Coset Space Research Articles

Spectral decompositions of spaces induced by spectral decompositions of acting groups

The presence on a space of an additional algebraic structure, which is compatible with its topology, in many cases imposes strong restrictions on the properties of the space itself. In the paper we discuss connections between spectral characterizations of groups and spaces with their actions when the latter are near to their coset spaces. The compatible system of maps on a space which is induced by a compatible system of maps on the acting group is constructed. This approach allows to give a unified possibility in the study of spaces with special actions of groups with suitable spectral decompositions. Among them are Čech complete groups, their products and subgroups. In particular, it is proved that a pseudocompact coset space of a Čech-complete group is ϰ-metrizable and a compactum which is a coset space of a subgroup of a product of Čech-complete groups is an openly generated compactum. We also show that an arbitrary d-open action of an ℵ0-balanced group on a pseudocompact space can be replaced by a d-open action of an ℵ0-bounded group.

Invariant proper metrics on coset spaces

It is known that for every second countable locally compact group G, there exists a proper G-invariant metric which induces the topology of the group. This is no longer true for coset spaces G/H viewed as G-spaces. We study necessary and sufficient conditions which ensure the existence of such metrics on G/H.

On twistor transformations and invariant differential operator of simple Lie group G2(2)

The twistor transformations associated to the simple Lie group G2 are described explicitly. We consider the double fibration \documentclass[12pt]{minimal}\begin{document}${\rm G}_2/P_2 \xleftarrow {\eta } {{\rm G}_2/B} \xrightarrow {\tau }{\rm G}_2/P_1$\end{document}G2/P2←ηG2/B→τG2/P1, where P1 and P2 are two parabolic subgroups of G2 and B is a Borel subgroup, and its local version: \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}^*_2 \xleftarrow {\eta } \mathcal {F} \xrightarrow {\tau } \mathcal {H}_1$\end{document}H2*←ηF→τH1, where \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_1$\end{document}H1 is the Heisenberg group of dimension 5 embedded in the coset space G2/P1, \documentclass[12pt]{minimal}\begin{document}$\mathcal {F} = \mathbb {CP}^1 \times \mathcal {H}_1$\end{document}F=CP1×H1 and \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}^*_2$\end{document}H2* contains the nilpotent Lie group \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_2$\end{document}H2 of step three. The Baker-Campbell-Hausdorff formula is used to parametrize the coset spaces, coordinates charts, their transition functions and the fibers of the projection η as complex curves. We write down the relative De-Rham sequence on \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F along the fibers and push it down to \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_1$\end{document}H1 to get a family of matrix-valued differential operators \documentclass[12pt]{minimal}\begin{document}${\mathscr D}_k$\end{document}Dk. Then we establish a kind of Penrose correspondence for G2: the kernel of \documentclass[12pt]{minimal}\begin{document}$\mathscr{D}_k$\end{document}Dk is isomorphic to the first cohomology of the sheaf \documentclass[12pt]{minimal}\begin{document}$\mathscr{O} (-k )$\end{document}O(−k) over \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}^*_2$\end{document}H2*. We also give the Penrose-type integral transformation u = Pf for \documentclass[12pt]{minimal}\begin{document}$f\in \mathscr{O} (-k )$\end{document}f∈O(−k), which gives solutions to equations \documentclass[12pt]{minimal}\begin{document}${\mathscr D}_ku=0$\end{document}Dku=0. When restricted to the real Heisenberg group, the differential operators are invariant under the action of G2(2). Exchanging P1 and P2, we derive corresponding results on \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_2$\end{document}H2.

Holographic RG flows in six dimensional F(4) gauged supergravity

We study critical points of F (4) gauged supergravity in six dimensions coupled to three vector multiplets. Scalar fields are described by $ {{\mathbb{R}}^{+}}\times \frac{{\mathrm{SO}\left( {4,3} \right)}}{{\mathrm{SO}(4)\times \mathrm{SO}(3)}} $ coset space, and the gauge group is given by SO(3)R × SO(3) with SO(3)R being the R-symmetry. We identify new non-supersymmetric critical points of the scalar potential. One of these new critical points is shown to be stable with all scalar masses above the BF bound and should correspond to a new non-supersymmetric CFT in five dimensions. On the other hand, the maximally supersymmetric critical point with all scalars vanishing is dual to an SCFT5 arising from a near horizon geometry of the D4-D8 brane system in type I′ theory with an enhanced global symmetry E 1 ~ SU(2). We give a numerical RG flow solution interpolating between this SCFT and the new stable critical point. The flow describes a non-supersymmetric deformation driven by relevant operators of dimension 3. We identify the dual operators with the mass terms for hypermultiplet scalars in the dual field theory. The solution provides another example of holographic RG flows in AdS6/CFT5 correspondence.

