Periods and Superstring Amplitudes
Periods and Superstring Amplitudes
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周期和超弦振幅
DOI：
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发表时间：
2014
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通讯作者：
S. Stieberger
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作者：
S. Stieberger
描述物理状态相互作用的散射振幅在确定物理可观测量中起着重要作用。在弦理论中，物理状态由开弦和闭弦的振动给出，它们的相互作用（在微扰论中的首阶）由分别由圆盘或球体拓扑给出的世界片描述。
形式上，对于N个弦的散射，这分别导致沿着紧化真实的轴的N-3维迭代真实的积分或N-3维复球面积分。因此，物理观测量可以用M_{0，N} --N个有序标点的黎曼球面的模空间上的周期来描述。
这些弦振幅的数学结构分享了算术代数几何和数论中的许多最新进展，如多重zeta值，单值多重zeta值，Drinfeld，Deligne关联子，与Grothendiffel Galois理论相关的Hopf代数和李代数结构。我们回顾了这些结果，重点是描述M_{0，N}（R）上真实的迭代积分的广义超几何函数与motivic多重zeta值分解之间的美丽联系。进一步，给出了M_{0，N}（C）上的复积分表示为M_{0，N}（R）上的真实的迭代积分的单值投影的关系.
Scattering amplitudes which describe the interaction of physical states play an important role in determining physical observables. In string theory the physical states are given by vibrations of open and closed strings and their interactions are described (at the leading order in perturbation theory) by a world-sheet given by the topology of a disk or sphere, respectively.
Formally, for scattering of N strings this leads to N-3-dimensional iterated real integrals along the compactified real axis or N-3-dimensional complex sphere integrals, respectively. As a consequence the physical observables are described by periods on M_{0,N} - the moduli space of Riemann spheres of N ordered marked points.
The mathematical structure of these string amplitudes share many recent advances in arithmetic algebraic geometry and number theory like multiple zeta values, single-valued multiple zeta values, Drinfeld, Deligne associators, Hopf algebra and Lie algebra structures related to Grothendiecks Galois theory. We review these results, with emphasis on a beautiful link between generalized hypergeometric functions describing the real iterated integrals on M_{0,N}(R) and the decomposition of motivic multiple zeta values. Furthermore, a relation expressing complex integrals on M_{0,N}(C) as single-valued projection of iterated real integrals on M_{0,N}(R) is exhibited.