#### DMCA

## Rigid 6D supersymmetry and localization (2012)

### Citations

368 | Localization of gauge theory on a four-sphere and supersymmetric Wilson loops,” arXiv:0712.2824 [hep-th
- Pestun
(Show Context)
Citation Context ... on curved backgrounds have regained considerable attention, in particular for the application of localization techniques in the computation of indices, partition functions and Wilson loops, see e.g. =-=[1, 2, 3, 4, 5]-=-. A systematic approach to the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in [7, 8, 9, 10, 11]. Holographic applications of these the... |

251 |
Exact Results for Wilson Loops
- Kapustin, Willett, et al.
(Show Context)
Citation Context ... on curved backgrounds have regained considerable attention, in particular for the application of localization techniques in the computation of indices, partition functions and Wilson loops, see e.g. =-=[1, 2, 3, 4, 5]-=-. A systematic approach to the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in [7, 8, 9, 10, 11]. Holographic applications of these the... |

189 |
NonKahler string backgrounds and their five torsion classes
- Cardoso, Curio, et al.
(Show Context)
Citation Context ...mediate consequence of the SKT condition is that M6 cannot be special hermitian [35].7 The first two lines of (5.27) also appear as necessary conditions for supersymmetric heterotic compactifications =-=[37, 38, 39]-=-. This is not surprising given the fact that the Killing spinor equations (2.1), (2.3) are of the same form as those for the vanishing of the gravitino and dilatino variation in the heterotic theory, ... |

151 |
Supersymmetries and their representations," Nucl. Phys. B135
- Nahm
- 1978
(Show Context)
Citation Context ...rsymmetric theory cannot be defined on the round six-sphere, at least for θr 6= 0. The latter conclusion is in accordance with the representation-theoretic classification of Euclidean supersymmetries =-=[42]-=-, as pointed out recently in [17]. Note however that the classification of [42] only considers “spacetimes” whose isometries are simple Lie groups. In particular it does not cover the cases without an... |

149 |
Real Killing spinors and holonomy
- Bär
- 1993
(Show Context)
Citation Context ...n be defined as sixdimensional manifolds admitting a Killing spinor, ∇mη = 14W1γmηc, cf. eq. (5.8), so that they are Einstein, Rmn = 5 4 |W1|2gmn, and the metric cone over M6 is a G2holonomy manifold =-=[40]-=-. In the case of a nearly KählerM6 it can immediately be seen from eqs. (5.10) that W1 is constant. Performing a constant phase redefinition of Ω we can take W1 = iw, with w a real constant. Eqs. (5.... |

82 |
Stable forms and special metrics, in Global Differential Geometry
- Hitchin
(Show Context)
Citation Context ....33) and: ⋆ H = 1 2 Im (W1Ω ∗) +W3 −W4 ∧ J . (5.34) 7A special hermitian six-manifold is a manifold which is both half-flat (and therefore can be lifted to a seven-dimensional manifold of G2 holonomy =-=[36]-=-) and hermitian. Equivalently, a special hermitian six-manifold is a manifold which admits an SU(3) structure whose only non-zero torsion class is W3. 14 Finally, the necessary and sufficient conditio... |

81 |
Proeyen, Yang-Mills theories with local supersymmetry: Lagrangian, transformation laws and superhiggs effect,
- Cremmer, Ferrara, et al.
- 1983
(Show Context)
Citation Context ...p independent solutions ǫ and q independent solutions ζ̃.3 Flat space with vanishing background fields would thus correspond to N = (4, 4). Note the unlike the analogous structures in four dimensions =-=[22, 6]-=- the Killing spinor equations (2.1), (2.3) for ǫ and ζ̃ are entirely decoupled. For the study of minimal supersymmetry we can thus consistently set ζ̃ = 0. While we will give a full-fledged analysis o... |

63 | Families of strong KT structures in six dimensions
- Fino, Parton, et al.
(Show Context)
Citation Context ...2) The third line of (5.27) is the condition that the Hermitian manifold M6 is strong Kähler with torsion (SKT). One immediate consequence of the SKT condition is that M6 cannot be special hermitian =-=[35]-=-.7 The first two lines of (5.27) also appear as necessary conditions for supersymmetric heterotic compactifications [37, 38, 39]. This is not surprising given the fact that the Killing spinor equation... |

