José Figueroa-O'Farrill - Academia.edu (original) (raw)

Papers by José Figueroa-O'Farrill

Advances in Theoretical and Mathematical Physics, 2017

A. We study the algebraic structure of the Killing superalgebra of a supersymmetric background of... more A. We study the algebraic structure of the Killing superalgebra of a supersymmetric background of 11-dimensional supergravity and show that it is isomorphic to a filtered deformation of a Z-graded subalgebra of the Poincaré superalgebra. We are able to map the classification problem for highly supersymmetric backgrounds (i.e., those which preserve more than half the supersymmetry) to the classification problem of a certain class of filtered deformations of graded subalgebras of the Poincaré superalgebra. We show that one can reconstruct a highly supersymmetric background from its Killing superalgebra; in so doing, we relate the bosonic field equations of 11-dimensional supergravity to the Jacobi identity of the Killing superalgebra and show in this way that preserving more than half the supersymmetry implies the bosonic field equations.

Journal of Physics A: Mathematical and Theoretical, 2016

A. We summarise recent results concerning the classification of filtered deformations of graded s... more A. We summarise recent results concerning the classification of filtered deformations of graded subalgebras of the Poincaré superalgebra in eleven dimensions, highlighting what could be considered a novel Lie-algebraic derivation of eleven-dimensional supergravity.

Journal of High Energy Physics, 2016

We determine the Killing superalgebras underpinning field theories with rigid unextended supersym... more We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of Z-graded subalgebras with maximum odd dimension of the N = 1 Poincaré superalgebra in four dimensions. Part of this calculation involves computing a Spencer cohomology group which, by analogy with a similar result in eleven dimensions, prescribes a notion of Killing spinor, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four dimensions. We prove that such Killing spinors always generate a Lie superalgebra, and that this Lie superalgebra is a filtered deformation of a subalgebra of the N = 1 Poincaré superalgebra in four dimensions. Demanding the flatness of the connection defining the Killing spinors, we obtain equations satisfied by the maximally supersymmetric backgrounds. We solve these equations, arriving at the classification of maximally supersymmetric backgrounds whose associated Killing superalgebras are precisely the filtered deformations we classify in this paper.

Classical and Quantum Gravity, 2016

We extend the link between Einstein Sasakian manifolds and Killing spinors to a class of η-Einste... more We extend the link between Einstein Sasakian manifolds and Killing spinors to a class of η-Einstein Sasakian manifolds, both in Riemannian and Lorentzian settings, characterising them in terms of generalised Killing spinors. We propose a definition of supersymmetric M-theory backgrounds on such a geometry and find a new class of such backgrounds, extending previous work of Haupt, Lukas and Stelle.

Journal of Geometry and Physics, 2018

Poincaré superalgebras in six dimensions: with and without R-symmetry. As the cases of four and e... more Poincaré superalgebras in six dimensions: with and without R-symmetry. As the cases of four and eleven dimensions taught us, we may read off from this calculation a Killing spinor equation which allows the determination of which geometries admit rigidly supersymmetric theories in this dimension. We prove that the resulting Killing spinors generate a Lie superalgebra and determine the geometries admitting the maximal number of such Killing spinors. They are divided in two branches. One branch consists of the lorentzian Lie groups with bi-invariant metrics and, as a special case, it includes the lorentzian Lie groups with a self-dual Cartan three-form which define the maximally supersymmetric backgrounds of (1, 0) Poincaré supergravity in six dimensions. The notion of Killing spinor on the other branch does not depend on the choice of a three-form but rather on a one-form valued in the R-symmetry algebra. In this case, we obtain three different (up to local isometry) maximally supersymmetric backgrounds, which are distinguished by the causal type of the one-form. C 1. Introduction 2. Conventions 3. Spencer complexes associated to the (1, 0) Poincaré superalgebra 3.1. Spencer complex of p 3.2. Spencer complex ofp 4. Calculation of H 2,2 (p − , p) 4.1. Solving the first cocycle condition 4.2. Solving the second cocycle condition 5. Calculation of H 2,2 (p − ,p) 6. The Killing superalgebra 6.1. Preliminaries 6.2. The Killing superalgebra. Case of self-dual 3-form. 6.3. The Killing superalgebra. Case of H not necessarily self-dual. 7. Killing superalgebras (alternative calculation with some indices) 8. Maximally supersymmetric backgrounds 8.1. The curvature of the superconnection 8.2. Zero curvature conditions 8.3. First branch: H = 0 8.4. Second branch: ϕ = 0 8.5. Killing superalgebras and filtered deformations

