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3
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449 views

A certain regularization and renormalization scheme

In a certain lecture of Witten's about some QFT in $1+1$ dimensions, I came across these two statements of regularization and renormalization, which I could not prove, (1) $\int ^\Lambda \frac{d^2 ...
3
votes
1answer
119 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 ...
3
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1answer
118 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 ...
3
votes
1answer
276 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. ...
3
votes
1answer
230 views

Massless fields in curved spacetimes

I read the following statement in one of Penrose's paper zero rest-mass field equations can, with suitable interpretations, be regarded as being conformally invariant. I take this to imply that ...
3
votes
1answer
297 views

2D Ghost CFT and two-point functions

For some reason I am suddenly confused over something which should be quit elementary. In two-dimensional CFT's the two-point functions of quasi-primary fields are fixed by global $SL(2,\mathbb ...
3
votes
1answer
83 views

Stress tensor in product of 2D CFTs

I was struggling with a question, hoping someone could point me in the right direction. I'm interested in 2D CFTs on a cylinder. I want to take the tensor product of two CFTs. My questions are these: ...
3
votes
1answer
355 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 ...
3
votes
1answer
258 views

Deriving Virasoro algebra question

I'm reading a book Lie groups, Lie algebras, cohomology and some applications in physics by Azcarraga and Izquierdo, and on page 347, when deriving the exact form of the central extension term I came ...
3
votes
1answer
114 views

Spectral properties of CFT

What are the general spectral properties of CFT? I mean what is the "spectrum"/eigenvalues of CFT in 2d and d>2 spacetime dimensions? I understand the "spectrum" and "Fock space" realization of Dirac ...
3
votes
1answer
257 views

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 ...
3
votes
1answer
180 views

How does the conformal Ward identity guarantee a vanishing 3-point function in this case?

I was looking through some conformal Ward identity related things when I noticed that this paper (arXiv:1212.3788) writes in their equation (33), a 3-point function between a conserved current and two ...
3
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1answer
185 views

Question on Section 9.1.3 in “Conformal Field Theory” by Philippe Di Francesco et. al

Question on Section 9.1.3 in "Conformal Field Theory" by Philippe Di Francesco et. al. The basic idea of the Coulomb-gas formalism is to place a background charge in the system, making the $U(1)$ ...
3
votes
1answer
42 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. ...
3
votes
1answer
44 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}\, ...
3
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1answer
78 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
3
votes
1answer
74 views

Renormalization of worldsheet energy-momentum tensor

At the end of section 2.3, Polchinski (in his volume 1) derives the energy-momentum tensor for free massless scalars on worldsheet. He adds a footnote that "the only possible ambiguity introduced by ...
3
votes
1answer
79 views

Explicit definition of the energy operator in the Ising model

I've simulated a few 2d Ising models at critical temperature on triangular lattice and I'm now trying to check that the correlation functions are right. I alraedy did it for the spin operator ...
3
votes
1answer
291 views

A key relation in Di Francesco's book on Conformal Field Theory

Recently I am reading Di Francesco's book Volume I on Conformal Field Theory. In order to reduce the number of fields in a correlator, the calculation of Operator Algebra is extremely important. To do ...
3
votes
1answer
107 views

Symmetries in Wilsonian RG (2)

This question is related to the paper http://arxiv.org/abs/1204.5221 and is a continuation of the previous question Symmetries in Wilsonian RG In the liked paper why do the equalities in equation ...
3
votes
1answer
163 views

Inclusion of information about external particles to calculate scattering amplitudes

In this (schematic) equation to calculate the scattering amplitude A by integrating over all possible world sheets and lifetimes of the bound states $$ A = \int\limits_{\rm{life time}} d\tau ...
3
votes
1answer
158 views

The neutrality condition and the (non)-vanishing of the one-point correlator for the bosonic vertex operator

Consider the massless scalar field Hamiltonian, \begin{align} H = \frac{1}{2}\int \Pi^2- (\partial_x\phi)^2 dx \end{align} with $\Pi \sim \partial_t\phi$ the conjugate field of $\phi$. This ...
3
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0answers
64 views

Non-abelian bosonization

Reading this review about non-abelian bosonization, Non-abelian bosonization by I.Karmazin, I stumbled about two questions Below equation 6, I don't get the final point in the statement about the ...
3
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0answers
103 views

Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two ...
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0answers
54 views
3
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0answers
107 views

Derivation of the Noether current

(c.f Di Francesco et al, Conformal Field Theory, pp40-41) I am trying to derive eqn (2.142) or $\delta S = \int d^d x \partial_{\mu}j^{\mu}_a \omega_a$ in the book CFT by Di Francesco et al. I have ...
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0answers
91 views

Virasoro Operators commutation relations

For the commutation relation in quantising the bosonic string $\left[L_n,L_{m}\right]=(n-m)L_{n+m}+\frac{D}{12}n(n^2-1)\delta_{n+m,0}$ we can then calculate this for $m=-n$ in between the vacuum ...
3
votes
0answers
119 views

Questions on entanglement entropy

If the spatial entangling surface is $M$ then it seems that one way to get the entanglement entropy is to think of the QFT on the manifold $S \times M$ where $S$ is a 2-manifold with the metric, ...
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votes
0answers
125 views

