Conformal killing vector
That identity is only true when integrating by parts (and off by an overall factor of 2). First prove that
[tex]
\nabla_a \nabla_b \xi^b = 0
[/tex]
for a conformal Killing vector. You then integrate by parts and use Ricci's identity plus the fact that a Ricci tensor in 2 dimensions is entirely determined by the Ricci scalar (and the metric).
I don't have Polchinski, so I'm curious. Is he trying to write down a variation principle for an "approximate conformal Killing vector?"