61 |
Riemannian Holonomy Groups and Calibrated Geometry
- Joyce
(Show Context)
Citation Context ...onnection ∇′ with torsion given by the background three-form H . Therefore the holonomy of ∇′ (in the spinor representation) is a subgroup of G, the stabiliser group of ǫ, and M6 admits a G-structure =-=[32]-=-. Since ǫ transforms in the 4 of the structure group Spin(6) ∼= SU(4) of the Riemannian spin manifold M6, the stabiliser is G = SU(3) as can be seen by the decomposition 4→ 3⊕ 1 under SU(4)→ SU(3). He... |

55 | Introducing Cadabra: A symbolic computer algebra system for field theory problems. arXiv:hep-th/0701238
- Peeters
- 2007
(Show Context)
Citation Context ...lobal U(1) acting on the fermions with parameter ξH that is constant due to (2.9). 5 4 Part of these and the following calculations have been facilitated by use of the computer algebra system Cadabra =-=[26, 27]-=-. 5As we will see in the following the full analysis of the Killing spinor equations shows that the constant ξH vanishes in the case of backgrounds for which E ◦ m = 0, but is generically non-vanishin... |

49 |
A field-theory motivated approach to symbolic computer algebra
- Peeters
(Show Context)
Citation Context ...lobal U(1) acting on the fermions with parameter ξH that is constant due to (2.9). 5 4 Part of these and the following calculations have been facilitated by use of the computer algebra system Cadabra =-=[26, 27]-=-. 5As we will see in the following the full analysis of the Killing spinor equations shows that the constant ξH vanishes in the case of backgrounds for which E ◦ m = 0, but is generically non-vanishin... |

47 | The intrinsic torsion of SU(3) and G2 structures
- Chiossi, Salamon
(Show Context)
Citation Context ...) and (0,1) parts. Moreover, it can be seen that Ω ∧ J = 0 Ω ∧ Ω∗ = 4i 3 J3 . (5.7) 10 The globally defined forms J , Ω subject to the above conditions can be seen to specify an SU(3) structure on M6 =-=[34]-=-. The intrinsic torsion parametrizes the failure of the spinor η to be covariantly constant. In the case of an SU(3)-structure manifoldM6, the intrinsic torsion decomposes into five modules (torsion c... |

47 | Geometric model for complex non-Kähler manifolds with SU(3) structure
- Goldstein, Prokushkin
(Show Context)
Citation Context ...s along the z3 direction), K is not ∂-closed: ∂K = dz1 ∧ dz2. Hence the conditions (5.46) for (1,2) supersymmetry are not satisfied. T2 bundles over noncompact K3 This example is based on the work of =-=[44]-=- (see also [45]). In that reference it was shown that six-dimensional manifolds with SU(3) structure can be constructed as T2 fibrations over K3 surfaces. The metric on the total space is given by ds2... |

43 | Deformations of generalized Calibrations and compact non-Kähler manifolds with vanishing first Chern class
- Gutowski, Ivanov, et al.
(Show Context)
Citation Context ...s are simple Lie groups. In particular it does not cover the cases without any isometries, such as e.g. Calabi-Yau manifolds. 17 The round S3 × S3 This example is based on an SU(3) structure given in =-=[43]-=-. We identify S3×S3 with the group manifold SU(2)×SU(2). The orthonormal frame is given by the SU(2)-invariant one-forms ea, fa, a = 1, 2, 3, satisying dea = 1 2 εabce b ∧ ec dfa = 1 2 εabcf b ∧ f c .... |

38 |
Topological Quantum Field Theory,” Commun.Math.Phys
- Witten
- 1988
(Show Context)
Citation Context ... leaves invariant the expectation values of Q-closed operators. Hence we may take the limit of zero coupling constant, e2 → 0, upon which the theory localizes to the set Σ of critical points of Q · U =-=[47]-=-. In this limit the path integral can be performed by restricting S to Σ and computing a one-loop determinant describing the fluctuations normal to Σ. This procedure has been carried out in detail in ... |

34 | Rigid supersymmetric theories in curved superspace
- Festuccia, Seiberg
- 2011
(Show Context)
Citation Context ...omputation of indices, partition functions and Wilson loops, see e.g. [1, 2, 3, 4, 5]. A systematic approach to the study of rigid supersymmetry in four-dimensional curved space has been initiated in =-=[6]-=- and further developed in [7, 8, 9, 10, 11]. Holographic applications of these theories have been studied in [12, 13] by embedding the curved space at the boundary of an asymptotically AdS space. A si... |

