Gauge Invariants at Arbitrary N and Trace Relations

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 Qb. 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.