0712.2318 (Makoto Sakurai)

Mixed anomalies of chiral algebras compactified to smooth
quasi-projective surfaces    [PDF]

Makoto Sakurai

Some time ago, the chiral algebra theory of Beilinson-Drinfeld is expected to a play 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 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 gluing affine patches, and the other is that of Kapranov-Vasserot by gluing the formal loop spaces. We 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 Pontrjagin 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 non-Calabi-Yau manifold compactification. The future direction towards the geometric Langlands program is also discussed in the last section.

View original: http://arxiv.org/abs/0712.2318