  Some time ago, the chiral algebra theory of Beilinson-Drinfeld was expected
to play a central role in the convergence of divergence in mathematical physics
of superstring theory for quantization of gauge theory and gravity. Naively,
this algebra plays an important role in a holomorphic conformal field theory
with a non-negative integer graded conformal dimension, whose target space does
not necessarily have the vanishing first Chern class. This algebra has two
definitions until now: one is that by Malikov-Schechtman-Vaintrob by gluing
affine patches, and the other is that of Kapranov-Vasserot by gluing the formal
loop spaces. I will use the new definition of Nekrasov by simplifying
Malikov-Schechtman-Vaintrob in order to compute the obstruction classes of
gerbes of chiral differential operators.
  In this paper, I will examine the two independent Ans$\"{a}$tze (or working
hypotheses) of Witten's $\mathcal{N}=(0,2)$ heterotic strings and Nekrasov's
generalized complex geometry, after Hitchin and Gualtieri, are consistent in
the case of $\mathbb{CP}^2$, which has $3$ affine patches and is expected to
have the "first Pontryagin anomaly".
  I also scrutinized the physical meanings of $2$ dimensional toric Fano
manifolds, or rather toric del Pezzo surfaces, obtained by blowing up the
non-colinear $1, 2, 3$ points of $\mathbb{CP}^2$. The obstruction classes of
gerbes of them coincide with the second Chern characters obtained by the
Riemann-Roch theorem and in particular vanishes for $1$ point blowup, which
means that one of the gravitational anomalies vanishes for a non-Calabi-Yau
manifold compactification.
  The future direction towards the geometric Langlands program is also
discussed in the last section.
