Symmetric Coset Space Research Articles

An exact symmetry in \u03bb-deformed CFTs

We consider λ-deformed current algebra CFTs at level k, interpolating between an exact CFT in the UV and a PCM in the IR. By employing gravitational techniques, we derive the two-loop, in the large k expansion, β-function. We find that this is covariant under a remarkable exact symmetry involving the coupling λ, the level k and the adjoint quadratic Casimir of the group. Using this symmetry and CFT techniques, we are able to compute the Zamolodchikov metric, the anomalous dimension of the bilinear operator and the Zamolodchikov C -function at two-loops in the large k expansion, as exact func- tions of the deformation parameter. Finally, we extend the above results to λ-deformed parafermionic algebra coset CFTs which interpolate between exact coset CFTs in the UV and a symmetric coset space in the IR.

Trigonometric Parity for Composite Higgs Models.

We identify trigonometric parity as the key ingredient behind models of neutral naturalness for the Higgs potential and show how to construct the minimal model realizing trigonometric parity. We show that any symmetric coset space readily includes such a trigonometric parity, which is simply a combination of a π/2 rotation along a broken direction and a Higgs parity transformation. The top sector can be extended such that this Z_{2} remains intact, which ensures the cancelation of the quadratic divergences in the Higgs potential, yielding the simplest model of neutral naturalness.

Topological interactions of Nambu-Goldstone bosons in quantum many-body systems

We classify effective actions for Nambu-Goldstone (NG) bosons assuming the absence of anomalies. Special attention is paid to Lagrangians that are invariant only up to a surface term, which are shown to be in a one-to-one correspondence with Chern-Simons (CS) theories for unbroken symmetry. Without making specific assumptions on the spacetime symmetry, we give explicit expressions for these Lagrangians, generalizing the Berry and Hopf terms in ferromagnets. Globally well-defined matrix expressions are derived for symmetric coset spaces of broken symmetry. The CS Lagrangians exhibit special properties, on both the perturbative and the global topological levels. The order-one CS term is responsible for the noninvariance of the canonical momentum density under internal symmetry, known as the linear momentum problem. The order-three CS term gives rise to a novel type of interaction among NG bosons. All the CS terms are robust against local variations of microscopic physics.

Spectra of coset sigma models

We compute the complete 1-loop spectrum of anomalous dimensions for the bulk fields of non-linear sigma models on symmetric coset (super)spaces G/H, both with and without world-sheet supersymmetry. In addition, we provide two new methods for the construction of partition functions in the infinite radius limit and demonstrate their efficiency in the case of (super)sphere sigma models. Our results apply to a large number of target spaces including superspheres and superprojective spaces such as the N=2 sigma model on CP3|4.

Holomorphic bisectional curvatures, supersymmetry breaking, and Affleck-Dine baryogenesis

Working in $D=4$, $N=1$ supergravity, we utilize relations between holomorphic sectional and bisectional curvatures of Kahler manifolds to constrain Affleck-Dine baryogenesis. We show the following no-go result: Affleck-Dine baryogenesis cannot be performed if the holomorphic sectional curvature at the origin is isotropic in tangent space; as a special case, this rules out spaces of constant holomorphic sectional curvature (defined in the above sense) and in particular maximally symmetric coset spaces. We also investigate scenarios where inflationary supersymmetry breaking is identified with the supersymmetry breaking responsible for mass splitting in the visible sector, using conditions of sequestering to constrain manifolds where inflation can be performed.

Creation of matter in the universe and groups of type E7

We relate the mechanism of matter creation in the universe after inflation to a simple and universal mathematical property of extended N > 1 supergravities and related compactifications of superstring theory. We show that in all such models, the inflaton field may decay into vector fields due to a nonminimal scalar-vector coupling. This coupling is compulsory for all scalars except N=2 hyperscalars. The proof is based on the fact that all extended supergravities described by symmetric coset spaces G/H have duality groups G of type E7, with exception of U(p,n) models. For N=2 we prove separately that special geometry requires a non-minimal scalar-vector coupling. Upon truncation to N=1 supergravity, extended models generically preserve the non-minimal scalar-vector coupling, with exception of U(p,n) models and hyperscalars. For some string theory/supergravity inflationary models, this coupling provides the only way to complete the process of creation of matter in the early universe.

