Tautological systems and quantum differential equations of homogeneous spaces

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This project aims at extending our knowledge of the mirror symmetry phenomenon for homogeneous spaces, which are a very important class of algebraic varieties defined by transitive group actions. Mirror symmetry is well understood in various aspects (enumerative, Hodge theoretic, homological) for toric varieties, however, many questions remains unanswered when stepping out of the toric world. The current proposal intends to make considerable progress for various classes of homogeneous spaces using several rather different approaches. On the one hand, we seek to extend the known mirror theorems to larger classes of homogeneous spaces, partly by concrete identification of the quantum-D-modules with the character D-modules of the mirror. This technique shall in particular be used in the case of cominuscule spaces. On the other hand, we plan to make systematic use of the theory of tautological systems, which are a vast generalization of the classical hypergeometric GKZ-D-modules. Building on the recently achieved functorial construction of tautological system, we seek to describe the quantum-D-modules of homogeneous spaces from them via a dimensional reduction, and to study related differential system such as Frenkel-Gross connections and generalized Kloosterman D-modules. We will also use Hodge theoretic results on tautological systems to study the irregular Hodge filtration for those D-modules. When relating the classical, Lie-theoretic approach to the mirror conjecture with these more involved considerations concerning tautological systems, we will re-interpret canonical Landau-Ginzburg models as dimensional reductions of Lefschetz fibrations of the mirror variety. We will also study the representation-theoretic constructions of canonical Landau-Ginzburg models further, in order to generalize this construction beyond the class of cominuscule spaces.