A note on the boson-fermion correspondence and infinite dimensional groups

We show how to construct irreducible projective representations of the infinite dimensional Lie group Map (S ¹, ), by embedding it into the group of Bogoliubov automorphisms of the CAR. Using techniques of G. Segal for extending certain representations of Map (S ¹, SU(2)) we show that our representations extend to give representations of a certain infinite dimensional superalgebra. We relate our work to the well known boson-fermion correspondence which exists in 1+1 dimensions.     

Unitary positive energy representations of the gauge group

We investigate the positive energy representations (also called highest weight representations) of the gauge groupC (T ^v,G ₀),G ₀ being a compact simple Lie group, and discuss their unitarity, using the technique of Verma modules constructed from generalized loop algebras (a simple generalization of Kac-Moody affine Lie algebras). We show that the unitarity of the representation imposes severa restrictions in it. In particular, we show, as a part of a more general result, that the gauge group does not admit faithful unitary positive energy representations.Allocataire du MRT.     

The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.     

Highly mobile Einstein spaces in the large

We consider an Einstein spaceV of the Petrov type II or III admitting a group of motionsG of high order. First we calculate the composition law and topological structure ofG. ThenV (or its submanifolds of transitivity) is represented as the homogeneous spaceG/H ofG,H being a subgroup ofG, and the actionG onV and the topology ofV are determined. The topologies of the spacesV are as follows: ⁴ (spaceT*₂), ⁴ of ³ T¹ (spaceT ₂), ⁴ (spaceT*₃), ³ (submanifolds of transitivity in spaceT ₃).In two cases (spacesT ₂ andT ₃) we have obtained metrics free of singularities.     

The extended phase space of the BRS approach

The origin of the classical BRS symmetry is discussed for the case of a first class constrained system consisting of a 2n-dimensional phase spaceS with free action of a Lie gauge groupG of dimensionm. The extended phase spaceS _ext of the Fradkin-Vilkovisky approach is identified with a globally trivial vector bundle overS with fibreL*(G)L(G), whereL(G) is the Lie algebra ofG andL*(G) its dual. It is shown that the structure group of the frame bundle of the supermanifoldS _ext is the orthosymplectic group OSp(m,m; 2n), which is the natural invariance group of the super Poisson bracket structure on the function spaceC (S _ext). The action of the BRS operator is analyzed for the caseS=R ²ⁿ with constraints given by pure momenta. The breaking of the osp(m,m; 2n)-invariance down to sp(2n–2m) occurs via an intermediate osp(m; 2n–m). Starting from a (2n+2m)-dimensional system with orthosymplectic invariance, different choices for the BRS operator correspond to choosing different 2n-dimensional constraint supermanifolds inS _ext, which in general characterize different constrained systems. There is a whole family of physically equivalent BRS operators which can be used to describe a particular constrained system.     

Temporal correlations and persistence in the kinetic Ising model: the role of temperature

We study the statistical properties of the sum S _t = dt'σ _t', that is the difference of time spent positive or negative by the spin σ _t, located at a given site of a D-dimensional Ising model evolving under Glauber dynamics from a random initial configuration. We investigate the distribution of S_t and the first-passage statistics (persistence) of this quantity. We discuss successively the three regimes of high temperature ( T > T _c), criticality ( T = T _c), and low temperature ( T < T _c). We discuss in particular the question of the temperature dependence of the persistence exponent , as well as that of the spectrum of exponents (x), in the low temperature phase. The probability that the temporal mean S _t/t was always larger than the equilibrium magnetization is found to decay as t ^{- - ?}. This yields a numerical determination of the persistence exponent in the whole low temperature phase, in two dimensions, and above the roughening transition, in the low-temperature phase of the three-dimensional Ising model. Received 4 December 2000     

Effective Action and Phase Transitions in Yang-Mills Theory on Spheres

We study the covariantly constant Savvidy-type chromomagnetic vacuum in finite-temperature Yang-Mills theory on the four-dimensional curved spacetime. Motivated by the fact that a positive spatial curvature acts as an effective gluon mass we consider the compact Euclidean spacetime S ¹ × S ¹ × S ², with the radius of the first circle determined by the temperature a ₁ = (2π T)⁻¹. We show that covariantly constant Yang-Mills fields on S ² cannot be arbitrary but are rather a collection of monopole-antimonopole pairs. We compute the heat kernels of all relevant operators exactly and show that the gluon operator on such a background has negative modes for any compact semi-simple gauge group. We compute the infrared regularized effective action and apply the result for the computation of the entropy and the heat capacity of the quark-gluon gas. We compute the heat capacity for the gauge group SU(2N) for a field configuration of N monopole-antimonopole pairs. We show that in the high-temperature limit the heat capacity per unit volume is well defined in the infrared limit and exhibits a typical behavior of second-order phase transition ~ (T-T_c)^-3/2{\sim(T-T_c)^{-3/2}} with the critical temperature T _c  = (2π a)⁻¹, where a is the radius of the 2-sphere S ².     