Brane solutions and integrability: a status report

We review the status of the integrability and solvability of the geodesics equations of motion on symmetric coset spaces that appear as sigma models of supergravity theories when reduced over respectively the timelike and spacelike direction. Such geodesic curves describe respectively timelike and spacelike brane solutions. We emphasize the applications to black holes.

Black holes in supergravity and integrability

Stationary black holes of massless supergravity theories are described by certain geodesic curves on the target space that is obtained after dimensional reduction over time. When the target space is a symmetric coset space we make use of the group-theoretical structure to prove that the second order geodesic equations are integrable in the sense of Liouville, by explicitly constructing the correct amount of Hamiltonians in involution. This implies that the Hamilton-Jacobi formalism can be applied, which proves that all such black hole solutions, including non-extremal solutions, possess a description in terms of a (fake) superpotential. Furthermore, we improve the existing integration method by the construction of a Lax integration algorithm that integrates the second order equations in one step instead of the usual two step procedure. We illustrate this technology with a specific example.

Algebraic integration of sigma-model field equations

We prove that the dualization algebra of the sigma model with a symmetric coset space is a Lie algebra and show that it generates an appropriate adjoint representation that allows integrating the field equations locally, which yields first-order equations.

Random matrices beyond the Cartan classification

It is known that hermitean random matrix ensembles can be identified with symmetric coset spaces of Lie groups, or else with tangent spaces of the same. This results in a classification of random matrix ensembles as well as applications in practical calculations of physical observables. In this paper, we show that a large number of non-hermitean random matrix ensembles defined by physically motivated symmetries—chiral symmetry, time-reversal invariance, space-rotation invariance, particle–hole symmetry or different reality conditions—can likewise be identified with symmetric spaces. We give explicit representations of the random matrix ensembles identified with lateral algebra subspaces, and of the corresponding symmetric subalgebras spanning the group of invariance. Among the ensembles listed we identify as special cases all the hermitean ensembles identified with the Cartan classes of symmetric spaces and the three Ginibre ensembles with complex eigenvalues.

Domain walls, black holes, and supersymmetric quantum mechanics

Supersymmetric solutions, such as BPS domain walls or black holes, in four- and five-dimensional supergravity theories with eight supercharges can be described by effective quantum mechanics with a potential term. We show how properties of the latter theory can help us to learn about the physics of supersymmetric vacua and BPS solutions in these supergravity theories. The general approach is illustrated in a number of specific examples where scalar fields of matter multiplets take values in symmetric coset spaces.

Supersymmetry breaking by dimensional reduction over coset spaces

We study the dimensional reduction of a ten-dimensional supersymmetric E 8 gauge theory over six-dimensional coset spaces. We find that the coset space dimensional reduction over a symmetric coset space leaves the four dimensional gauge theory without any track of the original supersymmetry. On the contrary the dimensional reduction over a non-symmetric coset space leads to a softly broken supersymmetric gauge theory in four dimensions. The SO 7/ SO 6 and G 2/ SU(3) are used as representative prototypes of symmetric and non symmetric coset spaces, respectively.

Construction and analysis of anomaly-free supersymmetric -models

This paper discusses a procedure for the consistent coupling of gauge- and matter superfields to supersymmetric sigma-models on symmetric coset spaces of Kaehler type. We exhibit the finite isometry transformations and the corresponding Kaehler transformations. These lead to the construction of a generalized type of Killing potentials. In certain cases a charge quantization condition needs to be imposed to guarantee the global existence of a line bundle on a coset space. The results are applied to the explicit construction of sigma-models on cosets SO(2N)/U(N). Only a finite number of these models can consistently incorporate matter in representations descending from the spinorial representations of SO(2N). We investigate in detail some aspects of the vacuum structure of the gauged SO(10)/U(5) theory, with surprising results: the fully gauged minimal anomaly-free model is shown be singular, as the kinetic terms of the quasi-Goldstone fermions vanish in the vacuum. Gauging only the linear isometry group SU(5)xU(1), or one of its subgroups, can give a physically well-behaved theory. With gauged U(1) this requires the Fayet-Iliopoulos term to take values in a specific limited range.

Modified symmetry generators and the geometric phase

Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately Sd (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.

