Curves of genus g which admit a map to P 1 with specified ramification profile µ over 0 ∈ P 1 and ν over ∞ ∈ P 1 define a double ramification cycle DR g (µ, ν) on the moduli space of curves. The study of the restrictions of these cycles to the moduli of nonsingular curves is a classical topic. In 2003, Hain calculated the cycles for curves of compact type. We study here double ramification cycles on the moduli space of Deligne-Mumford stable curves.The cycle DR g (µ, ν) for stable curves is defined via the virtual fundamental class of the moduli of stable maps to rubber. Our main result is the proof of an explicit formula for DR g (µ, ν) in the tautological ring conjectured by Pixton in 2014. The formula expresses the double ramification cycle as a sum over stable graphs (corresponding to strata classes) with summand equal to a product over markings and edges. The result answers a question of Eliashberg from 2001 and specializes to Hain's formula in the compact type case.When µ = ν = ∅, the formula for double ramification cycles expresses the top Chern class λ g of the Hodge bundle of M g as a push-forward of tautological classes supported on the divisor of nonseparating nodes. Applications to Hodge integral calculations are given.

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.

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 satisfy n i=1 a i = β c 1 (S). Consider 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 class [M ∼ g,A,β (X, S)] vir ∈ A * M ∼ g,A,β (X, S) in 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. 1 Contents 0 Introduction 2 1 Curves with an rth root 13 2 GRR for the universal line bundle 27 3 Localization analysis 36 4 Applications 51 We define the set G g,n,β (X) of X-valued stable graphs as follows. A graph Γ ∈ G g,n,β (X) consists of the data

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In this article, we establish foundations for a logarithmic compactification of general GLSM moduli spaces via the theory of stable log maps [15,2,25]. We then illustrate our method via the key example of Witten's r-spin class. In the subsequent articles [17,16], we will push the technique to the general situation. One novelty of our theory is that such a compactification admits two virtual cycles, a usual virtual cycle and a "reduced virtual cycle".A key result of this article is that the reduced virtual cycle in the r-spin case equals to the r-spin virtual cycle as defined using cosection localization by Chang-Li-Li [13]. The reduced virtual cycle has the advantage of being C * -equivariant for a non-trivial C *action. The localization formula has a variety of applications such as computing higher genus Gromov-Witten invariants of quintic threefolds [27] and the class of the locus of holomorphic differentials [18].