The study of gauge theory was initiated after the discovery of Yang-Mills equation in 70's. Since then, many mathematicians and physicists have been working on this field from many different perspectives. In the last decade, the fascinating development pioneered by Donaldson and his school has brought many disciplines, including differential geometry, differential topology and algebraic geometry into the study of gauge theory. In this talk, the speaker will provide a survey of recent development in gauge theory from algebraic geometric point of view.
To begin with, recall that for any Riemannian four-manifold $(X,g)$ and any principle ${\rm SU}(2)$-bundle $P$ of second Chern class $d$, one can form the space of gauge equivalent classes of irreducible anti-self-dual connections (i.e. whose curvatures are anti-self-dual with respect to $g$). As was demonstrated in the past ten years, these spaces inherited very rich structures.
One approach in studying this space is by algebraic geometric tool Note that a large class of four manifolds are provided by algebraic surfaces. It turns out that for algebraic surfaces, space of ASD-connections (modulo gauge equivalence) coincides with the moduli of stable vector bundles of second Chern class $d$. In the following, we denote this pace by $M_d$.
It was first established in early 70's that $M_d$ is indeed a projective variety. Later there has been some progress in understanding the geometry of the moduli $M_d$ mostly for surfaces of special types, say projective plane or K3 surfaces. It is only after the pioneeer work of Donaldson in using his new smooth invariants to find examples of homeomorphic but not diffeomorphic algebraic surfaces that sparked a surge of interest in studying the moduli space $M_d$. There are two (closely related) directions in studying $M_d$. One is in attempting to determine the Donaldson's polynomial invariants of algebraic surfaces and the other is to better understand the geometry of the moduli space. In this talk, the speaker will discuss the recent developments along these two directions, especially those that are achieved by using algebraic geometry techinque, which includes the comparison of Uhlenbeck's compactification and the Gieseker's compactification of the space of ASD-connections; two non-vanishing results of the polynomial invariants of algebraic surfaces by Donaldson and O'Grady; the connectedness of the moduli space $M_d$; the "positivity" of the Ricci curvature of $M_d$ and the Atiyah-Jones' conjecture concerning the topology of the moduli space $M_d$. In the end, the generalization of these results to moduli of high rank stable vector bundles will be discussed as well.