Existence and invariance of twisted products on a symplectic manifold

It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T ^*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T ^*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.     

NMR investigations of molecular motions inp-n-hexyloxybenzylidene-p′-n-propylaniline

Proton spin-lattice relaxation times,T ₁, have been measured in the smectic phases, S _G ² , S _G ¹ and S_A, and in the nematic phase of HxBPA, in the temperature range, 220K<T<360 K. In the S _G ¹ and S _G ² phases,T ₁ has been measured at 15 and 40 MHz. The (S _G ¹ →S _G ² ) and (S _G ² →S _G ¹ ) transitions are clearly seen as discontinuities inT ₁. The former transition is seen to be due to possible freezing or change of hydrocarbon chain motions of the molecule. Our data in the S _G ¹ phase have been fitted to a model in which anisotropic rotational diffusion of the end hydrocarbon chains as also that of the rigid part of the molecule are considered. In the nematic phase, at 351 K, the observed behaviour ofT ₁, measured in the frequency range, 5 to 40 MHz, agrees well with the theoretical predictions of Uklejaet al and Freed, who take into account long range collective order fluctuations and local reorientations. Using these theories, the correlation time for short range reorientations has been calculated from our results to be 4.3 × 10⁻¹⁰ and 1.1 × 10⁻⁹ s respectively.     

Boundary Scattering,Symmetric Spaces and the Principal Chiral Model on the Half-Line

We investigate integrable boundary conditions (BCs) for the principal chiral model on the half-line, and rational solutions of the boundary Yang-Baxter equation (BYBE). In each case we find a connection with (type I, Riemannian, globally) symmetric spaces G/H: there is a class of integrable BCs in which the boundary field is restricted to lie in a coset of H; these BCs are parametrized by G/H×G/H; there are rational solutions of the BYBE in the defining representations of all classical G parametrized by G/H; and using these we propose boundary S-matrices for the principal chiral model, parametrized by G/H×G/H, which correspond to our boundary conditions.An erratum to this article can be found at     

An explicit *-product on the cotangent bundle of a Lie group

We give explicit formulas for a ^*-product on the cotangent bundle T ^* G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of the Lie algebra G of G and on the other hand bidifferential operators analogous to the ones used by Moyal to define a ^*-product on IR²ⁿ.Chargé de recherches au FNRS, on leave of absence from Université libre de Bruxelles.     

On Majorana Representations of A6 and A7

The Majorana representations of groups were introduced in Ivanov (The Monster Group and Majorana Involutions, 2009) by axiomatising some properties of the 2A-axial vectors of the 196 884-dimensional Monster algebra, inspired by the sensational classification of such representations for the dihedral groups achieved by Sakuma (Int Math Res Notes, 2007). This classification took place in the heart of the theory of Vertex Operator Algebras and expanded earlier results by Miyamoto (J Alg 268:653–671, 2003). Every subgroup G of the Monster which is generated by its intersection with the conjugacy class of 2A-involutions possesses the (possibly unfaithful) Majorana representation obtained by restricting to G the action of the Monster on its algebra. This representation of G is said to be based on an embedding of G in the Monster. So far the Majorana representations have been classified for the groups G isomorphic to the symmetric group S ₄ of degree 4 (Ivanov et al. in J Alg 324:2432–2463, 2010), the alternating group A ₅ of degree 5 (Ivanov AA, Seress á in Majorana Representations of A ₅, 2010), and the general linear group GL ₃(2) in dimension 3 over the field of two elements (Ivanov AA, Shpectorov S in Majorana Representations of L ₃(2), 2010). All these representations are based on embeddings in the Monster of either the group G itself or of its direct product with a cyclic group of order 2. The dimensions and shapes of these representations are given in the following table:     

Spectroscopy properties of the amide group in valpromide and some derivatives with antiepileptic activity

Infrared spectra at 300 and 77 K and Raman spectra at 300 K of the valpromide (Vpd), N‐substituted derivatives, N‐ethylvalpromide (Etvpd), N‐isopropylvalpromide (Ipvpd) and the N,N‐disubstituted derivative, N,N‐dimethylvalpromide (Dmvpd) with antiepileptic activity, have been measured and analyzed with results derived from computational chemistry calculation. In agreement with theoretical predictions, experimental data indicate that while in Etvpd, Dmvpd and Ipvpd there are four different conformational co‐existing components (Etvpd: TTCG⁺, TCCG⁻, TTTC, G⁺G⁺C G⁺; Dmvpd: TTCC, G⁻TTA⁺, G⁺A⁻TC, G⁺A⁻C A⁺; Ipvpd: TTCT, TCCT, TCCC, G⁻ TTT) in the Vpd there are only three distinct stable conformations of C₁ symmetry group: TTC, TCT, G⁺G⁺T. Based on the accuracy of the B3LYP calculation, with the 6‐31 + G** basis set estimated by comparison between the predicted values of the vibrational modes and the available experimental data, we performed a structural and vibrational study of the amide group in the Vpd and their derivatives. We found that small nonplanarity deviations of C(O)N backbone induce significant changes on the structural and spectroscopic properties. These are not compatible with the decreasing of the resonance effect as it is produced when the twisting around the C(O) N increases. From the Natural Bond Orbital (NBO) analysis the existence of stabilizing electrostatic interactions of type C H···O/N and C H···H N/C, which induce significant structural changes and a complex electronic redistribution of charge on the π‐system in those structures becomes evident. We view this as a consequence of the filled electron density change Lewis‐type NBOs type lp_{O1, 2}, lp_N1, σ_{(C H)N acyl} and empty non‐Lewis NBOs type σ*_{(C H)N acyl}, σ*_{N H}. Copyright © 2009 John Wiley & Sons, Ltd.     

