Additional Symmetries and String Equation of the CKP Hierarchy

Based on the Orlov and Shulman’s M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on L ^l are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra , which is a subalgebra of W _1+∞.      

A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory, which follows from our results by taking the limit as q → 1. Our method similarly leads to a definition of quantum spheres, which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry.     

Cosmological Horizons and Reconstruction of Quantum Field Theories

As a starting point, we state some relevant geometrical properties enjoyed by the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds. Those properties are generalised to a larger class of expanding spacetimes M admitting a geodesically complete cosmological horizon common to all co-moving observers. This structure is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat at null infinity. Under suitable hypotheses on M, encompassing both the cosmological de Sitter background and a large class of other FRW spacetimes, the algebra of observables for a Klein-Gordon field is mapped into a subalgebra of the algebra of observables constructed on the cosmological horizon. There is exactly one pure quasifree state λ on which fulfills a suitable energy-positivity condition with respect to a generator related with the cosmological time displacements. Furthermore λ induces a preferred physically meaningful quantum state λ _M for the quantum theory in the bulk. If M admits a timelike Killing generator preserving , then the associated self-adjoint generator in the GNS representation of λ _M has positive spectrum (i.e., energy). Moreover λ _M turns out to be invariant under every symmetry of the bulk metric which preserves the cosmological horizon. In the case of an expanding de Sitter spacetime, λ _M coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case. Remarks on the validity of the Hadamard property for λ _M in more general spacetimes are presented. Dedicated to Professor Klaus Fredenhagen on the occasion of his 60th birthday.     

Chevalley Cohomology for linear graphs

The space of linear polyvector fields on is a Lie subalgebra of the (graded) Lie algebra , equipped with the Schouten bracket. In this paper, we compute the cohomology of this subalgebra for the adjoint representation in , restricting ourselves to the case of cochains defined with purely aerial Kontsevich’s graphs, as in Pac. J. Math. 218(2):201–239, 2005. We find a result which is very similar to the cohomology for the vector case Pac. J. Math. 229(2):257–292, 2007. This work was supported by the CMCU contract 06 S 1502. W. Aloulou and R. Chatbouri thank the Université de Bourgogne and D. Arnal the Faculté des Sciences de Monastir for their kind hospitalities during their stay.     

Topological Strings and (Almost) Modular Forms

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H ³(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP ₂ and IP ₁ × IP ₁. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold .     

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra on the groupoid Γ=E×G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of , is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein’s equation. The algebra , when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra ℳ of random operators representing the quantum sector of the model. The Tomita–Takesaki theorem allows us to define the dynamics of random operators which depends on the state φ. The same state defines the noncommutative probability measure (in the sense of Voiculescu’s free probability theory). Moreover, the state φ satisfies the Kubo–Martin–Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra , one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not “feel” singularities.     

Two-parameter Quantum Affine Algebra $$U_{r,s}(\widehat{\mathfrak {sl}_n})$$, Drinfel’d Realization and Quantum Affine Lyndon Basis

We further define two-parameter quantum affine algebra (n > 2) after the work on the finite cases (see [BW1,BGH1,HS,BH]), which turns out to be a Drinfel’d double. Of importance for the quantum affine cases is that we can work out the compatible two-parameter version of the Drinfel’d realization as a quantum affinization of and establish the Drinfel’d Isomorphism Theorem in the two-parameter setting, via developing a new combinatorial approach (quantum calculation) to the quantum affine Lyndon basis we present (with an explicit valid algorithm based on the use of Drinfel’d generators). N.H., supported in part by the NNSF (Grants 10431040, 10728102), the PCSIRT, the TRAPOYT and the FUDP from the MOE of China, the SRSTP from the STCSM, die Deutche Forschungsgemeinschaft (DFG), as well as an ICTP long-term visiting scholarship. H.Z., supported by a Ph.D. Program Scholarship Fund of ECNU 2006.     

Exact Results for Topological Strings on Resolved <Emphasis Type="Italic">Y</Emphasis><Superscript><Emphasis Type="Italic"> p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Superscript> Singularities

We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y ^p,q manifolds and give rise via M-theory compactification to SU(p) gauge theories on . As an application we present a detailed study of the local case and compute open and closed genus zero Gromov-Witten invariants of the orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A ₁ Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y ^p,q geometries.     

Spectral stochastic processes arising in quantum mechanical models with a non-L 2 ground state

A functional integral representation is given for a large class of quantum mechanical models with a non-L ² ground state. As a prototype, the particle in a periodic potential is discussed: a unique ground state is shown to exist as a state on the Weyl algebra, and a functional measure (spectral stochastic process) is constructed on trajectories taking values in the spectrum of the maximal Abelian subalgebra of the Weyl algebra isomorphic to the algebra of almost periodic functions. The thermodynamical limit of the finite-volume functional integrals for such models is discussed, and the superselection sectors associated to an observable subalgebra of the Weyl algebra are described in terms of boundary conditions and/or topological terms in the finite-volume measures.Supported by DFG, Nr. Al 374/1-2     

Jordanian twist quantization of D=4 Lorentz and Poincaré algebras and D=3 contraction limit

We describe in detail the two-parameter nonstandard quantum deformation of the D=4 Lorentz algebra , linked with a Jordanian deformation of . Using the twist quantization technique we obtain the explicit formulae for the deformed co-products and antipodes. Further extending the considered deformation to the D=4 Poincaré algebra we obtain a new Hopf-algebraic deformation of four-dimensional relativistic symmetries with a dimensionless deformation parameter. Finally, we interpret as the D=3 de Sitter algebra and calculate the contraction limit (R is the de Sitter radius) providing an explicit Hopf algebra structure for the quantum deformation of the D=3 Poincaré algebra (with mass-like deformation parameters), which is the two-parameter light-cone κ-deformation of the D=3 Poincaré symmetry.     

