On the splitting type of holomorphic vector bundles induced from regular systems of differential equation

We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

Symplectic and Kähler structures on $$\mathbb CP^1$$-bundles over $$\mathbb CP^2$$

We show that there exist symplectic structures on a $$\mathbb {CP}^1$$ CP 1 -bundle over $$\mathbb {CP}^2$$ CP 2 that do not admit a compatible Kähler structure. These symplectic structures were originally constructed by Tolman and they have a Hamiltonian $${\mathbb {T}}^2$$ T 2 -symmetry. Tolman’s manifold was shown to be diffeomorphic to a $$\mathbb CP^1$$ C P 1 -bundle over $$\mathbb {CP}^{2}$$ CP 2 by Goertsches, Konstantis, and Zoller. The proof of our result relies on Mori theory, and on classical facts about holomorphic vector bundles over $$\mathbb {CP}^{2}$$ CP 2 .

Twistor Actions for Integrable Systems

Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativity to two spacetime dimensions is an integrable coset sigma model coupled to a dilaton and 2d gravity. The dimensional reduction of supergravity to two spacetime dimensions is an integrable coset sigma model coupled to matter fermions, a dilaton, and 2d supergravity. We derive Lax operators and Lagrangians for these 2d integrable systems using the Chern-Simons theory on twistor space. In the supergravity example, we use an extended setup in which twistor Chern-Simons theory is coupled to a pair of matter fermions.

Atiyah’s work on holomorphic vector bundles and gauge theories

The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.

On 2k-Hitchin’s equations and Higgs bundles: A survey

We study the [Formula: see text]-Hitchin’s equations introduced by Ward from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and [Formula: see text]-Hitchin’s equations, we review some elementary facts on complex geometry and Yang–Mills theory. Then, we study some properties of holomorphic vector bundles and Higgs bundles and we review the Hermite–Yang–Mills equations together with two functionals related to such equations. Using some geometric tools we show that, as far as Higgs bundles are concerned, [Formula: see text]-Hitchin’s equations are reduced to a set of two equations. Finally, we introduce a functional closely related to [Formula: see text]-Hitchin’s equations and we study some of its basic properties.

Kobayashi—Hitchin corrispondence for twisted vector bundles

We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein.

Holomorphic vector field and topological sigma model on ℂP1 worldsheet

Witten suggested that fixed-point theorems can be derived by the supersymmetric sigma model on a Riemann manifold [Formula: see text] with potential terms induced from a Killing vector on [Formula: see text].3. One of the well-known fixed-point theorems is the Bott residue formula9 which represents the intersection number of Chern classes of holomorphic vector bundles on a Kähler manifold [Formula: see text] as the sum of contributions from fixed point sets of a holomorphic vector field [Formula: see text] on [Formula: see text]. In this paper, we derive the Bott residue formula by using the topological sigma model (A-model) that describes dynamics of maps from [Formula: see text] to [Formula: see text], with potential terms induced from the vector field [Formula: see text]. Our strategy is to restrict phase space of path integral to maps homotopic to constant maps. As an effect of adding a potential term to the topological sigma model, we are forced to modify the BRST symmetry of the original topological sigma model. Our potential term and BRST symmetry are closely related to the idea used in the paper by Beasley and Witten2 where potential terms induced from holomorphic section of a holomorphic vector bundle and corresponding supersymmetry are considered.