial H(0) + 2zT^H_B(0) + O(z^2).\tag{10.3.12a} $$ Now I tried to do the OPE, what I ...
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1answer
179 views

Conformal blocks in 2D CFTs

I have studied conformal field theories in two dimensions and I understand the basic idea behind conformal blocks too. But I never completely realized what they are when it comes to computing them. ...
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87 views

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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211 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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47 views

Large Spin operators and radial quantization

I've been reading the paper "Comments on operators with large Spin" (here) and I am having some trouble understanding the following: In section 2, they begin by studying, in a conformal field theory ...
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77 views

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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51 views

OPE in a general $d$-dimensional CFT

I am looking for a good reference which explains how to express an OPE in a general $d$-dimensional CFT for bosonic and fermionic fields. Precisely, I don't understand the reason of the appearance of ...
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49 views

Calculating OPE of Graviton Vertex Operator [duplicate]

Consider Exercise 2.8 in Polchinski's String Theory book. We are asked to compute the weight of $$f_{\mu \nu}:\partial X^{\mu} \bar{\partial}X^{\nu}e^{ik\cdot X}:$$ I have carried out the usual ...
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122 views

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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1answer
156 views

Operator product expansion energy momentum tensor

We have the following equation from Polchinski (2.4.6) $$ T(z)X^{\mu}(0) \sim \frac{1}{z}\partial X^{\mu}(0) , \tag{2.4.6} $$ where $T(z)$ is defined as $T(z) = -\frac{1}{\alpha'} :\partial X^{\mu} \...
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110 views

Anyonic Braiding and Conformal Field Theory

I am looking for resources (both pedagogical and newer research articles) on the connection between topological quantum computation and conformal field theory. In particular, a CFT description of ...
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234 views

Ishibashi states and Cardy states in CFT

What are the Ishibashi states and Cardy states in CFTs? I am familiar with conformal field theory language. It would be great if someone can discuss about the basic idea of these states and their ...
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197 views

Monstrous Moonshine outside of String Theory

My question concerns applications of monstrous moonshine, which is the connection between the $j$-function and the monster group. Recently, physicists have applied it to string theory and, ultimately, ...
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94 views

Special OPE in $\beta\gamma$ system

I would like to find the OPE $$\beta(z)\gamma(w)^{-1}\tag{1}$$ given $$\beta(z)\gamma(w)~\sim~\frac{1}{z-w}\tag{2}$$ from the $\beta\gamma$-system in CFT. Is it possible?
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240 views

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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192 views

Polchinski Equation (7.2.4)

On page 209 of Polchinski's string theory book he writes down the expectation value of a product of vertex operators on the torus; equation $(7.2.4)$. The derivation is analogous to an earlier ...
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1answer
98 views

Standard derivation of Witt algebra

I have been studying Conformal field theory for the past one week from the books by Blumenhagen and Di Francesco etal. If I understand correctly, whenever one talks of 'local (infinitesimal) ...
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351 views

Can you take the cutoff to infinity at a conformal fixed point?

A conformal fixed point is defined by $$\beta(g)=0$$ We hence know that couplings, masses and dimensions of operators do not flow in the effective Lagrangian when we change the renormalization ...
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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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94 views

For what values of $\lambda$ is the distribution $(x-i\varepsilon)^\lambda$ positive?

I've been reading the famous unpublished paper by Luescher and Mack "The energy momentum tensor of critical quantum field theories in 1+1 dimensions". In the proof of their main theorem, page 7 of the ...
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297 views

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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111 views

Deriving the energy-momentum tensor conservation equation in complex coordinates, Polchinski 2.4.2

I am trying to derive equation (2.4.2) in Polchinski's string theory textbook, $$\overline \partial T_{zz}=\partial T_{\overline z \overline z} = 0 \tag{2.4.2}.$$ Using the conservation equation, ...
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2answers
387 views

Vertex operator and normal ordering

The two point function, or propagator for a free massless boson, $\phi$ in 2 dimensions is given by, $$\begin{equation} \langle \phi (z,\bar{z})\phi(w, \bar{w})\rangle ~=~ -\frac{\alpha^{\prime}}{2\...
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194 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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1answer
65 views

Superstring vacuum amplitude on the torus

My question is how to obtain the superstring (Type II A and B) vacuum amplitudes on a torus. They are given in Polchinski's String Theory Vol. 2 equation (10.7.9): $$Z_\psi^{\pm}=\frac{1}{2}[Z^0_0(\...
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Is there a Ramond vacuum for real fermions?

When studying the CFT of a complex fermion $\Psi$ we know that if it's periodic, ie if $$\Psi(\sigma_1+2\pi,\sigma_2)=\Psi(\sigma_1,\sigma_2)$$ then there is a doubly degenerate Ramond vacuum which I ...
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Books or papers recommendation on orbifold and CFT [duplicate]

Could you recommend some references on orbifold CFT? I have found this paper "The conformal field theory of orbifolds"(1987)(http://inspirehep.net/record/230342) is very useful for me, so I want to ...
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1answer
121 views

Connections between Density Matrix Renormalization Group and Conformal Field Theory

Can we use the density matrix renormalization group (DMRG) method to understand problems in conformal field theory? I have been trying to find some connections, but nothing is coming up when I search. ...
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1answer
128 views

QFT calculations via holographic duality

Holographic duality tells us that there is a duality between anti-deSitter space and lower dimensional conformal field theory. However, what quantum phenomenon, exactly, can we calculate using the ...
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1answer
454 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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1answer
126 views

Universality classes

I would like to ask about the universality classes. I know that these are the statistical models which describes different phase transitions with different critical exponents. But I would like to know ...
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1answer
110 views

How can one prove that there cannot exist a conformal primary, in the case of free field theory, that doesn't saturate the unitarity bound?

In free field theory, the full list of conformal primaries, is given by the Twist-2 operators. These have $\Delta = l+2$, which is also the saturation condition for the unitarity bound for $l \neq 0$. ...
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2answers
163 views

Looking for intro to Conformal Bootstrap

I want to start looking at the conformal bootstrap. I've heard very interesting things about it but would like to clear some things up first. I taken QFT at the level of Peskin & Schroeder, ...
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Questions about the existence of 5d & 6d version of 4d ${\cal N}=2$ SCFTs

Given a 4d N=2 Superconfomal field theory (SCFT) with a global flavor symmetry ( $\mathfrak{f}$ as the corresponding lie algebra), can we clam that this theory can always flow from a 5d ${\cal N}=1$ ...
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1answer
86 views

Operator Dimension and Field Transformation under Rescaling

In conformal field theory the operator dimension $\Delta$ determines how fields and thus correlation functions behave under rescaling. I am having trouble seeing how this number arises from a scale ...
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1answer
73 views

Coset construction of Tricritical Ising CFT

In http://iopscience.iop.org/1742-5468/2008/03/P03010 the authors state that the Tricritical Ising Model (TIM) CFT can be obtained from a Wess Zumino Witten construction based in the coset $\frac{(...
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2answers
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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1answer
203 views

Connection of “spin” to conformal dimension

I have read The spin and weight of a primary field in CFT but it does not answer my question, short of a restatement of the question itself. So I hope this post does not risk being removed.. In ...