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

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.

References in corpus (76)

• Harmonic Polylogarithms
• Multiloop integrals in dimensional regularization made simple
• Differential Equations for Two-Loop Four-Point Functions
• Classical Polylogarithms for Amplitudes and Wilson Loops
• Connection between Feynman integrals having different values of the space-time dimension
• Numerical evaluation of multiple polylogarithms
• Kira - A Feynman Integral Reduction Program
• Local Integrals for Planar Scattering Amplitudes
• Reduze - Feynman Integral Reduction in C++
• Integral Reduction with Kira 2.0 and Finite Field Methods
• From polygons and symbols to polylogarithmic functions
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• PolyLogTools - Polylogs for the masses
• Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
• Calabi-Yau fourfolds for M- and F-Theory compactifications
• The elliptic dilogarithm for the sunset graph
• Topological Field Theory and Rational Curves
• Analytic treatment of the two loop equal mass sunrise graph
• Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials
• Towards a Basis for Planar Two-Loop Integrals
• Space-time dimensionality D as complex variable: calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D
• Uniqueness of two-loop master contours
• Explicit solutions of the multi-loop integral recurrence relations and its application
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• The massless higher-loop two-point function
• Feynman Integrals and Intersection Theory
• Analytical and numerical methods for massive two-loop self-energy diagrams
• Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms
• Explicit solutions of the 3--loop vacuum integral recurrence relations
• Mirror Manifolds in Higher Dimension
• On the maximal cut of Feynman integrals and the solution of their differential equations
• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• Elliptic integral evaluations of Bessel moments
• A Feynman integral via higher normal functions
• Cuts of Feynman Integrals in Baikov representation
• Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph
• Elliptic Feynman integrals and pure functions
• The kite integral to all orders in terms of elliptic polylogarithms
• Mirror Symmetry, N=1 Superpotentials and Tensionless Strings on Calabi-Yau Four-Folds
• A K3 in phi4
• Traintracks Through Calabi-Yaus: Amplitudes Beyond Elliptic Polylogarithms
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• The two-loop sunrise graph with arbitrary masses
• Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph
• Simple one-dimensional integral representations for two-loop self-energies: the master diagram
• Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• Enumerative geometry of Calabi-Yau 4-folds
• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• Analytic Structure of all Loop Banana Amplitudes
• An analytic solution for the equal-mass banana graph
• HandyG -- rapid numerical evaluation of generalised polylogarithms in Fortran
• Two-loop sunset diagrams with three massive lines
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• Duals of Feynman Integrals, I: Differential Equations
• The Two Loop Crossed Ladder Vertex Diagram with Two Massive Exchanges
• Maximal Cuts in Arbitrary Dimension
• Baikov-Lee Representations Of Cut Feynman Integrals
• On the reduction of generalized polylogarithms to ${\text{Li}}_n$ and ${\text{Li}}_{2,2}$ and on the evaluation thereof
• Taming Calabi-Yau Feynman integrals: The four-loop equal-mass banana integral
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals
• The three-loop equal-mass banana integral in $\epsilon$-factorised form with meromorphic modular forms
• Numerical Implementation of Harmonic Polylogarithms to Weight w = 8
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Traintrack Calabi-Yaus from Twistor Geometry
• Prescriptive Unitarity with Elliptic Leading Singularities
• An Elliptic Yangian-Invariant, `Leading Singularity'
• WKB Method and Quantum Periods beyond Genus One
• Bananas: multi-edge graphs and their Feynman integrals
• Integral representation for three-loop banana graph
• The Stratification of Rigidity

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