30 | Proeyen, “Superconformal tensor calculus and matter couplings in six dimensions”, Nucl. Phys. B264 - Bergshoeff, Sezgin, et al. - 1986 |

24 |
Rigid supersymmetric theories
- Samtleben, Tsimpis
(Show Context)
Citation Context ...ion functions and Wilson loops, see e.g. [1, 2, 3, 4, 5]. A systematic approach to the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in =-=[7, 8, 9, 10, 11]-=-. Holographic applications of these theories have been studied in [12, 13] by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories h... |

24 | Exploring Curved Superspace
- Dumitrescu, Festuccia, et al.
(Show Context)
Citation Context ...ion functions and Wilson loops, see e.g. [1, 2, 3, 4, 5]. A systematic approach to the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in =-=[7, 8, 9, 10, 11]-=-. Holographic applications of these theories have been studied in [12, 13] by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories h... |

23 |
On the instantons and the hypermultiplet mass
- Okuda, Pestun
(Show Context)
Citation Context ... on curved backgrounds have regained considerable attention, in particular for the application of localization techniques in the computation of indices, partition functions and Wilson loops, see e.g. =-=[1, 2, 3, 4, 5]-=-. A systematic approach to the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in [7, 8, 9, 10, 11]. Holographic applications of these the... |

21 |
Superstrings with Torsion” Nucl.Phys. B274
- Strominger
- 1986
(Show Context)
Citation Context ...mediate consequence of the SKT condition is that M6 cannot be special hermitian [35].7 The first two lines of (5.27) also appear as necessary conditions for supersymmetric heterotic compactifications =-=[37, 38, 39]-=-. This is not surprising given the fact that the Killing spinor equations (2.1), (2.3) are of the same form as those for the vanishing of the gravitino and dilatino variation in the heterotic theory, ... |

17 |
An index to count chiral primaries in N = 1 d = 4 superconformal field theories,” arXiv:hep-th/0510060. 59
- Romelsberger
(Show Context)
Citation Context |

17 |
A Possible constructive approach to (super-φ3)4. 1. Euclidean formulation of the model
- Nicolai
- 1978
(Show Context)
Citation Context ....) Again, the spinors λr and νr are not related by complex conjugation, such that the variation of the vector fields and the Lagrangian 5 are in fact complex, as usual in Euclidean supersymmetry, cf. =-=[23, 24, 25]-=-, see also the discussion in section 6. It is straightforward to check that the Lagrangian (3.1) is invariant under the supersymmetry transformations rules δAm r = ν̃rγmǫ− ζ̃γmλr , δλr = 1 4 γmnǫ Fmn ... |

16 |
New superconformal models in six dimensions: Gauge group and representation structure
- Samtleben, Sezgin, et al.
- 2013
(Show Context)
Citation Context ... allowing for minimal couplings between vector and tensor fields with the latter charged under a non-abelian gauge group. The corresponding supersymmetric system in flat space has been constructed in =-=[28, 29]-=- and allows for an action modulo the standard subtleties concerning self-dual three-forms. Replacing the constant supersymmetry parameters of that system by solutions to the Killing spinor equations (... |

14 |
Supersymmetric Gauge Theories on the Five-Sphere
- Hosomichi, Seong, et al.
- 2012
(Show Context)
Citation Context ...tudied in [12, 13] by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories has been performed for curved five-dimensional spaces in =-=[14, 15]-=- and in [16] for three-manifolds. The construction of these curved theories is based on the existence of an underlying off-shell supergravity in which the full off-shell supergravity multiplet is set ... |

14 | Gauge dyonic strings and their global limit
- Duff, Liu, et al.
- 1998
(Show Context)
Citation Context ...nsor-YM system à la [18] as well as its dual formulation, both in 6D Minkowski spacetime, were also obtained from different global limits of the (on-shell) heterotic supergravity compactified on K3, =-=[52]-=-. It should be straightforward to extend the localization results presented here to nonCalabi-Yau manifolds. This would entail modifying the analysis of section 6 to include non-vanishing background f... |