SCALAR FIELDS IN THE DIMENSIONAL REDUCTION SCHEME FOR SYMMETRIC SPACES

We study the general features of the dimensional reduction scheme for multi-dimensional spaces of the type M4 × S/R, S/R being a symmetric coset space. The properties of the scalar potentials of the reduced theories are investigated and an effective method of explicit calculation of these potentials is elaborated. We consider also a wide class of embeddings of Lie subalgebras into simple Lie algebras resulting in reduced theories of physical interst.

N = 1 supergravity models based on non-compact hermitian symmetric coset spaces

We study N = 1 supergravity models where the Kähler metric of the scalar manifold has a dilatation isometry. It is demonstrated that, if the Kähler potential transforms in a particular way under dilatations, the scalar potential is semipositive definite giving rise to a vanishing cosmological constant at the classical level. We find a class of supergravity models based on kählerian symmetric manifolds which represents a natural generalisation of no-scale models and SU( N, 1) supergravity. The mass spectrum of these theories exhibits manifest SUSY-breaking.

Almost paracontact and parahodge structures on manifolds

In this paper we study the paracomplex analogues of almost contact structures, and we introduce and study the notion of parahodge structures on manifolds. In particular, we construct new examples of paracomplex manifolds and we find all simply connected parahermitian symmetric coset spaces, which are the adjoint orbits of noncompact simple Lie groups, with parahodge structures induced by the Killing forms. This is done by (i) observing that a version of the results of A. Morimoto [4] on almost contact structures can be formulated and proved for almost paracontact structures, and by (ii) the methods of geometric quantization [3] applied to parahermitian symmetric triples [1] in conjunction with results of [7]. Two of the main results are Theorem 2.5 (which ties together the above structures) and Corollary 3.9.

Gauging the d = 5 Maxwell/Einstein supergravity theories: More on Jordan algebras

We discuss in detail the possible gaugings, abelian and non-abelian, of a class of d = 5 Maxwell/Einstein supergravity theories for which the manifold of scalar fields is a symmetric coset space. We show that a U(1) “gauged” d =5 supergravity theory is possible with a vanishing scalar field potential, and we give necessary and sufficient conditions for this to occur. We discuss the d = 5 Yang-Mills/Einstein supergravity theories for both compact and non-compact gauge groups. We show that the “irreducibililty” of the “magical” subclass of d = 5 Maxwell/Einstein supergravity theories is preserved if and only if the Yang-Mills gauge group is SU(3,1). We expect that the irreducibility property of the “exceptional” d = 4 Maxwell/Einstein supergravity theory can similarly be preserved by gauging an SO ∗(8) subgroup of its symmetry group E 7(−25). Throughout we make extensive use of the underlying Jordan algebraic structure of N = 2 supergravity which we have established in previous work.

Stability of higher-dimensional Einstein-Yang-Mills theories on symmetric spaces

We study (4 + d)-dimensional Einstein-Yang-Mills theories with arbitrary gauge groups, G YM. The theory is compactified on a d-dimensional symmetric coset space G H with a symmetric, topologically non-trivial classical gauge field, embedded in an H-subgroup of the Yang-Mills group. These theories are known to be classically stabilized by gravity if G YM = H, G H is a sphere and d ≠ 3. We study classical instabilities caused by embedding H in a larger gauge group. The small fluctuation spectrum is completely calculable, and leads to a stability condition. For two-dimensional spheres this condition is precisely the Brandt-Neri stability condition for non-abelian monopole fields. For four-spheres we find stability for SU(2) instantons embedded in arbitrary gauge groups and we reproduce the fluctuation spectrum around instantons. For higher-dimensional spheres the stable solutions of this type are completely classified, and occur only for d = 5, 6, 8, 9, 10, 12 and 16. The results show a remarkable agreement with expected topological stability. We also give a few examples with other symmetric spaces, such as CP n , where the stability criterion appears less restrictive.

Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces

On a symmetric coset space G/H the eigenvalues of the Laplacian and the Lichnerowicz operator acting on arbitrary tensor harmonics are given in terms of the eigenvalues of the quadratic Casimir operators of G and H. Explicit examples for Sn, CPn, and real (complex) Grassmann manifolds are analyzed.