Seven-dimensional Manifolds Research Articles

Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group.

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

Twistor Theory of Dancing Paths

Given a path geometry on a surface $\mathcal{U}$, we construct a causal structure on a four-manifold which is the configuration space of non-incident pairs (point, path) on $\mathcal{U}$. This causal structure corresponds to a conformal structure if and only if $\mathcal{U}$ is a real projective plane, and the paths are lines. We give the example of the causal structure given by a symmetric sextic, which corresponds on an ${\rm SL}(2,{\mathbb R})$-invariant projective structure where the paths are ellipses of area $\pi$ centred at the origin. We shall also discuss a causal structure on a seven-dimensional manifold corresponding to non-incident pairs (point, conic) on a projective plane.

Uplifting dyonic AdS4 black holes on seven-dimensional Sasaki-Einstein manifolds

We revisit non-rotating, dyonically charged, and supersymmetric AdS4 black holes, which are solutions of mathcal{N} = 2 gauged supergravity with vector- and hyper-multiplets. Uplifting the near horizon solutions to D = 11 supergravity on seven-dimensional Sasaki-Einstein manifolds, we show that dyonic AdS4 black holes correspond to AdS2 solutions with electric and magnetic baryonic fluxes in D = 11 supergravity. We identify the off-shell AdS4 black hole solutions with parameters of D = 11 AdS2 solutions without imposing the equations of motion. We calculate the entropy of dyonic black holes, carefully analyzing the Page charge quantization conditions.

Closed $\mathrm{G}_2$-structures with a transitive reductive group of automorphisms

We provide the complete classification of seven-dimensional manifolds endowed with a closed non-parallel G$_2$-structure and admitting a transitive reductive group G of automorphisms. In particular, we show that the center of G is one-dimensional and the manifold is the Riemannian product of a flat factor and a non-compact homogeneous six-dimensional manifold endowed with an invariant strictly symplectic half-flat SU(3)-structure.

Black holes with baryonic charge and mathcal{I} -extremization

Recently it was discovered that twisted superconformal index mathcal{I} can be used to understand the Bekenstein-Hawking entropy of magnetically charged black holes in AdS spacetime. In this paper we apply the so-called mathcal{I} -extremization procedure to three-dimensional gauge field theories and their geometric dual, focusing in particular on the seven-dimensional Sasaki-Einstein manifold M1,1,1. We generalize recent studies on relations among toric geometry, variational principles, and black hole entropy to the case of AdS2× Y9, where Y9 is a fibration of toric Sasaki-Einstein manifold M1,1,1 over a two-dimensional Riemann surface Σg. The nine-dimensional variational problem is given in terms of an entropy functional. In order to illustrate the computations as explicitly as possible, we consider cases where either only mesonic or baryonic fluxes are turned on. By employing the operator counting method, we calculate the S3 free energy and the topologically twisted index mathcal{I} at large-N. The result for mathcal{I} , it turns out, can be also obtained from the variational principle of the entropy functional with mesonic fluxes. We also study asymptotically AdS4 black holes which are magnetically charged with respect to the vector field in the Betti multiplet. By extremizing the entropy functional with baryonic flux, we compute the entropy and find that it agrees with the entropy of an explicit solution in a four-dimensional gauged supergravity which is a consistent truncation of eleven-dimensional supergravity in AdS4× M1,1,1.

Aspects of AdS2 classification in M-theory: solutions with mesonic and baryonic charges

We construct necessary and sufficient geometric conditions for a class of AdS2 solutions of M-theory with, at least, minimal supersymmetry to exist. We generalize previous results in the literature for mathcal{N} = (2, 0) supersymmetry in AdS2 to mathcal{N} = (1, 0). When the solution can be locally described as AdS2× Σg× SE7 with Σg a Riemann surface of genus g and SE7 a seven-dimensional Sasaki-Einstein manifold, we clarify and unify various solutions present in the literature. In the case of SE7 = Q1,1,1 we find a new solution with baryonic and mesonic charges turned on simultaneously.

