While dealing with General Sigma models $$ S = \frac{1}{2}\int dt g_{ij} \dot{X^i} \dot{X^j} $$ where the Riemann metric can be expanded as, $$g_{ij} = \delta_{ij} + C_{ijkl}X^kX^l$$ The Hamiltonian is given by, $$ H = \frac{1}{2} g^{ij} P_{i}P_{j}$$ The authors say that in quantum theory the above expression is ambiguous, because $X$ and $P$ don't commute. Hence there are many nonequivalent quantum choices for $H$ reduces to the same classical object. I am not able to figure this out.

Also this Hamiltonian is related to Laplacian, which I am not able to understand, why ? This Hamiltonian can be related to Laplacian if $g^{ij}$ is the usual $\eta^{ij}$. Do the authors want to say that in some atlas we can always find a local coordinates which reduces to $\eta^{ij}$ or is there a general definition of Laplacian which I am unaware of  ?

While dealing with General Sigma models (See e.g. Ref. 1)

$$\tag{10.67} S ~=~ \frac{1}{2}\int \! dt ~g_{ij}(X) \dot{X^i} \dot{X^j}, $$

where the Riemann metric can be expanded as,

$$\tag{10.68} g_{ij}(X) ~=~ \delta_{ij} + C_{ijkl}X^kX^l+ \ldots $$ The Hamiltonian is given by,

$$ H ~=~ \frac{1}{2} g^{ij}(X) P_{i}P_{j}.$$

The authors say that in quantum theory the above expression is ambiguous, because $X$ and $P$ don't commute. Hence there are many nonequivalent quantum choices for $H$ reduces to the same classical object. I am not able to figure this out.

Also this Hamiltonian is related to Laplacian, which I am not able to understand, why ? This Hamiltonian can be related to Laplacian if $g^{ij}$ is the usual $\eta^{ij}$. Do the authors want to say that in some atlas we can always find a local coordinates which reduces to $\eta^{ij}$ or is there a general definition of Laplacian which I am unaware of?

References:

1. K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror Symmetry, 2003, chapter 10, eqs. 10.67-10.68. The pdf file is available here or here.