[Paper Review] The ice cone family and iterated integrals for Calabi-Yau varieties

Claude Duhr, Albrecht Klemm+2 more|arXiv (Cornell University)|Dec 19, 2022

Geometry and complex manifolds4 citations

TL;DR

This paper presents the first fully analytic results for multi-loop equal-mass ice cone graphs in two dimensions, showing that their maximal cuts organize into periods of Calabi-Yau varieties identical to those of equal-mass banana integrals. The authors derive a conjectural basis of master integrals and solve their differential equations using iterated integrals tied to the moduli space geometry, extending the concepts of pure functions and transcendental weight to Calabi-Yau varieties and reducing period relations to shuffle identities.

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.

Motivation & Objective

• To provide fully analytic results for multi-loop equal-mass ice cone graphs in two dimensions.
• To identify the structure of maximal cuts and relate them to periods of Calabi-Yau varieties.
• To extend the framework of pure functions and transcendental weight to Calabi-Yau geometries via iterated integrals.
• To derive a conjectural basis of master integrals valid at arbitrary loop order.
• To show that period relations among Calabi-Yau periods reduce to shuffle relations among iterated integrals.

Proposed method

• Analyzing leading singularities of ice cone integrals to identify maximal cut structures.
• Mapping maximal cuts to periods of Calabi-Yau varieties isomorphic to those of equal-mass banana integrals.
• Deriving a system of differential equations for master integrals and solving them using iterated integrals of modular forms.
• Expressing results in terms of canonical coordinates on the moduli space of Calabi-Yau varieties.
• Utilizing the shuffle algebra of iterated integrals to prove quadratic period relations.
• Establishing a novel integral representation of Calabi-Yau periods using the same class of iterated integrals.

Experimental results

Research questions

• RQ1 Can multi-loop equal-mass ice cone integrals in two dimensions be evaluated analytically beyond one-loop approximations?
• RQ2 Do the maximal cuts of ice cone graphs organize into the same periods as those of equal-mass banana integrals?
• RQ3 Can the concept of transcendental weight and purity be generalized to Calabi-Yau varieties via iterated integrals?
• RQ4 How do the periods of Calabi-Yau varieties relate to the structure of Feynman integrals in this class?
• RQ5 Do quadratic relations among Calabi-Yau periods reduce to algebraic identities among iterated integrals?

Key findings

• The maximal cuts of ice cone integrals in two dimensions are isomorphic to the periods of Calabi-Yau varieties associated with equal-mass banana integrals.
• A conjectural basis of master integrals is proposed that applies at arbitrary loop order.
• The system of differential equations for the master integrals is solved using iterated integrals of modular forms associated with the moduli space of the Calabi-Yau geometry.
• When expressed in canonical coordinates on the moduli space, the results naturally decompose into iterated integrals involving geometric invariants of the Calabi-Yau variety.
• The well-known quadratic relations among Calabi-Yau periods are shown to reduce to simple shuffle relations among the iterated integrals.
• A new representation of the periods of Calabi-Yau varieties is derived, expressed directly in terms of the same class of iterated integrals used to solve the Feynman integrals.