Abstract:
We revisit the Hitchin integrable system [11, 21] whose phase space is the bundle cotangent to the moduli space of holomorphic SL 2-bundles over a smooth complex curve of genus 2. As shown in [18], may be identified with the 3-dimensional projective space of theta functions of the 2nd order, i.e. . We prove that the Hitchin system on possesses a remarkable symmetry: it is invariant under the interchange of positions and momenta. This property allows to complete the work of van Geemen–Previato [21] which, basing on the classical results on geometry of the Kummer quartic surfaces, specified the explicit form of the Hamiltonians of the Hitchin system. The resulting integrable system resembles the classic Neumann systems which are also self-dual. Its quantization produces a commuting family of differential operators of the 2nd order acting on homogeneous polynomials in four complex variables. As recently shown by van Geemen–deJong [11], these operators realize the Knizhnik–Zamolodchikov–Bernard–Hitchin connection for group SU(2) and genus 2 curves.
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Received: 24 October 1997 / Accepted: 21 January 1998
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Gawędzki, K., Tran-Ngoc-Bich, P. Self-Duality of the SL 2 Hitchin Integrable System at Genus 2 . Comm Math Phys 196, 641–670 (1998). https://doi.org/10.1007/s002200050438
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DOI: https://doi.org/10.1007/s002200050438