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This note is a short comment on central extensions of Lie groups of the form Map(X, G) when X=Sn and G is a compact simple Lie group. The cohomology of Map(X, G) is computed and cases where the Lie algebra is extended but the corresponding Lie group extension is trivial are considered. [ABSTRACT FROM AUTHOR]

Examines the application of the dimer method to Ising models on group lattices having nontrivial topological genus g. Effects of group extension and external and internal group isomorphisms on the number of Pffafians; Explicit topological formulas for sign and weight in the expansion of the partition function.

The moduli space of gapped Hamiltonians that are in the same topological phase is an intrinsic object that is associated with the topological order. The topology of these moduli spaces has been used recently in the construction of Floquet codes. We propose a systematical program to study the topology of these moduli spaces. In particular, we use effective field theory to study the cohomology classes of these spaces, which includes and generalizes the Berry phase. We discuss several applications for studying phase transitions. We show that a nontrivial family of gapped systems with the same topological order can protect isolated phase transitions in the phase diagram, and we argue that the phase transitions are characterized by screening of topological defects. We argue that the family of gapped systems obeys bulk-boundary correspondence. We show that a family of gapped systems in the bulk with the same topological order can rule out a family of gapped systems on the boundary with the topological order given by the topological boundary condition, constraining phase transitions on the boundary. [ABSTRACT FROM AUTHOR]

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular graphs, which have an interpretation as objects dual to triangulations of three-dimensional manifolds. Two different polynomial invariants are developed to characterize these graphs—one inspired by the Kauffman bracket relations and the other based on quandles. How the latter invariant changes under the Pachner moves acting on the graphs is then studied. [ABSTRACT FROM AUTHOR]

In this paper, we deduce the sketch of proof of the Duistermaat-Heckman formula and investigate how the known Duistermaat-Heckman result could be specialized to the symplectic structure on the orbit space. The theorems of localization in equivariant cohomology not only provide us with beautiful mathematical formulas and stimulate achievements in algorithmic computations but also promote progress in theoretical and mathematical physics. We present the elliptic genera and the characteristic q-series for the circle actions and twisted equivariant K-theory, with the case of the symmetric group of n symbols separately analyzed. We show that the Poincaré q-polynomials admit presentation in terms of the Patterson-Selberg (or the Ruelle-type) spectral functions. [ABSTRACT FROM AUTHOR]

There are important conjectures about logarithmic conformal field theories (LCFT’s), which are constructed as a kernel of screening operators acting on the vertex algebra of the rescaled root lattice of a finite-dimensional semisimple complex Lie algebra. In particular, their representation theory should be equivalent to the representation theory of an associated small quantum group. This article solves the case of the rescaled root lattice B n / 2 as a first working example beyond A 1 / p . We discuss the kernel of short screening operators, its representations, and graded characters. Our main result is that this vertex algebra is isomorphic to a well-known example: The even part of n pairs of symplectic fermions. In the screening operator approach, this vertex algebra appears as an extension of the vertex algebra associated with A 1 n / 2 , which are n copies of the even part of one pair of symplectic fermions. The new long screenings give the new global Cn-symmetry. The extension is due to a degeneracy in this particular case: Rescaled long roots still have an even integer norm. For the associated quantum group of divided powers, the first author has previously encountered matching degeneracies: It contains the small quantum group of type A 1 n and the Lie algebra Cn. Recent results by Farsad, Gainutdinov, and Runkel on symplectic fermions suggest finally the conjectured category equivalence to this quantum group. We also study the other degenerate cases of a quantum group, giving extensions of LCFT’s of type Dn, D4, A2 with larger global symmetry Bn, F4, G2. [ABSTRACT FROM AUTHOR]

General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is based on the principle of unitary evolution. In this article, I provide a possible answer to the black hole information paradox by means of homological algebra and pairings generated by the universal coefficient theorem. The unitarity of processes involving black holes is restored by the demanding invariance of the laws of physics to the change of coefficient structures in cohomology. [ABSTRACT FROM AUTHOR]

Highest weight modules for Uq(gl2) are endowed with a structure of modules for the quantum toriodal algebra Uκ of type sl2. Using this, we define Uκ actions on the space of vertex operators for irreducible highest weight Uq(gl2) modules. Highest or lowest weight vectors of the thus obtained Uκ modules are expressed in terms of an intertwiner for Uq(sl2) modules and an extra boson. The submodules generated by these vectors are investigated. [ABSTRACT FROM AUTHOR]

Twisted supercharge families on product manifolds T × M have been applied in the analysis of the odd twisted K-theory. We shall suspend these families to the even twisted K-theory and solve their twisted families index problem. This is applied to give analytic representatives of the twisted K-theory classes on tori--including all the torsion classes. [ABSTRACT FROM AUTHOR]

We propose a geometric formulation of the theory of effective Hamiltonians associated with active spaces. We analyze particularly the case of the time-dependent wave operator theory. This formulation is related to the geometry of the manifold of the active spaces, particularly to its Kählerian structure. We introduce the concept of quantum distance between active spaces. We show that the time-dependent wave operator theory is, in fact, a gauge theory, and we analyze its relationship with the geometric phase concept. [ABSTRACT FROM AUTHOR]

