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We introduce a quantization of the graded algebra of functions on the canonical cone of
an algebraic curve , based on the theory of formal pseudodifferential operators. When
is a complex curve with Poincaré uniformization, we propose another, equivalent
construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a
presentation of the quantum algebra when is a rational curve, and discuss the problem
of constructing algebraically “differential liftings”.
We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity of type our construction of the stack of -curves is canonically isomorphic to the stack of -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the...
A lattice model with exponential interaction, was proposed and integrated by M. Toda in the 1960s; it was then extensively studied as one of the completely integrable (differential-difference) equations by algebro-geometric methods, which produced both quasi-periodic solutions in terms of theta functions of hyperelliptic curves and periodic solutions defined on suitable Jacobians by the Lax-pair method. In this work, we revisit Toda’s original approach to give solutions of the Toda lattice in terms...
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