# Explicit calculation of bosonic string Weyl invariance at one loop

I have been trying to do all the calculations in the Green, Schwarz and Witten Superstring Theory textbook.

At the end of chapter 3, the author did one-loop calculation for Weyl invariance for the bosonic string, in section 3.4.2 and 3.4.5. In the latter section more fields were included, and not much calculation detail was given.

The actions are

$S_1=\frac{-1}{4\pi\alpha'}\int d^2\sigma\sqrt h h^{\alpha \beta}\partial_\alpha X^\mu \partial_\beta X^\nu g_{\mu \nu}$

$S_2=\frac{-1}{4\pi\alpha'}\int d^2\sigma \epsilon^{\alpha \beta}\partial_\alpha X^\mu \partial_\beta X^\nu B_{\mu \nu}$

$S_3=\frac{1}{4\pi}\int d^2\sigma \sqrt h \Phi (X^\rho) R^{(2)}$

I consider those one-loop calculations to be very good exercise, but as a beginner in string theory I find myself not able to do them.

Therefore may I ask if there are some notes/papers in literature that give explicit calculation or point out key steps? Thank you very much!

• I found a reference but it is not easy to read... I let you judge : Ref paragraph $3.2$ p $21$. Maybe somebody has a simplest reference. – Trimok Dec 14 '13 at 18:27