Double ramification cycles with target varieties

Abstract

Let X be a nonsingular projective algebraic variety over C, and let M g,n,β (X) be the moduli space of stable maps f : (C, x 1 , . . . , x n ) → X from genus g, n-pointed curves C to X of degree β. Let S be a line bundle on X. Let A = (a 1 , . . . , a n ) be a vector of integers which satisfyConsider the following condition: the line bundle f * S has a meromorphic section with zeroes and poles exactly at the marked points x i with orders prescribed by the integers a i . In other words, we require f * S (− n i=1 a i x i ) to be the trivial line bundle on C. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X and is denoted by M ∼ g,A,β (X, S). The moduli space carries a virtual fundamental classin Gromov-Witten theory. The main result of the paper is an explicit formula (in tautological classes) for the push-forward via the forgetful morphism of [M ∼ g,A,β (X, S)] vir to M g,n,β (X). In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle proven in [28]. Several applications of the new formula are given.

Gromov-Witten classes of K3 surfaces

“…The push-forward ǫ * [M ∼ ] vir of the virtual class is the E-valued double ramification cycle [6]; it is a tautological class on M g,n+2 (E, h). The Gromov-Witten classes of E are tautological by [7] and the proof is complete.…”

Gromov--Witten invariants of root stacks with mid-ages and the loop axiom

“…There is a natural C * -action on Y which induces a natural C * -action on the moduli space [M 0, k,,ka,k b , µ,β (Y D 0 ,r , D ∞ )] vir . Therefore, it can be computed by the virtual localization formula studied in [JPPZ18], [TY18] and [FWY19]. We refer readers to [JPPZ18] for details of the virtual localization formula.…”

“…Therefore, it can be computed by the virtual localization formula studied in [JPPZ18], [TY18] and [FWY19]. We refer readers to [JPPZ18] for details of the virtual localization formula. A component of the domain curve is called contracted if it lands on the zero section D 0 or the infinity section…”

Double ramification cycles with orbifold targets

“…Recently, people observe that it is closely related to the so called double ramification cycles (abbreviated as DR-cycles). DR-cycles for target space X as a point and a smooth manifold were studied and an elegant formula for DR-cycles was obtained by Janda-Pandharipande-Pixton-Zvonkine ( [15,16]). This is a break-through in this subject and it has many interesting applications, see for example [15], [16], [14], [23], [13], [12], etc.…”