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

Abstract

Five-dimensional Sp(N) supersymmetric Yang-Mills admits a Z 2 version of a theta angle θ. In this note, we derive a double quantization of the Seiberg-Witten geometry of N = 1 Sp(1) gauge theory at θ = π, on the manifold S 1 × R 4. Crucially, R 4 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 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 (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.

5d/6d Wilson loops from blowups

“…for θ = π. The first case for θ = 0 perfectly matches the result from the ADHM calculation in [55,56,59] and the case for θ = π agrees with the result in [75]. Higher order instanton corrections of these VEVs can be obtained in a similar fashion by solving the higher order equations.…”

On the quantization of Seiberg-Witten geometry

“…Indeed, such an angle exists in five dimensions, since π4(Sp(N )) = Z2. The analysis in that case is carried out in [53]. 16 This is in contrast to other physical quantities, such as the superconformal index, for example; see for instance the work [51], where the index is highly sensitive to this anomalous shift.…”

Non-Perturbative Schwinger-Dyson Equations for 3d ${\cal N} = 4$ Gauge Theories

“…The above construction can be generalized in many ways, for example by considering additional defects in the background [8,9], by studying different gauge groups [10,11], or by going away from four dimensions: the case of a five-dimensional gauge theory compactified on a circle has been an particularly fruitful area of research [12][13][14][15][16][17][18][19][20][21], where the qq-character observable arises not as an object defined in the representation theory of Yangians, but instead in the representation theory of quantum affine algebras. Likewise, in the case of a six-dimensional gauge theory compactified on a 2-torus [22,23], the qq-character observable becomes an object in the representation theory of quantum elliptic algebras.…”

“…An analogous Lorentz force was identified in a five-dimensional context, where the Wilson loop quark is moving instead in a nontrivial instanton background[12] 6. This program was recently carried out successfully in instanton physics in five dimensions[2,10,11,13,18,21]: there, the problem of counting instantons in the presence of a Wilson loop is not solved by localizing the loop on usual ADHM solutions[52], but by defining instead a more general "crossed instanton" moduli space from the onset. The fact that a similar problem arises in our context is not too surprising, since…”