Classical Weight-Four L-value Ratios as Sums of Calabi--Yau Invariants

TL;DR

This paper explores the summability of series solutions to attractor equations in 4d N=2 supergravity, revealing conjectural identities linking special L-values with Gromov--Witten invariants and introducing new rank-two attractors.

Contribution

It demonstrates Padé resummation for series solutions, derives conjectural identities relating L-values and Gromov--Witten invariants, and presents two new rank-two attractors with interesting moduli spaces.

Findings

Padé resummation effectively sums series solutions.

Identities linking L-values with Gromov--Witten invariants are conjectured.

Two new rank-two attractors with special moduli spaces are identified.

Abstract

We revisit the series solutions of the attractor equations of 4d N=2 supergravities obtained by Calabi--Yau compactifications previously studied in Candelas, Kuusela, and McGovern (2021). While only convergent for a restricted set of black hole charges, we find that they are summable with Pad\'e resummation providing a suitable method. By specialising these solutions to rank-two attractors, we obtain many conjectural identities of the type discovered in CKM. These equate ratios of weight-four special L-values with an infinite series whose summands are formed out of genus-0 Gromov--Witten invariants. We also present two new rank-two attractors which belong to moduli spaces each interesting in their own right. Each moduli space possesses two points of maximal unipotent monodromy. One has already been studied by Hosono and Takagi, and we discuss issues stemming from the associated L-function having nonzero rank.