On the L-infinity description of the Hitchin Map

Peter Dalakov, "On the L-infinity description of the Hitchin Map", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 46 (2014), pp.1-17We exhibit, for a G-Higgs bundle on a compact complex manifold, a subspace of the second cohomology of the controlling dg Lie algebra, containing the obstructions to smoothness. For this we construct an L-infinity morphism, which induces the Hitchin map and whose "toy version" controls the adjoint quotient morphism. This extends recent results of E.Martinengo

Seiberg-Witten differentials on the Hitchin base

In this note we describe explicitly, in terms of Lie theory and cameral data, the covariant (Gauss--Manin) derivative of the Seiberg--Witten differential defined on the weight-one variation of Hodge structures that exists on a Zariski open subset of the base of the Hitchin fibration. Dedicated to Tony Pantev on the occasion of his 60th birthday.Comment: 18 pages. Second version with minor modifications and corrected typo

Donagi-Markman cubic for the generalised Hitchin system

Donagi and Markman (1993) have shown that the infinitesimal period map for an algebraic completely integrable Hamiltonian system (ACIHS) is encoded in a section of the third symmetric power of the cotangent bundle to the base of the system. For the ordinary Hitchin system the cubic is given by a formula of Balduzzi and Pantev. We show that the Balduzzi\u2013Pantev formula holds on maximal rank symplectic leaves of the G-generalised Hitchin system

Higgs bundles and opers

In this thesis we address the question of determining the Higgs bundles on a Riemann surface which correspond to opers by the non-abelian Hodge theorem. We study one specific Higgs bundle (P, &thetas;) obtained by extension of structure group from a particular rank two Higgs bundle on K1/2 ⊕ K -1/2. For a simple complex group we identify the tangent space to the Dolbeault moduli space at that particular point as the Hitchin base plus its dual and exhibit harmonic representatives for the hypercohomology of the corresponding deformation complex. We study the Maurer-Cartan equation for (P, &thetas;) and give an explicit, recursive (terminating after finitely many steps) method for writing a family of deformations parametrised by the tangent space which can be interpreted as a holomorphic exponential map. Next, we identify opers with their tangent space at the uniformisation oper. By quaternionic linear algebra we determine the subspace of infinitesimal deformations of (P, &thetas;) corresponding to opers and apply the exponential map to those. In this way we obtain the germ of the family of Higgs bundles corresponding to opers. In the final chapter we sketch a plan for obtaining the complete solution, as well as certain questions for further investigation

Higgs bundles and opers

In this thesis we address the question of determining the Higgs bundles on a Riemann surface which correspond to opers by the non-abelian Hodge theorem. We study one specific Higgs bundle (P, &thetas;) obtained by extension of structure group from a particular rank two Higgs bundle on K1/2 ⊕ K -1/2. For a simple complex group we identify the tangent space to the Dolbeault moduli space at that particular point as the Hitchin base plus its dual and exhibit harmonic representatives for the hypercohomology of the corresponding deformation complex. We study the Maurer-Cartan equation for (P, &thetas;) and give an explicit, recursive (terminating after finitely many steps) method for writing a family of deformations parametrised by the tangent space which can be interpreted as a holomorphic exponential map. Next, we identify opers with their tangent space at the uniformisation oper. By quaternionic linear algebra we determine the subspace of infinitesimal deformations of (P, &thetas;) corresponding to opers and apply the exponential map to those. In this way we obtain the germ of the family of Higgs bundles corresponding to opers. In the final chapter we sketch a plan for obtaining the complete solution, as well as certain questions for further investigation