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首页> 外文期刊>Classical and Quantum Gravity: An Interantional Journal of Gravity Geometry of Field Theories Supergravity Cosmology >Some finite terms from ladder diagrams in three and four loop maximal supergravity
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Some finite terms from ladder diagrams in three and four loop maximal supergravity

机译：三环和四环最大超重力中梯形图的一些有限项

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摘要

We consider the finite part of the leading local interactions in the low energy expansion of the four graviton amplitude from the ladder skeleton diagrams in maximal supergravity on T-2, at three and four loops. At three loops, we express the (DR4)-R-8 and (DR4)-R-10 amplitudes as integrals over the moduli space of an underlying auxiliary geometry. These amplitudes are evaluated exactly for special values of the the moduli of the auxiliary geometry, where the integrand simplifies. We also perform a similar analysis for the (DR4)-R-8 amplitude at four loops that arise from the ladder skeleton diagrams for a special value of a parameter in the moduli space of the auxiliary geometry. While the dependence of the amplitudes on the volume of the T-2 is very simple, the dependence on the complex structure of the T-2 is quite intricate. In some of the cases, the amplitude consists of terms each of which factorizes into a product of two SL (2, Z) invariant modular forms. While one of the factors is a non-holomorphic Eisenstein series, the other factor splits into a sum of modular forms each of which satisfies a Poisson equation on moduli space with source terms that are bilinear in the Eisenstein series. This leads to several possible perturbative contributions unto genus 5 in type II string theory on S-1. Unlike the one and two loop supergravity analysis, these amplitudes also receive non-perturbative contributions from bound states of three D-(anti) instantons in the IIB theory.
机译：我们从T-2的最大超重力的梯形骨架图中，在三个回路和四个回路处，考虑了四个引力子振幅的低能膨胀中主要局部相互作用的有限部分。在三个循环中，我们将（DR4）-R-8和（DR4）-R-10振幅表示为基础辅助几何结构的模空间上的积分。这些振幅是针对辅助几何体模数的特殊值进行精确评估的，其中被积物简化了。我们还对四个循环中的（DR4）-R-8振幅执行了类似的分析，这四个循环是由梯形骨架图得出的，用于辅助几何图形的模空间中参数的特殊值。尽管振幅对T-2体积的依赖性非常简单，但对T-2复杂结构的依赖性却十分复杂。在某些情况下，振幅由项组成，每个项分解为两个SL（2，Z）不变模块形式的乘积。尽管其中一个因素是非亚纯的Eisenstein级数，但另一个因素分裂为模数形式的总和，每个形式都满足模空间上的Poisson方程，且源项在Eisenstein数列中是双线性的。这导致在S-1的II型弦理论中，对属5的几种可能的摄动作用。与一回路和两回路超重力分析不同，这些振幅还从IIB理论中的三个D-（反）实例的束缚态中获得非微扰作用。

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