Decomposable (4, 7) solutions in eleven-dimensional supergravity

We describe a class of decomposable eleven-dimensional supergravity backgrounds which are products of a four-dimensional Lorentzian manifold and a seven-dimensional Riemannian manifold, endowed with a flux form given in terms of the volume form on and a closed 4-form F4 on M7. We show that the Maxwell equation for such a flux form can be read in terms of the co-closed 3-form . Moreover, the supergravity equation reduces to the condition that is an Einstein manifold with negative Einstein constant and (M7,g,F) is a Riemannian manifold which satisfies the Einstein equation with a stress-energy tensor associated to the 3-form . Whenever this 3-form is generic, we show that the Maxwell equation induces a weak -structure on M7 and obtain decomposable supergravity backgrounds given by the product of a weak -manifold with a Lorentzian Einstein manifold . We also construct examples of compact homogeneous Riemannian 7-manifolds endowed with non-generic invariant 3-forms which satisfy the Maxwell equation, but the construction of decomposable homogeneous supergravity backgrounds of this type remains an open problem.

Gluing and deformation of asymptotically cylindrical Calabi–Yau manifolds in complex dimension three

We develop some consequences of the connection between Calabi–Yau structures and torsion-free G2 structures on compact and asymptotically cylindrical six- and seven-dimensional manifolds. Firstly, we improve the known proof that matching asymptotically cylindrical Calabi–Yau threefolds can be glued. Secondly, we give an alternative proof that the moduli space of Calabi–Yau structures on a six-dimensional real manifold is smooth, and extend it to the asymptotically cylindrical case. Finally, we prove that the gluing map of Calabi–Yau threefolds, extended between these moduli spaces, is a local diffeomorphism: that is, that every deformation of a glued Calabi–Yau threefold arises from an essentially unique deformation of the asymptotically cylindrical pieces.

G2-structures for N = 1 supersymmetric AdS4 solutions of M-theory

We study the N = 1 supersymmetric solutions of D = 11 supergravity obtained as a warped product of four-dimensional anti-de Sitter space with a seven-dimensional Riemannian manifold M. Using the octonion bundle structure on M we reformulate the Killing spinor equations in terms of sections of the octonion bundle on M. The solutions then define a single complexified G2-structure on M or equivalently two real G2-structures. We then study the torsion of these G2-structures and the relationships between them.

An elliptic boundary value problem for $G_{2}$-structures

We show that the G 2 holonomy equation on a seven-dimensional manifold with boundary, with prescribed 3-form on the boundary and modulo the action of diffeomorphisms, is elliptic. The main point is to set up a suitable linear elliptic theory. This result leads to a deformation theory, governed by a finite-dimensional obstruction space. We discuss conditions under which this obstruction space vanishes and as one application we establish the existence of certain G 2 cobordisms between two small deformations of a Calabi–Yau 3-fold.

$${\mathrm {SU}}(4)$$ SU ( 4 ) -holonomy via the left-invariant hypo and Hitchin flow

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M, g) with holonomy in $${\mathrm {Spin}}(7)$$ starting with a cocalibrated $$\mathrm {G}_2$$ -structure on a seven-dimensional manifold. As $${\mathrm {Sp}}(2)\subseteq {\mathrm {SU}}(4)\subseteq {\mathrm {Spin}}(7)$$ , one may also obtain Calabi–Yau fourfolds or hyperkahler manifolds via the Hitchin flow. In this paper, we show that the Hitchin flow on almost Abelian Lie algebras and on Lie algebras with one-dimensional commutator always yields Riemannian metrics g with $$Hol(g)\subseteq {\mathrm {SU}}(4)$$ but $$Hol(g)\ne {\mathrm {Sp}}(2)$$ . We investigate when we actually get $$Hol(g)={\mathrm {SU}}(4)$$ and obtain many new explicit examples of Calabi–Yau fourfolds. The results rely on the connection between cocalibrated $$\mathrm {G}_2$$ -structures and hypo $${\mathrm {SU}}(3)$$ -structures and between the Hitchin and the hypo flow and on a systematic study of hypo $${\mathrm {SU}}(3)$$ -structures and the hypo flow on Lie algebras. This study gives us many other interesting results: We obtain full classifications of hypo $${\mathrm {SU}}(3)$$ -structures with particular intrinsic torsion on Lie algebras. Moreover, we can exclude reducible or $${\mathrm {Sp}}(2)$$ -holonomy for the Riemannian manifolds obtained by the hypo flow on Lie algebras with initial values lying in certain intrinsic torsion classes and show that for initial values in other torsion classes we always get Riemannian metrics with holonomy equal to $${\mathrm {SU}}(4)$$ .

