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Finite $N$ Hilbert Spaces of Bilocal Holography

Robert de Mello Koch, Antal Jevicki, Junggi Yoon

Abstract

For vector/AdS and dS holography we establish the structure of the emergent Hilbert space. This is done through implementation of finite $N$ trace relations on the infinite collective space. For fermionic theories a finite Hilbert space is established, while for bosonic theories a space of freely acting primaries multiplied by a finite set of secondaries emerges. The Hilbert space of states obey finite $N$ cut off bounds, implying finiteness of traces and entropy.

Finite $N$ Hilbert Spaces of Bilocal Holography

Abstract

For vector/AdS and dS holography we establish the structure of the emergent Hilbert space. This is done through implementation of finite trace relations on the infinite collective space. For fermionic theories a finite Hilbert space is established, while for bosonic theories a space of freely acting primaries multiplied by a finite set of secondaries emerges. The Hilbert space of states obey finite cut off bounds, implying finiteness of traces and entropy.
Paper Structure (21 sections, 158 equations, 1 figure)

This paper contains 21 sections, 158 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Collective Representations
  3. Bosons
  4. Fermions
  5. Phase Space
  6. Trace Relations and Reduction of Hilbert Space
  7. Hilbert space from trace relations
  8. Counting: Molien-Weyl vs Hilbert space
  9. Molien-Weyl counting of fermionic singlets
  10. Kähler Quantization
  11. Trace and Partition Function
  12. Conclusions
  13. Other groups
  14. Sp(N) gauge group
  15. SO(N) gauge group
  16. ...and 6 more sections

Figures (1)

  • Figure 1: A plot of the logarithm of the normalized partition function, $\log\!(Z(\beta)/N_Z)$, as a function of the temperature $T$ (with $\beta \equiv 1/T$). The temperature is measured in units of $\omega^{-1}$. The example shown has $N=10$ and $K=12$.