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Fortuity in ABJM

Alexandre Belin, Palash Singh, Rita Vadala, Alberto Zaffaroni

TL;DR

This work analyzes fortuitous (non-monotone) BPS cohomologies in ABJM theory, focusing on the $1/12$-BPS sector and, at special levels, the $1/16$-BPS sector. It develops a framework based on Q-cohomology, monopole operators, and the superconformal index to identify and classify fortuitous states across $N=1,2$ and varying Chern–Simons level $k$, including nontrivial monopole sectors. A key finding is that fortuitous states can appear at much smaller quantum numbers than in $ ext{N}=4$ SYM, with explicit operators (such as $ ext{O}_f$ at large $k$) and, at $k=2$, a richer monopole sector giving additional fortuitous cohomologies tied to an SU(4) flavor symmetry. The authors formulate a one-loop exactness conjecture for ABJM cohomologies below a threshold, and demonstrate strong consistency with the superconformal index, shedding light on the black-hole-like microstates in AdS$_4$/CFT$_3$ and the role of trace relations in finite-$N$ spectra.

Abstract

We study $1/12$-BPS and $1/16$-BPS cohomologies and the fortuitous mechanism in ABJM theory. We first establish the existence of fortuitous states in the $N=1$ theory, where the theory is abelian and trace relations are extreme. We then provide explicit constructions of fortuitous states at $N=2$. We find fortuitous states both at weak coupling, in direct parallel to what has been done in $\mathcal{N}=4$ SYM, but we also find additional fortuitous states at $k=2$, which is in the strongly coupled regime. The extra fortuitous states that appear at $k=2$ are in non-trivial monopole sectors. A striking distinction from $\mathcal{N}=4$ SYM is that the fortuitous states appear at much smaller quantum numbers, making them easier to find. Along the way, we formulate a non-renormalization conjecture for cohomologies in ABJM.

Fortuity in ABJM

TL;DR

This work analyzes fortuitous (non-monotone) BPS cohomologies in ABJM theory, focusing on the -BPS sector and, at special levels, the -BPS sector. It develops a framework based on Q-cohomology, monopole operators, and the superconformal index to identify and classify fortuitous states across and varying Chern–Simons level , including nontrivial monopole sectors. A key finding is that fortuitous states can appear at much smaller quantum numbers than in SYM, with explicit operators (such as at large ) and, at , a richer monopole sector giving additional fortuitous cohomologies tied to an SU(4) flavor symmetry. The authors formulate a one-loop exactness conjecture for ABJM cohomologies below a threshold, and demonstrate strong consistency with the superconformal index, shedding light on the black-hole-like microstates in AdS/CFT and the role of trace relations in finite- spectra.

Abstract

We study -BPS and -BPS cohomologies and the fortuitous mechanism in ABJM theory. We first establish the existence of fortuitous states in the theory, where the theory is abelian and trace relations are extreme. We then provide explicit constructions of fortuitous states at . We find fortuitous states both at weak coupling, in direct parallel to what has been done in SYM, but we also find additional fortuitous states at , which is in the strongly coupled regime. The extra fortuitous states that appear at are in non-trivial monopole sectors. A striking distinction from SYM is that the fortuitous states appear at much smaller quantum numbers, making them easier to find. Along the way, we formulate a non-renormalization conjecture for cohomologies in ABJM.

Paper Structure

This paper contains 28 sections, 129 equations, 1 figure, 11 tables.

Table of Contents

  1. Introduction
  2. Summary of results
  3. The $U(N)_k \times U(N)_{-k}$ ABJM theory
  4. Quiver and supersymmetry
  5. Monopole operators
  6. BPS cohomology
  7. Superconformal Index
  8. A non-renormalization conjecture
  9. Monotones vs Fortuitous cohomologies
  10. Fortuity in $U(1)_k \times U(1)_{-k}$ ABJM
  11. Large $k$
  12. Finite $k$
  13. Fortuity in $U(2)_k \times U(2)_{-k}$ at large $k$
  14. Generalities at large $k$
  15. Single-trace cohomologies
  16. ...and 13 more sections

Figures (1)

  • Figure 1: Description of the $U(N)_k\times U(N)_{-k}$ as a 3d $\mathcal{N}=2$ quiver gauge theory. The $(\phi_i,(\lambda_i)_\alpha)$ (and their '$tilde$' counterparts) denote an $\mathcal{N}=2$ chiral multiplet. There are two of each, which explains the double arrows. We have also included the standard notation for ABJM bi-fundamentals $A_i$ and $B_i$, which we will use throughout the paper and that are defined in equation \ref{['eq:BPS_words']}.

Theorems & Definitions (1)

  • Conjecture 1