arXiv subjects

C. N. Pope

Publications and source records attributed to C. N. Pope.

At least 181 records · Page 10Linked to original sources

arXiv2002

M-theory PP-waves, Penrose Limits and Supernumerary Supersymmetries

We study supersymmetric pp-waves in M-theory, their dimensional reduction to D0-branes or pp-waves in type IIA, and their T-dualisation to solutions in the type IIB theory. The general class of pp-waves that we consider encompass the Penrose limits of AdS_p\times S^q with (p,q)=(4,7), (7,4), (3,3), (3,2), (2,3), (2,2), but includes also many other examples that can again lead to exactly-solvable massive strings, but which do not arise from Penrose limits. All the pp-waves in D=11 have 16 ``standard'' Killing spinors, but in certain cases one finds additional, or ``supernumerary,'' Killing spinors too. These give rise to linearly-realised supersymmetries in the string or matrix models. A focus of our investigation is on the circumstances when the Killing spinors are independent of particular coordinates (x^+ or transverse-space coordinates), since these will survive at the field-theory level in dimensional reduction or T-dualisation.

hep-th↗

arXiv2002

T-Duality in the Green-Schwarz Formalism, and the Massless/Massive IIA Duality Map

We derive a component-field expansion of the Green-Schwarz action for the type IIA string, in an arbitrary background of massless NS-NS and R-R bosonic fields, up to quadratic order in the fermionic coordinates θ. Using this action, we extend the usual derivation of Buscher T-duality rules to include not only NS-NS, but also R-R fields. Our implementation of the T-duality transformation rules makes use of adapted background-field parametrizations, which provide a more geometrically natural and elegant description for the duality maps than the ones previously presented. These T-duality rules allow us to derive the Green-Schwarz action for the type IIB string in an arbitrary background of massless NS-NS and R-R bosonic fields, up to O(θ^2). Implemention of another T-duality transformation on this type IIB action then allows us also to derive the Green-Schwarz action for the massive IIA string. By further considering T-duality transformations for backgrounds with the two U(1) isometries of a 2-torus, we give a string-theoretic derivation of the direct T-duality relation between the massless and massive type IIA strings. In addition, we give an explicit construction of the D=8 SL(3,R)xSL(2,R) invariant supergravity with two mass parameters that form a doublet under the SL(2,R) factor.

hep-th↗

arXiv2002

Bohm and Einstein-Sasaki Metrics, Black Holes and Cosmological Event Horizons

We study physical applications of the Bohm metrics, which are infinite sequences of inhomogeneous Einstein metrics on spheres and products of spheres of dimension 5 <= d <= 9. We prove that all the Bohm metrics on S^3 x S^2 and S^3 x S^3 have negative eigenvalue modes of the Lichnerowicz operator and by numerical methods we establish that Bohm metrics on S^5 have negative eigenvalues too. We argue that all the Bohm metrics will have negative modes. These results imply that higher-dimensional black-hole spacetimes where the Bohm metric replaces the usual round sphere metric are classically unstable. We also show that the stability criterion for Freund-Rubin solutions is the same as for black-hole stability, and hence such solutions using Bohm metrics will also be unstable. We consider possible endpoints of the instabilities, and show that all Einstein-Sasaki manifolds give stable solutions. We show how Wick rotation of Bohm metrics gives spacetimes that provide counterexamples to a strict form of the Cosmic Baldness conjecture, but they are still consistent with the intuition behind the cosmic No-Hair conjectures. We show how the Lorentzian metrics may be created ``from nothing'' in a no-boundary setting. We argue that Lorentzian Bohm metrics are unstable to decay to de Sitter spacetime. We also argue that noncompact versions of the Bohm metrics have infinitely many negative Lichernowicz modes, and we conjecture a general relation between Lichnerowicz eigenvalues and non-uniqueness of the Dirichlet problem for Einstein's equations.

hep-th↗

arXiv2002

Special Holonomy Spaces and M-theory

We review the construction of regular p-brane solutions of M-theory and string theory with less than maximal supersymmetry whose transverse spaces have metrics with special holonomy, and where additional fluxes allow for brane resolutions via transgression terms. We summarize properties of resolved M2-branes and fractional D2-branes, whose transverse spaces are Ricci flat eight-dimensional and seven-dimensional spaces of special holonomy. Recent developments in the construction of new G_2 holonomy spaces are also reviewed.

