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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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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82 views

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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0answers
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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78 views

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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226 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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1answer
144 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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78 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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2answers
87 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 ...
2
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40 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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34 views

Decoupling of Weyl factor in critical dimension

In his paper called "Quantum geometry of bosonic strings", A.M.Polyakov quantizes a bosonic string using path integrals over the space of all metrics on the worldsheet. A critical dimension renders $D ...
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49 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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72 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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47 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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116 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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0answers
134 views

Calculation of OPE in Polchinski

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 ...
2
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1answer
138 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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85 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?
3
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104 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 ...
4
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2answers
292 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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0answers
184 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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185 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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0answers
30 views

Why are closed strings with different perioidicities equivalent?

I was typing up some lecture notes the other day when I saw something unclear. While talking about bosonic open and closed strings and the Polyakov action, the notes say we don't need to distinguish ...
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1answer
340 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 ...
6
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1answer
683 views

What is the connection between Conformal Field Theory and Renormalization group in QFT?

As I know, the fundamental concept of QFT is Renormalization Group and RG flow. It is defined by making 2 steps: We introduce cutting-off and then integrating over "fast" fields $\widetilde{\phi}$, ...
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695 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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1answer
177 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 ...
2
votes
1answer
95 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) ...
6
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2answers
489 views

Conformal transformation equation

I am currently reading Kiritsis's string theory book, and something bugs in the CFT (fourth) chapter. He derives the equation that should satisfy an infinitesimal conformal transformation $$x^{\mu} ...
3
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150 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 $O_p$ ...
3
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122 views

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 ...
4
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1answer
702 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
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2answers
377 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 ~=~ ...
2
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1answer
101 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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1answer
105 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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1answer
118 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 ...
4
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1answer
371 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
62 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): ...
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50 views

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 ...
0
votes
1answer
108 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. ...
0
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1answer
109 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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votes
2answers
152 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, ...
3
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0answers
74 views

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 ${\cal N}=1$ ...
3
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1answer
82 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 ...
2
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1answer
69 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 ...
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1answer
188 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 ...
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2answers
169 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
217 views

OPE of fermionic field bosonization in string theory, in Polchinski 10.3.12

In Polchinski's String Theory Vol. 2, equations 10.3.12 are $$e^{iH(z)}e^{-iH(-z)}~=~\frac{1}{2z}+i\partial H(0)+2zT_B^H(0)+O(z^2)\tag{10.3.12a}$$ ...
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0answers
52 views

Local versus global conformal groups/algebras

I am reading Ginsparg's Conformal Field Theory Notes and I am somewhat puzzled by the use of global and local. Specifically, I understand that the generators of two-dimensional conformal ...
5
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1answer
473 views

Divergence theorem in complex coordinates

This question is related to Stokes' theorem in complex coordinates (CFT) but, I still don't understand :( Namely how to prove the divergence theorem in complex coordinate in Eq (2.1.9) in ...