Quantum geometry and θ-angle in five-dimensional super Yang-Mills

Five-dimensional Sp(N) supersymmetric Yang-Mills admits a ℤ2 version of a theta angle θ. In this note, we derive a double quantization of the Seiberg-Witten geometry of $$ \mathcal{N} $$ N = 1 Sp(1) gauge theory at θ = π, on the manifold S1× ℝ4. Crucially, ℝ4 is placed on the Ω-background, which provides the two parameters to quantize the geometry. Physically, we are counting instantons in the presence of a 1/2-BPS fundamental Wilson loop, both of which are wrapping S1. Mathematically, this amounts to proving the regularity of a qq-character for the spin-1/2 representation of the quantum affine algebra $$ {U}_q\left(\hat{A_1}\right) $$ U q A 1 ̂ , with a certain twist due to the θ-angle. We motivate these results from two distinct string theory pictures. First, in a (p, q)-web setup in type IIB, where the loop is characterized by a D3 brane. Second, in a type I′ string setup, where the loop is characterized by a D4 brane subject to an orientifold projection. We comment on the generalizations to the higher rank case Sp(N) when N > 1, and the SU(N) theory at Chern-Simons level κ when N > 2.