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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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0answers
24 views

Conformal Field Theory consistency

In Rational Conformal Field Theory the partition function is $Z(\tau,\overline{\tau})~=~\sum\limits_{p,\overline{q}} F_{p\overline{q}}~\chi_{p}(\tau)\chi_{q}(\overline{\tau})$ where $\chi_{p}$ are ...
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0answers
14 views

Two and Three point function of primary fields of arbitrary fields.

I was looking at some papers and I found the following equations, $$ \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
38 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
31 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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0answers
33 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
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2answers
52 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, ...
2
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0answers
45 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
48 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
34 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 ...
2
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1answer
73 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 ...
3
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2answers
95 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
118 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
19 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 ...
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0answers
43 views

Can a conformal field theory with chiral central charge be gapped out?

Consider a 2-dimensional conformal field theory with nonzero chiral central charge (that is, the central charges of the holomorphic and antiholomorphic sectors are different.) I think that ...
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5answers
4k views

A pedestrian explanation of conformal blocks

I would be very happy if someone could take a stab at conveying what conformal blocks are and how they are used in conformal field theory (CFT). I'm finally getting the glimmerings of understanding ...
5
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1answer
361 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 ...
3
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0answers
49 views

Are the following terms, related to scale invariance and renormalization in QFT, equivalent?

Which of the following terms are equivalent? and in what cases/limits do the non-equivalent terms become equivalent? A) a scale invariant quantum field theory. B) a conformal quantum field theory. ...
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1answer
34 views

Conformally invariant theory. Relationship between conformal transformations and conformal rescaling

So, I'm learning about Twistors, and in every book I've read they say the same: "If a flat theory is Poincaré-invariant and it is invariant under conformal rescaling (Weyl scaling), it is then ...
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1answer
29 views

Conformal group and stereographic projection

In Ginsparg's Applied Conformal Field Theory (http://arxiv.org/abs/hep-th/9108028, on the bottom of p. 5) the following remark is made: Indeed the conformal group admits a nice realization acting ...
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0answers
17 views

Correlation functions on $S^2$ (from OPE singularities)

Consider a closed string scattering (worldsheet topology $S^2$ sphere). Given that the OPE: $i \partial X^\mu(z) e^{ik X(w)} \sim \frac{k^\mu}{z - w} e^{ik X (w)} + ... \ \ , \ \ i \partial X^\mu (z) ...
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1answer
44 views

Notion of distance in a Conformal Field Theory

I'm confused about the how the notion of distance is used in Conformal Field Theory. Let's take for example the Operator Product Expansion (OPE). In a conformal field theory, due to the scale ...
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0answers
37 views

Dropping nonsingular term in OPE

Why (in 2D CFT; I don't know if this statement can be generalized to other theories), for the purpose of calculating the correlation functions, non-singular term in the OPE is always dropped?
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0answers
17 views

2D CFT for nontrivial topology

What is a systematic way to calculate a general $N$-points correlation function of 2D CFT for a nontrivial topology? Piece by piece of this can be found in many different CFT and String Theory ...
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0answers
30 views

SUSY preserved by an extended D-brane

There's a nice way to prove that for an extended D-brane half of SUSY is preserved, from perturbative string argument with SUSY Ward identities and the doubling trick (at least, at tree-level and ...
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0answers
26 views

Tadpole-free condition

Tadpole-free is a very important condition for perturbative string theory (which is equivalent to the theory to be expanded around the "right" vacuum). For simplicity, let's consider closed string ...
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0answers
33 views

Monodromy, Holonomy and Braiding Phase

In quantum Hall effect, especially in the context of CFT description, these words come up often. I think I understand the braiding phase - as the phase gained by the wave function when a quasi ...
2
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1answer
33 views

How to show OPE coefficients are symmetric in three indices ?

May it is very trivial, but I am stuck here, given (I have suppressed the conjugate coordinates) $$ \phi_i(x) \phi_j(y) \sim \sum_{k} c_{ijk} (x-y)^{h_k - h_i - h_j} \phi_k(y) $$ $$ \langle ...
3
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0answers
45 views

Conserved charge of a conformal transformation

From Becker, Becker and Schwarz String Theory and M-Theory: For the infinitesimal conformal transformation $$\tag{3.25}\delta z=\varepsilon(z)\quad\text{and}\quad \delta\bar ...
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1answer
76 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 ...
1
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1answer
24 views

Why is it that every locally conformal transformation can be extended to a global conformal transformation for D>2?

