Transcendental fiber functors, shift of argument algebras and Riemann-Hilbert correspondence for q-difference equations

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Quantum groups are deformations of the most basic symmetries of Nature. They were discovered during the 1980s in the study of one- and two-dimensional statistical mechanical models describing thin layers of ice. Amazingly, quantum groups have recently been shown to arise as the symmetries of 4-dimensional gauge theories, which describe the interaction of elementary particles such as quarks. Differential equations are another basic paradigm in science, and describe the evolution of physical, chemical, biological and economic systems. One of their striking aspects is that they can exhibit Stokes phenomena: their solutions are not entirely captured by the recursive, and often programmable methods used to solve them. The missing information, or Stokes data, can be considered as a hidden symmetry of the differential equation, as they relate different solutions possessing the same formal expansions. This project stems from the recent discovery that quantum groups naturally arise from the Stokes data of differential equations associated to classical symmetries. The main goals are to further explore this bridge between classical and quantum symmetries. Of particular interest is the extension to difference equations, which are natural discretisations of differential equations, and whose Stokes data are not well-understood beyond the one-variable case. Another important direction will the study of the integrable systems, or constants of motion, corresponding to these differential and difference equations. The project will provide research training opportunities for graduate students.In more detail, the project stems from transcendental construction of quantum groups from the Stokes data of the dynamical Knizhnik-Zamolodchikov equations for the corresponding Lie algebra due to the PI. The first component will extending the construction to numerical values of the deformation parameter, in particular to roots of unity, and to the difference setting. The second component will establish a Riemann-Hilbert correspondence for q-difference equations in several variables by defining an appropriate notion of regular singularities and capturing these by elliptic monodromy data, similar to the one-variable case treated by Birkhoff. The third component is concerned with the integrable systems arising from the Casimir connection, and their parametrisation in terms of sheets of the corresponding Lie algebra. The results of the project will have application in the study of Stokes phenomena, quantum integrable systems and geometric representation theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.