hep-th↗

arXiv2002

Bianchi IX Self-dual Einstein Metrics and Singular G_2 Manifolds

We construct explicit cohomogeneity two metrics of G_2 holonomy, which are foliated by twistor spaces. The twistor spaces are S^2 bundles over four-dimensional Bianchi IX Einstein metrics with self-dual (or anti-self-dual) Weyl tensor. Generically the 4-metric is of triaxial Bianchi IX type, with SU(2) isometry. We derive the first-order differential equations for the metric coefficients, and obtain the corresponding superpotential governing the equations of motion, in the general triaxial Bianchi IX case. In general our metrics have singularities, which are of orbifold or cosmic-string type. For the special case of biaxial Bianchi IX metrics, we give a complete analysis their local and global properties, and the singularities. In the triaxial case we find that a system of equations written down by Tod and Hitchin satisfies our first-order equations. The converse is not always true. A discussion is given of the possible implications of the singularity structure of these spaces for M-theory dynamics.

hep-th↗

arXiv2002

General Metrics of G_2 Holonomy and Contraction Limits

We obtain first-order equations for G_2 holonomy of a wide class of metrics with S^3\times S^3 principal orbits and SU(2)\times SU(2) isometry, using a method recently introduced by Hitchin. The new construction extends previous results, and encompasses all previously-obtained first-order systems for such metrics. We also study various group contractions of the principal orbits, focusing on cases where one of the S^3 factors is subjected to an Abelian, Heisenberg or Euclidean-group contraction. In the Abelian contraction, we recover some recently-constructed G_2 metrics with S^3\times T^3 principal orbits. We obtain explicit solutions of these contracted equations in cases where there is an additional U(1) isometry. We also demonstrate that the only solutions of the full system with S^3\times T^3 principal orbits that are complete and non-singular are either flat R^4 times T^3, or else the direct product of Eguchi-Hanson and T^3, which is asymptotic to R^4/Z_2\times T^3. These examples are in accord with a general discussion of isometric fibrations by tori which, as we show, in general split off as direct products. We also give some (incomplete) examples of fibrations of G_2 manifolds by associative 3-tori with either T^4 or K3 as base.

hep-th↗

arXiv2002

Almost Special Holonomy in Type IIA&M Theory

We consider spaces M_7 and M_8 of G_2 holonomy and Spin(7) holonomy in seven and eight dimensions, with a U(1) isometry. For metrics where the length of the associated circle is everywhere finite and non-zero, one can perform a Kaluza-Klein reduction of supersymmetric M-theory solutions (Minkowksi)_4\times M_7 or (Minkowksi)_3\times M_8, to give supersymmetric solutions (Minkowksi)_4\times Y_6 or (Minkowksi)_3\times Y_7 in type IIA string theory with a non-singular dilaton. We study the associated six-dimensional and seven-dimensional spaces Y_6 and Y_7 perturbatively in the regime where the string coupling is weak but still non-zero, for which the metrics remain Ricci-flat but that they no longer have special holonomy, at the linearised level. In fact they have ``almost special holonomy,'' which for the case of Y_6 means almost Kahler, together with a further condition. For Y_7 we are led to introduce the notion of an ``almost G_2 manifold,'' for which the associative 3-form is closed but not co-closed. We obtain explicit classes of non-singular metrics of almost special holonomy, associated with the near Gromov-Hausdorff limits of families of complete non-singular G_2 and Spin(7) metrics.

hep-th↗

arXiv2002

New Cohomogeneity One Metrics With Spin(7) Holonomy

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A_8, is complete and non-singular on R^8. The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S^4, and are denoted by B_8^+, B_8^- and B_8. The metrics on B_8^+ and B_8^- occur in families with a non-trivial parameter. The metric on B_8 arises for a limiting value of this parameter, and locally this metric is the same as the one for A_8. The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on CP^3. We construct the covariantly-constant spinor and calibrating 4-form. We also obtain an L^2-normalisable harmonic 4-form for the A_8 manifold, and two such 4-forms (of opposite dualities) for the B_8 manifold.

