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Trihyperkähler reduction and instanton bundles on
| Content Provider | Scilit |
|---|---|
| Author | Jardim, Marcos Verbitsky, Misha |
| Copyright Year | 2014 |
| Description | Journal: Compositio Mathematica A trisymplectic structure on a complex $2n$-manifold is a three-dimensional space ${\rm\Omega}$ of closed holomorphic forms such that any element of ${\rm\Omega}$ has constant rank $2n$, $n$ or zero, and degenerate forms in ${\rm\Omega}$ belong to a non-degenerate quadric hypersurface. 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 in the twistor space of a hyperkähler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkähler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkähler manifold $M$ is compatible with the hyperkähler reduction on $M$. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank $r$, charge $c$ framed instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth trisymplectic manifold of complex dimension $4rc$. In particular, it follows that the moduli space of rank two, charge $c$ instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth complex manifold dimension $8c-3$, thus settling part of a 30-year-old conjecture. |
| Ending Page | 1868 |
| Starting Page | 1836 |
| ISSN | 00221295 |
| e-ISSN | 15705846 |
| DOI | 10.1112/s0010437x14007477 |
| Journal | Compositio Mathematica |
| Issue Number | 11 |
| Volume Number | 150 |
| Language | English |
| Publisher | Wiley-Blackwell |
| Publisher Date | 2014-08-27 |
| Access Restriction | Open |
| Subject Keyword | Journal: Compositio Mathematica Mathematical Physics Twistor Space Algebraic Geometry Complex Manifold Differential Geometry Hyperkahler Manifold Symplectic Geometry |
| Content Type | Text |
| Resource Type | Article |
| Subject | Algebra and Number Theory |
