Recursion relations for Double Ramification Hierarchies

In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in [Bur15] using intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the Hamiltonians, the one associated to the first descendant of the unit of a cohomological field theory. Moreover, we introduce analogues of the topological recursion relations and the divisor equation both for the hamiltonian densities and for the string solution of the double ramification hierarchy. This machinery is very efficient and we apply it to various computations for the trivial and Hodge cohomological field theories, and for the $r$-spin Witten's classes. Moreover we prove the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line (extended Toda hierarchy).Comment: Revised version, to be published in Communications in Mathematical Physics, 27 page

Homogeneous components in the moduli space of sheaves and Virasoro characters

The moduli space $\mathcal M(r,n)$ of framed torsion free sheaves on the projective plane with rank $r$ and second Chern class equal to $n$ has the natural action of the $(r+2)$-dimensional torus. In this paper, we look at the fixed point set of different one-dimensional subtori in this torus. We prove that in the homogeneous case the generating series of the numbers of the irreducible components has a beautiful decomposition into an infinite product. In the case of odd $r$ these infinite products coincide with certain Virasoro characters. We also propose a conjecture in a general quasihomogeneous case.Comment: Published version, 19 page

Extended r-spin theory and the mirror symmetry for the Ar–1-singularity

By a famous result of K. Saito, the parameter space of the miniversal deformation of the Ar−1Ar−1 singularity carries a Frobenius manifold structure. The Landau–Ginzburg mirror symmetry says that, in the flat coordinates, the potential of this Frobenius manifold is equal to the generating series of certain integrals over the moduli space of rr spin curves. In this paper we show that the parameters of the miniversal deformation, considered as functions of the flat coordinates, also have a simple geometric interpretation using the extended rr spin theory, first considered by T. J. Jarvis, T. Kimura and A. Vaintrob, and studied in a recent paper of E. Clader, R. J. Tessler and the author. We prove a similar result for the singularity D4D4 and present conjectures for the singularities E6E6 and E8E8