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Two roads to fortuity in ABJM theory

Connor Behan, Leonardo Pipolo de Gioia

TL;DR

This paper investigates fortuitous operators in ABJM theory through the cohomology of a nilpotent supercharge $Q$, connecting microscopic operator structure to holographic black hole physics. It develops two complementary strategies: a brute-force level-by-level enumeration that yields 244 low-lying fortuitous operators organized into centralizer multiplets, and a BMN-type truncation that maps ABJM bilinears to a sector of ${ m N}=4$ SYM, producing infinite towers of fortuitous representatives. The fortuitous space is controlled by the centralizer ${ m C}(igrace Q,Q^aggerig) \, ext{simeq}\, { m osp}(4|2)oxplus{ m u}(1|1)$, and explicit representatives are constructed for ${ m U}(2)$ and ${ m U}(3)$ ABJM, with trace-relations enabling liftings to ${ m U}(N)$. A superfield approach further aligns ABJM bilinears with ${ m N}=4$ BMN structures, yielding a compact $(1|3)$-dimensional description of a monotone fortuitous sector akin to the SYM construction. The results support the holographic picture in AdS$_4$/CFT$_3$ where fortuitous operators model black hole microstates and point to promising avenues for higher-loop checks and explicit BPS representative construction.

Abstract

A recently proposed addition to the holographic dictionary connects extremal black holes to fortuitous operators -- those which are only supersymmetric for sufficiently small values of the central charge. The most efficient techniques for finding them come from studying the cohomology of a nilpotent supercharge. We explore two aspects of this problem in weakly-coupled ABJM theory, where the gauge group is $\mathrm{U}(N) \times \mathrm{U}(N)$ and the Chern-Simons level is taken to be large. Adapting an algorithm which has been used to great effect in $\mathcal{N} = 4$ Super Yang-Mills, we enumerate 244 low-lying fortuitous operators and sort them into multiplets of the centralizer algebra. This leads to the construction of two leading fortuitous representatives for $N = 3$ which are subleading for $N = 2$. In the second part of this work, we identify a truncation of ABJM theory where the action of the one-loop supercharge matches the one in the BMN subsector of $\mathcal{N} = 4$ Super Yang-Mills. This allows a known infinite tower of representatives to be lifted from one theory to the other.

Two roads to fortuity in ABJM theory

TL;DR

This paper investigates fortuitous operators in ABJM theory through the cohomology of a nilpotent supercharge , connecting microscopic operator structure to holographic black hole physics. It develops two complementary strategies: a brute-force level-by-level enumeration that yields 244 low-lying fortuitous operators organized into centralizer multiplets, and a BMN-type truncation that maps ABJM bilinears to a sector of SYM, producing infinite towers of fortuitous representatives. The fortuitous space is controlled by the centralizer , and explicit representatives are constructed for and ABJM, with trace-relations enabling liftings to . A superfield approach further aligns ABJM bilinears with BMN structures, yielding a compact -dimensional description of a monotone fortuitous sector akin to the SYM construction. The results support the holographic picture in AdS/CFT where fortuitous operators model black hole microstates and point to promising avenues for higher-loop checks and explicit BPS representative construction.

Abstract

A recently proposed addition to the holographic dictionary connects extremal black holes to fortuitous operators -- those which are only supersymmetric for sufficiently small values of the central charge. The most efficient techniques for finding them come from studying the cohomology of a nilpotent supercharge. We explore two aspects of this problem in weakly-coupled ABJM theory, where the gauge group is and the Chern-Simons level is taken to be large. Adapting an algorithm which has been used to great effect in Super Yang-Mills, we enumerate 244 low-lying fortuitous operators and sort them into multiplets of the centralizer algebra. This leads to the construction of two leading fortuitous representatives for which are subleading for . In the second part of this work, we identify a truncation of ABJM theory where the action of the one-loop supercharge matches the one in the BMN subsector of Super Yang-Mills. This allows a known infinite tower of representatives to be lifted from one theory to the other.
Paper Structure (21 sections, 6 theorems, 126 equations, 1 figure, 1 table)

This paper contains 21 sections, 6 theorems, 126 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Holographic Coverings and Black Hole States
  3. BPS operators and supercharge cohomology
  4. Specializing to traces
  5. BPS States at Weak Coupling
  6. Basics of ABJM theory
  7. The supercharge and BPS letters
  8. The centralizer of the BPS condition
  9. Comparison to $\mathcal{N} = 4$ SYM
  10. The question of renormalization
  11. Two Types of Searches
  12. Brute force enumeration
  13. Counting at low level
  14. Explicit representatives
  15. More representatives from $\mathcal{N} = 4$ SYM
  16. ...and 6 more sections

Key Result

Proposition 1

Let $f:{\cal X}\to {\cal X}$ be a mapping of letters and let $F:\tilde{\cal H}\to \tilde{\cal H}$ be the associated induced mapping of formal multi-traces on these letters. If $Q\in \operatorname{End}(\tilde{\cal H})$ is a linear derivation induced by its corresponding action on letters, and if $Q\c

Figures (1)

  • Figure 1: Distribution of file sizes in the search for the simplest fortuitous cohomology class in $\mathcal{N} = 4$ SYM up to $E + J_L = 12$ --- the 15864 files are associated with the different $(Y, J_L, J_R, H_1, H_2, H_3)$ charge sectors. Within a given file, each of the $L$ lines specifies a different admissible grouping of single-trace words into a multi-trace word. The single-trace words we use are limited to length $3$ instead of $2$ for the technical reason described in the text. The longest file corresponds to $\left ( 8, \frac{3}{2}, 0, \frac{5}{2}, \frac{5}{2}, \frac{5}{2} \right )$ and has 53546 lines. The shortest files have one line and there are 173 of them. The file containing the fortuitous class, labelled in red, is $\left ( 7, \frac{5}{2}, 0, \frac{3}{2}, \frac{3}{2}, \frac{3}{2} \right )$ with 1908 lines.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 8 more