  The Riemann surface for polylogarithms of half-integer index, which has the
topology of an infinite dimensional hypercube, is studied in relation to
one-dimensional KPZ universality in finite volume. Known exact results for
fluctuations of the KPZ height with periodic boundaries are expressed in terms
of meromorphic functions on this Riemann surface, summed over all the sheets of
a covering map to an infinite cylinder. Connections to stationary large
deviations, particle-hole excitations and KdV solitons are discussed.