12 | Rigidly supersymmetric gauge theories on curved superspace
- Jia, Sharpe
- 2012
(Show Context)
Citation Context ...ion functions and Wilson loops, see e.g. [1, 2, 3, 4, 5]. A systematic approach to the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in =-=[7, 8, 9, 10, 11]-=-. Holographic applications of these theories have been studied in [12, 13] by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories h... |

12 |
Couplings of selfdual tensor multiplet in six-dimensions
- Bergshoeff, Sezgin, et al.
- 1996
(Show Context)
Citation Context ... will study the coupling of off-shell Yang-Mills (YM) multiplets to a number of uncharged on-shell tensor multiplets. Such couplings have been constructed in 6D flat space-time of Minkowski signature =-=[18]-=-. In view of the applications mentioned above, we shall extend this model to curved background and in Euclidean signature. One way to construct these theories is to start from the Euclidean version of... |

11 |
Supersymmetry on curved spaces and holography
- Klare, Tomasiello, et al.
(Show Context)
Citation Context ...the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in [7, 8, 9, 10, 11]. Holographic applications of these theories have been studied in =-=[12, 13]-=- by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories has been performed for curved five-dimensional spaces in [14, 15] and in [1... |

11 |
Six-dimensional superconformal couplings of non-Abelian tensor and hypermultiplets
- Samtleben, Sezgin, et al.
- 2013
(Show Context)
Citation Context ...curved space. It would also be interesting to analyze if the curved space models can be generalized to include couplings to hypermultiplets, by applying the same procedure to the flat space models of =-=[30, 31]-=-. 5 Killing spinor equations II So far, we have derived the couplings and field equations for vector and tensor multiplets on a curved background under the assumption that the supersymmetry parameters... |

10 |
Supersymmetry in Lorentzian curved spaces and holography,” arXiv:1207.2181 [hep-th
- Cassani, Klare, et al.
(Show Context)
Citation Context ...the study of rigid supersymmetry in four-dimensional curved space has been initiated in [6] and further developed in [7, 8, 9, 10, 11]. Holographic applications of these theories have been studied in =-=[12, 13]-=- by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories has been performed for curved five-dimensional spaces in [14, 15] and in [1... |

8 |
Axioms for Euclidean Green’s functions Commun
- Osterwalder, Schrader
- 1973
(Show Context)
Citation Context ....) Again, the spinors λr and νr are not related by complex conjugation, such that the variation of the vector fields and the Lagrangian 5 are in fact complex, as usual in Euclidean supersymmetry, cf. =-=[23, 24, 25]-=-, see also the discussion in section 6. It is straightforward to check that the Lagrangian (3.1) is invariant under the supersymmetry transformations rules δAm r = ν̃rγmǫ− ζ̃γmλr , δλr = 1 4 γmnǫ Fmn ... |

6 | Rigid supersymmetric backgrounds of minimal off-shell supergravity
- Liu, Zayas, et al.
(Show Context)
Citation Context |

6 |
On Supersymmetric Gauge Theories on
- Terashima
(Show Context)
Citation Context ...tudied in [12, 13] by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories has been performed for curved five-dimensional spaces in =-=[14, 15]-=- and in [16] for three-manifolds. The construction of these curved theories is based on the existence of an underlying off-shell supergravity in which the full off-shell supergravity multiplet is set ... |

6 | Supersymmetric field theories on three-manifolds
- Closset, Dumitrescu, et al.
- 2013
(Show Context)
Citation Context ...3] by embedding the curved space at the boundary of an asymptotically AdS space. A similar analysis of supersymmetric theories has been performed for curved five-dimensional spaces in [14, 15] and in =-=[16]-=- for three-manifolds. The construction of these curved theories is based on the existence of an underlying off-shell supergravity in which the full off-shell supergravity multiplet is set to classical... |

5 | Global supersymmetry on curved spaces in various dimensions
- Kehagias, Russo
(Show Context)
Citation Context ...which in turn poses non-trivial constraints on the background fields.1 1 A recent approach for the construction of theories starting from an on-shell formulation of supergravity has been discussed in =-=[17]-=-. 1 In the present paper we will focus on rigid supersymmetric theories in six-dimensional Riemannian spin manifolds. We will study the coupling of off-shell Yang-Mills (YM) multiplets to a number of ... |

