## 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 G_{2}-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 G_{2}. We then focus our
attention on superconformal almost-complex
*f* : **C**
→ *S*^{6}
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.