The authors solve the problem stated by M. Atiyah [see his book Geometry of Yang-Mills fields, p. 94 (1979; Zbl 0435.58001)], namely they prove completeness of the AHDM (Atiyah, Hitchin, Drinfeld, Manin) linear algebra construction of instantons over \(S^4\), in terms only of ”classical” global differential geometry of \(S^4\) (i.e. without algebraic geometry of \(P^3C\) and the vanishing theorem \(H^1(E(- 2))=0)\). Moreover they prove in this way completeness of the Nahm construction of SU(2)-monopoles and discuss dualities in both constructions.
The proofs consist of three main parts: (1) the matrix operators corresponding to instantons in the AHDM and Nahm construction are expressed via ”moments” and asymptotics of solutions of the Dirac equation in instanton field, (2) symmetries reflecting self-duality in matrices are derived, (3) non-degeneracy of matrices which must be invertible is proved.
The main tools are: index theorem, linear elliptic equations with subtle conditions in infinity, Green’s function and Stokes’s theorem. The proof uses only notions and techniques relevant to elliptic operators theory.
If we compare this proof with the previous AHDM one (1977) then we see that calculations in the reviewed proof are longer (note that important equation 2.31 is proved in another paper by these authors). The intuitional reason is that the algebraic geometry allows us to work in ”smaller” and ”closer” spaces of functions.
In a separate point the authors review reciprocities between objects in d and 4-d dimensions. In the case of monopoles they present a duality between Bogomol’nyĭ and Nahm equations, dimension of the space of solutions of the Dirac equation and the number of gauge degrees of freedom and between other objects. In the case of instantons the reciprocities discussed (e.g. between space-time variables and discrete indices) do not look so explicit.
It seems to be very interesting to explain the “moment”, asymptotics and other similar formulae in the case of a general base 4-manifold. The main geometric objects used by the authors were introduced systematically by V. G. Drinfeld and Yu. I. Manin in [Sov. J. Nucl. Phys. 29, 845–849 (1979); translation from Yad. Fiz. 29, 1646–1654 (1979; Zbl 0536.58010)].