Algebraic Nahm transform for parabolic Higgs bundles on ℙ¹

Abstract

We formulate the Nahm transform in the context of parabolic Higgs bundles on P 1 and extend its scope in completely algebraic terms. This transform requires parabolic Higgs bundles to satisfy an admissibility condition and allows Higgs fields to have poles of arbitrary order and arbitrary behavior. Our methods are constructive in nature and examples are provided. The extended Nahm transform is established as an algebraic duality between moduli spaces of parabolic Higgs bundles. The guiding principle behind the construction is to investigate the behavior of spectral data near the poles of Higgs fields. 14H60; 14E05, 14J26

Singular Geometry and Higgs Bundles in String Theory

“…5 Given a stable bundle E, take Φ = 0, then the Higgs bundle (E, 0) is stable and trivially fixed by the S 1 -action. 6 The authors of [96] actually study the slightly more general situation of "twisted Higgs bundles" where the canonical bundle K is replaced by a line bundle L with deg(L) > 0. For this setting of L-twisted SL(2, C)-Higgs bundles, the monodromy action was considered in [24].…”

The birational geometry of unramified irregular Higgs bundles on curves

“…This structure may be conveniently expressed using the technique of proper transform of R-parabolic sheaves defined in Section 5 of [1]. Let us now describe this construction.…”

“…Irregular Dolbeault moduli space. Let us first treat the moduli space of stable irregular parabolic Higgs bundles on C with fixed semi-simple irregular part (3), (1) and residue in a fixed regular orbit: by [5] it carries an Atiyah-Bott hyper-Kähler structure, in particular for the Dolbeault holomorphic structure I it admits a holomorphic symplectic structure. These irregular Dolbeault moduli spaces M irr Dol may then be put into a family…”

“…The route we take is slightly different from Logares and Martens even in the logarithmic case, in that we encode the parabolic structure of the bundle in a (R-)parabolic structure of the spectral sheaf. Furthermore, as it was pointed out in Section 4 of [1] by K. Aker and the author, for such a correspondence in the irregular parabolic case one needs to perform iterated blow-ups along non-reduced 0-dimensional subschemes of the fibres of Z over the irregular singular points. In addition, the spectral sheaves transform by a so-called proper transform functor for coherent sheaves, whose definition and main properties are given in Section 5. loc.…”

Nahm transformation for parabolic Higgs bundles on the projective line—Case of non-semisimple residues

“…Let C be a different copy of the complex affine line with coordinate ζ, and P 1 the associated projective line. The aim of this paper is to construct a transformed harmonic bundle ( V ,∂ E , θ, h) on P 1 and study the properties of the mapping (1) N : (V, F j i ,∂ E , ∇, h) → ( V , F j i ,∂ E , ∇, h), called Nahm transformation, on moduli spaces.…”