Special Holonomy Research Articles

Conformal holonomy, symmetric spaces, and skew symmetric torsion

We consider the question: Can the isotropy representation of an irreducible pseudo-Riemannian symmetric space be realized as a conformal holonomy group? Using recent results by Čap, Gover and Hammerl, we study the representations of SO(2,1), PSU(2,1) and PSp(2,1) as isotropy groups of irreducible symmetric spaces of signature (3,2), (4,4) and (6,8), respectively, describing the geometry induced by a conformal holonomy reduction to the corresponding subgroups. In the case of SO(2,1), we show that conformal manifolds with such a holonomy reduction are always locally conformally flat and hence this group cannot be a conformal holonomy group. This result completes the classification of irreducible conformal holonomy groups in Lorentzian signature. In the case of PSU(2,1), we show that conformal manifolds of signature (3,3) with this holonomy reduction carry, on an open dense subset, a canonical nearly para-Kähler metric with positive Einstein constant. For PSp(2,1) we also show that there is an open dense subset endowed with a canonical Einstein metric in the conformal class. As a result, after restricting to an open dense subset, the conformal holonomy must be a proper subgroup of PSU(2,1) or of PSp(2,1), respectively. These are special cases of an interesting relationship between a class of special conformal holonomy groups, and non-integrable geometries with skew symmetric, parallel torsion, which we also explore. Finally, using a recent result of Graham and Willse we prove the following general non-existence result: for a real-analytic, odd-dimensional conformal manifold, the conformal holonomy group can never be given by the isotropy representation of an irreducible pseudo-Riemannian symmetric space unless the isotropy group is SO0(p+1,q+1).

Chiral de Rham Complex on Riemannian Manifolds and Special Holonomy

Interpreting the chiral de Rham complex (CDR) as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for the study of CDR on manifolds with special holonomy. We show how to systematically construct global sections of CDR from differential forms, and investigate the algebra of the sections corresponding to the covariantly constant forms associated with the special holonomy. As a concrete example, we construct two commuting copies of the Odake algebra (an extension of the N=2 superconformal algebra) on the space of global sections of CDR of a Calabi-Yau threefold and conjecture similar results for G_2 manifolds. We also discuss quasi-classical limits of these algebras.

Conformally flat Lorentzian manifolds with special holonomy groups

We obtain a local classification of conformally flat Lorentzian manifolds with special holonomy groups. The corresponding local metrics are certain extensions of Riemannian spaces of constant sectional curvature to Walker metrics.Bibliography: 28 titles.

Extrinsic hyperspheres in manifolds with special holonomy

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kähler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.

Closed forms and multi-moment maps

We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.

On Hexagonal Structures in Higher Dimensional Theories

We analyze the geometrical background under which many Lie groups relevant to particle physics are endowed with a (possibly multiple) hexagonal structure. There are several groups appearing, either as special holonomy groups on the compactification process from higher dimensions, or as dynamical string gauge groups; this includes groups like SU(2), SU(3), G 2, Spin(7), O(8) as well as E 8 and O(32). We emphasize also the relation of these hexagonal structures with the octonion division algebra, as we expect as well eventually some role for octonions in the interpretation of symmetries in High Energy Physics.

Instantons on special holonomy manifolds

We consider cones over manifolds admitting real Killing spinors and instanton equations on connections on vector bundles over these manifolds. Such cones are manifolds with special (reduced) holonomy. We generalize the scalar ansatz for a connection proposed by Harland and N\"olle [D. Harland and C. N\"olle, J. High Energy Phys. 03 (2012) 082.] in such a way that instantons are parametrized by constrained matrix-valued functions. Our ansatz reduces instanton equations to matrix model equations which can be further reduced to Newtonian mechanics with particle trajectories obeying first-order gradient flow equations. Generalizations to K\"ahler-Einstein manifolds and resolved Calabi-Yau cones are briefly discussed. Our construction allows one to associate quiver gauge theories with special holonomy manifolds via dimensional reduction.

Branes and fluxes in special holonomy manifolds and cascading field theories

We conduct a study of holographic RG flows whose UV is a theory in 2+1 dimensions decoupled from gravity, and the IR is the N=6,8 superconformal fixed point of ABJM. The solutions we consider are constructed by warping the M-theory background whose eight spatial dimensions are manifolds of special holonomies sp(1) times sp(1) and spin(7). Our main example for the spin(7) holonomy manifold is the A8 geometry originally constructed by Cvetic, Gibbons, Lu, and Pope. On the gravity side, our constructions generalize the earlier construction of RG flow where the UV was N=3 Yang-Mills-Chern-Simons matter system and are simpler in a number of ways. Through careful consideration of Page, Maxwell, and brane charges, we identify the discrete and continuous parameters characterizing each system. We then determine the range of the discrete data, corresponding to the flux/rank for which the supersymmetry is unbroken, and estimate the dynamical supersymmetry breaking scale as a function of these data. We then point out the similarity between the physics of supersymmetry breaking between our system and the system considered by Maldacena and Nastase. We also describe the condition for unbroken supersymmetry on class of construction based on a different class of spin(7) manifolds known as B8 spaces whose IR is different from that of ABJM and exhibit some interesting features.

