Elliptic hyperlogarithms

Abstract

Abstract Let script upper E ${\mathcal {E}}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi mathvariant="script">E</mml:mi> </mml:math> be a complex elliptic curve and S be a non-empty finite subset of script upper E ${\mathcal {E}}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi mathvariant="script">E</mml:mi> </mml:math> . We show that the functions upper Gamma overtilde $\tilde {\Gamma }$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mover accent="true"> <mml:mi>Γ</mml:mi> <mml:mo accent="true">~</mml:mo> </mml:mover> </mml:math> introduced in [BDDT] out of string theory motivations give rise to a basis (as a vector space) of the minimal algebra upper A Subscript script upper E minus upper S $A_{{\mathcal {E}}{\smallsetminus } S}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi mathvariant="script">E</mml:mi> <mml:mo>∖</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:msub> </mml:math> of holomorphic multivalued functions on script upper E minus upper S ${\mathcal {E}}{\smallsetminus } S$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mi mathvariant="script">E</mml:mi> <mml:mo>∖</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:math> which is stable under integration, introduced in [EZ]; this basis is alternative to the basis of upper A Subscript script upper E minus upper S $A_{{\mathcal {E}}{\smallsetminus } S}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi mathvariant="script">E</mml:mi> <mml:mo>∖</mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:msub> </mml:math> constructed in loc. cit. using elliptic analogs of the hyperlogarithm functions.

Topics & Concepts