math.DG↗

arXiv2002

AdS and dS Braneworld Kaluza-Klein Reduction

We obtain new results for consistent braneworld Kaluza-Klein reductions, showing how we can derive four-dimensional N=2 gauged supergravity ``localised on the AdS_4 brane'' as an exact embedding in five-dimensional N=4 gauged supergravity. Similarly, we obtain five-dimensional N=2 gauged supergravity localised on an AdS_5 brane as a consistent Kaluza-Klein reduction from six-dimensional N=4 gauged supergravity. These embeddings can be lifted to type IIB and massive type IIA supergravity respectively. The new AdS braneworld Kaluza-Klein reductions are generalisations of earlier results on braneworld reductions to ungauged supergravities. The lower-dimensional cosmological constant in our AdS braneworld reductions is an adjustable parameter, and so it can be chosen to be small enough to be phenomenologically realistic, even if the higher-dimensional one is of Planck scale. We also discuss analytic continuations to give a de Sitter gauged supergravity in four dimensions as a braneworld Kaluza-Klein reduction. We find that there are two distinct routes that lead to the same four-dimensional theory. In one, we start from a five-dimensional de Sitter supergravity, which itself arises from a Kaluza-Klein reduction of type IIB^* supergravity on the hyperbolic 5-sphere. In the other, we start from AdS gauged supergravity in five dimensions, with an analytic continuation of the two 2-form potentials, and embed the four-dimensional de Sitter supergravity in that. The five-dimensional theory itself comes from an O(4,3)/O(3,2) reduction of Hull's type IIB_{7+3} supergravity in ten dimensions.

hep-th↗

arXiv2001

A G_2 Unification of the Deformed and Resolved Conifolds

We find general first-order equations for G_2 metrics of cohomogeneity one with S^3\times S^3 principal orbits. These reduce in two special cases to previously-known systems of first-order equations that describe regular asymptotically locally conical (ALC) metrics \bB_7 and \bD_7, which have weak-coupling limits that are S^1 times the deformed conifold and the resolved conifold respectively. Our more general first-order equations provide a supersymmetric unification of the two Calabi-Yau manifolds, since the metrics \bB_7 and \bD_7 arise as solutions of the {\it same} system of first-order equations, with different values of certain integration constants. Additionally, we find a new class of ALC G_2 solutions to these first-order equations, which we denote by \wtd\bC_7, whose topology is an \R^2 bundle over T^{1,1}. There are two non-trivial parameters characterising the homogeneous squashing of the T^{1,1} bolt. Like the previous examples of the \bB_7 and \bD_7 ALC metrics, here too there is a U(1) isometry for which the circle has everywhere finite and non-zero length. The weak-coupling limit of the \wtd\bC_7 metrics gives S^1 times a family of Calabi-Yau metrics on a complex line bundle over S^2\times S^2, with an adjustable parameter characterising the relative sizes of the two S^2 factors.

hep-th↗

arXiv2001

M-theory Conifolds

Seven-manifolds of G_2 holonomy provide a bridge between M-theory and string theory, via Kaluza-Klein reduction to Calabi-Yau six-manifolds. We find first-order equations for a new family of G_2 metrics D_7, with S^3\times S^3 principal orbits. These are related at weak string coupling to the resolved conifold, paralleling earlier examples B_7 that are related to the deformed conifold, allowing a deeper study of topology change and mirror symmetry in M-theory. The D_7 metrics' non-trivial parameter characterises the squashing of an S^3 bolt, which limits to S^2 at weak coupling. In general the D_7 metrics are asymptotically locally conical, with a nowhere-singular circle action.

hep-th↗

arXiv2001

Orientifolds and Slumps in G_2 and Spin(7) Metrics

We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by C_8, which are complete on a complex line bundle over CP^3. The principal orbits are S^7, described as a triaxially squashed S^3 bundle over S^4. The behaviour in the S^3 directions is similar to that in the Atiyah-Hitchin metric, and we show how this leads to an M-theory interpretation with orientifold D6-branes wrapped over S^4. We then consider new G_2 metrics which we denote by C_7, which are complete on an R^2 bundle over T^{1,1}, with principal orbits that are S^3\times S^3. We study the C_7 metrics using numerical methods, and we find that they have the remarkable property of admitting a U(1) Killing vector whose length is nowhere zero or infinite. This allows one to make an everywhere non-singular reduction of an M-theory solution to give a solution of the type IIA theory. The solution has two non-trivial S^2 cycles, and both carry magnetic charge with respect to the R-R vector field. We also discuss some four-dimensional hyper-Kahler metrics described recently by Cherkis and Kapustin, following earlier work by Kronheimer. We show that in certain cases these metrics, whose explicit form is known only asymptotically, can be related to metrics characterised by solutions of the su(\infty) Toda equation, which can provide a way of studying their interior structure.