Moduli Space of $\mathcal{N}=4$ SYM on $\mathbb{R} \times S^3$

When we define $\mathcal{N}=4$ SYM on flat Minkowski space, the supersymmetric vacua are parametrized by scalars living in the cartan subalgebra of the gauge group. A generic point in the moduli space ...
3
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0answers
58 views

Free energy of the critical U(N) model

Can someone help explain how the equations 30, 31 and 34 were obtained in this paper. At a conceptual level I am wondering looking at equation 34 as to if they mean that $\lambda$ is somehow the ...
3
votes
0answers
82 views

Link between anomalous dimensions and fractal dimensions

I just realized that anomalous dimensions in quantum/statistical field theory is not that different from fractal dimensions of objects. They both describe how quantitaive objects transform under a ...
3
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0answers
390 views

A gentle introduction to CFT [closed]

Which is the definition of a conformal field theory? Which are the physical prerequisites one would need to start studying conformal field theories? (i.e Does one need to know supersymmetry? Does one ...
3
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0answers
348 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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0answers
144 views

Questions about classical and quantum scale invariance

This is kind of a continuation of this and this previous questions. Say one has a free "classical" field theory which is scale invariant and one develops a perturbative classical solution for an ...
3
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0answers
154 views

Conformal symmetry of Navier-Stokes?

This question is in reference to the paper arXiv:0810.1545 Can someone help understand this scaling argument and the proof(?) that there is a conformal symmetry in Navier-Stoke's equation? (..am I ...
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votes
0answers
126 views

Derivation of the enhancement of U(1)$_L$ x U(1)$_R$ to SU(2)$_L$ x SU(2)$_R$ at the self-dual radius

Towards the end of the paragraph with the title String theory's added value 2: enhanced non-Abelian symmetries at self-dual radii and abstract C with current algebras of this article, it is explained ...
3
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0answers
89 views

Relating the deformation of Calabi-Yau metrics and the conformal quantum field theories

(v2) As I read e.g. in this question, the nice holonomy group features of Calabi-Yau manifolds are valuable regarding supersymmetry (I suspect because it's a symmetry involving the target manifold, ...
3
votes
0answers
203 views

Central charge at the fixed point of the ${\cal N}=2$ Landau-Ginzburg theory in $1+1$ dimensions

Let me first believe that the ${\cal N}=2$ Landau-Ginzburg theory does in the IR flow to a non-trivial fixed point and that if the potential is of the form $\Phi ^k$ then the central charge of the CFT ...
3
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0answers
88 views

How to construct a 2D conformal field on a mirrored annulus with a looping pattern?

I would like to construct a 2D conformal field on an annulus in which the inner and outer boundaries are like mirrors, and can be approximated by regular polygons (with the same number n of mirror ...
2
votes
3answers
634 views

Question on Conformal Field Theory

Since every question has to be asked in a seperate topic, I'm asking a question refering to the following topic: Beginners questions concerning Conformal Field Theory In particula I'm refering to the ...
2
votes
2answers
136 views

A question about Virasoro algebra

(skip disclaimer) I have a question in Polchinski's string theory book volume 1 p54, related to the Virasoro algebra. Introducing complex coordinates $$w=\sigma^1 + i \sigma^2 $$ $$z=\exp (-i ...
2
votes
2answers
128 views

Identity of Operator Product Expansion (OPE)

I have one more s****d question in Polchinski's string theory book, Eqs. (2.3.14a) $$ j^{\mu}(z) :e^{ik \cdot X(0,0)}:~ \sim~ \frac{k^{\mu}}{2 z} :e^{ik \cdot X(0,0)}:,$$ where $j^{\mu}_a ...
2
votes
1answer
122 views

Are diffeomorphisms a proper subgroup of conformal transformations?

The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not ...
2
votes
1answer
133 views

representation of conformal group in d>2

In P. Di Francesco, P. Mathieu, D. Snchal they fix the generators of the conformal group acting on a scalar field by somewhat arbitrarily defining $$\Phi'(x)=\Phi(x)-i\omega_a G_a\Phi(x)$$ and by ...
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2answers
55 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.
2
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1answer
240 views

Stress-energy Trace of Massless Klein Gordon Field

I've calculated the trace of the stress-energy for a massless KG field and I keep getting $T = - (\partial \phi)^2$ in 3+1 dimensions. I'm using $$T_{\mu\nu} = \partial_\mu \phi \partial_\nu \phi - ...
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2answers
101 views

Factor of two differences for free field Green's functions in conformal field theory

I have a question about the expressions for free field Green's functions in conformal field theory. It comes from three origins 1) In Polchinski's string theory volume I p36, it is given $$ ...
2
votes
1answer
378 views

Energy-Momentum Tensor in Conformal Field Theory

Basically, I would really like it if somebody just explained to me what is going on here. Please use any physics lingo you feel is necessary, but explain what you mean. I am just having trouble ...
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2answers
382 views

Invariance of Maxwell's Equations under inverting variables - Reference and use

Some months ago, an ArXiv paper mentioned in passing that Maxwell's Equations were invariant under reciprocating the variables, or at least this results in a dual set of Maxwell Equations. (Actually I ...
2
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1answer
83 views

Number of zero-modes on the sphere

Is it true that a field of conformal dimension $h$ (integer or half integer) has $1-2h$ zero-modes on the sphere, if $1-2h \geq0$. This seems to be right for different ghost fields : $c$ has ...