Instantons, Hilbert Schemes and Integrability.pdfVIP

a r X i v : h e p - t h / 0 1 0 3 2 0 4 v 1 2 3 M a r 2 0 0 1 Instantons, Hilbert Schemes and Integrability? H. W. Braden1?, N. A. Nekrasov2,3? 1Department of Mathematics and Statistics, The University of Edinburgh, Edinburgh, UK 2Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France 3Institute for Theoretical and Experimental Physics, Moscow, Russia Abstract We review the deformed instanton equations making connection with Hilbert schemes and integrable systems. A single U(1) instanton is shown to be anti-self-dual with respect to the Burns metric. 1 Introduction The aim of the present review is to describe various settings surrounding the matrix equations [B2, B1] + IJ = ζc1V , (1) [B1, B ? 1] + [B2, B ? 2] + II ? ? J?J = 2ζr1V , (2) where B1,2 ∈ Mv(C), I ∈ Mv×w(C), J ∈ Mw×v(C). We will be interested in the space of solutions to these equations up to equivalence under an action of GL(α) := GL(v,C) × GL(w,C). These equations arise naturally in the context of integrable systems which will be recalled in the next section. The space of such matrices describes the phase space of the integrable system and we will refer to it as a “moduli space” as it describes the system at all energies and momenta. This space carries a natural hyper-Ka?hler structure and possesses several moment maps which are also reviewed. Now the same moduli space also describes other interesting phenomena. Indeed, if there is to be a simple motto describing this talk it is: “Phase spaces of completely integrable systems give interesting moduli spaces for field theories”. This is well known in the context of Seiberg- Witten theory [7, 40] but true more generally [49, 6, 13]. In the present setting the same moduli space parametrises the (semistable) torsion free sheaves on CP 2 whose restriction on the pro- jective line ?∞ at infinity is trivial, as was shown first by Nakajima. The connection between the Calogero-Moser systems and instanton/sheaf moduli was noted by Nekrasov [42] an