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Quantization of canonical cones of algebraic curves

Benjamin Enriquez, Alexander Odesskii (2002)

Annales de l’institut Fourier

We introduce a quantization of the graded algebra of functions on the canonical cone of an algebraic curve C , based on the theory of formal pseudodifferential operators. When C 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 C is a rational curve, and discuss the problem of constructing algebraically “differential liftings”.

Quantum Singularity Theory for A ( r - 1 ) and r -Spin Theory

Huijun Fan, Tyler Jarvis, Yongbin Ruan (2011)

Annales de l’institut Fourier

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the r -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity W of type A our construction of the stack of W -curves is canonically isomorphic to the stack of r -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an r -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the...

Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

Yuji Kodama, Shigeki Matsutani, Emma Previato (2013)

Annales de l’institut Fourier

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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