Manifold*

We study a continuum model of the weakly interacting Bose gas in the presence of an external field with minima forming a triangular lattice. The second lowest band of the single-particle spectrum (p-band) has three minima at non-zero momenta. We consider a metastable Bose condensate at these momenta and find that, in the presence of interactions that vary slowly over the lattice spacing, the order parameter space is isomorphic to S⁵. The enlarged symmetry manifests itself in an additional number of gapless excitation modes, as well as the loss of topologically stable vortices. We expect that this non-Abelian condensate is a “failed superfluid” that does not undergo a Berezinskii-Kosterlitz-Thouless (BKT) transition. Order-by-disorder splitting appears highly suppressed, implying that signatures of the S⁵ manifold ought to be observable at small temperatures.