Abstracts for Workshop on "Combinatorics, Resurgence and Algebraic Geometry in Quantum Field Theory"

K3 periods of banana integrals depending on 2 different masses.

Abstract: Maximal cuts of 3-loop banana integrals compute the periods of some 4-parameter family of K3 surfaces, and the moduli are the values of the masses of the particles.

It is well known that in the case that all four masses are equal, the Picard-Fuchs operator describing the periods is a symmetric square, and the periods can be written as products of modular forms.

M. Aguiar and F. Sottile proved nice formulas for the coproduct of the Malvenuto-Reutenauer Hopf algebra, resp. for the coproduct of the Hopf algebra of planar binary rooted trees, in terms of the Möbius function of the weak Bruhat order, resp. of the Tamari order.  The free magmatic algebra $Mag(\vert)$ on one element $\vert$ is the vector space spanned by the set of planar binary rooted trees equipped with the grafting of trees. We prove that the Möbius formula for the Tamari order may be described recursively in terms of the grafting.

The perturbative expansion of quantum field theory associates numbers (or functions) to combinatorial graphs. These Feynman integrals are often transcendental and hard to evaluate. I will review various combinatorial invariants of graphs that behave similar to these integrals. In particular, I will explain a relation between spanning tree partitions and circuit partitions. It allows for efficient counting of these partitions, producing for every graph an integer sequence that determines the Feynman integral, conjecturally through an Apery-like limit.

We introduce and study a category of algebras, which, on the one hand, contains such important categories as those of BV-infinity (homotopy Batalin-Vilkovisky) algebras, IBL-infinity (homotopy involutive Lie bi-) algebras and L-infinity (homotopy Lie) algebras, and on the other hand, is homotopically trivial, in particular allowing for a simple solution of the quantum master equation.