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摘要翻译：
前一段时间，Beilinson-Drinfeld的手征代数理论被期望在超弦理论的数学物理中对规范理论和引力量子化的发散收敛方面发挥核心作用。该代数在目标空间不一定具有消失的第一Chern类的非负整数分次共形维数的全纯共形场论中起着重要作用。到目前为止，该代数有两种定义：一种是Malikov-Schechtman-Vaintrob通过粘合仿射贴片定义的，另一种是Kapranov-Vasserot通过粘合形式环空间定义的。我将通过简化Malikov-Schechtman-Vaintrob的Nekrasov的新定义来计算手征微分算子的gerbes阻塞类。本文将考察Witten的$\mathcal{N}=(0,2)$异形串和Nekrasov的广义复几何的两个独立的假设，在具有$3$仿射斑块且预期具有“第一庞特里亚金异常”的$\mathbb{CP}^2$情形下是一致的。我还仔细研究了$2$维toric Fano流形，或者更确切地说是toric del Pezzo曲面的物理意义，它是通过炸毁$\MathBB{CP}^2$的非共线性的$1,2,3$点而得到的。它们的gerbes阻塞类与Riemann-Roch定理得到的第二Chern特征一致，特别是在1$点爆破时消失，这意味着在非Calabi-Yau流形压缩时，其中一个引力异常消失。最后一节还讨论了几何Langlands程序的未来发展方向。
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英文标题：
《Mixed anomalies of chiral algebras compactified to smooth
quasi-projective surfaces》
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作者：
Makoto Sakurai
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最新提交年份：
2015
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分类信息：
一级分类：Physics 物理学
二级分类：High Energy Physics - Theory 高能物理-理论
分类描述：Formal aspects of quantum field theory. String theory, supersymmetry and supergravity.
量子场论的形式方面。弦理论，超对称性和超引力。
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一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
Some time ago, the chiral algebra theory of Beilinson-Drinfeld was expected to play a central role in the convergence of divergence in mathematical physics of superstring theory for quantization of gauge theory and gravity. Naively, this algebra plays an important role in a holomorphic conformal field theory with a non-negative integer graded conformal dimension, whose target space does not necessarily have the vanishing first Chern class. This algebra has two definitions until now: one is that by Malikov-Schechtman-Vaintrob by gluing affine patches, and the other is that of Kapranov-Vasserot by gluing the formal loop spaces. I will use the new definition of Nekrasov by simplifying Malikov-Schechtman-Vaintrob in order to compute the obstruction classes of gerbes of chiral differential operators. In this paper, I will examine the two independent Ans$\"{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.
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PDF链接：
https://arxiv.org/pdf/0712.2318
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