Journal of High Energy Physics, 2009

We prove a structure theorem for finite-dimensional indefinite-signature metric 3-Lie algebras ad... more We prove a structure theorem for finite-dimensional indefinite-signature metric 3-Lie algebras admitting a maximally isotropic centre. This algebraic condition indicates that all the negative-norm states in the associated Bagger-Lambert theory can be consistently decoupled from the physical Hilbert space. As an immediate application of the theorem, new examples beyond index 2 are constructed. The lagrangian for the Bagger-Lambert theory based on a general physically admissible 3-Lie algebra of this kind is obtained. Following an expansion around a suitable vacuum, the precise relationship between such theories and certain more conventional maximally supersymmetric gauge theories is found. These typically involve particular combinations of N = 8 super Yang-Mills and massive vector supermultiplets. A dictionary between the 3-Lie algebraic data and the physical parameters in the resulting gauge theories will thereby be provided.

Journal of High Energy Physics, 2009

We classify Freund-Rubin backgrounds of eleven-dimensional supergravity of the form AdS 4 ×X 7 wh... more We classify Freund-Rubin backgrounds of eleven-dimensional supergravity of the form AdS 4 ×X 7 which are at least half BPS -equivalently, smooth quotients of the round 7-sphere by finite subgroups of SO(8) which admit an (N > 3)-dimensional subspace of Killing spinors. The classification is given in terms of pairs consisting of an ADE subgroup of SU(2) and an automorphism defining its embedding in SO(8). In particular we find novel half-BPS quotients associated with the subgroups of type D n≥6 , E 7 and E 8 and their outer automorphisms.

Journal of High Energy Physics, 2008

We recast physical properties of the Bagger-Lambert theory, such as shiftsymmetry and decoupling ... more We recast physical properties of the Bagger-Lambert theory, such as shiftsymmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3algebraic terms, thus motivating the study of metric Lie 3-algebras and their Lie algebras of derivations. We prove a structure theorem for metric Lie 3-algebras in arbitrary signature showing that they can be constructed out of the simple and one-dimensional Lie 3-algebras iterating two constructions: orthogonal direct sum and a new construction called a double extension, by analogy with the similar construction for Lie algebras. We classify metric Lie 3-algebras of signature (2, p) and study their Lie algebras of derivations, including those which preserve the conformal class of the inner product. We revisit the 3-algebraic criteria spelt out at the start of the paper and select those algebras with signature (2, p) which satisfy them, as well as indicate the construction of more general metric Lie 3-algebras satisfying the ghost-decoupling criterion.

Journal of High Energy Physics, 2005

We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneou... more We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneous geodesics the limit is known to be homogeneous and we exhibit the limiting metric in terms of Lie algebraic data. This simplifies many calculations and we illustrate this with several examples. We also investigate the behaviour of (reductive) homogeneous structures under the plane-wave limit.

We classify maximally supersymmetric backgrounds (vacua) of chiral (1,0) and (2,0) supergravities... more We classify maximally supersymmetric backgrounds (vacua) of chiral (1,0) and (2,0) supergravities in six dimensions and, by reduction, also those of the minimal N=2 supergravity in five dimensions. Up to R-symmetry, the (2,0) vacua are in one-to-one correspondence with (1,0) vacua, and these in turn are locally isometric to Lie groups admitting a bi-invariant lorentzian metric with anti-selfdual parallelising torsion,

