Supersymmetry on curved spaces and holography

167

415

We consider supersymmetric field theories on compact manifolds M and obtain constraints on the parameter dependence of their partition functions Z M . Our primary focus is the dependence of Z M on the geometry of M, as well as background gauge fields that couple to continuous flavor symmetries. For N = 1 theories with a U(1) R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. We find that Z M is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over M and Z M is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional N = 2 theories with a U(1) R symmetry, where the necessary geometric structure on M is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that Z M is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S 3 ×S 1 or S 2 ×S 1 , which are related to supersymmetric indices, and manifolds diffeomorphic to S 3 (squashed spheres). In examples where Z M has been calculated explicitly, our results explain many of its observed properties.

Section: Jhep01(2014)124mentioning

confidence: 99%

Section: Jhep01(2014)124mentioning

confidence: 99%

The geometry of supersymmetric partition functions

Closset

Dumitrescu

Festuccia

et al. 2014

167

415

“…4 The half CS unit in (2.12) has the effect to cancel the quadratic factor in (2.10) so that the anomaly free result can be written in a block factorised form 5 14) in terms of holomorphic variables…”

Section: Factorisationmentioning

confidence: 99%

“…These 4-manifolds can preserve 2 supercharges with opposite R-charge and a holomorphic Killing vector generating the torus action on M 4 [2][3][4]. 1 General results [6,7] state that partition functions on these spaces do not depend on the Hermitian metric but are holomorphic functions of the complex structure parameters and of the background gauge fields through the corresponding vector bundles.…”

Section: Introductionmentioning

confidence: 99%

Factorisation and holomorphic blocks in 4d

Nieri

Pasquetti

2015

165

We study N = 1 theories on Hermitian manifolds of the form M 4 = S 1 × M 3 with M 3 a U(1) fibration over S 2 , and their 3d N = 2 reductions. These manifolds admit an Heegaard-like decomposition in solid tori D 2 × T 2 and D 2 × S 1 . We prove that when the 4d and 3d anomalies are cancelled, the matrix integrands in the Coulomb branch partition functions can be factorised in terms of 1-loop factors on D 2 × T 2 and D 2 × S 1 respectively. By evaluating the Coulomb branch matrix integrals we show that the 4d and 3d partition functions can be expressed as sums of products of 4d and 3d holomorphic blocks.

“…Most authors have focused on supersymmetric theories with an R-symmetry. Riemannian fourmanifolds that admit rigid supersymmetry for N = 1 theories with a U (1) R symmetry were classified in [9,10]. In this paper, we will complete the classification by analyzing theories that do not possess a U (1) R symmetry.…”

Section: Introductionmentioning

confidence: 99%

Exploring curved superspace (II)

Dumitrescu

Festuccia

2013

129

We extend our previous analysis of Riemannian four-manifolds M admitting rigid supersymmetry to N = 1 theories that do not possess a U (1) R symmetry. With one exception, we find that M must be a Hermitian manifold. However, the presence of supersymmetry imposes additional restrictions. For instance, a supercharge that squares to zero exists, if the canonical bundle of the Hermitian manifold M admits a nowhere vanishing, holomorphic section. This requirement can be slightly relaxed if M is a torus bundle over a Riemann surface, in which case we obtain a supercharge that squares to a complex Killing vector. We also analyze the conditions for the presence of more than one supercharge.The exceptional case occurs when M is a warped product S 3 × R, where the radius of the round S 3 is allowed to vary along R. Such manifolds admit two supercharges that generate the superalgebra OSp(1|2).