arXiv : Supersymmetric partition functions and the three-dimensional A-twist

Closset, Cyril; Kim, Heeyeon; Willett, Brian

doi:10.1007/JHEP03(2017)074

Article
Report number	arXiv:1701.03171 ; CERN-TH-2017-006
Title	Supersymmetric partition functions and the three-dimensional A-twist
Related title	Supersymmetric partition functions and the three-dimensional A-twist
Author(s)	Closset, Cyril (CERN) ; Kim, Heeyeon (Perimeter Inst. Theor. Phys.) ; Willett, Brian (Santa Barbara, KITP)
Publication	2017-03-14
Imprint	2017-01-11
Number of pages	112
Note	84 pages plus appendix, 8 figures; v2: added references
In:	JHEP 03 (2017) 74
DOI	10.1007/JHEP03(2017)074
Subject category	hep-th ; Particle Physics - Theory
Abstract	We study three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on $\mathcal{M}_{g,p}$, an oriented circle bundle of degree $p$ over a closed Riemann surface, $\Sigma_g$. We compute the $\mathcal{M}_{g,p}$ supersymmetric partition function and correlation functions of supersymmetric loop operators. This uncovers interesting relations between observables on manifolds of different topologies. In particular, the familiar supersymmetric partition function on the round $S^3$ can be understood as the expectation value of a so-called "fibering operator" on $S^2 \times S^1$ with a topological twist. More generally, we show that the 3d $\mathcal{N}=2$ supersymmetric partition functions (and supersymmetric Wilson loop correlation functions) on $\mathcal{M}_{g,p}$ are fully determined by the two-dimensional A-twisted topological field theory obtained by compactifying the 3d theory on a circle. We give two complementary derivations of the result. We also discuss applications to F-maximization and to three-dimensional supersymmetric dualities.
Copyright/License	arXiv nonexclusive-distrib. 1.0 preprint: © 2017-2026 CERN (License: CC-BY-4.0) publication: © 2017-2026 The Author(s) (License: CC-BY-4.0), sponsored by SCOAP³

$Poles of integrand for $U(1)_{-1/2}$ with one chiral, for $g=0,\;p=1$. Poles due to the positively charged chiral multiplet are shown in blue, and the ``poles at infinity'' from the negatively charged monopole, $T_+$, are denoted by the red line.$ $The $p \neq 0$ JK contour, for $\eta >0$ and $\eta<0$, respectively. For $\eta>0$ the contour surrounds the poles due to positively charged chirals in an anti-clockwise manner, and for $\eta<0$ it surrounds the poles due to negatively charged chirals in a clockwise manner. Only the part of the contour at infinity that satisfies the condition $ \sign\left(\text{Im}(\d_u \CW)\right) =- \sign(\eta) $ should be included in the respective contours.$ $JK contour $\CC_{-1}^\eta$ shown for $\eta>0$ in blue, and $\CC_{0}^\eta$ for $\eta<0$ in red. Here we assume $\text{Im}(\tau)<0$, which implies that the contribution at $\text{Im}(u) \rightarrow \infty$ is included in the $\eta>0$ contour.$ Show more plots

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