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We study a continuum model of the weakly interacting Bose gas in the presenceof an external field with minima forming a triangular lattice. The secondlowest band of the single-particle spectrum ($p$-band) has three minima atnon-zero momenta. We consider a metastable Bose condensate at these momenta andfind that, in the presence of interactions that vary slowly over the latticespacing, the order parameter space is isomorphic to $S^{5}$. We show that theenlarged symmetry leads to the loss of topologically stable vortices, as wellas two extra gapless modes with quadratic dispersion. The former featureimplies that this non-Abelian condensate is a "failed superfluid" that does notundergo a Berezinskii-Kosterlitz-Thouless (BKT) transition. Order-by-disordersplitting appears suppressed, implying that signatures of the $S^5$ manifoldought to be observable at low temperatures.

Intertwined Superfluid and Density Wave Order in a $p$-Orbital Bose Condensate