Quantum field theory over F_q

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 ¶^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.

26 pages, 1 figure

References in corpus (3)

• On the periods of some Feynman integrals
• A K3 in phi4
• On one example and one counterexample in counting rational points on graph hypersurfaces

Cited by in corpus (13)

• Feynman graph polynomials
• Numbers and Functions in Quantum Field Theory
• On the periods of some Feynman integrals
• A K3 in phi4
• The Galois coaction on ${\varphi }^4$ periods
• On one example and one counterexample in counting rational points on graph hypersurfaces
• Matrices with restricted entries and q-analogues of permutations
• Quantum field theory over F1
• A special case of completion invariance for the $c_2$ invariant of a graph
• Further investigations into the graph theory of ${\varphi }^4$-periods and the $c_2$ invariant
• Forbidden minors for graphs with no first obstruction to parametric Feynman integration
• Feynman graphs in perturbative quantum field theory
• Potts models with magnetic field: arithmetic, geometry, and computation