Symmetries and moduli spaces in algebraic geometry and physics – SMAGP

The project "Symmetries and moduli spaces in algebraic geometry and physics" (SMAGP) is an interdisciplinary research project between algebraic geometry, theory of automorphic forms and theoretical physics. The main goal of the project is the description of geometric, algebraic and arithmetic properties of moduli spaces of geometric or physical objects linked by the presence of the same type of symmetry. Our study passes through the discovery of their groups of hidden symmetries and the exploitation of the interplay between geometry and physics. Symmetries manifest themselves in various ways. It may be an automorphism group of a manifold preserving some structure on it, for example, a group of symplectic automorphisms, or the global monodromy group of a moduli space, or a symmetry group of an integrable system. In theoretical physics it may be a group of transformations of a partition function in quantum field theory. All these groups appear in the theory of automorphic forms either as modular groups, or as their subgroups of finite index, or else as parabolic subgroups, though the proof of such realization may be completely nontrivial.

In the scope of the project are the following classes of objects from the above three areas: (a) special varieties with prescribed symmetries in algebraic geometry, (b) indefinite lattices and automorphic forms and (c) quantum field theories, strings, integrable systems and Feynman integrals in physics.

The three areas are interconnected in several ways. Firstly, the moduli spaces of objects in (a) and (c) are at the same time arithmetic quotients associated to indefinite lattices in (b). Secondly, the most interesting objects in (c) are constructed by means of algebraic geometry—as algebraic varieties with additional structures. And thirdly, symmetries between physical objects in (c) translate themselves into correspondences between moduli spaces in (b) or into special automorphisms acting on algebraic varieties in (a). A spectacular manifestation of this correspondence is mirror symmetry.

The relevant varieties in (a) include K3, abelian, Enriques or Del Pezzo surfaces, moduli spaces of sheaves, as well as hyperkähler, irreducible symplectic, Calabi–Yau, and Fano varieties. The moduli spaces related to these varieties are arithmetic quotients of the complex ball or of Siegel's symmetric domains. Some of these quotients are moduli spaces of quantum field theories or parameter spaces of certain types of integrable systems. Others that still have no clear moduli interpretation can be thought of as awaiting the discovery of such interpretation. This justifies the interest in such quotients for their own sake.

The automorphic forms also appear in the description of the string scattering amplitudes, which links them with the classical tool of study of the scattering amplitudes—the Feynman integrals. From the point of view of algebraic geometry, Feynman integrals are periods of algebraic varieties with very rich geometrical, algebraic and arithmetic properties that are studied by application of various tools as automorphic forms, variations of mixed Hodge structure and theory of motifs.

The proposal aims at putting together the efforts of researchers having different perspectives on all these objects at the interface of the three domains—algebraic geometry, automorphic forms and theoretical physics—which will open new research directions and new points of view. It is expected that the proposed research will have a fundamental impact and will result in a significant progress in this interdisciplinary field.