Geometry of semi-Riemannian group manifold and its applications in spacetime admitting Ricci solitons

In this work, we show that semi-Riemannian group manifold admits Ricci solitons and satisfies the dynamical cosmology equation of spacetime. In Sec. 2, we introduce and provide some geometric properties of semisymmetric nonmetric connection in semi-Riemannian space. In Sec. 3, we define and show some geometric properties of group manifold endowed with semisymmetric nonmetric connection in semi-Riemannian space. In the section that follows, we give a condition of a group manifold to be Ricci solitons and gradient Ricci soliton. In Sec. 5, we provide the applications of group manifold admitting Ricci solitons in the theory of general relativity.

Complexity growth in integrable and chaotic models

We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O($$ \sqrt{N} $$ N ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

Lorentz invariance of basis tensor gauge theory

Basis tensor gauge theory (BTGT) is a vierbein analog reformulation of ordinary gauge theories in which the vierbein field describes the Wilson line. After a brief review of the BTGT, we clarify the Lorentz group representation properties associated with the variables used for its quantization. In particular, we show that starting from an SO(1,3) representation satisfying the Lorentz-invariant U(1,3) matrix constraints, BTGT introduces a Lorentz frame choice to pick the Abelian group manifold generated by the Cartan subalgebra of U(1,3) for the convenience of quantization even though the theory is frame independent. This freedom to choose a frame can be viewed as an additional symmetry of BTGT that was not emphasized before. We then show how an [Formula: see text] permutation symmetry and a parity symmetry of frame fields natural in BTGT can be used to construct renormalizable gauge theories that introduce frame-dependent fields but remain frame independent perturbatively without any explicit reference to the usual gauge field.

Generalized spinning particles on $${\mathcal {S}}^2$$ in accord with the Bianchi classification

Motivated by recent studies of superconformal mechanics extended by spin degrees of freedom, we construct minimally superintegrable models of generalized spinning particles on $${\mathcal {S}}^2$$ S 2 , the internal degrees of freedom of which are represented by a 3-vector obeying the structure relations of a three-dimensional real Lie algebra. Extensions involving an external field of the Dirac monopole, or the motion on the group manifold of SU(2), or a scalar potential giving rise to two quadratic constants of the motion are discussed. A procedure how to build similar models, which rely upon real Lie algebras with dimensions $$d=4,5,6$$ d = 4 , 5 , 6 , is elucidated.

Limits of JT gravity

We construct various limits of JT gravity, including Newton-Cartan and Carrollian versions of dilaton gravity in two dimensions as well as a theory on the three-dimensional light cone. In the BF formulation our boundary conditions relate boundary connection with boundary scalar, yielding as boundary action the particle action on a group manifold or some Hamiltonian reduction thereof. After recovering in our formulation the Schwarzian for JT, we show that AdS-Carroll gravity yields a twisted warped boundary action. We comment on numerous applications and generalizations.

E6(6) exceptional Drinfel’d algebras

The exceptional Drinfel’d algebra (EDA) is a Leibniz algebra introduced to provide an algebraic underpinning with which to explore generalised notions of U-duality in M-theory. In essence, it provides an M-theoretic analogue of the way a Drinfel’d double encodes generalised T-dualities of strings. In this note we detail the construction of the EDA in the case where the regular U-duality group is E6(6). We show how the EDA can be realised geometrically as a generalised Leibniz parallelisation of the exceptional generalised tangent bundle for a six-dimensional group manifold G, endowed with a Nambu-Lie structure. When the EDA is of coboundary type, we show how a natural generalisation of the classical Yang-Baxter equation arises. The construction is illustrated with a selection of examples including some which embed Drinfel’d doubles and others that are not of this type.

Extended Drinfel’d algebras and non-Abelian duality

The Drinfel’d algebra provides a method to construct generalized parallelizable spaces and this allows us to study an extended T-duality, known as the Poisson–Lie T-duality. Recently, in order to find a generalized U-duality, an extended Drinfel'd algebra (ExDA), called the Exceptional Drinfel'd algebra (EDA), was proposed and a natural extension of Abelian U-duality was studied both in the context of supergravity and membrane theory. In this paper, we clarify the general structure of ExDAs and show that an ExDA always gives a generalized parallelizable space, which may be regarded as a group manifold with generalized Nambu–Lie structures. We then discuss the non-Abelian duality that is based on a general ExDA. For a coboundary ExDA, this non-Abelian duality reduces to a generalized Yang–Baxter deformation and we find a general formula for the twist matrix. In order to study the non-Abelian U-duality, we particularly focus on the En(n) EDA for n ≤ 8 and study various aspects, both in terms of M-theory and type IIB theory.

Poisson-Lie T-duality of WZW model via current algebra deformation

Poisson-Lie T-duality of the Wess-Zumino-Witten (WZW) model having the group manifold of SU(2) as target space is investigated. The whole construction relies on the deformation of the affine current algebra of the model, the semi-direct sum $$ \mathfrak{su}(2)\left(\mathrm{\mathbb{R}}\right)\overset{\cdot }{\oplus}\mathfrak{a} $$ su 2 ℝ ⊕ ⋅ a , to the fully semisimple Kac-Moody algebra $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right)\left(\mathrm{\mathbb{R}}\right) $$ sl 2 ℂ ℝ . A two-parameter family of models with SL(2, ℂ) as target phase space is obtained so that Poisson-Lie T-duality is realised as an O(3, 3) rotation in the phase space. The dual family shares the same phase space but its configuration space is SB(2, ℂ), the Poisson-Lie dual of the group SU(2). A parent action with doubled degrees of freedom on SL(2, ℂ) is defined, together with its Hamiltonian description.

SCENE CLASSIFICATION BASED ON THE INTRINSIC MEAN OF LIE GROUP

Remote Sensing scene classification aims to identify semantic objects with similar characteristics from high resolution images. Even though existing methods have achieved satisfactory performance, the features used for classification modeling are still limited to some kinds of vector representation within a Euclidean space. As a result, their models are not robust to reflect the essential scene characteristics, hardly to promote classification accuracy higher. In this study, we propose a novel scene classification method based on the intrinsic mean on a Lie Group manifold. By introducing Lie Group machine learning into scene classification, the new method uses the geodesic distance on the Lie Group manifold, instead of Euclidean distance, solving the problem that non-euclidean space samples could not be calculated by Euclidean distance directly. The experiments show that our method produces satisfactory performance on two public and challenging remote sensing scene datasets, UC Merced and SIRI-WHU, respectively.