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Montonen–Olive electric–magnetic self-duality in N=4 SU(2) supersymmetric Yang–Mills

Prove that N=4 SU(2) supersymmetric Yang–Mills theory is self-dual under electric–magnetic duality by constructing the required non-perturbative magnetic states and demonstrating the duality for all relevant quantities beyond the BPS sector.

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Background

The text reviews evidence from the supersymmetry algebra indicating a match between lowest-energy electric particle states and magnetic solitonic states in N=4 SYM, suggesting a self-duality (Montonen–Olive duality).

However, the authors stress that while BPS-sector evidence is strong, a complete proof requires constructing non-perturbative magnetic states and establishing duality across the full spectrum and for all quantities, which has not been achieved with mathematical rigor.

References

These lowest-energy states are called BPS states, and for these states the evidence points to the existence of a duality. However, the duality involves constructing non-perturbative magnetic states, and there is no rigorous mathematical proof that it can be done. Also, the duality needs to be demonstrated for all other quantities.

— Dualities in Physics (2509.15866 - Haro et al., 19 Sep 2025) in Section 4.1 Electric-magnetic duality in supersymmetric Yang-Mills theory (First example: N=4 SU(2))