Two roads to fortuity in ABJM theory

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.

💡 Research Summary

The paper investigates “fortuitous” operators in weakly‑coupled ABJM theory, i.e. operators that are supersymmetric only when the central charge is sufficiently small. The authors adopt the cohomology of a nilpotent supercharge Q as the primary tool for identifying such operators, following the framework introduced in earlier work on holographic coverings. They first formalize the notion of a holographic covering: a universal vector space of formal multi‑trace monomials ˜H together with an ideal I_N that encodes the trace relations specific to N×N matrices. The physical Hilbert space H_N is obtained as the quotient ˜H/I_N. Within this setting, a state is called monotone if its lift ˜ψ is annihilated by the lifted supercharge ˜Q, while it is fortuitous if ˜Q ˜ψ is non‑zero but lies inside the ideal I_N. This definition captures the idea that the operator is Q‑closed only because of matrix identities that disappear in the large‑N limit.

The authors then turn to the field content of ABJM theory (𝒩=6 Chern‑Simons‑matter with gauge group U(N)×U(N) at large level k). They compute the action of Q on the elementary fields (bifundamental scalars C_I, fermions ψ_I, and gauge fields) and extract the “BPS letters” that survive in Q‑cohomology. Using these letters they construct all possible single‑ and multi‑trace operators up to a given total degree. The centralizer algebra – the set of operators commuting with Q – is identified and used to organize the cohomology classes into irreducible multiplets.

To enumerate the fortuitous operators, the authors adapt a brute‑force algorithm that was previously successful in 𝒩=4 SYM. The algorithm proceeds in three stages: (1) generate all trace monomials up to a chosen degree, (2) test the Q‑closure condition Q O = 0 (or Q O ∈ I_N) using symbolic manipulation, and (3) decompose the surviving operators into representations of the centralizer algebra. The implementation is made publicly available. Applying it to ABJM with N≤3 and degree ≤ 6 yields 244 distinct low‑lying fortuitous operators. These operators are sorted into multiplets of the centralizer algebra, revealing a rich structure that depends sensitively on N.

A particularly interesting outcome is the construction of two leading fortuitous representatives for N = 3: one of trace degree four and another of degree five. Both operators become subleading (i.e. trivial in cohomology) when N = 2 because the corresponding trace relations already force them into the ideal I_2. This illustrates how the size of the gauge group controls the existence of fortuitous states.

In the second major part of the work, the authors identify a truncation of ABJM theory whose one‑loop supercharge action coincides exactly with that of the BMN subsector of 𝒩=4 SYM. By redefining the bifundamental fields into BMN‑like combinations (essentially forming complex scalars that mimic the Z‑field of SYM) they show that the Q‑action on these composites matches the SYM expression. Consequently, the infinite tower of BMN operators known to be fortuitous in SYM can be lifted directly to ABJM. This provides an explicit infinite family of fortuitous operators in ABJM, extending far beyond the finite set obtained by brute force.

The paper concludes with a discussion of the holographic interpretation. Fortuitous operators are proposed to correspond to microstates of supersymmetric black holes that exist only when the bulk Newton constant is non‑zero (i.e. at finite N). As N→∞, the trace identities disappear and the operators lift above the BPS bound, mirroring the disappearance of black‑hole microstates in the classical gravity limit. The authors suggest several future directions: extending the enumeration to higher degrees and larger N, exploring the strong‑coupling regime via AdS/CFT, and applying the same methodology to related three‑dimensional Chern‑Simons‑matter theories such as ABJ or BLG.

Overall, the study provides a systematic classification of fortuitous operators in ABJM, demonstrates a concrete bridge to the well‑understood BMN sector of 𝒩=4 SYM, and offers new tools for probing the microscopic structure of holographic black holes.