hep-th↗

arXiv2001

Cohomogeneity One Manifolds of Spin(7) and G(2) Holonomy

In this paper, we look for metrics of cohomogeneity one in D=8 and D=7 dimensions with Spin(7) and G_2 holonomy respectively. In D=8, we first consider the case of principal orbits that are S^7, viewed as an S^3 bundle over S^4 with triaxial squashing of the S^3 fibres. This gives a more general system of first-order equations for Spin(7) holonomy than has been solved previously. Using numerical methods, we establish the existence of new non-singular asymptotically locally conical (ALC) Spin(7) metrics on line bundles over \CP^3, with a non-trivial parameter that characterises the homogeneous squashing of CP^3. We then consider the case where the principal orbits are the Aloff-Wallach spaces N(k,\ell)=SU(3)/U(1), where the integers k and \ell characterise the embedding of U(1). We find new ALC and AC metrics of Spin(7) holonomy, as solutions of the first-order equations that we obtained previously in hep-th/0102185. These include certain explicit ALC metrics for all N(k,\ell), and numerical and perturbative results for ALC families with AC limits. We then study D=7 metrics of $G_2$ holonomy, and find new explicit examples, which, however, are singular, where the principal orbits are the flag manifold SU(3)/(U(1)\times U(1)). We also obtain numerical results for new non-singular metrics with principal orbits that are S^3\times S^3. Additional topics include a detailed and explicit discussion of the Einstein metrics on N(k,\ell), and an explicit parameterisation of SU(3).

hep-th↗

arXiv2001

Supersymmetric Domain Walls from Metrics of Special Holonomy

Supersymmetric domain-wall spacetimes that lift to Ricci-flat solutions of M-theory admit generalized Heisenberg (2-step nilpotent) isometry groups. These metrics may be obtained from known cohomogeneity one metrics of special holonomy by taking a "Heisenberg limit", based on an Inönü-Wigner contraction of the isometry group. Associated with each such metric is an Einstein metric with negative cosmological constant on a solvable group manifold. We discuss the relevance of our metrics to the resolution of singularities in domain-wall spacetimes and some applications to holography. The extremely simple forms of the explicit metrics suggest that they will be useful for many other applications. We also give new but incomplete inhomogeneous metrics of holonomy SU(3), $G_2$ and Spin(7), which are $T_1$, $T_2$ and $T_3$ bundles respectively over hyper-Kähler four-manifolds.

hep-th↗

arXiv2001

Resolved Branes and M-theory on Special Holonomy Spaces

We review the construction of regular p-brane solutions of M-theory and string theory with less than maximal supersymmetry whose transverse spaces have metrics with special holonomy, and where additional fluxes allow brane resolutions via transgression terms. We focus on the properties of resolved M2-branes and fractional D2-branes, whose transverse spaces are Ricci flat eight-dimensional and seven-dimensional spaces of special holonomy. We also review fractional M2-branes with transverse spaces corresponding to the new two-parameter Spin(7) holonomy metrics, and their connection to fractional D2-branes with transverse spaces of G_2 holonomy.

hep-th↗

arXiv2001

A New Fractional D2-brane, G_2 Holonomy and T-duality

Recently, a new example of a complete non-compact Ricci-flat metric of G_2 holonomy was constructed, which has an asymptotically locally conical structure at infinity with a circular direction whose radius stabilises. In this paper we find a regular harmonic 3-form in this metric, which we then use in order to obtain an explicit solution for a fractional D2-brane configuration. By performing a T-duality transformation on the stabilised circle, we obtain the type IIB description of the fractional brane, which now corresponds to D3-brane with one of its world-volume directions wrapped around the circle.

hep-th↗

arXiv2001

Massless 3-brane in M-theory

We construct supersymmetric M3-brane solutions in D=11 supergravity. They can be viewed as deformations of backgrounds taking the form of a direct product of four-dimensional Minkowski spacetime and a non-compact Ricci-flat manifold of G_2 holonomy. Although the 4-form field strength is turned on it carries no charge, and the 3-branes are correspondingly massless. We also obtain 3-branes of a different type, arising as M5-branes wrapped over S^2.

hep-th↗

arXiv2001

Supersymmetric M3-branes and G_2 Manifolds

We obtain a generalisation of the original complete Ricci-flat metric of G_2 holonomy on R^4\times S^3 to a family with a non-trivial parameter λ. For generic λthe solution is singular, but it is regular when λ={-1,0,+1}. The case λ=0 corresponds to the original G_2 metric, and λ={-1,1} are related to this by an S_3 automorphism of the SU(2)^3 isometry group that acts on the S^3\times S^3 principal orbits. We then construct explicit supersymmetric M3-brane solutions in D=11 supergravity, where the transverse space is a deformation of this class of G_2 metrics. These are solutions of a system of first-order differential equations coming from a superpotential. We also find M3-branes in the deformed backgrounds of new G_2-holonomy metrics that include one found by A. Brandhuber, J. Gomis, S. Gubser and S. Gukov, and show that they also are supersymmetric.

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