Beilinson-Drinfeld chiral algebras for del Pezzo surfaces

Recently the chiral algebra of Beilinson-Drinfeld draws much attention in the mathematical physics of superstring theory. Naively, this is a holomorphic conformal field theory with integer graded conformal dimension, whose target space not necessarily has the vanishing first Chern class. This algebra has two ways of definition: one is that of Malikov-Schechtman-Vaintrob by gluing affine patches, and the other is that of Kapranov-Vasserot by gluing the formal loop space. We will use the method of Malikov-Schechtman-Vaintrob in order to compute the gerbes of chiral differential operators. In this paper, we will examine the two independent ansatzes of Witten's (0,2) heterotic strings and Nekrasov's generalized complex geometry are consistent in the case of $\mathbb{CP}^2$, which has 3 affine patches and is expected to has the 1st Pontrjagin anomaly. We also extend this direction to the case of 2 dimensional toric Fano manifolds (toric del Pezzo surfaces) of all degrees, by blowing up the generic 1,2,3 points of $\mathbb{CP}^2$. These coincide with the computation of the Hirzebruch Riemann-Roch theorem. The most notable case is the 1 point blowup, where the total gauge invariant anomaly vanishes. The significant future direction towards its application to the geometric Langlands program is also discussed in the last section