We classify and construct all the smooth Kaluza-Klein reductions to ten dimensions of the M2-and ... more We classify and construct all the smooth Kaluza-Klein reductions to ten dimensions of the M2-and M5-brane configurations which preserve some of the supersymmetry. In this way we obtain a wealth of new supersymmetric IIA backgrounds describing composite configurations of D-branes, NS-branes and flux/nullbranes; bound states of D2-branes and strings, D4-branes and NS5-branes, as well as some novel configurations in which the quotient involves nowhere-vanishing transverse rotations to the brane twisted by a timelike or lightlike translation. From these results there also follow novel M-theory backgrounds locally isometric to the M-branes, some of which are time-dependent and all of which are asymptotic to discrete quotients of eleven-dimensional Minkowski spacetime. We emphasise the universality of the formalism by briefly discussing analogous analyses in type IIA/IIB dual to the ones mentioned above. Some comments on the dual gauge theory description of some of our configurations are also included. e-print archive: http://xxx.lanl.gov/hep-th/0208107

The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained... more The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of WKp admitting a central extension for generic values of the parameter, reducing naturally to W, for special values of the parameter, and contracting to the centrally extended W1 +o~, Woo and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic to w~:p. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of V~/~o which contracts to a new nonlinear algebra of the W~-type.

Journal of High Energy Physics, 2005

We investigate the gauging of the Wess-Zumino term of a sigma model with boundary. We derive a se... more We investigate the gauging of the Wess-Zumino term of a sigma model with boundary. We derive a set of obstructions to gauging and we interpret them as the conditions for the Wess-Zumino term to extend to a closed form in a suitable equivariant relative de Rham complex ...

Journal of High Energy Physics, 1998

This is the first of a series of papers devoted to the group-theoretical analysis of the conditio... more This is the first of a series of papers devoted to the group-theoretical analysis of the conditions which must be satisfied for a configuration of intersecting M5-branes to be supersymmetric. In this paper we treat the case of static branes. We start by associating (a maximal torus of) a different subgroup of Spin 10 with each of the equivalence classes of supersymmetric configurations of two M5-branes at angles found by Ohta & Townsend. We then consider configurations of more than two intersecting branes. Such a configuration will be supersymmetric if and only if the branes are G-related, where G is a subgroup of Spin 10 contained in the isotropy of a spinor. For each such group we determine (a lower bound for) the fraction of the supersymmetry which is preserved.

Gauge Theories, Applied Supersymmetry and Quantum Gravity II, 1997

A method is presented by which a hidden N =2 superconformal symmetry can be exhibited in a string... more A method is presented by which a hidden N =2 superconformal symmetry can be exhibited in a string theory or indeed in a topological conformal field theory. More precisely, we present strong evidence, based on calculations with string theories, in favour of the conjecture that any topological conformal field theory can be obtained by twisting an N =2 superconformal field theory. (Talk given at the Workshop on Gauge Theories, Applied Supersymmetry and Quantum Gravity held at Imperial College, London, 5-10 July 1996.)

Nuclear Physics B, 1994

It has been realised recently that there is no unique way to describe the physical states of a gi... more It has been realised recently that there is no unique way to describe the physical states of a given string theory. In particular, it has been shown that any bosonic string theory can be embedded in a particular N =1 string background in such a way that the spectrum and the amplitudes of both theories agree. Similarly, it is also known that the amplitudes of any N =1 string theory can be obtained from a particular N =2 string background. When rephrased in the language of BRST cohomology, these results suggest a close connection to the theory of induced representations. The purpose of this note is to investigate this connection further and at the same time to reveal the mechanism behind these embeddings between string theories. We will first analyze the embedding of an affine algebra g in the N =1 affine algebra associated to g. Given any BRST cohomology theory for g we will be able to construct one for the N =1 affine algebra associated to g such that the cohomologies agree as operator product algebras. This is proven in two different ways. This example is the simplest in its kind and, in a sense that is made precise in the paper, all other similar embeddings are deformations of this one. We conclude the paper with a brief treatment of the general case, where we prove that for a particular class of "good" embeddings, the cohomologies are again isomorphic. †

