David H. Wohl | Searcharxiv

arXiv2002

Analysis of Zeta Functions, Multiple Zeta Values, and Related Integrals

In this work, we begin to uncover the architecture of the general family of zeta functions and multiple zeta values as they appear in the theory of integrable systems and conformal field theory. One of the key steps in this process is to recognize the roles that zeta functions play in various arenas using transform methods. Other logical connections are provided by the the appearance of the Drinfeld associator, Hopf algebras, and techniques of conformal field theory and braid groups. These recurring themes are subtly linked in a vast scheme of a logically woven tapestry. An immediate application of this framework is to provide an answer to a question of Kontsevich regarding the appearance of Drinfeld type integrals and in particular, multiple zeta values in: a) Drinfeld's work on the KZ equation and the associator; b) Etingof-Kazhdan's quantization of Poisson-Lie algebras; c) Tamarkin's proof of formality theorems; d) Kontsevich's quantization of Poisson manifolds. Combinatorial arguments relating Feynman diagrams to Selberg integrals, multiple zeta values, and finally Poisson manifolds provide an additional step in this framework. Along the way, we provide additional insight into the various papers and theorems mentioned above. This paper represents an overall introduction to work currently in progress. More details to follow. See our paper Math.QA/[ ] for a proof of the Connes Kreimer Conjecture.

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arXiv2002

Selberg Integrals, Multiple Zeta Values and Feynman Diagrams

We prove that there is an isomorphism between the Hopf Algebra of Feynman diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring H in C< > . In other words, Feynman diagrams evaluate to Multiple Zeta Values in all cases. This proves a recent conjecture of Connes-Kreimer, and others including Broadhurst and Kontsevich. The key step of our theorem is to present the Selberg integral as discussed in Terasoma [22] as a Functional from the Rooted Trees Operad to the Hopf algebra of Multiple Zeta Values. This is a new construction which provides illumination to the relations between zeta values, associators, Feynman diagrams and moduli spaces. An immediate implication of our Main Theorem is that by applying Terasoma's result and using the construction of our Selberg integral-rooted trees functional, we prove that the Hermitian matrix integral as discussed in Mulase [18] evaluates to a Multiple Zeta Value in all 3 cases: Asymptotically, the Limit as N goes to infinity, and in general. Furthermore, this construction provides for a positive resolution to Goncharov's conjecture (see [7] pg. 30). The Selberg integral functional can be extended to map the special values to depth m multiple polylogarithms on X.

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