Gauge theory in higher dimensions. II (Q2914210)

This is an extended verbatim quotation from the introduction of the paper:NEWLINENEWLINEWe follow up some of the ideas discussed by \textit{S. K. Donaldson} and \textit{R. P. Thomas} [in: S. A. Huggett (ed.) et al., The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science, Oxford, UK, June 1996 in honour of Roger Penrose in his 65th year. Oxford: Oxford University Press. 31--47 (1998; Zbl 0926.58003)]. The theme of that article was the possibility of extending familiar constructions in gauge theory, associated to problems in low-dimensional topology, to higher dimensional situations, in the presence of an appropriate special geometric structure. The starting point for this was the ``holomorphic Casson invariant'', counting holomorphic bundles over a Calabi-Yau \(3\)-fold, analogous to the Casson invariant which counts flat connections over a differentiable \(3\)-manifold. (\dots)NEWLINENEWLINEIn the familiar gauge theory picture one views the Casson invariant as the Euler characteristic of the instanton Floer homology groups. Thus it is natural to hope for some analogous structure associated to a Calabi-Yau \(3\)-fold. This was discussed in a general way in [loc. cit.] but the discussion there did not pin down exactly what structure one could expect. That is the goal of the present paper. In brief, we will argue that one should hope to find a holomorphic bundle over the moduli space of Calabi-Yau \(3\)-folds, of rank equal to the holomorphic Casson invariant (sometimes called the DT invariant) defined by Thomas.NEWLINENEWLINEJust as in [loc. cit.], many of the arguments here are tentative and speculative since the fundamental analytical results that one would need to develop theory properly are not yet in place. These have to do with the compactness of the moduli spaces of solutions. While considerable progress has been made in this direction by \textit{G. Tian} [Ann. Math. (2) 151, No. 1, 193--268 (2000; Zbl 0957.58013)], and by \textit{T. Tao} and \textit{G. Tian} [J. Am. Math. Soc. 17, No. 3, 557--593 (2004; Zbl 1086.53043)], a detailed theory -- in either the gauge theory or submanifold setting -- seems still to be fairly distant. The issues are similar to those involved by ``counting'' special Lagrangian submanifolds in Calabi-Yau manifolds, which have been considered by \textit{D. Joyce} [Contemp. Math. 314, 125--151 (2002; Zbl 1060.53059)], but where, again, a final theory is still lacking.NEWLINENEWLINEThe core of this article is Section 4, where we explain how to construct holomorphic bundles over Calabi-Yau moduli spaces, assuming favourable properties of a ``\((6+1)\)-dimensional'' differential-geometric theory. The preceding two sections 2 and 3 develop background material, mostly fairly standard but introducing a point of view involving ``taming forms''. In section 5, we explain how our construction matches up with standard algebraic topology, following the familiar Floer-theory philosophy. In section 6, we go back to discuss the central, unresolved, compactness issues. We explain the relevance of recent work of Haydys which brings in a version of the ``Fueter equation''. This perhaps points the way to a unification of the gauge theory and calibrated geometry discussions and connections with the more algebro-geometric approach. (\dots)NEWLINENEWLINEFor the entire collection see [Zbl 1230.53007].