Instantons on multi-Taub-NUT spaces. II: Bow construction (Q6562495)

This is the second part of a series consisting of three papers aimed to describe the detailed construction and properties of Yang-Mills instantons and their moduli spaces over the multi-Taub-NUT gravitational instantons.\N\NOn the one hand recall that a gravitational instanton (in the narrow sense) by definition is a complete hyper-Kähler \(4\)-manifold; only the non-compact case is interesting and in this case depending on the asymptotics of the metric the so-called ALE-ALF-ALG-ALH hierarchy has been introduced. Here the ALF, i.e., the asymptotically locally flat case contains the Gibbons-Hawking or multi-Taub-NUT spaces as classical examples (there is another family in it, the \(D_k\)-series with \(k=0,1,2,\dots\)). On the other hand, one can consider Yang-Mills instantons, i.e., anti-self-dual connections with finite \(L^2\)-curvature, and their moduli spaces in non-abelian Yang-Mills theories over oriented Riemannian \(4\)-manifolds and in particular over gravitational instantons. It turned out during the past decades that the problem of solving a nonlinear PDE problem of geometric origin in various dimensions can often be converted into a certain lower-dimensional problem which, depending on the particular situation, contains algebraic data only, i.e., gives rise to a zero-dimensional problem, or contains nonlinear ODE's only, i.e., gives rise to a one-dimensional problem. The prototypical example for the former case is the ADHM construction where an instanton over the flat \({\mathbb R}^4\) is converted into a so-called quiver, i.e., a zero-dimensional object while the prototypical example for the latter is the Nahm transform where a magnetic monopole over the flat \({\mathbb R}^3\) is converted into an ODE over a real interval, i.e., a one-dimensional object. A sort of combination of these is Cherkis' bow construction which allows one to convert instantons over ALF spaces into bows.\N\NThe main results of the paper are as follows. First, the rank of the Hermitian vector bundle over a multi-Taub-NUT space accommodating an instanton connection can be read off from the bow data (see Theorem 1 in the paper); second, there exists an asymptotically abelian gauge along infinity in which the asymptotics of the instanton connection can be read off from the bow data (see Theorem 2 in the paper). Two more theorems are also announced here (and going to appear in the third part) dealing with the construction of what the authors call the up transform, i.e., the way how to obtain an instanton over the multi-Taub-NUT space out of a bow (see Upcoming Theorems 1-2 in the paper). Finally, one can compute (the first and) the second Chern classes (in the asymptotic limit) of instantons using bow data (see Theorem 3 in the paper).