Abstract
We study 3d mathcal{N}=2 supersymmetric gauge theories on closed oriented Seifert manifolds — circle bundles over an orbifold Riemann surface —, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the “fibering operators.” These operators are half-BPS line defects, whose insertion along the S1 fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L(p, q)b with rational squashing parameter b2 ∈ ℚ, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.
Highlights
Any local quantum field theory (QFT) can be studied on a non-trivial space-time geometry
As described above (1.19), we expect that the supersymmetric partition function on a general Seifert manifold, M3, can be computed as the expectation value of a suitable “geometry changing line operator” LM3 inserted along the circle on the A-twisted S2 × S1 geometry
See [74] for some interesting recent work in that direction. Another interesting research direction concerns the existence of many supersymmetric backgrounds that admit a topologically-trivial canonical line bundle (in addition to Sb3 and the lens space L(p, −1)b), which can be used to study N = 2 superconformal field theories
Summary
Any local quantum field theory (QFT) can be studied on a non-trivial space-time geometry. This is done by coupling the stress-energy tensor to a background metric.. In the case of 3d N = 2 supersymmetric theories with an exact U(1)R symmetry [4,5,6], one finds a large class of compact half-BPS geometries, M3, which preserve two supercharges, Q and Q, satisfying the curved-space supersymmetry algebra: Q2 = 0 ,. We define the geometry-changing line operator for any Seifert manifold, in the case of 3d N = 2 gauge theories This gives us a compact formula for the supersymmetric partition functions, (1.2), for supersymmetric gauge theories on any half-BPS Seifert geometry. In the remainder of this introduction, we review some necessary background material and spell out our main results in some detail
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