Gravity, strings, modular and quasimodular forms

Petropoulos, P. Marios; Vanhove, Pierre

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 2, pp. 379-430.

Les formes modulaires et quasimodulaires ont joué un rôle important dans la théorie de la gravité et la théorie des cordes. Les séries d’Eisenstein sont apparues de façon systématique dans la détermination des spectres. Les fonctions de partitions sont apparues de façon systématique dans la description des effets non perturbatifs, dans les corrections d’ordre supérieur des espaces de champs scalaires,... Ces dernières apparaissent souvent comme des instantons gravitationnels, c’est-à-dire des solutions particulières des équations d’Einstein. Dans ces notes de cours, nous présentons une classe de telles solutions en dimension quatre, obtenues en exigeant l’autodualité (conforme) et l’homogénéité Bianchi IX. Dans ce cas, un large ensemble de configurations existe qui exhibent d’intéressantes propriétés modulaires. Nous donnons d’autres exemples d’espaces d’Einstein qui bien que n’ayant pas de symétrie Bianchi IX possèdent des caractéristiques similaires. Enfin, nous discutons de l’émergence et du rôle des séries d’Eisenstein dans le cadre des développements perturbatifs de la théorie des champs et des cordes. Nous motivons le besoin d’étudier dans ce cadre de nouvelles structures modulaires.

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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