  Generalized-unitarity calculations of two-loop amplitudes are performed by
expanding the amplitude in a basis of master integrals and then determining the
coefficients by taking a number of generalized cuts. In this paper, we present
a complete classification of the solutions to the maximal cut of integrals with
the double-box topology. The ideas presented here are expected to be relevant
for all two-loop topologies as well. We find that these maximal-cut solutions
are naturally associated with Riemann surfaces whose topology is determined by
the number of states at the vertices of the double-box graph. In the case of
four massless external momenta we find that, once the geometry of these Riemann
surfaces is properly understood, there are uniquely defined master contours
producing the coefficients of the double-box integrals in the basis
decomposition of the two-loop amplitude. This is in perfect analogy with the
situation in one-loop generalized unitarity.
  In addition, we point out that the chiral integrals recently introduced by
Arkani-Hamed et al. can be used as master integrals for the double-box
contributions to the two-loop amplitudes in any gauge theory. The infrared
finiteness of these integrals allow for their coefficients as well as their
integrated expressions to be evaluated in strictly four dimensions, providing
significant technical simplification. We evaluate these integrals at four
points and obtain remarkably compact results.
