Harmonic maps into the exceptional symmetric space G₂ /SO(4)

Abstract

We show that a harmonic map from a Riemann surface into the exceptional symmetric space G 2 /SO(4) has a J 2 -holomorphic twistor lift into one of the three flag manifolds of G 2 if and only if it is 'nilconformal', i.e., has nilpotent derivative. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into G 2 /SO(4) which are not of finite uniton number, and which have lifts into any of the three twistor spaces. Harmonic maps of finite uniton number are all nilconformal; for such maps, we show that our lifts can be constructed explicitly from extended solutions.2000 Mathematics Subject Classification. 53C43, 58E20.HARMONIC MAPS INTO THE EXCEPTIONAL SYMMETRIC SPACE G 2 /SO(4) 27 2 isotropic subspace of C 7 which is not complex-coassociative, justifying our alternative description of T 2 as the set of all such.That the map (4.14)(s=2) intertwines both J 1 and J 2 is proved as in case (i); this time,. Now T is the stabilizer in G 2 of the flags (4.12)(s = 3) and (4.13)(s = 3); as in case (ii), for convenience we work with the latter. Thenthe proof that the map (4.14)(s=3) intertwines both J 1 and J 2 proceeds as before.7.4. Proof of Lemma 3.4. We need the following facts. As in §4.1, we identify smooth maps ϕ : M → G k (C n ) and rank k subbundles of C n . As described in that section, we give C n the connection D ϕ and the Koszul-Malgrange structure with∂-operator given over each coordinate domain (U, z) by D φ z . From §4.1, the connection D ϕ is the direct sum of the connections ∇ ϕ on ϕ and ∇ ϕ ⊥ on ϕ ⊥ ; those connections are given by projection of the flat connection on C n .We now prove Lemma 3.4 for s = 1. The cases s = 2 and s = 3 are similar. Suppose that ψ : M → T 1 is J 2 -holomorphic. We need to show:(7.2) W is a holomorphic subbundle of (C n , D φ z ) which lies in ker A ϕ z . By J 2 -holomorphicity, the horizontal component of ψ z has values in H 1,0 J 2 T 1 = H 1,0 J 1 T 1 = α(ξ)/i >0, odd g α = g α 2 + g α 1 +α 2 + g 2α 1 +α 2 + g 3α 1 +α 2 , and the vertical component has values in V 1,0 J 2 T 1 = V 0,1 J 1 T 1 = g −3α 1 −2α 2 . Hence the horizontal component of ψ z maps W = ψ 1 = ℓ α 1 +α 2 ⊕ ℓ 2α 1 +α 2 to 0, and the vertical component has zero component in End(ψ i , ψ j ) for j > i so that A ′ ϕ ⊥ ⊖W,W = 0. As in [10], this means W is a holomomorphic subbundle of (C n , D φ z ) and (7.2) follows. Conversely, suppose that ψ : M → T 1 is not J 2 -holomorphic. Then (i) either the horizontal component of ψ z has a non-zero component in one of the components of H 0,1is non-zero: for example, if ψ z has a non-zero component in g −α 2 , then A ϕ z (W ) ⊃ A ϕ z (ℓ α 1 +α 2 ) = ℓ α 1 = 0. Similarly, in case (ii), A ′ W ,W = A ′ ψ −1 ,ψ 1 is non-zero, so that W is not a holomorphic subbundle of (C n , D φ z ). Hence one of the conditions in (7.2) is violated.

Twistor theory for exceptional holonomy

“…For the former see [9] (and the references therein) whilst the latter is wellknown as a (complex) Wolf space. Furthermore, the embedding g 2 ⊆ so (7) induces on it a Hermitian and a quaternionic-like structure (compare [11] ) corresponding to the exact sequence of vector bundles…”

Harmonic maps of finite uniton number and their canonical elements

“…When the Lie algebra g is the compact real form of one of the exceptional Lie algebras g 2 , f 4 and e 8 , the corresponding simply connected Lie group has trivial centre and the I-canonical element is ξ I = i∈I H i . Harmonic maps of finite uniton number into G 2 and its unique inner symmetric space G 2 /SO(4) were studied in [9], where a description of the extended solutions associated to the different canonical elements was given in terms of Frenet frames (see also [20] for a different twistorial approach to harmonic maps into G 2 /SO(4)). Next we focus our attention in the symmetric canonical elements of F 4 , which is simply connected and has trivial centre.…”

Harmonic surfaces in the Cayley plane

“…In this paper, we consider the twistor theory of nilconformal harmonic maps from a Riemann surface into the Cayley plane , the 16‐dimensional exceptional Riemannian symmetric space . Exhibiting this manifold as a submanifold of the Grassmannian of 10‐dimensional subspaces of the fundamental representation of allows us to use techniques and constructions similar to those used in earlier works on twistor constructions of nilconformal harmonic maps into classical Grassmannians [29], as well as the exceptional Grassmannian [30].…”

“…Following the general twistor theory for harmonic maps into compact symmetric spaces as developed in [7], we see that there are three canonical twistor fibrations over : , of dimension 30; of dimension 40; of dimension 42. As the dimensions of these twistor fibrations are far higher than, for example, those of the exceptional Grassmannian and too high to be handled in a straightforward fashion as in [30], we have chosen an approach significantly different from that of previous works in our study of the twistor lifts of nilconformal harmonic maps, making use of the classification of nilpotent orbits in Lie algebras as described in [11]. And while there are three different canonical twistor fibrations over , only one of these, , a submanifold of the isotropic lines in the fundamental representation of , is needed to describe the twistor theory of nilconformal harmonic maps.…”