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Kontsevich graphs and their weights in deformation quantization of Poisson structures

Buring, R.T. (2017) Kontsevich graphs and their weights in deformation quantization of Poisson structures. Master's Thesis / Essay, Mathematics.

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To show the existence of deformation quantizations for arbitrary Poisson structures, M. Kontsevich gave in 1997 an explicit universal formula: a formal power series in the deformation parameter ℏ with a sum of weighted graphs (wherein a Poisson structure can be implanted) at each order in ℏ. We outline a systematic and graphical approach, implemented in software, to the problem of expanding the power series for this ⋆-product, particularly the problem of finding the universal coefficients (weights of graphs) in terms of as few parameters as possible, and the problem of pictorially proving the associativity of ⋆ up to a given order in ℏ. We obtain the expansion of the star-product up to the order 4 in ℏ in terms of 10 parameters (6 parameters modulo gauge-equivalence) and we verify that the star-product expansion is associative modulo ō(ℏ⁴) for every value of the 10 parameters. Jointly with A. Bouisaghouane and A.V. Kiselev, we confirm at the infinitesimal level the existence of a universal flow on the space of Poisson structures.

Item Type: Thesis (Master's Thesis / Essay)
Degree programme: Mathematics
Thesis type: Master's Thesis / Essay
Language: English
Date Deposited: 15 Feb 2018 08:26
Last Modified: 15 Feb 2018 08:26

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