Gauge invariants at arbitrary N and trace relations

A bstract We investigate conformal field theories with gauge group U( N ) at arbitrary rank N , focusing on the role of trace relations in determining the structure of the Hilbert space. Working in the free trace algebra without imposing relations, we identify a class of evanescent states that vanish at finite N . Using the Koszul complex of [1], we implement trace relations systematically via ghosts and a fermionic charge Q b . This framework allows us to define and compute transition amplitudes between evanescent and physical states, which we show correspond precisely to ordinary CFT amplitudes analytically continued in N . Our results provide a direct algebraic realization of the proposals which realize trace relations in the bulk as over-maximal giant gravitons [1–3] and establish analytic continuation in N as a powerful tool for understanding finite- N effects.