  Arkani-Hamed and collaborators have recently shown that scattering amplitudes
for colored theories can be expressed as integrals over combinatorial objects
simply constructed from surfaces decorated by kinematic data. In this paper we
extend the curve integral formalism to theories with colored fermionic matter
and present a compact formula for the all-loop, all-genus, all-multiplicity
amplitude integrand of a colored Yukawa theory. The curve integral formalism
makes certain properties of the amplitudes manifest and repackages non-trivial
numerators into a single combinatorial object. We also present an efficient
formula for $L$-loop integrated amplitudes in terms of a sum over $2^L$
combinatorial determinants.