Knot invariants and M-theory: Hitchin equations, Chern-Simons actions, and surface operators

Recently Witten introduced a type IIB brane construction with certain boundary conditions to study knot invariants and Khovanov homology. The essential ingredients used in his work are the topologically twisted N = 4 Yang-Mills theory, localization equations and surface operators. In this paper we extend his construction in two possible ways. On one hand we show that a slight modification of Witten's brane construction could lead, using certain well defined duality transformations, to the model used by Ooguri-Vafa to study knot invariants using gravity duals. On the other hand, we argue that both these constructions, of Witten and of Ooguri-Vafa, lead to two different seven-dimensional manifolds in M-theory from where the topological theories may appear from certain twisting of the G-flux action. The non-abelian nature of the topological action may also be studied if we take the wrapped M2-brane states in the theory. We discuss explicit constructions of the seven-dimensional manifolds in M-theory, and show that both the localization equations and surface operators appear naturally from the Hamiltonian formalism of the theories. Knots and link invariants are then constructed using M2-brane states in both the models.

T4 fibrations over Calabi–Yau two-folds and non-Kähler manifolds in string theory

We construct a geometric model of eight-dimensional manifolds and realize them in the context of type II string theory. These eight-manifolds are constructed by non-trivial T4 fibrations over Calabi–Yau two-folds. These give rise to eight-dimensional non-Kähler Hermitian manifolds with SU(4) structure. The eight-manifold is also a circle fibration over a seven-dimensional G2 manifold with skew torsion. The eight-manifolds of this type appear as internal manifolds with SU(4) structure in type IIB string theory with F3 and F7 fluxes. These manifolds have generalized calibrated cycles in the presence of fluxes.

Spinor fields classification in arbitrary dimensions and new classes of spinor fields on 7-manifolds

A classification of spinor fields according to the associated bilinear covariants is constructed in arbitrary dimensions and metric signatures, generalizing Lounesto's 4D spinor field classification. In such a generalized classification a basic role is played by the geometric Fierz identities. In 4D Minkowski spacetime the standard bilinear covariants can be either null or non-null -- with the exception of the current density which is invariably different from zero for physical reasons -- and sweep all types of spinor fields, including Dirac, Weyl, Majorana and more generally flagpoles, flag-dipoles and dipole spinor fields. To obtain an analogous classification in higher dimensions we use the Fierz identities, which constrain the covariant bilinears in the spinor fields and force some of them to vanish. A generalized graded Fierz aggregate is moreover obtained in such a context simply from the completeness relation. We analyze the particular and important case of Riemannian 7-manifolds, where the Majorana spinor fields turn out to have a quite special place. In particular, at variance with spinor fields in 4D Minkowski spacetime that are classified in six disjoint classes, spinors in Riemannian 7-manifolds are shown to be classified, according to the bilinear covariants: (a) in just one class, in the real case of Majorana spinors; (b) in four classes, in the most general case. Much like new classes of spinor fields in 4D Minkowski spacetime have been evincing new possibilities in physics, we think these new classes of spinor fields in seven dimensions are, in particular, potential candidates for new solutions in the compactification of supergravity on a seven-dimensional manifold and its exotic versions.

Invariant classification of second-order conformally flat superintegrable systems

In this paper we continue the work of Kalnins et al in classifying all second-order conformally-superintegrable (Laplace-type) systems over conformally flat spaces, using tools from algebraic geometry and classical invariant theory. The results obtained show, through Stäckel equivalence, that the list of known nondegenerate superintegrable systems over three-dimensional conformally flat spaces is complete. In particular, a seven-dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems. This manifold is foliated by the nonlinear action of the conformal group in three dimensions. Two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation. This foliation is explicitly described using algebraic varieties formed from representations of the conformal group. The proof of these results rely heavily on Gröbner basis calculations using the computer algebra software packages Maple and Singular.

