  We study the low energy effective theory for a non-Fermi liquid state in 2+1
dimensions, where a transverse U(1) gauge field is coupled with a patch of
Fermi surface with N flavors of fermion in the large N limit. In the low energy
limit, quantum corrections are classified according to the genus of the 2d
surface on which Feynman diagrams can be drawn without a crossing in a double
line representation, and all planar diagrams are important in the leading
order. The emerging theory has the similar structure to the four dimensional
SU(N) gauge theory in the large N limit. Because of strong quantum fluctuations
caused by the abundant low energy excitations near the Fermi surface, low
energy fermions remain strongly coupled even in the large N limit. As a result,
there are infinitely many quantum corrections that contribute to the leading
frequency dependence of the Green's function of fermion on the Fermi surface.
On the contrary, the boson self energy is not modified beyond the one-loop
level and the theory is stable in the large N limit. The non-perturbative
nature of the theory also shows up in correlation functions of gauge invariant
operators.
