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2
votes
1answer
33 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 ...
2
votes
0answers
51 views

Problem with OPE (from Polchinski) [on hold]

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 ...
1
vote
0answers
20 views

Correlator $bc$ system

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 ...
3
votes
0answers
25 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. ...
5
votes
1answer
130 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 ...
1
vote
1answer
39 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. ...
4
votes
0answers
96 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 ...
4
votes
0answers
35 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 ...
3
votes
2answers
78 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
votes
0answers
21 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 ...
0
votes
0answers
31 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 ...
1
vote
0answers
28 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 ...
4
votes
0answers
56 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)?
1
vote
0answers
42 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 ...
3
votes
0answers
96 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 ...
0
votes
0answers
63 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
votes
1answer
77 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} ...
0
votes
1answer
61 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
votes
0answers
67 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
votes
2answers
260 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 ...
0
votes
0answers
20 views

How does the energy-momentum tensor transform under parity in 2D CFT?

I'm trying to get that if parity is conserved, the two represenations of Virasoro algebra which we get from $L_n$'s and $\bar{L}_n$'s starting from dilatation invariant Wightman theory have the same ...
0
votes
0answers
50 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 ...
6
votes
0answers
121 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, ...
0
votes
0answers
28 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 ...
10
votes
1answer
256 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
votes
1answer
428 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 $\widetilde{\phi}$, ...
3
votes
2answers
386 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 ...
0
votes
1answer
126 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
58 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
votes
2answers
412 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
votes
0answers
141 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
votes
0answers
113 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
votes
1answer
508 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
votes
2answers
260 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 ~=~ ...
3
votes
1answer
2k views

Why/How is this Wick's theorem?

Let $\phi$ be a scalar field and then I see the following expression for the square of the normal ordered version of $\phi^2(x)$. $$T(:\phi^2(x)::\phi^2(0):) ~=~ 2<0|T(\phi(x)\phi(0))|0>^2 $$ ...
2
votes
0answers
35 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 ...
2
votes
1answer
72 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 $l \neq 0$. ...
0
votes
1answer
81 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, ...
2
votes
1answer
68 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 ...
1
vote
1answer
88 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
votes
1answer
186 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 ...
1
vote
1answer
44 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): ...
0
votes
0answers
39 views

Books or papers recommendation on orbifold and CFT

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
67 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
votes
1answer
61 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 ...
0
votes
0answers
61 views

Constructing dual gravity theory from general CFT

AdS/CFT is a little over my head at the moment, particularly the AdS side but I'll ask anyway. Has there been any work done on starting with a mathematically rigorous conformal field theory defined ...
2
votes
2answers
98 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
votes
0answers
61 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
votes
1answer
63 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 ...