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3
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1answer
127 views

Even-branes in IIA and odd-branes in IIB

The R-R sector of IIA and IIB are respectively given as, $8_s \otimes 8_c = [1]\oplus [3] = 8_v \oplus 56_t$ $8_s \otimes 8_s = [0]\oplus [2] \oplus [4]_+ = 1 \oplus 28 \oplus 35_+$ Now looking at ...
3
votes
1answer
337 views

Some questions about the free Fermionic partition function on a circle (Ginsparg's CFT lectures)

The following questions are based on these lectures, http://arxiv.org/abs/hep-th/9108028 I would like to know what is the relationship between the last equation on page 82 ($(L_0)_{cyl} = L_0 - ...
3
votes
2answers
384 views

How are functional determinants of Laplace-type operators used in physics?

Many mathematical papers concerning the $\zeta$-regularized Determinant of Laplace-type operators refer for motivation to the broad use of such determinants in mathematical physics, especially in ...
3
votes
1answer
165 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
110 views

Normal ordering of the identity operator

I'm puzzled about what should be the normal ordering of the identity operator (or any proportional operator): looking at it from the "Fock space operators POV",the prescription is to move all the ...
3
votes
1answer
257 views

What are the conserved charges related to the Virasoro generators?

I have just learned from reconsidering my demystified book, that when conformally maping the worldsheet of a closed string to the complex plain by using the transformation $z = e^{\tau + i\sigma}$ ...
3
votes
1answer
266 views

Conformal Quantum Mechanics

I heard the term Conformal Quantum Mechanics used today. What exactly does this mean? Why would one want to study this?
3
votes
2answers
470 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
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) ...
3
votes
1answer
320 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
255 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
333 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
97 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
308 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
123 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
271 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
211 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
votes
1answer
205 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
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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 ...
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}\, ...
3
votes
1answer
83 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
77 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
86 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
318 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
112 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
168 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
votes
1answer
479 views

What is the meaning of the concepts of “operator mixing” (and anomalous dimensions) [closed]

I am looking for an explanation about the idea of "operator mixing" and its associated concept about when anomalous dimension has to be thought of as a matrix. For example this idea is slightly ...
3
votes
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. ...
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 ...
3
votes
0answers
83 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
108 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 ...
3
votes
1answer
88 views

What's the conserved quantities correspond to the generator of conformal transformation

What's the conserved quantity corresponding to the generator of conformal transformations?
3
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0answers
120 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 ...
3
votes
0answers
111 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
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0answers
128 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, ...
3
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0answers
145 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
votes
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
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0answers
89 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
411 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
votes
0answers
407 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
160 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
votes
0answers
172 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 ...
3
votes
0answers
127 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
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0answers
209 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
90 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
661 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 ...
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2answers
144 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 ...
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2answers
142 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 ...