Quantum field theory over F_q

TL;DR

This paper explores the point counts of graph hypersurfaces over finite fields, identifying minimal graphs with non-polynomial counts and analyzing their properties, while also connecting these findings to quantum field theory over finite fields.

Contribution

It identifies the smallest graphs with non-polynomial point counts over finite fields and demonstrates how Feynman rules over _q can cause the perturbation series to terminate.

Findings

Smallest graphs with non-polynomial N(q) have 14 edges.

Examples depend on prime 2 and cube roots of unity.

Perturbation series terminate over _q for certain quantum field theories.

Abstract

We consider the number \bar N(q) of points in the projective complement of graph hypersurfaces over \F_q and show that the smallest graphs with non-polynomial \bar N(q) have 14 edges. We give six examples which fall into two classes. One class has an exceptional prime 2 whereas in the other class \bar N(q) depends on the number of cube roots of unity in \F_q. At graphs with 16 edges we find examples where \bar N(q) is given by a polynomial in q plus q^2 times the number of points in the projective complement of a singular K3 in \P^3. In the second part of the paper we show that applying momentum space Feynman-rules over \F_q lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories.