  When cold atoms are trapped in a square or cubic optical lattice, it should
be possible to pump the atoms into excited $p-$level orbitals within each well.
Following earlier work, we explore the metastable equilibrium that can be
established before the atoms decay into the $s-$wave orbital ground state. We
will discuss the situation with integer number of bosons on every site, and
consider the strong correlation "insulating" regime. By employing a spin-wave
analysis together with a new duality transformation, we establish the existence
and stability of a novel gapless "critical phase", which we refer to as a "bond
algebraic liquid". The gapless nature of this phase is stabilized due to the
emergence of symmetries which lead to a quasi-one dimensional behavior. Within
the algebraic liquid phase, both bond operators and particle flavor occupation
number operators have correlations which decay algebraically in space and time.
Upon varying parameters, the algebraic bond liquid can be unstable to either a
Mott insulator phase which spontaneously breaks lattice symmetries, or a
$\mathbb{Z}_2$ phase. The possibility of detecting the algebraic liquid phase
in cold atom experiments is addressed. Although the momentum distribution
function is insufficient to distinguish the algebraic bond liquid from other
phases, the density correlation function can in principle be used to detect
this new phase of matter.
