Counting chiral primaries in , superconformal field theories

168

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: Applicationsmentioning

confidence: 99%

“…The supersymmetric index I(p, q, u) is defined by a trace over states on S 3 in Hamiltonian quantization [1,30,31],…”

Section: S 3 × S 1 and The Supersymmetric Indexmentioning

confidence: 99%

The geometry of supersymmetric partition functions

Closset

Dumitrescu

Festuccia

et al. 2014

168

415

“…The general expression for the superconformal index of an N = 1 theory defined on a primary Hopf surface is given by [3,4] …”

Section: The Indexmentioning

confidence: 99%

“…The rise of localization as a tool to compute exact partition functions, especially in four-dimensional theories [1], has led to many interesting results and allowed the field to flourish (see [2] for a recent collection of reviews). Another quantity of interest is the superconformal index [3,4], which can be defined for theories on Hopf surfaces, complex manifolds with S 1 × S 3 topology. It is not affected by supersymmetry-preserving deformations and as such is invariant under the renormalization group flow.…”

Section: Introductionmentioning

confidence: 99%

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

Brünner

Regalado

Spiridonov

2017

We provide a recipe to extract the supersymmetric Casimir energy of theories defined on primary Hopf surfaces directly from the superconformal index. It involves an SL(3, Z) transformation acting on the complex structure moduli of the background geometry. In particular, the known relation between Casimir energy, index and partition function emerges naturally from this framework, allowing rewriting of the latter as a modified elliptic hypergeometric integral. We show this explicitly for N = 1 SQCD and N = 4 supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it holds more generally. We also use our method to derive an expression for the Casimir energy of the nonlagrangian N = 2 SCFT with E 6 flavour symmetry. Furthermore, we predict an expression for Casimir energy of the N = 1 SP(2N) theory with SU(8) × U(1) flavour symmetry that is part of a multiple duality network, and for the doubled N = 1 theory with enhanced E 7 flavour symmetry.

“…(The details of the remaining cases are discussed in appendix C.) If these two supercharges anticommute, then M is locally conformal to a Calabi-Yau manifold or to H 3 ×R, where H 3 is the three-dimensional hyperbolic space of constant negative curvature. 2 If M is compact, we can say more. In this case it must be globally conformal to a flat torus T 4 or to a K3…”

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).