Splitting formulas for the logarithmic double ramification cycle

TL;DR

The paper addresses the long-standing problem of obtaining gluing-type pullback formulas for logarithmic Gromov–Witten invariants, focusing on the logarithmic double ramification cycle and its pierced refinement. It leverages the pierced-curve framework to define and manipulate pullbacks along loop gluing, producing a recursive splitting formula that expresses the pulled-back LogDR cycle as a finite sum of smaller LogDR pieces twisted by piecewise-linear functions. The authors prove a compatibility of virtual fundamental classes across the splitting, and derive explicit pushforward formulas in terms of banana graphs, thus connecting logarithmic and classical DR theories and enabling computations in log Gromov–Witten theory and related areas like Hurwitz numbers and strata of differentials. This work provides a concrete, computable mechanism for splitting logarithmic invariants, with potential applications to toric log Gromov–Witten theory and the broader program of reconstructing invariants from gluing data.

Abstract

The logarithmic double ramification cycle is roughly a logarithmic Gromov--Witten invariant of $\mathbb{P}^1$. For classical Gromov--Witten invariants, formulas for the pullback along the gluing maps have been invaluable to the theory. For logarithmic Gromov--Witten invariants, such formulas have not yet been found. One issue is the fact that log stable maps cannot be glued. In this paper, we use the framework from [HS23] for gluing pierced log curves (a refinement of classical log curves) to give formulas for the pullback of the (log) (twisted) double ramification cycle along the loop gluing map.

Figures (3)

• Figure 1: A curve $C$ in $\mathrm{gl}^*\operatorname{DRL}_1{(4,-4)} \subset \overline{{\mathcal{M}}}_{0,4}$. The glued curve $C^\mathrm{gl} \in \operatorname{DRL}_1(4,-4) \subset \overline{{\mathcal{M}}}_{1,2}$ admits a rational function $f$ with divisor $\mathop{\mathrm{div}}\nolimits f = 4p_1 - 4p_2$ shown in blue. The pulback $\mathrm{gl}^*f$ on $C$ is a rational function with divisor $4p_1 - 4p_2$, hence we have $C \in \operatorname{DRL}_0(4,-4,0,0)$.
• Figure 2: A logarithmic curve $C$ in $\mathop{\mathrm{LogDRL}}\nolimits_{(4,-4),b=1}^\mathrm{gl} \subset \mathrm{gl}^*\operatorname{DRL}_1{(4,-4)} \subset \overline{{\mathcal{M}}}_{0,4}$ and its tropicalisation $C^\mathrm{trop}$. The glued curve $C^\mathrm{gl} \in \operatorname{DRL}_1(4,-4) \subset \overline{{\mathcal{M}}}_{1,2}$ admits a map $f: C^\mathrm{gl} \to \mathbb{G}_\mathrm{log}$ whose tropicalisation is the piecewise linear function on $C^{\mathrm{gl},\mathrm{trop}}$ with slopes shown in blue. The pulback $\mathrm{gl}^*f$ on $C$ is a rational function whose tropicalisation has slopes $(4,-4,1,-1)$ along the legs, hence we have $C \in \operatorname{DRL}_0(4,-4,1,-1)$.
• Figure 3: The piecewise linear function $\delta_A$ on $\mathbb{M}_{0,4}^\mathrm{trop} = \overline{{\mathcal{M}}}_{0,4}^\mathrm{trop} \times \mathbb{R}_{\geq 0}^4$. The cone complex in the middle is $\overline{{\mathcal{M}}}_{0,4}^\mathrm{trop}$. For each maximal stratum, a corresponding pierced tropical curve is shown. In blue are the slopes of $\alpha_A^\theta$, and in red the value of $\delta_A$ on this pierced tropical curve.

Theorems & Definitions (66)

• Definition 1.1: \ref{['def:pierceddr']}
• Theorem 1: \ref{['thm:firstformula']}
• Theorem 2: \ref{['def:banana']},\ref{['thm:secondformula']}
• Proposition 3: costantini2025integralspsiclassestwisteddouble, \ref{['prop:splitpsi']}
• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Example 2.6