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Gauge Invariants at Arbitrary $N$ and Trace Relations

Pawel Caputa, Robert de Mello Koch

TL;DR

This work develops a robust framework to study U(N) conformal field theories at arbitrary N by working in the free trace algebra and then systematically imposing trace-relations via a Koszul complex with ghosts and a fermionic charge Q_b. Evanescent states, which exist for non-integer N and vanish at finite N, are identified with bulk ghost-brane states, and transition amplitudes between evanescent and physical states precisely match ordinary CFT amplitudes analytically continued in N. The analysis unifies finite-N holographic physics in the 1/2-BPS sector, reproduces the giant-graviton expansion from an algebraic construction, and establishes analytic continuation in N as a powerful tool to understand finite-N effects. The paper provides concrete calculations in Schur/basis language, generalized sub-determinants, and HHL correlators, and demonstrates the equivalence of the evanescent-state formalism with bulk thimble/girant-brane pictures. These results pave the way for extending the approach to less supersymmetric sectors, multi-matrix theories, and broader gauge groups, offering a coherent link between boundary analytic continuation and bulk ghost/giant-graviton constructions.

Abstract

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.

Gauge Invariants at Arbitrary $N$ and Trace Relations

TL;DR

This work develops a robust framework to study U(N) conformal field theories at arbitrary N by working in the free trace algebra and then systematically imposing trace-relations via a Koszul complex with ghosts and a fermionic charge Q_b. Evanescent states, which exist for non-integer N and vanish at finite N, are identified with bulk ghost-brane states, and transition amplitudes between evanescent and physical states precisely match ordinary CFT amplitudes analytically continued in N. The analysis unifies finite-N holographic physics in the 1/2-BPS sector, reproduces the giant-graviton expansion from an algebraic construction, and establishes analytic continuation in N as a powerful tool to understand finite-N effects. The paper provides concrete calculations in Schur/basis language, generalized sub-determinants, and HHL correlators, and demonstrates the equivalence of the evanescent-state formalism with bulk thimble/girant-brane pictures. These results pave the way for extending the approach to less supersymmetric sectors, multi-matrix theories, and broader gauge groups, offering a coherent link between boundary analytic continuation and bulk ghost/giant-graviton constructions.

Abstract

We investigate conformal field theories with gauge group at arbitrary rank , 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 . Using the Koszul complex of [1], we implement trace relations systematically via ghosts and a fermionic charge . 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 . 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 as a powerful tool for understanding finite- effects.

Paper Structure

This paper contains 21 sections, 125 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Holography of the $\frac{1}{2}$-BPS Sector at finite $N$
  3. Correlators with Schurs
  4. Generalized sub-determinants
  5. Correlators with generalized sub-determinants
  6. HHL three-point functions in AdS/CFT
  7. HHL correlators from DBI action
  8. HHL correlators from LLM geometries
  9. Evanescent States and the Giant Graviton Expansion
  10. Evanescent states
  11. $\frac{1}{2}$-BPS index
  12. Giant Graviton Expansion
  13. Transition Amplitudes
  14. Free half-BPS ring $R$
  15. The physical half-BPS ring at finite integer $N$
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Plots of the two-point correlator \ref{['eq:fh1jN']} for sub-determinants labelled by a single column with k boxes $(1^k)$, as function of $k$ (left for $N=7$) and $N$ (right for $k=6$).
  • Figure 2: Plots of $\mathcal{H}/N$ for $l<1$ (left) and $l>1$ (right). Here we chose $l=19/20$ and $l=21/20$.