3

I have a question in the section 3.2 "The Polyakov path integral" in Polchinski's string theory p. 83. Given

$$ \chi=\frac{1}{4 \pi} \int_M d^2 \sigma g^{1/2} R + \frac{1}{2 \pi} \int_{\partial M} ds \,\, k \,\,\,\, (3.2.3b) $$

It is said

We have noted $\chi$ is locally a total derivative in two dimensions and therefore depends only on the topology of the world-sheet -- it is the Euler number of the world sheet

I have two questions related to this sentence...

1) Where did the book note that $\chi$ is locally a total derivative?

2) Does the Euler number for $\chi$ come from the Gauss-Bonnet theorem? I never learned that theorem in my differential geometry course :( and Eq. (3.2.3b) is in different form with wiki
by different factors $\frac{1}{4\pi}$ and $\frac{1}{2\pi}$ (why? convention of normalization?). Would you recommend any reference for the proof of Gauss-Bonnet theorem for physicist?

user26143
  • 6,571

1 Answers1

2

The Euler number (often called the Euler characteristic) is given a in terms of nice integral formulas in the Gauss-Bonnet theorem, but it can be defined in other ways. The difference in the factors simply comes from the fact that the two-dimensional scalar curvature $R$ is twice the Gaussian curvature $K$ (see towards the end of the first paragraph here). Since you're reading Polchinski, you'll probably have no problem with most proofs of the Gauss-Bonnet theorem (which are all over the internet), but I think Do Carmo's differential geoemetry has a rather nice, elementary proof.

joshphysics
  • 58,991