闭合弦振幅的Rankin-Selberg方法
Rankin-Selberg methods for closed string amplitudes
摘要：
在对超模和顶点算子位置进行积分后，超弦理论中亏格为$h\leq 3$的散射振幅被简化为Siegel上半平面基本域上的$h$次Siegel模函数的积分.直接计算通常是笨拙的，但如果被积函数可以表示为Siegel模群的适当子群下的图像之和，则变得可行：如果是这样，积分域可以扩展到一个更简单的域，代价是在每个轨道中保持一个单独的项-一种称为Rankin-Selberg方法的技术。出于对BPS饱和振幅的应用，Angelantonj，Florakis和我已经将这种技术应用于单圈模积分，其中被积函数是Siegel-Narain θ函数乘以弱的几乎全纯模形式的乘积。我调查我们的主要结果，并采取一些步骤，在推广这种方法的亏格大于一。
英文摘要：
After integrating over supermoduli and vertex operator positions, scattering amplitudes in superstring theory at genus $h\leq 3$ are reduced to an integral of a Siegel modular function of degree $h$ on a fundamental domain of the Siegel upper half plane. A direct computation is in general unwieldy, but becomes feasible if the integrand can be expressed as a sum over images under a suitable subgroup of the Siegel modular group: if so, the integration domain can be extended to a simpler domain at the expense of keeping a single term in each orbit -- a technique known as the Rankin-Selberg method. Motivated by applications to BPS-saturated amplitudes, Angelantonj, Florakis and I have applied this technique to one-loop modular integrals where the integrand is the product of a Siegel-Narain theta function times a weakly, almost holomorphic modular form. I survey our main results, and take some steps in extending this method to genus greater than one.
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