Double Ramification Research Articles

Towards a description of the double ramification hierarchy for Witten's r-spin class

The double ramification hierarchy is a new integrable hierarchy of Hamiltonian PDEs introduced recently by the first author. It is associated to an arbitrary given cohomological field theory. In this paper we study the double ramification hierarchy associated to the cohomological field theory formed by Witten's r-spin classes. Using the formula for the product of the top Chern class of the Hodge bundle with Witten's class, found by the second author, we present an effective method for a computation of the double ramification hierarchy. We do explicit computations for r=3,4,5 and prove that the double ramification hierarchy is Miura equivalent to the corresponding Dubrovin–Zhang hierarchy. As an application, this result together with a recent work of the first author with Paolo Rossi gives a quantization of the r-th Gelfand–Dickey hierarchy for r=3,4,5.

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 Buryak (CommunMath Phys 336(3):1085–1107, 2015) 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).

Double Ramification Cycles and Quantum Integrable Systems

In this paper we define a quantization of the Double Ramification Hierarchies of [Bur15b] and [BR14], using intersection numbers of the double ramification cycle, the full Chern class of the Hodge bundle and psi-classes with a given cohomological field theory. We provide effective recursion formulae which determine the full quantum hierarchy starting from just one Hamiltonian, the one associated with the first descendant of the unit of the cohomological field theory only. We study various examples which provide, in very explicit form, new $(1+1)$-dimensional integrable quantum field theories whose classical limits are well-known integrable hierarchies such as KdV, Intermediate Long Wave, Extended Toda, etc. Finally we prove polynomiality in the ramification multiplicities of the integral of any tautological class over the double ramification cycle.

Integrals of ψ-classes over double ramification cycles

A double ramification cycle, or DR-cycle, is a codimension $g$ cycle in the moduli space $\overline{\mathcal M}_{g,n}$ of stable curves. Roughly speaking, given a list of integers $(a_1,\ldots,a_n)$, the DR-cycle ${\rm DR}_g(a_1,\ldots,a_n) \subset\overline{\mathcal M}_{g,n}$ is the locus of curves $(C,x_1,\ldots,x_n)$ such that the divisor $\sum a_ix_i$ is principal. We compute the intersection numbers of DR-cycles with all monomials in $\psi$-classes.

Double Ramification Cycles and Integrable Hierarchies

It this paper we present a new construction of a hamiltonian hierarchy associated to a cohomological field theory. We conjecture that in the semisimple case our hierarchy is related to the Dubrovin-Zhang hierarchy by a Miura transformation and check it in several examples.

Geometric Perspective on Piecewise Polynomiality of Double Hurwitz Numbers

AbstractWe describe doubleHurwitz numbers as intersection numbers on the moduli space of curves Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the ψ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).

The double ramification cycle and the theta divisor

We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of M\uller. We also obtain a formula for the class in the Chow ring of the moduli space of curves of compact type of the double ramification locus, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain. Our approach for computing the theta divisor is more direct, via test curves and the geometry of the theta divisor, and works easily over the entire Deligne-Mumford compactification. We use our extended result in another paper to study the partial compactification of the double ramification cycle.

The zero section of the universal semiabelian variety and the double ramification cycle

We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties (ppav). We describe the subring generated by divisor classes, and we compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family. The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of Ag. Our formula provides a first step in a program to understand the Chow groups of A¯g, especially of the perfect cone compactification, by induction on genus. By restricting to the image of Mg under the Torelli map, our results extend the results of Hain on the double ramification cycle, answering Eliashberg’s question.

A new proof of Faberʼs intersection number conjecture

We give a new proof of Faberʼs intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves M g . The proof is based on a very straightforward geometric and combinatorial computation with double ramification cycles.

Intersection numbers with Witten’s top Chern class

Witten's top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with r-spin structures. It plays a key role in Witten's conjecture relating to the intersection theory on these moduli spaces. Our first goal is to compute the integral of Witten's class over the so-called double ramification cycles in genus 1. We obtain a simple closed formula for these integrals. This allows us, using the methods of [15], to find an algorithm for computing the intersection numbers of the Witten class with powers of the \psi-classes (or tautological classes) over any moduli space of r-spin structures, in short, all numbers involved in Witten's conjecture.

Tautological relations for stable maps to a target variety

We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda-Pandharipande-Pixton-Zvonkine, we construct nontrivial tautological relations parallel to Pixton's double ramification cycle relations for the moduli of curves. Examples and applications are discussed.