Construction of Lax connections by exponentiation

We propose a method for constructing the Lax connection of two-dimensional relativistic integrable sigma models on coset spaces by means of exponentiation of a suitable operator. We derive a simple quadratic relation that this operator must satisfy for an entire one-parameter family of connections to be flat.

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.

On Static Charged Black Holes in Type IIA on a Nearly-Kahler Coset

We study static, spherically symmetric black hole solutions in four-dimensional N = 2 gauged supergravity with one vector multiplet and one hyper-tensor multiplet. This is derived from massive type IIA theory compactified on the nearly-Kähler coset space G2/SU(3). It is well known that the Romans' mass parameter yields the Stückelberg-type deformation of the gauge field strengths in the four-dimensional system. This deformation requires that all the (covariant) derivatives of the scalar fields must vanish and the two-form field is closed. It turns out that charged solutions are forbidden. This implies that only AdS vacua or Schwarzschild-AdS black holes are allowed as the static, spherically symmetric solutions.

All homogeneous N = 2 M-theory truncations with supersymmetric AdS4 vacua

We study consistent truncations of M-theory to gauged N=2 supergravity in four dimensions, based on a large class of SU(3)-structures in seven dimensions. We show that the gauging involves isometries of the vector multiplet scalar manifold as well as the Heisenberg algebra and a special isometry of the hyperscalar manifold. As a result, non-abelian gauge groups and new non-trivial scalar potentials are generated. Then we specialize to all homogeneous SU(3)-structures supporting supersymmetric AdS4 vacua. These are the Stiefel manifold V52, the Aloff-Wallach spaces N(k,l), the seven-sphere (seen as SU(4)/SU(3) or Sp(2)/Sp(1)) and the M110 and Q111 coset spaces. For each of these cases, we describe in detail the N=2 model and discuss its peculiarities.

A Stochastic Difference Equation with Stationary Noise on Groups

AbstractWe consider the stochastic difference equation on a locally compact group G, where is an automorphism of G, ξk are given G-valued random variables and ηk are unknown G-valued random variables. This equation was considered by Tsirelson and Yor on a one-dimensional torus. We consider the case when ξk have a common law μ and prove that if G is a distal group and is a distal automorphism of G and if the equation has a solution, then extremal solutions of the equation are in one-to-one correspondence with points on the coset space K\G for some compact subgroup K of G such that μ is supported on for any z in the support of μ. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

Schramm-Loewner evolution martingales in coset conformal field theory

Schramm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice models and is suitable for strict proofs, it lacks computational and predictive power of conformal field theory. To provide a way for the concurrent use of SLE and CFT, CFT correlation functions, which are martingales with respect to SLE, are considered. A relation between parameters of Schramm-Loewner evolution on coset space and algebraic data of coset conformal field theory is revealed. The consistency of this approach with the behavior of parafermionic and minimal models is tested. Coset models are connected with off-critical massive field theories and implications of SLE are discussed.

VISIBLE ACTIONS ON FLAG VARIETIES OF TYPE B AND A GENERALISATION OF THE CARTAN DECOMPOSITION

AbstractWe give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of Kobayashi [‘A generalized Cartan decomposition for the double coset space $(U(n_{1})\times U(n_{2})\times U(n_{3})) \backslash U(n)/ (U(p)\times U(q))$’, J. Math. Soc. Japan59 (2007), 669–691] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma $ is a Chevalley–Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma }H$ holds if and only if either (I) both $H$ and $L$ are maximal and of type A, or (II) $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switching $H$ and $L$. This classification gives a visible action of $L$ on the generalised flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find an explicit ‘slice’ $B$ with $\dim B=\mathrm {rank}\, G$ in case I, and $\dim B=2$ or $3$ in case II, such that a generalised Cartan decomposition $G=LBH$holds. An application to multiplicity-free theorems of representations is also discussed.

Gaugings of N = 4 three dimensional gauged supergravity with exceptional coset manifolds

