In this paper some recent topological applications of Riemann surface theory and especially of their associated theta functions (in different geometric incarnations) are surveyed, taking the circle of ideas around geometric quantization as a vantage point. They include classical and quantum monodromy of 2d-integrable systems and the construction of unitary Riemann surface braid group representations (aimed, in particular, at devising a mathematical interpretation of the Laughlin wave functions emerging in condensed matter physics). The noncommutative version of theta functions due to A. Schwarz is briefly discussed, showing in particular its efficacy in Fourier-Mukai-Nahm computations.
Spera, M., Some Topological Applications of Theta Functions, in Donagi, R., Shaska, T. (ed.), Integrable Systems and Algebraic Geometry (vol. 2), Cambridge University Press, Cambridge 2020: <<LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES>>, 459 440- 484 [http://hdl.handle.net/10807/149955]
Some Topological Applications of Theta Functions
Spera, Mauro
2020
Abstract
In this paper some recent topological applications of Riemann surface theory and especially of their associated theta functions (in different geometric incarnations) are surveyed, taking the circle of ideas around geometric quantization as a vantage point. They include classical and quantum monodromy of 2d-integrable systems and the construction of unitary Riemann surface braid group representations (aimed, in particular, at devising a mathematical interpretation of the Laughlin wave functions emerging in condensed matter physics). The noncommutative version of theta functions due to A. Schwarz is briefly discussed, showing in particular its efficacy in Fourier-Mukai-Nahm computations.
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