Associative Cones in the Imaginary Octonions

Emma Carberry

Abstract

There are three types of almost-complex curves in the nearly-Kähler 6-sphere: they are totally geodesic, pseudo-holomorphic or superconformal, the last case being generic. This paper concerns superconformal almost-complex curves. We begin by giving a geometric construction of a particularly natural G₂-framing for such curves. This framing can easily be shown to agree with that in [BVW:94]; the exposition here can be viewed as giving a geometric interpretation of and motivation for this framing together with a simpler proof that it indeed lies in G₂. We then focus our attention on superconformal almost-complex f : C → S⁶ and use the above framing to construct a spectral curve for maps of finite type (which include all doubly-periodic examples). This curve is reducible, and we additionally obtain a linear flow in the Jacobian of the "main component" of the spectral curve. This linear flow is in fact restricted to the real slice of a sub-torus of this Jacobian and it is notable that the sub-torus is the intersection of two Prym varieties, rather than a single Prym variety as has arisen in spectral curve descriptions of other harmonic maps. This later part of the paper is a report on joint work with Erxiao Wang.