2007/12/14 by Makoto Sakurai, Sakurai, Makoto
Mathematics · Physics and Astronomy · #14D21 #14F43 #14J26 #55N05 #81T40 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · pdf · doi:10.48550/arxiv.0712.2318
openalex publication_date 2007/12/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 Ansätze (or working hypotheses) of Witten's N=(0,2) heterotic strings and Nekrasov's generalized complex geometry, after Hitchin and Gualtieri, are consistent in the case of \mathbbCP2, 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 \mathbbCP2. 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.