Hitchin systems on singular curves II. Gluing subschemes

TL;DR

This paper extends the study of Hitchin systems to a broad class of singular curves formed by gluing subschemes, providing explicit geometric descriptions and proving integrability using an $r$-matrix formalism.

Contribution

It introduces a general framework for Hitchin systems on singular curves created by gluing subschemes, including explicit geometric and algebraic descriptions and integrability proofs.

Findings

Explicit description of moduli space of vector bundles on singular curves

Calculation of the genus for these singular curves

Proof of Hitchin system integrability on such curves

Abstract

In this paper we continue our studies of Hitchin systems on singular curves (started in hep-th/0303069). We consider a rather general class of curves which can be obtained from the projective line by gluing two subschemes together (i.e. their affine part is: Spec $\{f \in \CC[z]: f(A(\ep))=f((B(\ep)); \ep^N=0 \}$, where $A(\ep), B(\ep)$ are arbitrary polynomials) . The most simple examples are the generalized cusp curves which are projectivizations of Spec $\{f \in \CC[z]: f'(0)=f''(0)=...f^{N-1}(0)=0 \}$). We describe the geometry of such curves; in particular we calculate their genus (for some curves the calculation appears to be related with the iteration of polynomials $A(\ep), B(\ep)$ defining the subschemes). We obtain the explicit description of moduli space of vector bundles, the dualizing sheaf, Higgs field and other ingredients of the Hitchin integrable systems; these results may deserve the independent interest. We prove the integrability of Hitchin systems on such curves. To do this we develop $r$-matrix formalism for the functions on the truncated loop group $GL_n(\CC[z]), z^N=0$. We also show how to obtain the Hitchin integrable systems on such curves as hamiltonian reduction from the more simple system on some finite-dimensional space.