Pixton’s double ramification cycle relations

Abstract

We prove a conjecture of Pixton, namely that his proposed formula for the double ramification cycle on M g,n vanishes in codimension beyond g. This yields a collection of tautological relations in the Chow ring of M g,n . We describe, furthermore, how these relations can be obtained from Pixton's 3-spin relations via localization on the moduli space of stable maps to an orbifold projective line.

Tautological relations for stable maps to a target variety

“…When X is a point, the relations constructed here specialize to the double ramification cycle relations for M g,n conjectured by Pixtion [28] and proven by Clader and Janda in [7]. In fact, our proof follows the strategy of [7].…”

“…However, we assumed that x, y ≪ 0 so the degree of the line bundle is also negative. The proof of [7,Lemma 4.2] implies −Rπ * L A ′ is a locally free sheaf of rank g.…”

“…R-matrix interpretation. In [7,Section 5], DR relations are packaged into cohomological field theory(CohFT). Similarly, there is a way to understand Xvalued DR relations by using the X-valued CohFT, first illustrated in [23, Appendix A.].…”

“…where ω g,n,β (v 1 ⊗ ζ a1 , · · · , v n ⊗ ζ an ) = r g ev * 1 (v 1 ) · · · ev * n (v n )[M g,n,β (X)] vir if A satisfies the condition (3.1) and 0 otherwise. Following [7], let R r (z) be the exponential of diagonal r × r matrix:…”

“…Following the computation in [7,Lemma 5.3], it is straight forward to check that (3.12) P g,A,β,k = R.ω g,n,β | r=0 .…”