  The identification, description, and classification of topological features
is an engine of discovery and innovation in several fields of physics. This
research encompasses a broad variety of systems, from the integer and
fractional Chern insulators in condensed matter, to protected states in complex
photonic lattices in optics, and the structure of the QCD vacuum. Here, we
introduce another playground for topology: the dissipative dynamics of
pseudo-Hermitian many-body quantum systems. For that purpose, we study two
different systems, the dissipative Sachdev-Ye-Kitaev (SYK) model, and a quantum
chaotic dephasing spin chain. For the two different many-body models, we find
the same topological features for a wide range of parameters suggesting that
they are universal. In the SYK model, we identify four universality classes,
related to pseudo-Hermiticity, characterized by a rectangular block
representation of the vectorized Liouvillian that is directly related to the
existence of an anomalous trace of the unitary operator implementing fermionic
exchange. As a consequence of this rectangularization, we identify a
topological index $\nu$ that only depends on symmetry. Another distinct
consequence of the rectangularization is the observation, for any coupling to
the bath, of purely real topological modes in the Liouvillian. The level
statistics of these real modes agree with that of the corresponding random
matrix ensemble and therefore can be employed to characterize the four
topological symmetry classes. In the limit of weak coupling to the bath,
topological modes govern the approach to equilibrium, which may enable a direct
path for experimental confirmation of topology in dissipative many-body quantum
chaotic systems.