Advances in Theoretical and Mathematical Physics, 2017

A. We study the algebraic structure of the Killing superalgebra of a supersymmetric background of... more A. We study the algebraic structure of the Killing superalgebra of a supersymmetric background of 11-dimensional supergravity and show that it is isomorphic to a filtered deformation of a Z-graded subalgebra of the Poincaré superalgebra. We are able to map the classification problem for highly supersymmetric backgrounds (i.e., those which preserve more than half the supersymmetry) to the classification problem of a certain class of filtered deformations of graded subalgebras of the Poincaré superalgebra. We show that one can reconstruct a highly supersymmetric background from its Killing superalgebra; in so doing, we relate the bosonic field equations of 11-dimensional supergravity to the Jacobi identity of the Killing superalgebra and show in this way that preserving more than half the supersymmetry implies the bosonic field equations.

Journal of Physics A: Mathematical and Theoretical, 2016

A. We summarise recent results concerning the classification of filtered deformations of graded s... more A. We summarise recent results concerning the classification of filtered deformations of graded subalgebras of the Poincaré superalgebra in eleven dimensions, highlighting what could be considered a novel Lie-algebraic derivation of eleven-dimensional supergravity.

Journal of High Energy Physics, 2016

We determine the Killing superalgebras underpinning field theories with rigid unextended supersym... more We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of Z-graded subalgebras with maximum odd dimension of the N = 1 Poincaré superalgebra in four dimensions. Part of this calculation involves computing a Spencer cohomology group which, by analogy with a similar result in eleven dimensions, prescribes a notion of Killing spinor, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four dimensions. We prove that such Killing spinors always generate a Lie superalgebra, and that this Lie superalgebra is a filtered deformation of a subalgebra of the N = 1 Poincaré superalgebra in four dimensions. Demanding the flatness of the connection defining the Killing spinors, we obtain equations satisfied by the maximally supersymmetric backgrounds. We solve these equations, arriving at the classification of maximally supersymmetric backgrounds whose associated Killing superalgebras are precisely the filtered deformations we classify in this paper.

Classical and Quantum Gravity, 2016

We extend the link between Einstein Sasakian manifolds and Killing spinors to a class of η-Einste... more We extend the link between Einstein Sasakian manifolds and Killing spinors to a class of η-Einstein Sasakian manifolds, both in Riemannian and Lorentzian settings, characterising them in terms of generalised Killing spinors. We propose a definition of supersymmetric M-theory backgrounds on such a geometry and find a new class of such backgrounds, extending previous work of Haupt, Lukas and Stelle.

Journal of Geometry and Physics, 2018

Poincaré superalgebras in six dimensions: with and without R-symmetry. As the cases of four and e... more Poincaré superalgebras in six dimensions: with and without R-symmetry. As the cases of four and eleven dimensions taught us, we may read off from this calculation a Killing spinor equation which allows the determination of which geometries admit rigidly supersymmetric theories in this dimension. We prove that the resulting Killing spinors generate a Lie superalgebra and determine the geometries admitting the maximal number of such Killing spinors. They are divided in two branches. One branch consists of the lorentzian Lie groups with bi-invariant metrics and, as a special case, it includes the lorentzian Lie groups with a self-dual Cartan three-form which define the maximally supersymmetric backgrounds of (1, 0) Poincaré supergravity in six dimensions. The notion of Killing spinor on the other branch does not depend on the choice of a three-form but rather on a one-form valued in the R-symmetry algebra. In this case, we obtain three different (up to local isometry) maximally supersymmetric backgrounds, which are distinguished by the causal type of the one-form. C 1. Introduction 2. Conventions 3. Spencer complexes associated to the (1, 0) Poincaré superalgebra 3.1. Spencer complex of p 3.2. Spencer complex ofp 4. Calculation of H 2,2 (p − , p) 4.1. Solving the first cocycle condition 4.2. Solving the second cocycle condition 5. Calculation of H 2,2 (p − ,p) 6. The Killing superalgebra 6.1. Preliminaries 6.2. The Killing superalgebra. Case of self-dual 3-form. 6.3. The Killing superalgebra. Case of H not necessarily self-dual. 7. Killing superalgebras (alternative calculation with some indices) 8. Maximally supersymmetric backgrounds 8.1. The curvature of the superconnection 8.2. Zero curvature conditions 8.3. First branch: H = 0 8.4. Second branch: ϕ = 0 8.5. Killing superalgebras and filtered deformations

