I am asking this question after reading this post: What is Euler Density?.
For a two dimensional manifold, the Euler density is given by:
\begin{equation}
E_2=2R_{1212}
\end{equation}
(note that $R_{1212}$ is the only independent component of the Riemann tensor in 2d).
Now, integrating over the 2d manifold, we should get the Euler characteristic
\begin{equation}
\chi=\int d^2x \sqrt {(\textrm{det }g)} E_2,
\end{equation}
where $(\textrm{det }g)$ is determinant of the metric. But $E_2=2R_{1212}=R(g_{11}g_{22}-g_{12}g_{21})=R \textrm{ det }g$, where $R$ is the Ricci scalar of the 2d manifold. This gives
\begin{equation}
\chi=\int d^2x (\textrm{det }g)^{\frac{3}{2}} R,
\end{equation}
which contradicts the first term of equation 3.2.3b of Polchinski's 'String Theory', volume 1. What's the reason for this contradiction?
This post imported from StackExchange Physics at 2015-09-15 19:02 (UTC), posted by SE-user Meer Ashwinkumar