Counting statistics for arbitrary cycles in quantum pumps

We study noise-related properties of current in adiabatic pumps. A symmetry of the problem allows us to relate the statistics of charge transport in the case of one-channel leads to the geometry of loops on a sphere (for many channels on a higher-dimensional manifold). This-provides a unifying framework, which simplifies analysis of transport in various realizations of pumps. For each pumping cycle, the average current and its minimal variance are given by the areas enclosed by the corresponding loop on the sphere and on a minimal surface (soap film) spanned by this loop. We formulate conditions for quantization of the pumped charge.