Supermanifolds from Feynman graphs

TL;DR

This paper extends Feynman integral computations to graphs with fermionic and bosonic edges, representing them as periods on supermanifolds and introducing a Grothendieck group for supermanifolds.

Contribution

It introduces a novel framework for analyzing Feynman graphs with mixed edges using supermanifolds and defines a Grothendieck group to classify these structures.

Findings

Feynman integrals expressed as periods on supermanifolds

Definition of a supergraph hypersurface via the Berezinian

Construction of a Grothendieck group for supermanifolds

Abstract

We generalize the computation of Feynman integrals of log divergent graphs in terms of the Kirchhoff polynomial to the case of graphs with both fermionic and bosonic edges, to which we assign a set of ordinary and Grassmann variables. This procedure gives a computation of the Feynman integrals in terms of a period on a supermanifold, for graphs admitting a basis of the first homology satisfying a condition generalizing the log divergence in this context. The analog in this setting of the graph hypersurfaces is a graph supermanifold given by the divisor of zeros and poles of the Berezinian of a matrix associated to the graph, inside a superprojective space. We introduce a Grothendieck group for supermanifolds and we identify the subgroup generated by the graph supermanifolds. This can be seen as a general procedure to construct interesting classes of supermanifolds with associated periods.