Instantons and Four-Manifolds

Universal growth law for knot energy of Faddeev type in general dimensions

“…Although both homotopy groups are isomorphic to the set of all integers, Z, the dependence relationships between the corresponding minimum energies and topologies are drastically different, which lead to the existence of different types of solitons: point-like ones in the Skyrme theory but knot-like ones in the Faddeev theory. More precisely, let us use E and Q to collectively denote the energy and topological invariant in either the Skyrme theory or the Faddeev theory, u is any static field configuration, N is a given integer, and Such a property is also commonly seen in previously well-studied gauge field theory soliton configurations including vortices and monopoles (Bogomol'nyi 1976;Actor 1979;Jaffe & Taubes 1980;Yang 2001) and instantons (Witten 1977;Atiyah et al 1978;Actor 1979;Rajaraman 1982;Nash & Sen 1983;Freed & Uhlenbeck 1991;Yang 2001). On the other hand, however, for the Faddeev theory case, we have, instead, the sublinear asymptotics E N wjN j 3=4 ; ð1:3Þ which is analogous to the ropelength energy, crossing number relation EwN p (3/4%p%1) stated earlier but is uncommonly seen in quantum field theory.…”

Multi-Instanton Measure from Recursion Relations in N = 2 Supersymmetric Yang-Mills Theory

“…On the other hand, the simple way in which the recursion relations have been derived, strongly suggests that there may be some mechanism which should make the explicit calculations possible. The investigation of such mechanism would provide important information on the structure of the instanton moduli space (of which only the boundaryà la Donaldson-Uhlenbeck is known for generic winding number [12,13,14]) and of the associated volume form. In particular, even if the integrals seem impossible to compute, (actually, as we stated before we know neither the structure of the space nor the volume form), the existence of recursion relations and the simple way in which they arise, seem to suggest that these integrals could be easy to compute because of some underlying geometrical recursive structure.…”

Morse theory for the Yang-Mills functional via equivariant homotopy theory

“…Since H 2 (M 2g ; Q) = 0 and since the bundles ∆ ± have non-zero second Chern classes by Theorem 3.1 in [FU,p. 47] (see also [Pa1,Lemma 3.3,p.…”

“…The following argument (given to me by J. Råde) shows that Uhlenbeck's generic metrics theorem ( [FU,Theorem,3.17,p. 59]) and its equivariant versions in [Ba] and [Cho] are also true in our Morrey space completions.…”