In D=2, we can have locally analytic transformations that cannot be globally well-defined. However, for CFTs in D>2, we have only the global group. Why is that? Also, is it a statement that depends ...
5
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0answers
157 views

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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0answers
23 views

Quasi-primary fields and usual fields

How do i see that the way quasi-primary/primary fields transform contain the transformation rule for fields as we know it (scalar, vector fields) in QFT?
4
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1answer
92 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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0answers
22 views

Modular S matrix in 2D rational conformal field theory

The modular $S$ matrix in 2D CFT is defined by $$ \chi_a(-\frac{1}{\tau})=S_{ab} \chi_b(\tau). $$ Can $S$ depend on the modular parameter $\tau$?
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1answer
27 views

In Rational Gaussian Model, why must $R^2$ be rational?

I am reading Fusion Rules and Modular Transformations in 2D Conformal Field Theory and Verlinde talks about the rational gaussian model. The simplest class of RCFT's are the rational gaussian ...
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1answer
48 views

Question about correlation functions of 2d CFTs

I have a question regarding equation (2.22) in Ginsparg's lecture notes on CFTs. Equation (2.22) is $$ \langle T(z) \phi_1(w_1, {\bar w}_1) \cdots \rangle = \sum_{i=1}^n \left( \frac{h_i}{(z-w_i)^2} ...
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0answers
76 views

Stack presentations and massive non-conformal theories

In the paper, Cluster Decomposition, T-duality, and Gerby CFT’s , by Hellerman, Henriques, Pantev and Sharpe, in the introduction it says: "Briefly, the idea is that nearly every stack has a ...
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0answers
33 views

What is the central charge of the disordered $q$-state Potts model, for large $q$?

The central charge of a model, is, heuristically related to the number of microscopic degrees of freedom. Is there a simple argument for the asymptotic behavior of the central charge for the ...
4
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0answers
59 views

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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2answers
79 views

What are marginal fields in CFT?

In this article they call weight $(h,\bar{h})=(1,1)$ fields marginal. Why are these fields called marginal? Why are they to be distinguished.
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3answers
323 views

Modular invariance for higher genus

As far as I understand, there are roughly 2 "common" kinds of 2D conformal field theories: Theories that are defined only on the plane, more precisely, on any surface of vanishing genus. Such a ...
5
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1answer
305 views

Thermal AdS and the Hawking Page phase transition

I have some difficulty understanding the concept of pure thermal radiation, as described in Hawking and Page's paper on the Hawking-Page phase transition. The four-dimensional thermal AdS solution ...
4
votes
1answer
74 views

Reading list and book recommendation on Conformal Field Theory

I have a background in QFT, GR and differential geometry at the level of a master student in theoretical physics. I would like to touch the area of CFT. I know the textbook of Philippe Di Francesco. ...
2
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1answer
59 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 ...
3
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1answer
61 views

The contribution to the one loop beta function for the WZW model

When the Wess-Zumino-Witten model $$S_{WZW}=\frac{k}{4\pi}\int d^2 z \, \, \mathrm{Tr}[\partial u \bar{\partial}u^{-1} ]+ \frac{k}{12\pi}\int d^3 \sigma \epsilon^{ijk}\, ...
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1answer
986 views

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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1answer
260 views

Why do we assume local conformal transformations are symmetries in 2D CFT

The global conformal group in 2D is $SL(2,\mathbb{C})$. It consists of the fractional linear transforms that map the Riemann sphere into itself bijectively and is finite dimensional. However, when ...
2
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1answer
170 views

Massless Dirac equation is Weyl covariant

Does somebody know how to show that the following equation is Weyl invariant? $$\gamma^ae_a^\mu D_\mu \Psi=0$$ where: $D_\mu \Psi=\partial_\mu\Psi+A_\mu^{ab}\Sigma_{ab}\Psi$ is the spin-covariant ...