The double ramification cycle formula

Abstract

The double ramification cycle ${\mathrm{DR}}_g(A)={\mathrm{DR}}_g(\mu ,\nu )$ is a cycle in the moduli space of stable curves parametrizing genus $g$ curves admitting a map to ${\mathbb{P}}^1$ with specified ramification profiles $\mu ,\nu$ over two points. In 2016, Janda, Pandharipande, Zvonkine, and the author proved a formula expressing the double ramification cycle in terms of basic tautological classes, answering a question of Eliashberg from 2001. This formula has an intricate combinatorial shape involving an unusual way to sum divergent series using polynomial interpolation. Here we give some motivation for where this formula came from, relating it both to an older partial formula of Hain and to Givental’s R-matrix action on cohomological field theories.