Journal of High Energy Physics, 2009

We prove a structure theorem for finite-dimensional indefinite-signature metric 3-Lie algebras ad... more We prove a structure theorem for finite-dimensional indefinite-signature metric 3-Lie algebras admitting a maximally isotropic centre. This algebraic condition indicates that all the negative-norm states in the associated Bagger-Lambert theory can be consistently decoupled from the physical Hilbert space. As an immediate application of the theorem, new examples beyond index 2 are constructed. The lagrangian for the Bagger-Lambert theory based on a general physically admissible 3-Lie algebra of this kind is obtained. Following an expansion around a suitable vacuum, the precise relationship between such theories and certain more conventional maximally supersymmetric gauge theories is found. These typically involve particular combinations of N = 8 super Yang-Mills and massive vector supermultiplets. A dictionary between the 3-Lie algebraic data and the physical parameters in the resulting gauge theories will thereby be provided.

Journal of High Energy Physics, 2009

We classify Freund-Rubin backgrounds of eleven-dimensional supergravity of the form AdS 4 ×X 7 wh... more We classify Freund-Rubin backgrounds of eleven-dimensional supergravity of the form AdS 4 ×X 7 which are at least half BPS -equivalently, smooth quotients of the round 7-sphere by finite subgroups of SO(8) which admit an (N > 3)-dimensional subspace of Killing spinors. The classification is given in terms of pairs consisting of an ADE subgroup of SU(2) and an automorphism defining its embedding in SO(8). In particular we find novel half-BPS quotients associated with the subgroups of type D n≥6 , E 7 and E 8 and their outer automorphisms.

Journal of High Energy Physics, 2008

We recast physical properties of the Bagger-Lambert theory, such as shiftsymmetry and decoupling ... more We recast physical properties of the Bagger-Lambert theory, such as shiftsymmetry and decoupling of ghosts, the absence of scale and parity invariance, in Lie 3algebraic terms, thus motivating the study of metric Lie 3-algebras and their Lie algebras of derivations. We prove a structure theorem for metric Lie 3-algebras in arbitrary signature showing that they can be constructed out of the simple and one-dimensional Lie 3-algebras iterating two constructions: orthogonal direct sum and a new construction called a double extension, by analogy with the similar construction for Lie algebras. We classify metric Lie 3-algebras of signature (2, p) and study their Lie algebras of derivations, including those which preserve the conformal class of the inner product. We revisit the 3-algebraic criteria spelt out at the start of the paper and select those algebras with signature (2, p) which satisfy them, as well as indicate the construction of more general metric Lie 3-algebras satisfying the ghost-decoupling criterion.

Journal of High Energy Physics, 2005

We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneou... more We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneous geodesics the limit is known to be homogeneous and we exhibit the limiting metric in terms of Lie algebraic data. This simplifies many calculations and we illustrate this with several examples. We also investigate the behaviour of (reductive) homogeneous structures under the plane-wave limit.

We classify maximally supersymmetric backgrounds (vacua) of chiral (1,0) and (2,0) supergravities... more We classify maximally supersymmetric backgrounds (vacua) of chiral (1,0) and (2,0) supergravities in six dimensions and, by reduction, also those of the minimal N=2 supergravity in five dimensions. Up to R-symmetry, the (2,0) vacua are in one-to-one correspondence with (1,0) vacua, and these in turn are locally isometric to Lie groups admitting a bi-invariant lorentzian metric with anti-selfdual parallelising torsion,