Spin(7)-manifolds in compactifications to four dimensions

We describe off-shell $\mathcal{N}=1$ M-theory compactifications down to four dimensions in terms of eight-dimensional manifolds equipped with a topological $Spin(7)$-structure. Motivated by the exceptionally generalized geometry formulation of M-theory compactifications, we consider an eight-dimensional manifold $\mathcal{M}_{8}$ equipped with a particular set of tensors $\mathfrak{S}$ that allow to naturally embed in $\mathcal{M}_{8}$ a family of $G_{2}$-structure seven-dimensional manifolds as the leaves of a codimension-one foliation. Under a different set of assumptions, $\mathfrak{S}$ allows to make $\mathcal{M}_{8}$ into a principal $S^{1}$ bundle, which is equipped with a topological $Spin(7)$-structure if the base is equipped with a topological $G_{2}$-structure. We also show that $\mathfrak{S}$ can be naturally used to describe regular as well as a singular elliptic fibrations on $\mathcal{M}_{8}$, which may be relevant for F-theory applications, and prove several mathematical results concerning the relation between topological $G_{2}$-structures in seven dimensions and topological $Spin(7)$-structures in eight dimensions.

$${\mathcal{N} = 2}$$ N = 2 Supersymmetric AdS4 Solutions of M-theory

We analyse the most general N=2 supersymmetric solutions of D=11 supergravity consisting of a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold Y_7. We show that the necessary and sufficient conditions for supersymmetry can be phrased in terms of a local SU(2)-structure on Y_7. Solutions with non-zero M2-brane charge also admit a canonical contact structure, in terms of which many physical quantities can be expressed, including the free energy and the scaling dimensions of operators dual to supersymmetric wrapped M5-branes. We show that a special class of solutions is singled out by imposing an additional symmetry, for which the problem reduces to solving a second order non-linear ODE. As well as recovering a known class of solutions, that includes the IR fixed point of a mass deformation of the ABJM theory, we also find new solutions which are dual to cubic deformations. In particular, we find a new supersymmetric warped AdS_4 x S^7 solution with non-trivial four-form flux.

EINSTEIN MANIFOLDS WITH SKEW TORSION

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection ∇ with skew-symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar ∇-curvature, it is necessarily compact and it has finite first fundamental group π1. The longest part of the paper is devoted to the systematic construction of large families of examples. We discuss when a Riemannian Einstein manifold can be Einstein with skew torsion. We give examples of almost Hermitian, almost metric contact, and G2 manifolds that are Einstein with skew torsion. For example, we prove that any Einstein–Sasaki manifold and any seven-dimensional 3-Sasakian manifolds admit deformations into an Einstein metric with parallel skew torsion.

Cocalibrated [formula omitted]-structures on products of four- and three-dimensional Lie groups

Cocalibrated G2-structures are structures naturally induced on hypersurfaces in Spin(7)-manifolds. Conversely, one may start with a seven-dimensional manifold M endowed with a cocalibrated G2-structure and construct via the Hitchin flow a Spin(7)-manifold which contains M as a hypersurface. In this article, we consider left-invariant cocalibrated G2-structures on Lie groups G which are a direct product G=G4×G3 of a four-dimensional Lie group G4 and a three-dimensional Lie group G3. We achieve a full classification of the Lie groups G=G4×G3 which admit a left-invariant cocalibrated G2-structure.

Quaternionic Kähler and Spin(7) metrics arising from quaternionic contact Einstein structures

We construct left-invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven-dimensional manifold, equipped with a certain qc structure, has a quaternionic Kahler metric as well as a metric with holonomy contained in Spin(7). As a consequence, we determine explicit quaternionic Kahler metrics and Spin(7)-holonomy metrics, which seem to be new. Moreover, we give explicit non-compact eight dimensional almost quaternion hermitian manifolds that are not quaternionic Kahler with either a closed fundamental four form or fundamental two forms defining a differential ideal.