5 |
Spinor algebras, J.Geom.Phys
- D’Auria, Ferrara, et al.
(Show Context)
Citation Context ...generators may be expected. In particular, the analogs of the Poincaré, AdS and conformal superalgebras for space-times in arbitrary dimensions and signatures have been systematically constructed in =-=[50]-=-. Another interesting open question concerns possible anomalies of our model. In Minkowski space-time and with all background fields set to zero, the model reduces to that of [18]. In that model there... |

4 |
Proeyen, Higher derivative extension of 6D chiral gauged supergravity
- Bergshoeff, Coomans, et al.
(Show Context)
Citation Context ...t is the off-shell formulation of six-dimensional supergravity obtained from superconformal tensor calculus with the dilaton Weyl multiplet coupled to a linear multiplet after particular gauge fixing =-=[19, 20, 21]-=-. The resulting bosonic fields including the space-time metric will constitute the background fields in the rigidly supersymmetric field theories to be constructed in this paper. As discussed in the i... |

4 |
On Euclidean spinors and Wick rotations, Phys.Lett. B389
- Nieuwenhuizen, Waldron
- 1996
(Show Context)
Citation Context ....) Again, the spinors λr and νr are not related by complex conjugation, such that the variation of the vector fields and the Lagrangian 5 are in fact complex, as usual in Euclidean supersymmetry, cf. =-=[23, 24, 25]-=-, see also the discussion in section 6. It is straightforward to check that the Lagrangian (3.1) is invariant under the supersymmetry transformations rules δAm r = ν̃rγmǫ− ζ̃γmλr , δλr = 1 4 γmnǫ Fmn ... |

4 |
The gauge invariant N = 2 supersymmetric sigma model with general scalar potential, Nucl. Phys. B233
- Sierra, Townsend
- 1984
(Show Context)
Citation Context ...curved space. It would also be interesting to analyze if the curved space models can be generalized to include couplings to hypermultiplets, by applying the same procedure to the flat space models of =-=[30, 31]-=-. 5 Killing spinor equations II So far, we have derived the couplings and field equations for vector and tensor multiplets on a curved background under the assumption that the supersymmetry parameters... |

4 | Metric bundles of split signature and type II supergravity, Recent developments in pseudo-Riemannian geometry
- Witt
(Show Context)
Citation Context ...4)→ SU(3). Hence M6 must admit an SU(3) structure. 9 The topological obstruction for an oriented Riemannian six-dimensional manifold M6 to admit an SU(3) structure is that it should be spin (see e.g. =-=[33]-=-). Since in the present paper we are assuming that M6 is spin, there is no additional topological condition imposed by the existence of an SU(3) structure on M6. However, as we will see in the followi... |

4 |
Orthogonal complex structures on
- LeBrun
- 1987
(Show Context)
Citation Context ... still an open question. It is known however that the six-dimensional sphere does not admit an orthogonal complex structure, i.e. one that obeys eq. (5.28) with respect to the standard ‘round’ metric =-=[46]-=-. In other words the round sphere S6 is not a hermitian manifold. This implies that S6 violates the first of the necessary and sufficient conditions in (5.27) therefore our rigid-supersymmetric theory... |

3 | Anomaly-free tensor-Yang-Mills system and its dual formulation, Phys. Lett. B440
- Howe, Sezgin
- 1998
(Show Context)
Citation Context ...these anomalies can be cancelled by Green-Schwarz mechanism involving the addition of a one-loop counterterm of the form ~B∧F ∧F . Supersymmetrization of the model in the presence of this counterterm =-=[51]-=- leads to ~-corrections to the Yang-Mills field equation, which however is anomalous, as its divergence does not vanish, as expected. This also implies an anomaly term in the closure of the superalgeb... |

1 |
Euclidean superalgebras for 3 ≤ D ≤ 10, in Supersymmetry and its applications: superstrings, anomalies and supergravity
- Lukierski
- 1986
(Show Context)
Citation Context ...space limit in which the dynamical fields and the background 2-form potential vanishes, yields the rigid superalgebra {Q̃α, Qβ} = (Cγµ)αβPµ , (7.1) which is the Euclidean Poincaré superalgebra in 6D =-=[49]-=-. If we set only the dynamical fields equal to zero, the interpretation of the resulting superalgebra depends on the details of the background fields, such as the Killing spinors and and isometries th... |