We classify and construct all the smooth Kaluza-Klein reductions to ten dimensions of the M2-and ... more We classify and construct all the smooth Kaluza-Klein reductions to ten dimensions of the M2-and M5-brane configurations which preserve some of the supersymmetry. In this way we obtain a wealth of new supersymmetric IIA backgrounds describing composite configurations of D-branes, NS-branes and flux/nullbranes; bound states of D2-branes and strings, D4-branes and NS5-branes, as well as some novel configurations in which the quotient involves nowhere-vanishing transverse rotations to the brane twisted by a timelike or lightlike translation. From these results there also follow novel M-theory backgrounds locally isometric to the M-branes, some of which are time-dependent and all of which are asymptotic to discrete quotients of eleven-dimensional Minkowski spacetime. We emphasise the universality of the formalism by briefly discussing analogous analyses in type IIA/IIB dual to the ones mentioned above. Some comments on the dual gauge theory description of some of our configurations are also included. e-print archive: http://xxx.lanl.gov/hep-th/0208107

The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained... more The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting W-algebra is a one-parameter deformation of WKp admitting a central extension for generic values of the parameter, reducing naturally to W, for special values of the parameter, and contracting to the centrally extended W1 +o~, Woo and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic to w~:p. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of V~/~o which contracts to a new nonlinear algebra of the W~-type.

Journal of High Energy Physics, 2005

We investigate the gauging of the Wess-Zumino term of a sigma model with boundary. We derive a se... more We investigate the gauging of the Wess-Zumino term of a sigma model with boundary. We derive a set of obstructions to gauging and we interpret them as the conditions for the Wess-Zumino term to extend to a closed form in a suitable equivariant relative de Rham complex ...

Journal of High Energy Physics, 1998

This is the first of a series of papers devoted to the group-theoretical analysis of the conditio... more This is the first of a series of papers devoted to the group-theoretical analysis of the conditions which must be satisfied for a configuration of intersecting M5-branes to be supersymmetric. In this paper we treat the case of static branes. We start by associating (a maximal torus of) a different subgroup of Spin 10 with each of the equivalence classes of supersymmetric configurations of two M5-branes at angles found by Ohta & Townsend. We then consider configurations of more than two intersecting branes. Such a configuration will be supersymmetric if and only if the branes are G-related, where G is a subgroup of Spin 10 contained in the isotropy of a spinor. For each such group we determine (a lower bound for) the fraction of the supersymmetry which is preserved.

Gauge Theories, Applied Supersymmetry and Quantum Gravity II, 1997

A method is presented by which a hidden N =2 superconformal symmetry can be exhibited in a string... more A method is presented by which a hidden N =2 superconformal symmetry can be exhibited in a string theory or indeed in a topological conformal field theory. More precisely, we present strong evidence, based on calculations with string theories, in favour of the conjecture that any topological conformal field theory can be obtained by twisting an N =2 superconformal field theory. (Talk given at the Workshop on Gauge Theories, Applied Supersymmetry and Quantum Gravity held at Imperial College, London, 5-10 July 1996.)

Nuclear Physics B, 1994

It has been realised recently that there is no unique way to describe the physical states of a gi... more It has been realised recently that there is no unique way to describe the physical states of a given string theory. In particular, it has been shown that any bosonic string theory can be embedded in a particular N =1 string background in such a way that the spectrum and the amplitudes of both theories agree. Similarly, it is also known that the amplitudes of any N =1 string theory can be obtained from a particular N =2 string background. When rephrased in the language of BRST cohomology, these results suggest a close connection to the theory of induced representations. The purpose of this note is to investigate this connection further and at the same time to reveal the mechanism behind these embeddings between string theories. We will first analyze the embedding of an affine algebra g in the N =1 affine algebra associated to g. Given any BRST cohomology theory for g we will be able to construct one for the N =1 affine algebra associated to g such that the cohomologies agree as operator product algebras. This is proven in two different ways. This example is the simplest in its kind and, in a sense that is made precise in the paper, all other similar embeddings are deformations of this one. We conclude the paper with a brief treatment of the general case, where we prove that for a particular class of "good" embeddings, the cohomologies are again isomorphic. †