MSCIH · Moduli spaces of curves and integrable hierarchies

Moduli spaces of curves and integrable hierarchies

Objective O1. Proving the DR/DZ equivalence conjecture, saying that the hierarchy of topological type is Miura equivalent to the double ramification hierarchy, is the main step needed to prove the Dubrovin-Zhang conjecture about a polynomial bihamiltonian structure for the hierarchies of topological type. In my paper I made a crucial step for the proof of the DR/DZ equivalence conjecture. I reduced this conjecture to a system of relations in the cohomology of the moduli spaces of stable algebraic curves and proved these relations up to genus 2. This proves the DR/DZ conjecture at the approximation up to genus 2. Another important result was obtained in my other paper, where I proved the DR/DZ equivalence conjecture for cohomological field theories of rank one at the approximation up to genus 5. Objective O2. My two papers are devoted to the Frobenius manifolds associated to the curve singularities of types A and D. More precisely, I constructed solutions of the associated open WDVV equations that should correspond to open FJRW invariants of the curve singularities of types A and D. I proved that the coefficients of the solutions of the open WDVV equations coincide with the transition functions between two natural coordinate systems on the Frobenius manifolds associated to the curve singularities of types A and D. In other words, the solutions of the open WDVV equations describe the mirror map for these Frobenius manifolds. Objective O3. In my paper I constructed a tau-structure for the double ramification hierarchy, and in my other paper I constructed its generalization for the quantum double ramification hierarchy. As a result, this gives a construction of a quantum tau-function for a large class of integrable systems, which are important in mathematical physics. Objective O4. A construction from my paper defines a quantum tau-function for the KdV hierarchy and, more generally, for higher Gelfand-Dickey reductions of the KP hierarchy. From another perspective, in my other paper I generalized the open Virasoro equations for the open extension of the Gelfand-Dickey hierarchy, finding open Virasoro equations for an arbitrary homogeneous solution of the open WDVV equations.

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