The ice cone family and iterated integrals for Calabi-Yau varieties

TL;DR

The paper presents fully analytic results for the ice cone family of multi-loop Feynman integrals in $d=2$ with equal masses, revealing that maximal cuts decompose into two copies of Calabi-Yau periods tied to the $(l-1)$-loop banana geometry. It constructs a conjectural master-integral basis and solves the Gauss-Manin system using iterated integrals of Calabi-Yau periods, both in the original Landau variable and in the canonical moduli-space coordinate $q$ via the mirror map. It further expresses the periods and related integrals through $Y$-invariants, establishing a shuffle-algebra structure that converts Griffiths-type quadratic relations among periods into simple shuffle relations. The results generalize the use of pure functions and transcendental weight to Calabi-Yau settings and provide a novel representation of CY periods, with potential extensions to multi-moduli and higher-loop regimes. Overall, the work extends the class of Feynman integrals expressible in terms of CY-period iterated integrals and lays groundwork for future proofs via motivic or $\widehat{ ext{Γ}}$-class formalisms.

Abstract

We present for the first time fully analytic results for multi-loop equal-mass ice cone graphs in two dimensions. By analysing the leading singularities of these integrals, we find that the maximal cuts in two dimensions can be organised into two copies of the same periods that describe the Calabi-Yau varieties for the equal-mass banana integrals. We obtain a conjectural basis of master integrals at an arbitrary number of loops, and we solve the system of differential equations satisfied by the master integrals in terms of the same class of iterated integrals that have appeared earlier in the context of equal-mass banana integrals. We then go on and show that, when expressed in terms of the canonical coordinate on the moduli space, our results can naturally be written as iterated integrals involving the geometrical invariants of the Calabi-Yau varieties. Our results indicate how the concept of pure functions and transcendental weight can be extended to the case of Calabi-Yau varieties. Finally, we also obtain a novel representation of the periods of the Calabi-Yau varieties in terms of the same class of iterated integrals, and we show that the well-known quadratic relations among the periods reduce to simple shuffle relations among these iterated integrals.

Figures (3)

• Figure 1: The $l$-loop equal-mass ice cone graph build up from a $(l-1)$-loop banana graph and a one-loop triangle.
• Figure 2: Sub-topologies of the ice cone graph with corresponding master integrals $I_{0,0}$ (left), $I_{0,1,},\hdots, I_{0,\lfloor l/2 \rfloor}$ (middle) and $I_0$ (right).
• Figure 3: Global analytic structure of the master integral $\mathcal{I}^{+}_{l,1}$ compared to the singularities of the differential operator $\mathcal{L}_{\text{ice},l}$. $\mathcal{I}^{+}_{l,1}$ has a branch cut from $x=0$ to $x=-\infty$.