Monopole Equations on 8-Manifolds with Spin(7) Holonomy

Abstract

We construct a consistent set of monopole equations on eight-manifolds with Spin(7) holonomy. These equations are elliptic and admit nontrivial solutions including all the 4-dimensional Seiberg-Witten solutions as a special case. 0

Seiberg-Witten type monopole equations on 8-manifolds with Spin(7) holonomy as minimizers of a quadratic action

“…For the specific spin c structure above, our monopole equations corresponding to the coupling to a positive spinor [12] are…”

“…In our previous paper [12], we defined a generalization of the monopole equations by interpreting the right hand sides of the Seiberg-Witten equations as a projection onto a subspace determined by the map ρ + and coupled positive spinors to the curvature. However, it was not possible to express these equations as absolute minimizers of an action.…”

“…We have previously given [12] an elliptic set of monopole equations using a projection with ρ + (F (7) ), where we denoted F (7) as F + , and ω 1j by f j−1 .…”

Seiberg--Witten-like equations on $5$-dimensional contact metric manifolds

“…The solution space of these equations gives differential topological invariants for 4-manifolds [1,11]. Some generalizations were given later on higher dimensional manifolds [4,7,10].…”

Seiberg-Witten-like Equations on 7-Manifolds

“…There are some analogues of these equations in higher dimensions (see [1,9,12,13]). All of the higher dimensional equations are stated for even dimensional manifolds.…”