Structural distortions in families of perovskite-like crystals

The crystallographic and group theoretical analysis of the structural phase transitions in perovskite and perovskite-like crystals is reviewed. We include ABX₃ perovskites and their relative crystals of ReO₃ type (G₀ = O¹ _h), elpasolites, cryolites and their relatives (G₀ = O⁵ _h), layered crystals of T1AlF₄ series (G₀ = D¹ _4h), Aurivillius and Ruddlesden-Popper series (Go = D¹⁷ _4h). The structures in their initial phase G₀ often contain n layers (n = 1,2,3) of vertex linked octahedra. The distorted phases produced by one kind of tilt and by superposition of tilts in the slabs are enumerated. Most of the tilts correspond to symmetry changes, which can be associated to definite librational lattice modes irreducible representations of the G₀ group. The softening of modes associated to the PT has been found experimentally in many perovskites, elpasolites and layered crystals with n = 1. In contrast, no such soft modes have been found yet for even-layered (n = 2) crystals. Examples of successive phase transitions due to the superposition of tilts in these types of crystals have been collected.     

High-colour and mass hierarchies

We present a model based on the gauge group G = G_HC × G_S × SU(2)_L × U(1), where the hypercolour gauge group G_HC is responsible for the dynamical breaking of the strong group G_S to SU(3)_C of QCD. Chiral symmetry breaking of high-colour representations produces dynamical breaking of the electroweak SU(2)_L × U(1) gauge group. Fermion masses and flavour mixing are dynamically generated from the condensations of high-colour representations. A phenomenological analysis of the model is also presented.     

Modular Categories and Orbifold Models

In this paper, we try to answer the following question: given a modular tensor category ? with an action of a compact group G, is it possible to describe in a suitable sense the “quotient” category ?/G? We give a full answer in the case when ?=?ℯ? is the category of vector spaces; in this case, ?ℯ?/G turns out to be the category of representation of Drinfeld's double D(G). This should be considered as the category theory analog of the topological identity {pt}/G=BG. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if ? is a vertex operator algebra which has a unique irreducible module, ? itself, and G is a compact group of automorphisms of ?, and some not too restrictive technical conditions are satisfied, then G is finite, and the category of representations of the algebra of invariants, ? ^G , is equivalent as a tensor category to the category of representations of Drinfeld's double D(G). We also get some partial results in the non-holomorphic case, i.e. when ? has more than one simple module. Received: 27 August 2001 / Accepted: 1 March 2002     

Temperature dependence of the lattice parameters in the S1G and S2G phases of HxBPA

The temperature dependence of the monoclinic lattice constants in the two smectic phases, S¹_G and S²_G, of HxBPA, has been obtained from X-ray diffraction data, in the temperature range 300 K < T < 240 K. The variation of b (unique axis) is consistent with the chain ordering in the S²_G phase indicated by Raman and NMR measurements.     

Orientifolds and slumps in G2 and Spin(7) metrics

We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by , which are complete on a complex line bundle over . The principal orbits are S⁷, described as a triaxially squashed S³ bundle over S⁴. The behaviour in the S³ directions is similar to that in the Atiyah–Hitchin metric, and we show how this leads to an M-theory interpretation with orientifold D6-branes wrapped over S⁴. We then consider new G₂ metrics which we denote by , which are complete on an bundle over T^1,1, with principal orbits that are S³×S³. We study the metrics using numerical methods, and we find that they have the remarkable property of admitting a U(1) Killing vector whose length is nowhere zero or infinite. This allows one to make an everywhere non-singular reduction of an M-theory solution to give a solution of the type IIA theory. The solution has two non-trivial S² cycles, and both carry magnetic charge with respect to the R–R vector field. We also discuss some four-dimensional hyper-Kähler metrics described recently by Cherkis and Kapustin, following earlier work by Kronheimer. We show that in certain cases these metrics, whose explicit form is known only asymptotically, can be related to metrics characterised by solutions of the su(∞) Toda equation, which can provide a way of studying their interior structure.     

Representations of Nets of C*-Algebras over S 1

In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C^*-algebras over S ¹ admits faithful representations, and when the net is covariant under Diff(S ¹), it admits representations covariant under any amenable subgroup of Diff(S ¹).