1
vote
0answers
42 views

How to derive the scale factor for special conformal transformation? [closed]

By definition a conformal transformation of the coordinates is an invertible mapping $x\rightarrow x'$ which leaves the metric invariant upto a scale factor: \begin{equation} g_{\mu\nu}'(x') = ...
3
votes
0answers
73 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 ...
5
votes
2answers
241 views

A calculation problem on Conformal Field Theory

This problem is on Di Francesco's book I. It's exercise 7.1: Calculate the norm of the following vector, where $\lvert h\rangle$ is the state of highest weight. $$L_{-1}^n\lvert h\rangle$$ I have ...
3
votes
1answer
233 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 ...
2
votes
1answer
143 views

About the conserved charge for the ghost number current in $bc$ conformal field theory

(skip disclaimer) I have a question about the conserved charge for the ghost number current in $bc$ conformal field theory in Polchinski's string theory p62. It is said For the ghost number ...
1
vote
1answer
186 views

Compute the central charge of $bc$ conformal field theory

I have a s****d question, how to calculate the central charge of $bc$ conformal-field theory in Polchinski's string theory, Eq. (2.5.12)? For a $bc$ CFT given by $$S=\frac{1}{2\pi } \int d^2 z \,\,b ...
1
vote
1answer
149 views

How to derive Eq. (2.4.23) in Polchinski's string theory book

Given the Operator Product Expansion (OPE) of a product of the energy momentum tensors $$T(z)T(0) = \frac{ \eta^{\mu}_{\mu} }{2z^4} - \frac{2}{\alpha' z^2} :\partial X^{\mu} \partial X_{\mu}(0): + ...
1
vote
1answer
75 views

Identify the weight of operator under conformal transformation

I have a stupid (homework tag may be suitable =_=) question about the problem 2.7 in Polchinski's string theory volume 1. Why the weight of operator $:e^{ik\cdot X}:$ is $(\frac{\alpha'k^2}{4}, ...
2
votes
2answers
117 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 ...
5
votes
1answer
137 views

A question about conformal transformation in Polchinski's string theory

I have one more stupid question in Polchinski's string theory book. P. 46, it is said It is convenient to take a basis of local operators that are eigenstates under rigid transformation (2.4.9) ...
1
vote
0answers
129 views

How to prove Eq. (2.4.5) in Polchinski's string theory book?

I got one more stupid question in Polchinski's string theory book. In p. 44, it is said The currents $$j(z)=i v(z) T(z), \tilde{j}(\bar{z}) = i v(z)^* \tilde{T}(\bar{z}) \tag{2.4.5}$$ are ...
5
votes
1answer
278 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 ...
1
vote
0answers
122 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 ...
4
votes
1answer
282 views

Wick Order and Radial Ordering in CFT

I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case): ...