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BMS Characters and Modular Invariance

Arjun Bagchi, Amartya Saha, Zodinmawia

TL;DR

The paper advances flat-space holography by constructing characters for BMS$_3$ highest-weight modules, enabling a partition function and a BMS-modular analysis to count primary states. It demonstrates two consistent routes to the same character formula: intrinsic BMS methods via Gram matrices and limiting procedures from 2d CFT Virasoro characters, with a surprising equivalence between NR and UR limits that is traced to a novel automorphism in the parent 2d CFT. The resulting BMS-Cardy analysis shows primaries account for the principal part of the FSC entropy, strengthening the microscopic interpretation of flat-space horizons. The work also opens avenues to understand representation theory subtleties, automorphisms of 2d CFTs, and potential string-theoretic realizations of FSCs.

Abstract

We construct the characters for the highest weight representations of the 3d Bondi-Metzner-Sachs (BMS$_3$) algebra. We then use these to construct the partition function and show how to use BMS modular transformations to obtain a density of primary states. The entropy thus obtained accounts for the principle part of the entropy obtained from the BMS-Cardy formula. This suggests that BMS primaries capture most of the entropy of Flat Space Cosmologies, which are the flatspace analogues of BTZ black holes in AdS$_3$. We reproduce our character formula by looking at singular limits from 2d CFT characters and find that our answers are identical to the characters obtained for the very different induced representations. We offer an algebraic explanation to this arising from a (to the best of our knowledge) novel automorphism in the parent 2d CFT.

BMS Characters and Modular Invariance

TL;DR

The paper advances flat-space holography by constructing characters for BMS highest-weight modules, enabling a partition function and a BMS-modular analysis to count primary states. It demonstrates two consistent routes to the same character formula: intrinsic BMS methods via Gram matrices and limiting procedures from 2d CFT Virasoro characters, with a surprising equivalence between NR and UR limits that is traced to a novel automorphism in the parent 2d CFT. The resulting BMS-Cardy analysis shows primaries account for the principal part of the FSC entropy, strengthening the microscopic interpretation of flat-space horizons. The work also opens avenues to understand representation theory subtleties, automorphisms of 2d CFTs, and potential string-theoretic realizations of FSCs.

Abstract

We construct the characters for the highest weight representations of the 3d Bondi-Metzner-Sachs (BMS) algebra. We then use these to construct the partition function and show how to use BMS modular transformations to obtain a density of primary states. The entropy thus obtained accounts for the principle part of the entropy obtained from the BMS-Cardy formula. This suggests that BMS primaries capture most of the entropy of Flat Space Cosmologies, which are the flatspace analogues of BTZ black holes in AdS. We reproduce our character formula by looking at singular limits from 2d CFT characters and find that our answers are identical to the characters obtained for the very different induced representations. We offer an algebraic explanation to this arising from a (to the best of our knowledge) novel automorphism in the parent 2d CFT.

Paper Structure

This paper contains 18 sections, 144 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. BMS$_3$ holography: a lightning review
  3. Aspects of dual 2d field theory
  4. 3d bulk physics
  5. Highest Weight Characters for BMS
  6. Formula for character and partition function
  7. Character in terms of inner product
  8. Limiting analysis from character of 2d CFT
  9. The Virasoro character
  10. The two limits and the limiting characters
  11. Equal limiting characters: the problem
  12. Hints of a solution
  13. BMS-Cardy formula for primary states
  14. Conclusions
  15. Structure of the BMS Gram matrix
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Phase space of Einstein gravity in asymptotically (a) AdS$_3$ and (b) Mink$_3$.
  • Figure 2: Penrose diagram for Flat Space Cosmologies.