[Paper Review] A double integral of dlog forms which is not polylogarithmic

Francis H. Brown, Claude Duhr|arXiv (Cornell University)|Jun 16, 2020

Black Holes and Theoretical Physics6 citations

TL;DR

This paper presents a double iterated integral of dlog forms that evaluates to a period of a cusp form, demonstrating it is not expressible in terms of multiple polylogarithms. Using a geometric configuration of a modular elliptic curve and lines intersecting at torsion points in ℙ², the authors show the motivic version of the integral is algebraically independent from all multiple polylogarithms at algebraic arguments, challenging the folklore belief that such integrals are always polylogarithmic.

ABSTRACT

Feynman integrals are central to all calculations in perturbative Quantum Field Theory. They often give rise to iterated integrals of dlog-forms with algebraic arguments, which in many cases can be evaluated in terms of multiple polylogarithms. This has led to certain folklore beliefs in the community stating that all such integrals evaluate to polylogarithms. Here we discuss a concrete example of a double iterated integral of two dlog-forms that evaluates to a period of a cusp form. The motivic versions of these integrals are shown to be algebraically independent from all multiple polylogarithms evaluated at algebraic arguments. From a mathematical perspective, we study a mixed elliptic Hodge structure arising from a simple geometric configuration in $\mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines which meet it at torsion points, which may provide an interesting worked example from the point of view of periods, extensions of motives, and L-functions.

Motivation & Objective

• To challenge the widespread belief in quantum field theory that all iterated integrals of dlog forms evaluate to multiple polylogarithms.
• To construct a concrete example of a double iterated integral of dlog forms that is not polylogarithmic.
• To demonstrate that the motivic version of this integral is algebraically independent from all multiple polylogarithms evaluated at algebraic arguments.
• To study the mixed elliptic Hodge structure arising from a configuration of a modular elliptic curve and lines meeting it at torsion points in ℙ².
• To provide a worked example for periods, motives, and L-functions in the context of Feynman integrals.

Proposed method

• Construct a geometric setup in ℙ² consisting of a modular elliptic curve E_k and 12 lines intersecting it at torsion points.
• Define a double iterated integral of dlog forms over a path in the complement of the lines and the curve.
• Use the modular parametrization φ: Γ(6)\ℋ → E\C to pull back the differential forms to modular forms on the upper half-plane.
• Express the pullback of each dlog form as a linear combination of Eisenstein series of weight two and the cusp form f(τ) of weight two for Γ(6).
• Compute the iterated integral via integration of modular forms, showing it yields a period of the cusp form f(τ).
• Establish algebraic independence of the motivic version of the integral from all multiple polylogarithms using motivic Galois theory and transcendence results.

Experimental results

Research questions

• RQ1 Can a double iterated integral of dlog forms be constructed that is not expressible in terms of multiple polylogarithms?
• RQ2 Is the motivic version of such an integral algebraically independent from all multiple polylogarithms evaluated at algebraic arguments?
• RQ3 What is the geometric and arithmetic structure underlying a Feynman integral that yields a period of a cusp form rather than a polylogarithm?
• RQ4 How do mixed elliptic Hodge structures arise from configurations of elliptic curves and lines intersecting at torsion points in ℙ²?
• RQ5 What is the role of modular parametrization in transforming dlog forms into modular forms for integration?

Key findings

• The double iterated integral of two dlog forms evaluates to a period of a cusp form of weight two for Γ(6), specifically the modular form f(τ).
• The motivic version of the integral is algebraically independent from all multiple polylogarithms evaluated at algebraic arguments, proving it lies outside the polylogarithmic framework.
• The pullback of each dlog form under the modular parametrization φ is expressed as a linear combination of Eisenstein series of weight two and the cusp form f(τ), with explicit q-series coefficients.
• The holomorphic differential −3dx/y pulls back to the normalized cusp form f(τ), confirming the modular nature of the period.
• The logarithmic differentials pull back to combinations involving Eisenstein series and f(τ), with the coefficient of f(τ) involving π², indicating transcendental contributions beyond polylogarithmic structure.
• The entire construction yields a non-polylogarithmic period, providing a counterexample to the folklore belief that all such integrals are polylogarithmic.