Dimension of monopoles on asymptotically conic 3-manifolds: Figure 1.

Abstract

The virtual dimensions of both framed and unframed SU(2) magnetic monopoles on asymptotically conic 3-manifolds are obtained by computing the index of a Fredholm extension of the associated deformation complex. The unframed dimension coincides with the one obtained by Braam for conformally compact 3-manifolds. The computation follows from the application of a Callias-type index theorem.

Monopoles on AC 3-manifolds

“…where v ∈ T x P and dH x (v) = d/dt| t=0 H(exp x (tv)). It follows from [9] that d * 1 ⊕ d 2 has index i = 4k + b 1 (X) − 1 (our weight β = − 1 2 corresponds in that reference to the weight γ = −β − 3 2 = −1). Moreover, the same proof as that of Proposition 5 gives that d * 1 ⊕ d 2 is surjective, and so we have dim(ker(d * 1 ⊕ d 2 )) = dim(T x P ).…”

“…a usual vector potential whose strength (magnetic field) vanishes inside the superconductor filling in X. [7], Kottke computes the virtual dimension of the moduli space of monopoles with fixed mass on an AC 3-manifold and obtains the formula dim(M m,k ) = 4k + 1 2 b 1 (Σ) − b 0 (Σ). The long exact sequence on cohomology H c (X) → H(X) → H(Σ) and the duality H * c ∼ = (H 3− * ) * , can be used to rewrite Kottke's formula as…”

“…The nomenclature of Definition 1 is in agreement with that of [9], and should be thought of as moduli space of unframed monopoles. The main result of this paper is the following.…”

“…(2) In [9], Kottke computes the virtual dimension of the moduli space of monopoles with fixed mass on an AC 3-manifold, and obtains the formula dim(M m,k ) = 4k + 1 2 b 1 (Σ) − b 0 (Σ). The long exact sequence on cohomology H c (X) → H(X) → H(Σ) and the duality H * c ∼ = (H 3− * ) * can be used to rewrite Kottke's formula as dim(M m,k ) = 4k + b 1 (X) − b 2 (X) − 1, and the construction described here gives a good geometric interpretation of all these parameters in an open set in M m,k .…”

Asymptotics of finite energy monopoles on AC $3$-manifolds

“…The study of monopoles on general AC 3-manifolds was initiated only recently by the works of Oliveira [Oli14,Oli16] and Kottke [Kot15]; see also [FO19] and Remark 1.4. In particular, Kottke [Kot15] computed the virtual dimension of the moduli spaces of monopoles on AC 3-manifolds and Oliveira [Oli16] tackled the problem of existence by proving an AC version of Taubes' original gluing theorem of well-separated multi-monopoles on R 3 , giving a construction which covers a smooth open set in the moduli space of monopoles on any AC 3-manifold with vanishing second Betti number.…”