arXiv subjects

Thomas Nevins

Publications and source records attributed to Thomas Nevins.

20 records · Page 2Linked to original sources

arXiv2003

From solitons to many-body systems

We present a bridge between the KP soliton equations and the Calogero-Moser many-body systems through noncommutative algebraic geometry. The Calogero-Moser systems have a natural geometric interpretation as flows on spaces of spectral curves on a ruled surface. We explain how the meromorphic solutions of the KP hierarchy have an interpretation via a noncommutative ruled surface. Namely, we identify KP Lax operators with vector bundles on quantized cotangent spaces (formulated technically in terms of D-modules). A geometric duality (a variant of the Fourier-Mukai transform) then identifies the parameter space for such vector bundles with that for the spectral curves and sends the KP flows to the Calogero-Moser flows. It follows that the motion and collisions of the poles of the rational, trigonometric, and elliptic solutions of the KP hierarchy, as well as of its multicomponent analogs, are governed by the (spin) Calogero-Moser systems on cuspidal, nodal, and smooth genus one curves. This provides a geometric explanation and generalizations of results of Airault-McKean-Moser, Krichever, and Wilson.

math.AG↗

arXiv2003

Cusps and $\D$-Modules

We study interactions between the categories of $\D$-modules on smooth and singular varieties. For a large class of singular varieties $Y$, we use an extension of the Grothendieck--Sato formula to show that $\D_Y$-modules are equivalent to stratifications on $Y$, and as a consequence are unaffected by a class of homeomorphisms, the {\em cuspidal quotients}. In particular, when $Y$ has a smooth bijective normalization $X$, we obtain a Morita equivalence of $\D_Y$ and $\D_X$ and a Kashiwara theorem for $\D_Y$, thereby solving conjectures of Hart-Smith and Berest-Etingof-Ginzburg (generalizing results for complex curves and surfaces and rational Cherednik algebras). We also use this equivalence to enlarge the category of induced $\D$-modules on a smooth variety $X$ by collecting induced $\D_X$-modules on varying cuspidal quotients. The resulting {\em cusp-induced} $\D_X$-modules possess both the good properties of induced $\D$-modules (in particular, a Riemann-Hilbert description) and, when $X$ is a curve, a simple characterization as the generically torsion-free $\D_X$-modules.

math.AG↗