Single-valued multiple polylogarithms and a proof of the zig-zag conjecture. (English) Zbl 1395.11115
Summary: A long-standing conjecture in quantum field theory due to D. J. Broadhurst and D. Kreimer [Int. J. Mod. Phys. C 6, No. 4, 519–524 (1995; Zbl 0940.81520)] states that the periods of the zig-zag graphs are a certain explicit rational multiple of the odd values of the Riemann zeta function. In this paper we prove this conjecture by constructing a certain family of single-valued multiple polylogarithms which correspond to multiple zeta values \(\zeta(2, \ldots, 2, 3, 2, \ldots 2)\) and using the method of graphical functions. The zig-zag graphs are the only infinite family of primitive graphs in \(\phi_4^4\) theory (in fact, in any renormalisable quantum field theory in four dimensions) whose periods are now known.
MSC:
| 11M32 | Multiple Dirichlet series and zeta functions and multizeta values |
| 81T18 | Feynman diagrams |
| 11G55 | Polylogarithms and relations with \(K\)-theory |
Citations:
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