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negru
#6
Mar8-10, 04:24 PM
P: 308
Ok I think I got the general idea now, thanks! I can get it to this form
[tex]
2\nabla_a\xi_b\nabla^a\xi^b+2\nabla_a\xi_b\nabla^b\xi^a-2\nabla_a\xi^a\nabla_b\xi^b
[/tex]
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
[tex]
\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
[/tex]
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
\mu-k=-3\chi
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.
http://www.phys.columbia.edu/~kabat/...8/handout3.pdf