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 $Q_b$. 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.

💡 Research Summary

The paper investigates U(N) gauge theories, focusing on the half‑BPS sector of N=4 super‑Yang‑Mills, and develops a systematic algebraic framework to treat trace relations at arbitrary (including non‑integer) rank N. The authors begin by considering the free trace algebra in which no trace relations are imposed; in this enlarged Hilbert space all multi‑trace operators built from a complex scalar Z are independent and can be organized using Schur polynomials χ_R(Z) labeled by Young diagrams. Two‑point functions are diagonal, ⟨χ_R(Z)χ_S( Z̄)⟩ = δ_{RS} f_R(N), where f_R(N) is a product of linear factors (N−i+j) that can be written as ratios of Gamma functions. Because Gamma functions admit analytic continuation, these correlators can be continued to arbitrary complex N without obstruction.

At finite integer N, however, the matrix size imposes non‑trivial trace relations: operators whose Young diagram contains more than N boxes vanish, reflecting the stringy exclusion principle and the truncation of the Hilbert space. The authors call the states that disappear at finite N “evanescent states”. To implement the trace relations in a controlled way they import the Koszul complex introduced in earlier work