Quantum Geometry and $\theta$-Angle in Five-Dimensional Super Yang-Mills

Abstract

Five-dimensional $Sp(N)$ supersymmetric Yang-Mills admits a ${\mathbb{Z}}_2$ version of a theta angle $\theta$. In this note, we derive a double quantization of the Seiberg-Witten geometry of $\mathcal{N}=1$ $Sp(1)$ gauge theory at $\theta =\pi$, on the manifold $S^1\times {\mathbb{R}}^4$. Crucially, ${\mathbb{R}}^4$ is placed on the $\Omega$-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 $S^1$. Mathematically, this amounts to proving the regularity of a $qq$-character for the spin-1/2 representation of the quantum affine algebra $U_q(\widehat{A_1})$, with a certain twist due to the $\theta$-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 $\kappa$ when $N>2$.

36 pages, 4 figures

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