We present a uniform treatment of rigid supersymmetric field theories in a curved spacetime $ \mathcal{M} $, focusing on four-dimensional theories with four supercharges. Our discussion is significantly simpler than earlier treatments, because we use classical background values of the auxiliary fields in the supergravity multiplet. We demonstrate our procedure using several examples. For $ \mathcal{M} = Ad{S_4} $ we reproduce the known results in the literature. A supersymmetric Lagrangian for $ \mathcal{M} = {\mathbb{S}^4} $ exists, but unless the field theory is conformal, it is not reflection positive. We derive the Lagrangian for $ \mathcal{M} = {\mathbb{S}^3} \times \mathbb{R} $ and note that the time direction $ \mathbb{R} $ can be rotated to Euclidean signature and be compactified to $ {\mathbb{S}^1} $ only when the theory has a continuous R-symmetry. The partition function on $ \mathcal{M} = {\mathbb{S}^3} \times {\mathbb{S}^1} $ is independent of the parameters of the flat space theory and depends holomorphically on some complex background gauge fields. We also consider R-invariant $ \mathcal{N} = 2 $ theories on $ {\mathbb{S}^3} $ and clarify a few points about them.

Rigid Supersymmetric Theories in Curved Superspace

The three sphere partition function, Z , of three dimensional theories with four supercharges and an R-symmetry is computed using localization, resulting in a matrix integral over the Cartan of the gauge group. There is a family of couplings to the curved background, parameterized by a choice of R-charge, such that supersymmetry is preserved; Z is a function of those parameters. The magnitude of the result is shown to be extremized for the superconformal R-charge of the infrared conformal field theory, in the absence of mixing of the R-symmetry with accidental symmetries. This exactly determines the IR superconformal R-charge.

The exact superconformal R-symmetry extremizes Z

The partition function of $${\mathcal{N}=6}$$
 supersymmetric Chern–Simons-matter theory (known as ABJM theory) on $${\mathbb{S}^3}$$
 , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on $${{\rm AdS}_4\times\mathbb{C}\mathbb{P}^3}$$
 and gives the correct N
 3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in $${\mathbb{C}\mathbb{P}^3}$$
 . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.

/pdf/from-weak-to-strong-coupling-in-abjm-theory-1kf6pbv8t6.pdf

From weak to strong coupling in ABJM theory

We study Euclidean 3D $ \mathcal{N} = 2 $ supersymmetric gauge theories on squashed three-spheres preserving isometries SU(2) × U(1) or U(1) × U(1) We show that, when a suitable background U(1) gauge field is turned on, these squashed spheres support charged Killing spinors and therefore $ \mathcal{N} = 2 $ supersymmetric gauge theories We present the Lagrangian and supersymmetry rules for general gauge theories The partition functions are computed using localization principle, and are expressed as integrals over Coulomb branch For the squashed sphere with U(1) × U(1) isometry, its measure and integrand are identified with the building blocks of structure constants in Liouville or Toda conformal field theories with b ≠ 1

SUSY gauge theories on squashed three-spheres

We extend theformulaforpartitionfunctions of $$\mathcal{N}=2 $$
 superconformalgauge theories on S
 3 obtained recently by Kapustin, Willett and Yaakov, to incorporate matter fields with arbitrary R-charge assignments. We use the result to check that the self-mirror property of $$\mathcal{N}=4 $$
 SQED with two electron hypermultiplets is preserved under a certain mass deformation which breaks the supersymmetry to $$\mathcal{N}=2 $$
 .

Notes on SUSY gauge theories on three-sphere

We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N = 2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.

/pdf/exact-results-for-wilson-loops-in-superconformal-chern-1rdc9v6ens.pdf

Exact results for Wilson loops in superconformal Chern-Simons theories with matter

We test several conjectural dualities between strongly coupled superconformal field theories in three dimensions by computing their exact partition functions on a three
sphere as a function of Fayet-Iliopoulos and mass parameters. The calculation is carried out using localization of the path integral and the matrix model previously derived for superconformal N = 2 gauge theories. We verify that the partition functions of quiver theories related by mirror symmetry agree provided the mass parameters and the Fayet-Iliopoulos parameters are exchanged, as predicted. We carry out a similar calculation for the mirror of N = 8 super-Yang-Mills theory and show that its partition function agrees with that of the ABJM theory at unit Chern-Simons level. This provides a nonperturbative test of the conjectural equivalence of the two theories in the conformal limit.

/pdf/nonperturbative-tests-of-three-dimensional-dualities-5ay2vl1yd2.pdf

Nonperturbative Tests of Three-Dimensional Dualities

We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N=2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.

/pdf/exact-results-for-wilson-loops-in-superconformal-chern-57kwvp6208.pdf

Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter

We consider 3d $ \mathcal{N}\geq 2 $
 superconformal field theories on a branched covering of a three-sphere. The Renyi entropy of a CFT is given by the partition function on this space, but conical singularities break the supersymmetry preserved in the bulk. We turn on a compensating R-symmetry gauge field and compute the partition function using localization. We define a supersymmetric observable, called the super Renyi entropy, parametrized by a real number q. We show that the super Renyi entropy is duality invariant and reduces to entanglement entropy in the q → 1 limit. We provide some examples.

/pdf/supersymmetric-renyi-entropy-1kuabl4sdv.pdf

Supersymmetric Rényi entropy

We define a class of supersymmetric defect loop operators in N = 2 gauge theories in 2 + 1 dimensions. We give a prescription for computing the expectation value of such operators in a generic N = 2 theory on the three-sphere using localization. We elucidate the role of defect loop operators in IR dualities of supersymmetric gauge theories, and write down their transformation properties under the SL(2, Z ) action on conformal theories with abelian global symmetries.

/pdf/exact-results-for-supersymmetric-abelian-vortex-loops-in-2-1-4sozum3wad.pdf

Itamar Yaakov

Papers

Exact results for Wilson loops in superconformal Chern-Simons theories with matter

Nonperturbative Tests of Three-Dimensional Dualities

Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter

Supersymmetric Rényi entropy

Exact results for supersymmetric abelian vortex loops in 2 + 1 dimensions