|XXXVII Workshop on Geometric Methods in Physics||1-7.07.2018|
|VII School on Geometry and Physics||25-29.06.2018|
Twistor Geometry and Gauge Fields
The main goal of this course is to present the basics of twistor theory and its applications to the solution of equations of gauge theory such as selfdual Yang-Mills equations. The first part of the course, devoted to the twistor theory, starts from the construction of the twistor model of Minkowsky space. Then we turn to the study of the twistor correspondence between geometric objects in Minkowsky space and their twistor counterparts. We pay attention also to the Klein model of Minkowsky space in which this space is identified with a quadric in the 5-dimensional complex projective space $\mathbb C\mathbb P^5$. In the second part of the course we apply twistor theory to the study of gauge field equations. As a first example we consider the Yang-Mills duality equations in the Euclidean space $\mathbb R^4$ and their solutions called instantons. Atiyah-Ward theorem gives a twistor interpretation of instantons and ADHM(Atiyah-Drinfeld-Hitchin-Manin)-construction, based on this theorem, yields a complete description of the moduli space of instantons. The next example is provided by the monopole equations in $\mathbb R^3$ otherwise called the Bogomolny-Prasad equations. Their twistor interpretation was proposed by Nahm. At last we turn to the 2-dimensional models which are represented by the Yang-Mills-Higgs equations in $\mathbb R^2$ and Hitchin equations on Riemann surfaces. The moduli space of solutions of selfdual Yang-Mills-Higgs equations is completely described by the theorem of Taubes. All considered equations have a deep physical meaning and their study is important both for mathematicians and physicists.
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