Non-Coulomb $Q$-Cohomology and Quantum Lifting

Abstract: We study quantum lifting of classical BPS cohomology classes in four-dimensional $\mathcal N=4$ super-Yang-Mills theory. Certain powers of the gaugino bilinear $\mathcal S=\text{Tr}(λ<em>+λ</em>-)$ in orthogonal theories provide non-Coulomb monotones: multi-graviton classes that vanish upon Cartan restriction. For $SO(7)$, we identify $\mathcal S<sup>4$ as a superconformal descendant of a known lifted monotone class, explaining the latter's non-Coulomb character, and explicitly construct a fortuitous lifting partner for $\mathcal S<sup>4$. This construction provides a strategy for finding analogous partners for higher powers at higher ranks. Our main result establishes restrictions on the quantum lifting of classical BPS cohomologies for general gauge groups. We show that the one-loop correction $Q_1$ to the supercharge annihilates every monotone class in classical cohomology, while every non-zero $Q_1$ image is of non-Coulomb type. Thus every non-zero one-loop pairing has a fortuitous source and a non-Coulomb target, so that Coulomb monotones survive at this order at finite rank. In the BMN sector, our results imply that non-Coulomb monotones survive at one loop order as well. A crucial step in the argument shows that every non-Coulomb class, fortuitous or monotone, is annihilated in classical cohomology by a sufficiently high power of a scalar discriminant, which we interpret as a partial no-hair property.