Spin(7)-structures on complex linear bundles and explicit Riemannian metrics with holonomy group SU(4)

A system of differential equations with 5 unknowns is fully investigated; this system is equivalent to the existence of a parallel Spin(7)-structure on a cone over a 3-Sasakian manifold. A continuous one-parameter family of solutions to this system is explicitly constructed; it corresponds to metrics with a special holonomy group, SU(4), which generalize Calabi's metrics.Bibliography: 10 titles.

A scan of universal breathing‐mode reductions

AbstractThe near horizon limits of M2 and D3 branes probing special holonomy cones give rise to supersymmetric AdS4 × M7 and AdS5 × M5 solutions of D = 11 and type IIB supergravity, respectively, where M7 and M5 are special types of Einstein spaces. Here we will show the existence of consistent Kaluza‐Klein truncations of D = 11 and type IIB supergravity on such M7 and M5 down to matter‐coupled supergravities in D = 4, 5. We explicitly discuss some details of the particular cases when M7 and M5 are Sasaki‐Einstein.

Quaternionic Kähler Detour Complexes and $${\mathcal{N} = 2}$$ Supersymmetric Black Holes

We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional N= 2 supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kaehler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generating special holonomy transformations. (An interesting byproduct of the construction is a novel Dirac operator on the superghost extended Hilbert space.) The resulting quantized models are gauge invariant field theories with fields equaling sections of special quaternionic vector bundles. They underly and generalize the quaternionic version of Dolbeault cohomology discovered by Baston. In fact, Baston's complex is related to the BPS sector of the models we write down. Our results rely on a calculus of operators on quaternionic Kaehler manifolds that follows from BRST machinery, and although directly motivated by black hole physics, can be broadly applied to any model relying on quaternionic geometry.

Covariantly constant forms on torsionful geometries from world-sheet and spacetime perspectives

The symmetries of two-dimensional supersymmetric sigma models on target spaces with covariantly constant forms associated to special holonomy groups are analysed. It is shown that each pair of such forms gives rise to a new one, called a Nijenhuis form, and that there may be further reductions of the structure group. In many cases of interest there are also covariantly constant one-forms which also give rise to symmetries. These geometries are of interest in the context of heterotic supergravity solutions and the associated reductions are studied from a spacetime point of view via the Killing spinor equations.

Heterotic black horizons

We show that the supersymmetric near horizon geometry of heterotic black holes is either an AdS 3 fibration over a 7-dimensional manifold which admits a G 2 structure compatible with a connection with skew-symmetric torsion, or it is a product $ {\mathbb{R}^{1,1}} \times {\mathcal{S}^8} $ , where $ {\mathcal{S}^8} $ is a holonomy Spin(7) manifold, preserving 2 and 1 supersymmetries respectively. Moreover, we demonstrate that the AdS 3 class of heterotic horizons can preserve 4, 6 and 8 supersymmetries provided that the geometry of the base space is further restricted. Similarly $ {\mathbb{R}^{1,1}} \times {\mathcal{S}^8} $ horizons with extended supersymmetry are products of $ {\mathbb{R}^{1,1}} $ with special holonomy manifolds. We have also found that the heterotic horizons with 8 supersymmetries are locally isometric to AdS 3 × S 3 × T 4, AdS 3 × S 3 × K 3 or $ {\mathbb{R}^{1,1}} \times {T^4} \times {K_3} $ , where the radii of AdS 3 and S 3 are equal and the dilaton is constant.

