Buring, R.T. (2017) Kontsevich graphs and their weights in deformation quantization of Poisson structures. Master's Thesis / Essay, Mathematics.
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Abstract
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) |
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| 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 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/14909 |
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