A Fock sheaf for Givental quantization
A Fock sheaf for Givental quantization
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DOI：
10.1215/21562261-2017-0036
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发表时间：
2014-11
影响因子：
0.6
通讯作者：
T. Coates;H. Iritani
中科院分区：
文献类型：
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作者：
T. Coates;H. Iritani
我们给出了Gromov-Witten不变量及其对应的b模型的一个全局的、内在的、无坐标的量化形式，它同时推广了Witten、Givental和Aganagic-Bouchard-Klemm描述的量化形式。后代势存在于一个Fock束中，由满足Eguchi-Xiong的(3g-2)-射流条件的gigiental Lagrangian锥上的局部函数组成；它们还满足某一异常方程，该异常方程推广了Bershadsky-Cecotti-Ooguri-Vafa全纯异常方程。在这种情况下，我们解释了与半简单Frobenius流形相关的高格势的Givental公式，表明在半简单情况下，存在典型的Fock束整体截面。这个规范节自动具有某些模块化属性。当X是具有半单量子上同调的变体时，Teleman的一个定理表明正则截面与X的Gromov-Witten不变量所定义的几何子代势相一致。我们利用我们的形式证明了紧环轨道的阮氏Crepant变换猜想的一个高格版本。当结合我们之前与Jiang的联合工作时，这表明紧环轨道X的总后代势是X的派生范畴的某一组自等价的模函数。
We give a global, intrinsic, and co-ordinate-free quantization formalism for Gromov-Witten invariants and their B-model counterparts, which simultaneously generalizes the quantization formalisms described by Witten, Givental, and Aganagic-Bouchard-Klemm. Descendant potentials live in a Fock sheaf, consisting of local functions on Givental's Lagrangian cone that satisfy the (3g-2)-jet condition of Eguchi-Xiong; they also satisfy a certain anomaly equation, which generalizes the Holomorphic Anomaly Equation of Bershadsky-Cecotti-Ooguri-Vafa. We interpret Givental's formula for the higher-genus potentials associated to a semisimple Frobenius manifold in this setting, showing that, in the semisimple case, there is a canonical global section of the Fock sheaf. This canonical section automatically has certain modularity properties. When X is a variety with semisimple quantum cohomology, a theorem of Teleman implies that the canonical section coincides with the geometric descendant potential defined by Gromov-Witten invariants of X. We use our formalism to prove a higher-genus version of Ruan's Crepant Transformation Conjecture for compact toric orbifolds. When combined with our earlier joint work with Jiang, this shows that the total descendant potential for compact toric orbifold X is a modular function for a certain group of autoequivalences of the derived category of X.