  A number of experiments on the hyperkagome iridate, Na$_4$Ir$_3$O$_8$,
suggest existence of a gapless quantum spin liquid state at low temperature.
Circumventing the slave particle approach commonly used in theoretical analyses
of frustrated magnets, we provide a more intuitive, albeit more
phenomenological, construction of a quantum spin liquid state for the
hyperkagome Heisenberg model. An effective monomer-dimer model on the
hyperkagome lattice is proposed a la Hao and Tchernyshyov's approach cultivated
from the Husimi cactus model. Employing an arrow representation for the
monomer-dimer model, we obtain a compact $U(1)$ gauge theory with a finite
density of Fermionic spinons on the hyperoctagon lattice. The resulting theory
and its mean field treatments lead to remarkable agreement with previous
slave-particle construction of a quantum spin liquid state on the hyperkagome
lattice. Our results offer novel insights on theoretical understanding of the
emergence of spinon Fermi surfaces and useful predictions for future
experiments.
