Scattering Diagrams, Sheaves, and Curves

Sheaves on Curves and Their Local Terms

Scattering equations and Feynman diagrams

We show a direct matching between individual Feynman diagrams and integration measures in the scattering equation formalism of Cachazo, He and Yuan. The connection is most easily explained in terms of triangular graphs associated with planar Feynman diagrams in φ3-theory. We also discuss the generalization to general scalar field theories with φp interactions, corresponding to polygonal graphs ...

Moduli of metaplectic bundles on curves and Theta-sheaves

Historically θ-series have been one of the major methods of constructing automorphic forms. A representation-theoretic appoach to the theory of θ-series, as discoved by A. Weil [18] and extended by R. Howe [12], is based on the oscillator representation of the metaplectic group (cf. [17] for a recent survey). In this paper we propose a geometric interpretation this representation (in the nonram...

Torsion-free Sheaves and Moduli of Generalized Spin Curves

This article treats the compactification of the space of higher spin curves, i.e. pairs (X,L) with L an r root of the canonical bundle of X. More precisely, for positive integers r and g, with g > 2, r dividing 2g − 2, and for a flat family of smooth curves f : X → T, an r-spin structure on X is a line bundle L such that L ∼= ωX /T . And an r-spin curve over T is a flat family of smooth curves ...

Genus Formula for Modular Curves of D-elliptic Sheaves

We prove a genus formula for modular curves of D-elliptic sheaves. We use this formula to show that the reductions of modular curves of D-elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of D tends to infinity.