弦论中的算术和双曲结构
Arithmetic and Hyperbolic Structures in String Theory
摘要：
这本专著是作者博士论文的更新和扩展版本。它包括一个介绍性的文本，后面是两个独立的部分，这两个部分之间的关系很松散，但可以相互独立阅读。在第一部分中，我们分析了在类空奇点（“BKL极限”）附近研究引力时产生的某些双曲结构。在这个限制，空间点解耦和动力学表现出超局部的行为，可以描述在一个（可能是混乱的）双曲台球。在作为弦论或M理论的低能极限而出现的所有超引力中，台球动力学发生在某些双曲卡茨-穆迪代数的基本外尔腔中，这表明这些代数产生了理论的隐藏的无限维对称性。论文的第二部分致力于研究弦论中的（U-）对偶如何对微扰和非微扰量子修正提供强有力的约束。这些对偶由某些算术群G（Z）来描述，这些算术群G（Z）被证明在有效作用中保持不变。精确耦合由二重商G（Z）\G /K上的自守形式给出.我们详细讨论了各种方法的自守形式，特别强调非全纯爱森斯坦系列。我们提供了SL（2，Z）和SL（3，Z）的物理相关的情况下，我们构建各自的爱森斯坦级数和计算其（非阿贝尔）傅立叶展开的详细例子。我们还展示了如何将这些技术可以应用到超多重模空间的II型卡-丘紧化，我们提供了一个详细的分析，为普遍的超多重。
英文摘要：
This monograph is an updated and extended version of the author's PhD thesis. It consists of an introductory text followed by two separate parts which are loosely related but may be read independently of each other. In Part I we analyze certain hyperbolic structures arising when studying gravity in the vicinity of a spacelike singularity (the "BKL-limit"). In this limit, spatial points decouple and the dynamics exhibits ultralocal behaviour which may be described in terms of a (possibly chaotic) hyperbolic billiard. In all supergravities arising as low-energy limits of string theory or M-theory, the billiard dynamics takes place within the fundamental Weyl chambers of certain hyperbolic Kac-Moody algebras, suggesting that these algebras generate hidden infinite-dimensional symmetries of the theory. Part II of the thesis is devoted to a study of how (U-)dualities in string theory provide powerful constraints on perturbative and non-perturbative quantum corrections. These dualities are described by certain arithmetic groups G(Z) which are conjectured to be preserved in the effective action. The exact couplings are given by automorphic forms on the double quotient G(Z)\G/K. We discuss in detail various methods of constructing automorphic forms, with particular emphasis on non-holomorphic Eisenstein series. We provide detailed examples for the physically relevant cases of SL(2,Z) and SL(3,Z), for which we construct their respective Eisenstein series and compute their (non-abelian) Fourier expansions. We also show how these techniques can be applied to hypermultiplet moduli spaces in type II Calabi-Yau compactifications, and we provide a detailed analysis for the universal hypermultiplet.
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