引力，弦，模和拟模形式
Gravity, strings, modular and quasimodular forms
摘要：
模和拟模形式在引力和弦论中起着重要的作用。爱森斯坦系列已出现系统的确定频谱和配分函数，在描述非微扰效应，在高阶修正标量场空间，.后者经常以引力瞬子的形式出现，即爱因斯坦方程的特殊解。在本讲义中，我们提出了一类这样的解决方案，在四个层面上，通过要求（共形）的自对偶性和比安奇IX齐性。在这种情况下，存在大量的配置，表现出有趣的模块化特性。其他爱因斯坦空间的例子，没有比安奇IX对称性，但具有类似的功能也给出了。最后，我们讨论了爱森斯坦系列在场和弦论微扰展开的框架中的出现和作用，并激发了解开新的模块化结构的需要。
英文摘要：
Modular and quasimodular forms have played an important role in gravity and string theory. Eisenstein series have appeared systematically in the determination of spectrums and partition functions, in the description of non-perturbative effects, in higher-order corrections of scalar-field spaces, ... The latter often appear as gravitational instantons i.e. as special solutions of Einstein's equations. In the present lecture notes we present a class of such solutions in four dimensions, obtained by requiring (conformal) self-duality and Bianchi IX homogeneity. In this case, a vast range of configurations exist, which exhibit interesting modular properties. Examples of other Einstein spaces, without Bianchi IX symmetry, but with similar features are also given. Finally we discuss the emergence and the role of Eisenstein series in the framework of field and string theory perturbative expansions, and motivate the need for unravelling novel modular structures.
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