  We construct and classify chiral topological phases in driven (Floquet)
systems of strongly interacting bosons, with finite-dimensional site Hilbert
spaces, in two spatial dimensions. The construction proceeds by introducing
exactly soluble models with chiral edges, which in the presence of many-body
localization (MBL) in the bulk are argued to lead to stable chiral phases.
These chiral phases do not require any symmetry, and in fact owe their
existence to the absence of energy conservation in driven systems.
Surprisingly, we show that they are classified by a quantized many-body index,
which is well defined for any MBL Floquet system. The value of this index,
which is always the logarithm of a positive rational number, can be interpreted
as the entropy per Floquet cycle pumped along the edge, formalizing the notion
of quantum-information flow. We explicitly compute this index for specific
models, and show that the nontrivial topology leads to edge thermalization,
which provides an interesting link between bulk topology and chaos at the edge.
We also discuss chiral Floquet phases in interacting fermionic systems and
their relation to chiral bosonic phases.