Half-flat structures and special holonomy

It was proven by Hitchin that any solution of his evolution equations for a half-flat SU(3)-structure on a compact six-manifold M defines an extension of M to a seven-manifold with holonomy in G_2. We give a new proof, which does not require the compactness of M. More generally, we prove that the evolution of any half-flat G-structure on a six-manifold M defines an extension of M to a Ricci-flat seven-manifold N, for any real form G of SL(3,C). If G is noncompact, then the holonomy group of N is a subgroup of the noncompact form G_2^* of G_2^C. Similar results are obtained for the extension of nearly half-flat structures by nearly parallel G_2- or G_2^*-structures, as well as for the extension of cocalibrated G_2- and G_2^*-structures by parallel Spin(7)- and Spin(3,4)-structures, respectively. As an application, we obtain that any six-dimensional homogeneous manifold with an invariant half-flat structure admits a canonical extension to a seven-manifold with a parallel G_2- or G_2^*-structure. For the group H_3 \times H_3, where H_3 is the three-dimensional Heisenberg group, we describe all left-invariant half-flat structures and develop a method to explicitly determine the resulting parallel G_2- or G_2^*-structure without integrating. In particular, we construct three eight-parameter families of metrics with holonomy equal to G_2 and G_2^*. Moreover, we obtain a strong rigidity result for the metrics induced by a half-flat structure (\omega,\rho) on H_3 \times H_3 satisfying \omega(Z,Z)=0 where Z denotes the centre. Finally, we describe the special geometry of the space of stable three-forms satisfying a reality condition. Considering all possible reality conditions, we find four different special K\"ahler manifolds and one special para-K\"ahler manifold.

Embedding into manifolds with torsion

We introduce a class of special geometries associated to the choice of a differential graded algebra contained in \({\Lambda^*\mathbb{R}^n}\). We generalize some known embedding results, that effectively characterize the real analytic Riemannian manifolds that can be realized as submanifolds of a Riemannian manifold with special holonomy, to this more general context. In particular, we consider the case of hypersurfaces inside nearly-Kähler and α-Einstein–Sasaki manifolds, proving that the corresponding evolution equations always admit a solution in the real analytic case.

Special symplectic connections

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.

Globally hyperbolic Lorentzian spaces with special holonomy groups

We prove that each special Lorentzian holonomy group (with the exception of those including the isotropy groups of Kahler symmetric spaces) can be realized as the holonomy group of a globally hyperbolic Lorentzian manifold.

Curvature, Lagrangian and holonomy of Cantorian-fractal spacetime

The special holonomy of the Cantorian-fractal manifold of spacetime is considered in some detail. The expression for spacetime curvature, energy and the corresponding Lagrangian formulation is given. Numerous special aspects of the theory are discussed.

Supersymmetry, holonomy and Kundt spacetimes

Supersymmetric solutions of supergravity theories, and consequently metrics with special holonomy, have played an important role in the development of string theory. We describe how a Lorentzian manifold is either completely reducible, and thus essentially known, or not completely reducible so that there exists a degenerate holonomy invariant lightlike subspace and consequently admits a covariantly constant or a recurrent null vector and belongs to the higher-dimensional Kundt class of spacetimes. These Kundt spacetimes (which contain the vanishing and constant curvature invariant spacetimes as special cases) are genuinely Lorentzian and have a number of interesting and unusual properties, which may lead to novel and fundamental physics.

Spacetime Singularity Resolution by M-Theory Fivebranes: Calibrated Geometry, Anti-de Sitter Solutions and Special Holonomy Metrics

The supergravity description of various configurations of supersymmetric M-fivebranes wrapped on calibrated cycles of special holonomy manifolds is studied. The description is provided by solutions of eleven-dimensional supergravity which interpolate smoothly between a special holonomy manifold and an event horizon with Anti-de Sitter geometry. For known examples of Anti-de Sitter solutions, the associated special holonomy metric is derived. One explicit Anti-de Sitter solution of M-theory is so treated for fivebranes wrapping each of the following cycles: Kähler cycles in Calabi-Yau two, three- and four-folds; special lagrangian cycles in three- and four-folds; associative three- and co-associative four-cycles in G 2 manifolds; complex lagrangian four-cycles in Sp(2) manifolds; and Cayley four-cycles in Spin(7) manifolds. In each case, the associated special holonomy metric is singular, and is a hyperbolic analogue of a known metric. The analogous known metrics are respectively: Eguchi-Hanson, the resolved conifold and the four-fold resolved conifold; the deformed conifold, and the Stenzel four-fold metric; the Bryant-Salamon-Gibbons-Page-Pope G 2 metrics on an $${\mathbb{R}^4}$$ bundle over S 3, and an $${\mathbb{R}^3}$$ bundle over S 4 or $${\mathbb{CP}^2}$$ ; the Calabi hyper-Kähler metric on $${T^*\mathbb{CP}^2}$$ ; and the Bryant-Salamon-Gibbons-Page-Pope Spin(7) metric on an $${\mathbb{R}^4}$$ bundle over S 4. By the AdS/CFT correspondence, a conformal field theory is associated to each of the new singular special holonomy metrics, and defines the quantum gravitational physics of the resolution of their singularities.