Some admissible gauge groups of N=4 Chern-Simons gauged supergravity in three dimensions with exceptional scalar manifolds $G_{2(2)}/SO(4)$, $F_{4(4)}/USp(6)\times SU(2)$, $E_{6(2)}/SU(6)\times SU(2)$, $E_{7(-5)}/SO(12)\times SU(2)$ and $E_{8(-24)}/E_7\times SU(2)$ are identified. In particular, a complete list of all possible gauge groups is given for the theory with $G_{2(2)}/SO(4)$ coset space. We also study scalar potentials for all of these gauge groups and find some critical points. In the case of $F_{4(4)}/USp(6)\times SU(2)$ target space, we give some semisimple gauge groups which are maximal subgroups of $F_{4(4)}$. Most importantly, we construct the $SO(4)\ltimes \mathbf{T}^6$ gauged supergravity which is equivalent to N=4 SO(4) Yang-Mills gauged supergravity. The latter is proposed to be obtained from an $S^3$ reduction of $(1,0)$ six dimensional supergravity coupled to two vector and two tensor multiplets. The scalar potential of this theory on the scalar fields which are invariant under SO(4) is explicitly computed. Depending on the value of the coupling constants, the theory admits both dS and AdS vacua when all of the 28 scalars vanish. The maximal N=4 supersymmetric $AdS_3$ should correspond to the $AdS_3\times S^3$ solution of the $(1,0)$ six dimensional theory. Finally, some gauge groups of the theories with $E_{6(2)}/SU(6)\times SU(2)$, $E_{7(-5)}/SO(12)\times SU(2)$ and $E_{8(-24)}/E_7\times SU(2)$ scalar manifolds are identified.

Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance

Using the effective Lagrangian approach, we clarify general issues about Nambu-Goldstone bosons without Lorentz invariance. We show how to count their number and study their dispersion relations. Their number is less than the number of broken generators when some of them form canonically conjugate pairs. The pairing occurs when the generators have a nonzero expectation value of their commutator. For non-semi-simple algebras, central extensions are possible. The underlying geometry of the coset space in general is partially symplectic.

Nambu–Goldstone dynamics and generalized coherent-state functional integrals

This paper gives a new method of attack on the Nambu–Goldstone dynamics in spontaneously broken theories. Since the target space of the Nambu–Goldstone fields is a group coset space, their effective quantum dynamics can be naturally phrased in terms of generalized coherent-state functional integrals. As an explicit example of this line of reasoning, we construct a low-energy effective Lagrangian for the Heisenberg ferromagnet in a broken phase. The leading field configuration in the WKB approximation leads to the Landau–Lifshitz equation for quantum ferromagnet. The corresponding linearized equations allow one to identify the Nambu–Goldstone boson with a ferromagnetic magnon.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Consistent truncation of d = 11 supergravity on AdS 4 × S 7

We study the system of equations derived twenty five years ago by B. de Wit and the first author [Nucl. Phys. B281 (1987) 211] as conditions for the consistent truncation of eleven-dimensional supergravity on AdS_4 x S^7 to gauged N = 8 supergravity in four dimensions. By exploiting the E_7(7) symmetry, we determine the most general solution to this system at each point on the coset space E_7(7)/SU(8). We show that invariants of the general solution are given by the fluxes in eleven-dimensional supergravity. This allows us to both clarify the explicit non-linear ansatze for the fluxes given previously and to fill a gap in the original proof of the consistent truncation. These results are illustrated with several examples.

Yang-Mills fields in flux compactifications on homogeneous manifolds with SU(4)-structure

The SU(4)-instanton equations are natural BPS equations for instantons on 8-manifolds. We study these equations on nearly Kaehler and Calabi-Yau torsion manifolds of the form M x G/H, with G/H a coset space and M a product of a torus with Euclidean space. By imposing G-invariance the instanton equations reduce to interesting equations on M; for example, equations used by Kapustin and Witten in the geometric Langlands program arise in this way. We carry out reductions in a number of examples, and where possible present simple solutions.

Equivariant extension properties of coset spaces of locally compact groups and approximate slices

We prove that for a compact subgroup H of a locally compact Hausdorff group G, the following properties are mutually equivalent: (1) G/H is finite-dimensional and locally connected, (2) G/H is a smooth manifold, (3) G/H satisfies the following equivariant extension property: for every paracompact proper G-space X having a paracompact orbit space, every G-map A→G/H from a closed invariant subset A⊂X extends to a G-map U→G/H over an invariant neighborhood U of A. A new version of the Approximate Slice Theorem is also proven in the light of these results.

Cohomology of line bundles: Applications

Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for the computation of line bundle valued cohomology classes over toric varieties are presented. For the heterotic string, the prime examples are so-called monad constructions on Calabi-Yau manifolds. In the context of Type II orientifolds, one often needs to compute cohomology for line bundles on finite group action coset spaces, necessitating us to generalize our algorithm to this case. Moreover, we exemplify that the different terms in Batyrev's formula and its generalizations can be given a one-to-one cohomological interpretation. Furthermore, we derive a combinatorial closed form expression for two Hodge numbers of a codimension two Calabi-Yau fourfold.

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.

Exactly marginal parafermions

We describe several families of non-unitary coset conformal field theories that possess truly marginal couplings. These generalize the known examples of Wess-Zumino-Witten models on supergroups such as PSU(n|n) or OSP(2n+2|2n). Our extension includes coset space sigma models, affine Toda theories or Gross-Neveu models which are believed to arise in certain limits.