Quantum fields, periods and algebraic geometry

Abstract

We discuss how basic notions of graph theory and associated graph polynomials define questions for algebraic geometry, with an emphasis given to an analysis of the structure of Feynman rules as determined by those graph polynomials as well as algebraic structures of graphs. In particular, we discuss the appearance of renormalization scheme independent periods in quantum field theory. Graphs and algebras2.1. Wheels in wheels. It is the purpose of this section to completely analyse an example. We choose wheels with three or four spokes, inserted at most once into each other. Results for them are available by methods which were recently developed [6,7,8,20] and which are presented elsewhere [18,19].

Filtrations in Dyson–Schwinger equations: Next-to{j}-leading log expansions systematically

“…and [x i , x j ] ∈ L. Thus, Ψ R (a i a j − a j a i )) ∝ L 1 although Ψ R (a i a j ) also contains terms ∝ L 2 [1]. Finally, we filter the word a i a j to…”

“…Throughout, we assume we leave those scattering angles unchanged for the renormalization point. A discussion of this point can be found in [1,2]. Therefore, any renormalized Green function G R can be written as a triangular expansion…”

“…M(u) = ∞ i=1 q M i;j u i is a series in u = αL/2 [101] with q M i;j ∈ Q. These series are determined from their counterparts in the universal enveloping algebra of Feynman graphs that we detail below, see also [1].…”

“…• [a 1 ,Θ(a 1 ,a 1 )] (u) = − x 3 8 + x 5 8 − x 3 8 log x + 3x 5 8 log x • [a 1 ,Θ(a 1 ,a 1 )] (u) = 0 TABLE I: The results obtained in this paper: a renormalized Green function GR is given by the log-expansion in Eq. (1). We calculate GR up to next-to-next-to-leading log order (j ≤ 2).…”

What can we learn from Knizhnik-Zamolodchikov Equations?

“…The sum over Feynman graphs which appears as a solution of a combinatorial DSE then dualizes to a series in the dual universal enveloping algebra U (L ) for a Lie algebra L . Terms of highest order in the leading log expansion -elements of maximal coradical degreecorrespond to highest symmetric powers in U (L ), while the linear terms (in ln S/S 0 ) are dual to elements of L ⊂ U (L ), and can be filtered themselves according to the lower central series filtration of L , such that angle dependence is relegated to commutators, as in the example below (see also [4] for a review of these properties of field theory).…”