Explicit calculation of bosonic string Weyl invariance at one loop

Question

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!

Answer 1

Thanks to @Trimok , the reference he provided gives some detailed calculation. To see one of the missing Feynman diagrams (you can draw others once you understand that) and more detailed calculation of the leading order (not one loop), you can also check http://www.itp.phys.ethz.ch/research/qftstrings/archive/13FSProseminar/LEEre_Guns which gives more useful math tricks.