Bootstrapping ABJM theory

Supersymmetric localization reduces the computation of protected observables in ABJM theory to finite-dimensional matrix integrals. Building on the techniques introduced in arXiv:2512.02119, we develop a bootstrap framework for the systematic calculation of instanton corrections to the free energy and to supersymmetric Wilson loops. Exploiting exact functional relations and consistency conditions satisfied by grand-canonical observables, in the Fermi-gas formulation of the ABJM matrix model, we provide analytic derivations of several relations for the free energy that were previously known only conjecturally, either from refined topological string theory or from high-precision numerical studies. We apply the same framework to determine the nonperturbative corrections to $1/2$ and $1/6$ BPS Wilson loops, elucidating their qualitative differences and uncovering novel structural features of the instanton effects. These results further highlight the intricate nonperturbative structure and network of dualities underlying ABJM theory.

💡 Research Summary

The paper develops a first‑principles bootstrap framework for computing non‑perturbative instanton corrections to the free energy and supersymmetric Wilson loops in ABJM theory. Starting from supersymmetric localization, the authors recast the finite‑dimensional matrix integrals for the S³ partition function and circular Wilson loops into the Fermi‑gas formulation, where the canonical partition function Z(N,k) is expressed as a Laplace transform of the grand potential J(µ,k). In the large‑N limit the grand potential admits a decomposition into a perturbative cubic polynomial in the chemical potential µ with coefficients C(k), B(k), A(k), and an infinite series of exponentially suppressed non‑perturbative terms ∑{m,ℓ≥0} f{m,ℓ}(µ) e^{-(4mk+2ℓ)µ}. The coefficients f_{m,ℓ}(µ) are polynomials in µ whose physical interpretation corresponds to world‑sheet and membrane instantons in the dual M‑theory description.

The novelty of the work lies in exploiting exact functional relations and consistency conditions satisfied by the grand‑canonical partition function Ξ(µ,k)=∑{N≥0} e^{Nµ} Z(N,k) and the generating functions of Wilson loops Wₙ(µ,k). These observables are periodic under µ→µ+2πi, obey pole‑cancellation constraints, and satisfy a master equation that links the non‑perturbative coefficients f{m,ℓ}(µ) of the free energy to analogous coefficients w_{m,ℓ}(µ) appearing in the expansion of the Wilson loop grand‑canonical expectation value. By inserting the ansätze for J(µ,k) and Wₙ(µ,k) into the integral representations of Z(N,k) and Wₙ(N,k), the authors derive two equivalent expressions for the Wilson loop expectation value. Equating these expressions yields a closed system of functional equations that can be solved recursively.

The bootstrap proceeds by first fixing the perturbative data C(k)=2π²/k and B(k)=1/(3k)+k/24 from the semiclassical WKB analysis of the Fermi‑gas. The remaining perturbative constant A(k) is taken from its all‑orders conjectured form. The non‑perturbative sector is then built order by order: starting from the lowest instanton numbers (m=0,ℓ=1) the functional constraints uniquely determine f_{m,ℓ}(µ) and w_{m,ℓ}(µ) as explicit polynomials. The solution naturally organizes the results into infinite sums of Airy functions and their derivatives, reproducing the known Airy‑function structure of the perturbative series and extending it to the full instanton series.

Applying the method to Wilson loops, the authors distinguish the 1/2‑BPS and 1/6‑BPS cases. For the 1/2‑BPS loop, the bootstrap confirms the Ooguri‑Vafa (OV) type structure previously inferred from topological string dualities, with world‑sheet and membrane instantons appearing in a combined exponential factor. For the 1/6‑BPS loop, which lacks a known topological‑string interpretation, the bootstrap yields the first analytic expressions for the instanton coefficients w_{m,ℓ}(µ), revealing qualitative differences: the 1/6‑BPS loop is dominated by world‑sheet instantons and exhibits a richer polynomial dependence on µ. The relation W_{1/2}ⁿ = W_{1/6}ⁿ − (−1)ⁿ W_{1/6}ⁿ* emerges automatically from the consistency conditions.

The paper also provides analytic proofs of several conjectured relations for the free energy, such as the exact pole‑cancellation mechanism at special Chern‑Simons levels and the precise form of the constant A(k). These results, previously supported only by high‑precision numerics or refined topological‑string arguments, are now derived directly from the matrix model.

In conclusion, the bootstrap framework offers a systematic, fully analytic route to the complete non‑perturbative expansion of ABJM observables, bypassing the need for dual string constructions or numerical fitting. It deepens the understanding of instanton effects in the M‑theory regime, clarifies the interplay between different BPS Wilson loops, and suggests that similar bootstrap strategies could be applied to other three‑dimensional supersymmetric gauge theories.