tailieunhanh - Báo cáo toán học: "Quantum Field Theory over Fq"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Quantum Field Theory over Fq. | Quantum Field Theory over Fq Oliver Schnetz Department Mathematik Bismarkstrafie 12 91054 Erlangen Germany schnetz@ Submitted Sep 4 2009 Accepted Apr 23 2011 Published May 8 2011 Mathematics Subject Classification 05C31 Abstract We consider the number N q of points in the projective complement of graph hypersurfaces over Fq and show that the smallest graphs with non-polynomial 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 N q depends on the number of cube roots of unity in Fq. At graphs with 16 edges we find examples where N q is given by a polynomial in q plus q2 times the number of points in the projective complement of a singular K3 in P3. In the second part of the paper we show that applying momentum space Feyn-man-rules over Fq lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories. Contents 1 Introduction 2 2 Kontsevich s Conjecture 3 Fundamental Definitions and Identities. 3 Methods. 12 Results. 14 3 Outlook Quantum Fields over Fq 20 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P102 1 1 Introduction Inspired by the appearance of multiple zeta values in quantum field theories 4 17 Kontsevich informally conjectured in 1997 that for every graph the number of zeros of the graph polynomial see Sect. for a definition over a finite field Fq is a polynomial in q 16 . This conjecture puzzled graph theorists for quite a while. In 1998 Stanley proved that a dual version of the conjecture holds for complete as well as for nearly complete graphs 18 . The result was extended in 2000 by Chung and Yang 8 . On the other hand in 1998 Stembridge verified the conjecture by the Maple-implementation of a reduction algorithm for all graphs with at most 12 edges 19 . However in 2000 Belkale and Brosnan were able to disprove the conjecture in fact the conjecture is maximally false in a certain sense 2 . .