''Tracing the origins of exciton shift and topology"

Characterizing the topology of bound electron-hole states is essential for understanding the charge distribution and properties of excitons. In this work, I present a framework that reveals how electron-hole interactions introduce nontrivial topological features to two-dimensional exciton states, beyond those inherited from single-particle bands. A key result is the introduction of the singular connection of the exciton, which isolates the role of interaction and allows us to define a new, gauge-invariant, shift vector for excitons. This quantity characterizes the topology of excitons, providing an intuitive criterion for identifying whether excitons in a system are topological beyond the topology inherited by the single particle bands.