Abstract
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.
Highlights
Modular forms often appear in physics as a consequence of duality properties
Recent developments on the perturbative expansions in quantum field theory reveal how relevant the spaces of quasimodular forms are for understanding the ultraviolet behaviour and its connections with string theory acting as a ultraviolet regulator[54]
M4 topologically equivalent to R×M3 have been investigated extensively in the cases where M3 are homogeneous of Bianchi type
Summary
Modular forms often appear in physics as a consequence of duality properties. This comes either as an invariance of a theory or as a relationship among two different theories, under some discrete transformation of the parameters. Instantons have finite action and enter the description of quantummechanical processes, which are not captured by perturbative expansions, as their magnitude is controlled by exp −1/g at small coupling g (in electrodynamics g = e2/~c) These phenomena include quantum-mechanical tunneling and, more generally, decay and creation of bound states. Recent developments on the perturbative expansions in quantum field theory reveal how relevant the spaces of quasimodular forms are for understanding the ultraviolet behaviour and its connections with string theory acting as a ultraviolet regulator[54]. They call for introducing new objects, which stand beyond the realm of Eisenstein series [53]
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