GDAHA representations with Teichmüller, Stokes, and Painlevé

Dal Martello, Davide ORCID: https://orcid.org/0000-0002-4975-8774 (2024). GDAHA representations with Teichmüller, Stokes, and Painlevé. University of Birmingham. Ph.D.

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Abstract

The Painlevé VI equation governs the isomonodromic deformation problem of both 2-dimensional Fuchsian and 3-dimensional irregular types of linear systems of ODEs. Through Harnad duality, this feature turns into a map between the two systems, which translates to monodromy as a middle convolution operation. This thesis studies the quantum algebraic manifestation of the systems’ monodromy data by introducing a noncommutative analogue of the middle convolution functor. The Fuchsian data are known to quantize as the \(CC^{v}_1\) DAHA; we construct a quantization of the irregular ones that match the \(\tilde{E}_6\)-type GDAHA, provided a specialization of the algebra parameters. Both quantum data are then shown to exhibit an alternative realization in higher Teichmüller terms. In particular, this framework advances the GDAHA representation theory by providing the first explicit representation of the universal GDAHA of type \(\tilde{E}_6\), which can be reduced to the quantum irregular monodromy data by a new quiver-theoretical operation.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):