337. Twistors and non-linear differential equations. Curved spacetime

Let my continue our micro-discussion of twistors. Last time I explained how using the language of twistors allows to express solutions of the linear differential equations (massless free fields propagating in the flat spacetime) in a different way – not too deep to really call it a result.

As it turns out, twistors also help dealing with non-linear differential equations. For example, Ward and Sir Michael Atiyah have used the language of twistors to construct self-dual solutions of the Yang-Mills equations – instantons. I guess I don’t need to explain how much instantons are important for the quantum Yang-Mills theory – they are non-trivial topological configurations of the Y.-M. field that extremize the Y.-M. action and (which is much more important) give contribution into the overall Y.-M. partition function of non-zero measure.

Instantons – solutions of the self-duality equation – defined in the 4-dimensional Euclidean space turn out to correspond to complex vector bundles in twistor space. This fact allowed Atiyah, Hitchin, Drinfeld and Manin to find all possible instanton solutions for the Y.-M.

Another direction of research where twistor formalism proved to be useful was finding new solutions of the Einstein equations. The point is that instead of Minkowski space we can of course consider general curved spacetime. While 4-dimensional Minkowski spacetime corresponds to the set of straight lines in the 6-dimensional complex space, it is natural to expect that generic curved spacetime should naturally correspond to the set of curved lines in the same complex space. Of course, as usual, reality is more complicated than the first naive guess about it. Not all curved manifolds can be realized as sets of curves in the twistor space, but only those which satisfy to vacuum Einstein equations and additional conformal condition of autoduality (autodual part of the Weyl tensor is equal to zero). Such manifolds indeed correspond to sets of curves in a curved twistor space, and the language of twistors allowed us to find many self-dual solutions of the Einstein equations.

Next time, let me discuss a couple of examples which will hopefully make all this twistor ideology clear.

Some basic literature

1. Yu. Manin, Gauge field theory and complex geometry. A very strong book about complex geometry, holomorphic bundle theory and some physical applications.
2. R. Penrose, W. Rindler, Spinors and Space-Time, Vol. 1. In this volume authors discuss twistor space corresponding to Minkowski space and free massless fields.
3. R. Penrose, W. Rindler, Spinors and Space-Time, Vol. 2. Here, the authors describe twistor description of curved spacetimes and also discuss many physical applications.