We construct the action of the quantum double of Uq(su(2)) on the standard Podles sphere and interpret it as the quantum projective formula generalizing to the q-deformed setting the action of the Lorentz group of global conformal transformations on the ordinary Riemann sphere. [ABSTRACT FROM AUTHOR]

The minimal coupling principle is revisited under the quantum perspectives of the space–time symmetry. This revision is better realized on a group approach to quantization (GAQ) where group cohomology and extensions of groups play a preponderant role. We first consider the case of the electromagnetic potential; the Galilei and/or Poincaré group is (noncentrally) extended by the “local” U(1) group. The resulting group can also be seen as a central extension, parametrized by both the mass and the electric charge, of an infinite-dimensional group, on which GAQ leads to the dynamics of a particle moving in the presence of an electromagnetic field. Then we try the gravitational interaction of a particle by making the space–time translations “local.” However, promoting to “local” the space–time subgroup of the true symmetry of the quantum free relativistic particle, i.e., the centrally extended by U(1) Poincaré group, results in a new electromagneticlike force of pure gravitational origin. This is a consequence of the space–time translations not being an invariant subgroup of the extended Poincaré group and constitutes a preliminary attempt to a nontrivial mixing of space–time and internal gauge interactions. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Describes a symmetry based quantization method of reparametrization invariant systems (RIS). Description of the mathematical apparatus for the study of RIS; Assumptions and equations concerning the 2 + 1 gravity model; Application of the time evolution idea of Dirac to the model.

Describes a systematic study of finite-dimensional nonrelativistic conformal groups performed under two complementary points of view. Solution of the conformal Killing equation to obtain a whole family of finite-dimensional conformal algebras corresponding to the Galilei and Newton-Hooke kinematical groups.

Discusses several aspects of second quantized scattering operators for fermions in external time-dependent fields. Existence of quantum scattering operators; Phase of quantum scattering operator; Quantum phase and parallel transport.

Examines a system with Gal(N) as symmetry group with the Hamiltonian H as one of the generators. Generic time dependence for any element in the universal enveloping algebra U(g); Free Schrodinger equation from Ray representations.

Focuses on the construction of the unitary irreducible representations of the proper Galilean group in 2+1 space-time dimensions using the Mackey theory of induced representations. Galilean group and its central extensions; Induced representations; Representations of the (2+1) Galilean group; Representations of the Lie algebra.

The evolution of a quantum harmonic oscillator, forced by a stationary ergodic process, is considered. Under suitable conditions on the ‘‘dynamics’’ of this process, it is shown that generically the spectrum of the associated quasienergy operator is continuous. These results also apply to kicked harmonic oscillators where the amplitudes of the kicks are governed by a stationary ergodic process. [ABSTRACT FROM AUTHOR]

A q-symplecton realization of the quantum group SUq(2) is constructed, giving explicitly the product law for the Weyl-ordered characteristic polynomials. A new form of the q-Racah coefficients is obtained, which is especially adapted to proving a new symmetry for these functions. A surprising result is that unlike the SU(2) case there are two distinct invariant q-triangle functions in the q-symplecton extension. [ABSTRACT FROM AUTHOR]

Affine Wigner functions are phase space representations based on the affine group in place of the usual Weyl–Heisenberg group of quantum mechanics. Such representations are relevant to the time–frequency analysis of real signals. An interesting family is singled out by the requirement of covariance with respect to each solvable three-parameter group containing the affine group. Explicit forms are given in each case and properties such as unitarity and localization are discussed. Some particular distributions are recovered. [ABSTRACT FROM AUTHOR]

We determine the fusion rules of the equivariantization of a fusion category C under the action of a finite group G in terms of the fusion rules of C and group-theoretical data associated to the group action. As an application we obtain a formula for the fusion rules in an equivariantization of a pointed fusion category in terms of group-theoretical data. This entails a description of the fusion rules in any braided group-theoretical fusion category. [ABSTRACT FROM AUTHOR]

We study the quantization of chiral fermions coupled to generalized Dirac operators arising in NCG Yang–Mills theory. The cocycles describing chiral symmetry breaking are calculated. In particular, we introduce a generalized locality principle for the cocycles. Local cocycles are by definition expressions which can be written as generalized traces of operator commutators. In the case of pseudodifferential operators, these traces lead in fact to integrals of ordinary local de Rham forms. As an application of the general ideas we discuss the case of noncommutative tori. We also develop a gerbe theoretic approach to the chiral anomaly in the Hamiltonian quantization of NCG field theory. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Hypercubic groups in any dimension are defined and their conjugate classifications and representation theories are derived. Double group and spinor representation are introduced. A detailed calculation is carried out on the structures of four-dimensional cubic group O[sub 4] and its double group, as well as all inequivalent single-valued representations and spinor representations of O[sub 4]. All representations are derived adopting Clifford theory of decomposition of induced representations. Based on these results, single-valued and spinor representations of the orientation-preserved subgroup of O[sub 4] are calculated. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré, and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles, along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, to stability theory, integrable systems, as well as geometric phases. This paper surveys progress in selected topics in reduction theory, especially those of the last few decades as well as presenting new results on non-Abelian Routh reduction. We develop the geometry of the associated Lagrange-Routh equations in some detail. The paper puts the new results in the general context of reduction theory and discusses some future directions. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]