PIRSA:19050001 Scientific Series Mathematical physics

DOI10.48660/19050001

Series: Mathematical Physics

Pym, B. (2019). Multiple zeta values in deformation quantization. Perimeter Institute. https://pirsa.org/19050001

Pym, Brent. Multiple zeta values in deformation quantization. Perimeter Institute, May. 06, 2019, https://pirsa.org/19050001 

@misc{ pirsa_PIRSA:19050001,
  doi = {10.48660/19050001},
  url = {https://pirsa.org/19050001},
  author = {Pym, Brent},
  keywords = {Mathematical physics},
  language = {en},
  title = {Multiple zeta values in deformation quantization},
  publisher = {Perimeter Institute},
  year = {2019},
  month = {may},
  note = {PIRSA:19050001 see, \url{https://pirsa.org}}
}

Abstract

In 1997, Kontsevich gave a universal solution to the "deformation quantization" problem in mathematical physics: starting from any Poisson manifold (the classical phase space), it produces a  noncommutative algebra of quantum observables by deforming the ordinary multiplication of functions.  His formula is a Feynma  expansion, involving an infinite sum over graphs, weighted by volume integrals on the moduli space of marked holomorphic disks. The precise values of these integrals are currently unknown.  I will describe recent joint work with Banks and Panzer, in which we develop a theory of integration on these moduli spaces via suitable sheaves of polylogarithms, and use it to prove that Kontsevich's integrals evaluate to integer-linear combinations of special transcendental constants called multiple zeta values, yielding the first algorithm for their calculation.

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