Quantum principal commutative subalgebra in the nilpotent part ofU_q \widehat{sl}_2 and lattice KdV variables

We propose a quantum lattice version of B. Feigin and E. Frenkel's constructions, identifying the KdV differential polynomials with functions on a homogeneous space under the nilpotent part of . We construct an action of the nilpotent part of on their lattice counterparts, and embed the lattice variables in a , coinduced from a quantum version of the principal commutative subalgebra, which is defined using the identification of with its dual algebra.     

General quantum measurements: Local transition maps

On the basis of four physically motivated assumptions, it is shown that a general quantum measurement of commuting observables can be represented by a local transition map, a special type of positive linear map on a von Neumann algebra. In the case that the algebra is the bounded operators on a Hilbert space, these local transition maps share two properties of von Neumann-type measurements: they decrease matrix elements of states and increase their entropy. It is also shown that local transition maps have all the properties of a conditional expectation of a von Neumann algebra onto a subalgebra except that their range is not restricted to the subalgebra. The notion of locality arises from requiring that a quantum measurement may be treated classically when restricted to the commutative algebra generated by the measured observables. The formalism established applies to observables with arbitrary spectrum. In the case that the spectrum is continuous we have only incomplete measurements.     

Quantum Orbifolds

This is a study of orbifold-quotients of quantum groups (quantum orbifolds \({\Theta } \rightrightarrows G_{q}\)). These structures have been studied extensively in the case of the quantum S U ₂ group. A generalized theory of quantum orbifolds over compact simple and simply connected quantum groups is developed. Associated with a quantum orbifold there is an invariant subalgebra and a crossed product algebra. For each spin quantum orbifold, there is a unitary equivalence class of Dirac spectral triples over the invariant subalgebra, and for each effective spin quantum orbifold associated with a finite group action, there is a unitary equivalence class of Dirac spectral triples over the crossed product algebra. A Hopf-equivariant Fredholm index problem is studied as an application.     

Braided Hopf Algebras Obtained from Coquasitriangular Hopf Algebras

In this paper, we study the generalized quantum double construction for paired Hopf algebras with particular attention to the case when the generalized quantum double is a Hopf algebra with projection. Applying our theory to a coquasitriangular Hopf algebra (H, σ), we see that H has an associated structure of braided Hopf algebra in the category of Yetter-Drinfeld modules over , where H _σ is a subHopf algebra of H ⁰, the finite dual of H. Specializing to the quantum group H = SL _q (N), we find that H _σ is , so that the duality between these quantum groups is just the evaluation map. Furthermore, we obtain explicit formulas for the braided Hopf algebra structure of SL _q (N) in the category of left Yetter-Drinfeld modules over . The second author held a postdoctoral fellowship at Mount Allison University from 2005 to 2007 and would like to thank Mount Allison for their warm hospitality. Support for the first author’s research and partial support for the postdoctoral position of the second author came from an NSERC Discovery Grant. The second author now holds research support from Grant 434/1.10.2007 of CNCSIS.     

Local Asymptotic Normality in Quantum Statistics

The theory of local asymptotic normality for quantum statistical experiments is developed in the spirit of the classical result from mathematical statistics due to Le Cam. Roughly speaking, local asymptotic normality means that the family consisting of joint states of n identically prepared quantum systems approaches in a statistical sense a family of Gaussian state ϕ _u of an algebra of canonical commutation relations. The convergence holds for all “local parameters” such that parametrizes a neighborhood of a fixed point . In order to prove the result we define weak and strong convergence of quantum statistical experiments which extend to the asymptotic framework the notion of quantum sufficiency introduces by Petz. Along the way we introduce the concept of canonical state of a statistical experiment, and investigate the relation between the two notions of convergence. For the reader’s convenience and completeness we review the relevant results of the classical as well as the quantum theory. Dedicated to Slava Belavkin on the occasion of his 60th anniversary     

Analytic fields on Riemann surfaces. II

The properties of analytic fields on a Riemann surface represented by a branch covering of ¹ are investigated in detail. Branch points are shown to correspond to the vertex operators with simple conformal properties. As applications we compute determinants of operators forZ _n -symmetric surfaces and obtain various representations for the two-loop measure in the bosonic string theory together with various identities for theta-functions of hyperelliptic surfaces. We also present an integral representation for the quantum part of the twist field correlation functions, which describe propagation of the string on the orbifold background. We also calculate the quantum part of the structure constants of the twist-field operator algebra, generalizing the results of Dixon, Friedan, Martinec, and Shenker.     

A remark on the motivic Galois group and the quantum coadjoint action

It has been suggested that the Grothendieck–Teichmüller group GT should act on the Duflo isomorphism of su(2), but the corresponding realization of GT turned out to be trivial. We show that a solvable quotient of the motivic Galois group – which is supposed to agree with GT – is closely related to the quantum coadjoint action on for q a root of unity, i.e. in the quantum group case one has a nontrivial realization of a quotient of the motivic Galois group. From a discussion of the algebraic properties of this realization we conclude that in more general cases than it should be related to a quantum version of the motivic Galois group. Finally, we discuss the relation of our construction to quantum field and string theory and explain what we believe to be the ‘physical reason’ behind this relation between the motivic Galois group and the quantum coadjoint action. This might be a starting point for the generalization of our construction to more involved examples.