弦行列式的模不变正则化与Serre GAGA原理
Modular Invariant Regularization of String Determinants and the Serre GAGA Principle
摘要：
由于任何弦理论都涉及到世界片度规上的路径积分，它们的配分函数是亏格g黎曼曲面M_g的模空间上的体积形式，或者是它的超级类似物上的体积形式。众所周知，模不变性修复了在某些情况下只出现在更高亏格的强约束。在这里，我们分类的所有外尔和模不变配分函数给出的路径积分的世界片度规，连同时空坐标，b-c和/或β-γ系统，对应于体积形式的M_g。这是一个长期存在的问题，由Belavin和Knizhnik倡导，受到Serre GAGA原则的启发，并基于芒福德形式的属性。关键的观察是，伯格曼再生核提供了一种外尔和模不变的方式来消除出现在上述字符串行列式中的点依赖性，这是一个基于超级伯格曼再生核的超级类似性质。这与时空坐标的传播子的性质密切相关。这样的配分函数Z [J]具有定义良好的渐近行为，可以被认为是代表广泛的一类弦论的基础。特别地，由于非临界玻色弦配分函数Z_D是M_g上的体积形式，基于玻色化和退化技术，我们认为存在一个从Liouville扇区到一阶系统的映射，可以将Z_D识别为Z [J]的子类.任何共形权的b-c和beta-gamma系统的出现表明这些理论与W代数有关。在一个大的类N't Hooft极限下，二维W_N极小模型CFTs与AdS_3上的高自旋引力理论有关，这一事实表明这里引入的弦配分函数可能导致在弦背景下的高自旋引力理论的公式化。
英文摘要：
Since any string theory involves a path integration on the world-sheet metric, their partition functions are volume forms on the moduli space of genus g Riemann surfaces M_g, or on its super analog. It is well known that modular invariance fixes strong constraints that in some cases appear only at higher genus. Here we classify all the Weyl and modular invariant partition functions given by the path integral on the world-sheet metric, together with space-time coordinates, b-c and/or beta-gamma systems, that correspond to volume forms on M_g. This was a long standing question, advocated by Belavin and Knizhnik, inspired by the Serre GAGA principle and based on the properties of the Mumford forms. The key observation is that the Bergman reproducing kernel provides a Weyl and modular invariant way to remove the point dependence that appears in the above string determinants, a property that should have its superanalog based on the super Bergman reproducing kernel. This is strictly related to the properties of the propagator associated to the space-time coordinates. Such partition functions Z[J] have well-defined asymptotic behavior and can be considered as a basis to represent a wide class of string theories. In particular, since non-critical bosonic string partition functions Z_D are volume forms on M_g, we suggest that there is a mapping, based on bosonization and degeneration techniques, from the Liouville sector to first order systems that may identify Z_D as a subclass of the Z[J]. The appearance of b-c and beta-gamma systems of any conformal weight shows that such theories are related to W algebras. The fact that in a large N 't Hooft-like limit 2D W_N minimal models CFTs are related to higher spin gravitational theories on AdS_3, suggests that the string partition functions introduced here may lead to a formulation of higher spin theories in a string context.
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