The cusp anomalous dimension in N=4 super-Yang-Mills theory governs the ultraviolet divergence when two Wilson lines meet at an angle. Its perturbative expansion is known to extraordinary precision. Its nonperturbative completion — the full transseries that includes exponentially suppressed contributions — has resisted systematic treatment.
Bajnok, Boldis, and le Plat (arXiv:2603.17943) solve this. The complete transseries is expressed as a ratio of two determinants. The perturbative series sits in one. The nonperturbative corrections organize into sectors classified by partitions of distinct non-negative odd integers — the same combinatorial structure that appears in fermionic systems, where the Pauli exclusion principle forbids repeats.
Each nonperturbative sector comes with a Stokes constant that determines its relative weight when crossing anti-Stokes lines in the complex coupling plane. These constants are computed iteratively from the perturbative data using the resurgence relations that connect different sectors. The structure is self-consistent: the perturbative expansion encodes all nonperturbative information, and vice versa.
The fermionic character of the transseries is striking. The sectors are labeled by partitions into distinct odd integers — exactly the states of a free fermion system. This is not a coincidence of notation but a structural feature: the nonperturbative contributions obey an exclusion principle, with each “instanton number” occupied at most once.
A ratio of determinants. Fermionic selection rules. The full nonperturbative structure of a fundamental gauge theory observable, reduced to linear algebra and combinatorics. The complexity of the physics resolves into the simplicity of the mathematics — as it sometimes does when enough symmetry is present.