DR/DZ equivalence conjecture and tautological relations

Abstract

In this paper we present a family of conjectural relations in the tautological ring of the moduli spaces of stable curves which implies the strong double ramification/Dubrovin-Zhang equivalence conjecture introduced in [3]. Our tautological relations have the form of an equality between two different families of tautological classes, only one of which involves the double ramification cycle. We prove that both families behave the same way upon pullback and pushforward with respect to forgetting a marked point. We also prove that our conjectural relations are true in genus 0 and 1 and also when first pushed forward from M g,n+m to M g,n and then restricted to M g,n , for any g, n, m ≥ 0. Finally we show that, for semisimple CohFTs, the DR/DZ equivalence only depends on a subset of our relations, finite in each genus, which we prove for g ≤ 2. As an application we find a new formula for the class λ g as a linear combination of dual trees intersected with kappa and psi classes, and we check it for g ≤ 3.14H10; 37K10

Double ramification cycles with target varieties

“…Pixton's formula's opened new directions in the subject: new formulas for Hodge classes [35, Section 3], new relations in the tautological ring of [17], new connections to the loci of meromorphic differentials [24, Appendix], and connections to new integrable hierarchies [9]. For a sampling of the subsequent study and applications, see [6–8, 10, 12, 16, 23, 31, 32, 45, 56, 57]. We refer the reader to [35, Section 0] and [50, Section 5] for more leisurely introductions to the subject.…”

Integrability, Quantization and Moduli Spaces of Curves

“…In particular we managed to further reduce this family to a finite number (equal to the number of partitions of 2g) of relations for each genus [5].…”

“…Its properties, quantization and relation with the DZ hierarchy were studied and clarified in the series of joint papers [2,3,5,8,9], partly guided by our previous investigations of the classical and quantum integrable systems arising in SFT [19,29,30,31,32,33].…”

“…While the general proof of such conjecture is the object of an ongoing work, we managed to show its validity in a number of examples and classes of interesting special cases. Our present approach to the general statement reduces it to proving a finite number of relations in the tautological ring of each M g,n with n ≤ 2g [5].…”

“…After a self contained introduction to the language of integrable systems in the formal loop space and the needed notions from the geometry of the moduli space of stable curves we will explain the double ramification hierarchy construction and present its main features, with an accent on the quantization procedure, concluding with a list of examples worked out in detail. This paper does not contain new results with respect to the series of papers [2,3,5,8,9]. It is however a complete reorganization and, in part, a rephrasing of those results with the aim of showcasing the power of our methods and making them more accessible to the mathematical physics community.…”