Marcos Jardim

A survey on Nahm transform

Abstract We review the construction known as the Nahm transform in a generalised context, which includes all the examples of this construction already described in the literature. The Nahm transform for translation invariant instantons on R 4 is presented in an uniform manner. We also briefly analyse two new examples, the second of which being the first example involving a four-manifold that is not hyperkahler.

Construction of doubly-periodic instantons

Abstract: We construct finite-energy instanton connections over ℝ4 which are periodic in two directions via an analogue of the Nahm transform for certain singular solutions of Hitchins equations defined over a 2-torus.

Nahm Transform and Spectral Curves¶for Doubly-Periodic Instantons

We present the Nahm transform of the doubly-periodic instantons previously introduced by the author, converting them into certain meromorphic solutions of Hitchin’s equations over an elliptic curve.

Trihyperkähler reduction and instanton bundles on CP^3

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has rank 2n, n or 0. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves

On the Semistability of Instanton Sheaves Over Certain Projective Varieties

We show that instanton bundles of rank r ≤ 2n − 1, defined as the cohomology of certain linear monads, on an n-dimensional projective variety with cyclic Picard group are semistable in the sense of Mumford–Takemoto. Furthermore, we show that rank r ≤ n linear bundles with nonzero first Chern class over such varieties are stable. We also show that these bounds are sharp.

Nahm transform of doubly-periodic instantons

We present the Nahm transform of the doubly-periodic instantons previously introduced by the author, converting them into certain meromorphic solutions of Hitchin’s equations over an elliptic curve.

Linear and Steiner Bundles on Projective Varieties

We generalize the theory of Horrocks monads to ACM varieties, and use the generalization to establish a cohomological characterization of linear and Steiner bundles on projective space and on quadric hypersurfaces. We also characterize Steiner bundles on the Grassmannian G(1, 4) of lines in ℙ4. Finally, we study linear resolutions of bundles on ACM varieties, and characterize linear homological dimension on quadric hypersurfaces.

Quantum Instantons with Classical Moduli Spaces

Abstract: We introduce a quantum Minkowski space-time based on the quantum group SU(2)q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penroses twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective space ℙ3.

Nonsingular solutions of Hitchin's equations for noncompact gauge groups

We consider a general ansatz for solving the 2-dimensional Hitchins equations, which arise as dimensional reduction of the 4-dimensional self-dual Yang–Mills equations, with remarkable integrability properties. We focus on the case when the gauge group G is given by a real form of . For G = SO(2,1), the resulting field equations are shown to reduce to either the Liouville, elliptic sinh-Gordon or elliptic sine-Gordon equations. As opposed to the compact case, given by G = SU(2), the field equations associated with the noncompact group SO(2,1) are shown to have smooth real solutions with nonsingular action densities, which are furthermore localized in some sense. We conclude by discussing some particular solutions, defined on , S2 and T2, that come out of this ansatz.

ADHM CONSTRUCTION OF PERVERSE INSTANTON SHEAVES

We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalises the one on projective spaces. This is done by generalising the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterisation of these sheaves in the particular case of projective spaces.