View Single Post
Mar8-10, 04:24 PM
P: 308
Ok I think I got the general idea now, thanks! I can get it to this form
and I assume that now I should use integration by parts. However some details are still escaping me.

For example, how does integrating by parts precisely work? Naively I assumed from the start it should somehow work like
\int d^2 x \sqrt{g} \nabla_a\xi^b\nabla_b\xi^a=\nabla_a\xi^b\xi^a|-\int d^2 x \sqrt{g} \nabla_b\nabla_a\xi^b\xi^a
But is it really just that simple?

And now is it immediately clear that the surface term is zero?

And third, I'm not very sure how or why your expression
[tex]\nabla_a \nabla_b \xi^b = 0[/tex],
appears. Should this be the surface term and what I did above is wrong? :-?

I don't have Polchinski, so I'm curious. Is he trying to write down a variation principle for an "approximate conformal Killing vector?"
Unless I'm mistaken, I think Polchinski does something like that in some other chapter, but here his goal is to find the number of conformal killing vectors (k=dim ker h_{ab} )and number of metric moduli (\mu=dim ker f_{ab}^T) for a surface of genus g. It turns out that by the Riemann-Roch theorem
He needs this when computing a path integral for strings.
You can find a better description of what is going on here if you're interested.