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Asymptotic representation theory and applications to Yang-Mills theory

Abstract : This thesis is devoted to the study of Yang–Mills measure on a compact surface, with structure group the unitary group U(N) or special unitary group SU(N). This study is more precisely about this measure’s asymptotic behaviour in the largeN limit, using asymptotic representations of the unitary group. The first chapter, as an introduction to the subject, explains in details the construction of Yang–Mills measure after having developed the several theories it is based on: gauge theory, noncommutativeprobability and group representations. We show in Chapter 2 that the partition function of this Yang–Mills measure converges, for orientable surfaces of genus greater or equal to 1 and non-orientable surfaces of genus greater or equal to 2, to a finite limit that only depends on the genus, the orientability and the area of the underlying surface. In Chapter 3 we partially construct the so-called master field on orientable compact surfaces of genus greater or equal to 1, which is the limit – in a non-commutative probabilistic sense – to the random field on the underlying surface whose distribution is given by Yang–Mills measure.
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Submitted on : Friday, November 19, 2021 - 2:34:12 PM
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  • HAL Id : tel-03096870, version 2

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Thibaut Lemoine. Asymptotic representation theory and applications to Yang-Mills theory. Representation Theory [math.RT]. Sorbonne Université, 2020. English. ⟨NNT : 2020SORUS343⟩. ⟨tel-03096870v2⟩

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