125 research outputs found

    Linear independence of Gamma values in positive characteristic

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    We investigate the arithmetic nature of special values of Thakur's function field Gamma function at rational points. Our main result is that all linear independence relations over the field of algebraic functions are consequences of the known relations of Anderson and Thakur arising from the functional equations of the Gamma function.Comment: 51 page

    On moment maps associated to a twisted Heisenberg double

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    We review the concept of the (anomalous) Poisson-Lie symmetry in a way that emphasises the notion of Poisson-Lie Hamiltonian. The language that we develop turns out to be very useful for several applications: we prove that the left and the right actions of a group GG on its twisted Heisenberg double (D,κ)(D,\kappa) realize the (anomalous) Poisson-Lie symmetries and we explain in a very transparent way the concept of the Poisson-Lie subsymmetry and that of Poisson-Lie symplectic reduction. Under some additional conditions, we construct also a non-anomalous moment map corresponding to a sort of quasi-adjoint action of GG on (D,κ)(D,\kappa). The absence of the anomaly of this "quasi-adjoint" moment map permits to perform the gauging of deformed WZW models.Comment: 52 pages, LaTeX, introduction substantially enlarged, several explanatory remarks added, final published versio

    Finite multiplicity theorems for induction and restriction

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    We find upper and lower bounds of the multiplicities of irreducible admissible representations π\pi of a semisimple Lie group GG occurring in the induced representations IndHGτInd_H^G\tau from irreducible representations τ\tau of a closed subgroup HH. As corollaries, we establish geometric criteria for finiteness of the dimension of HomG(π,IndHGτ)Hom_G(\pi,Ind_H^G \tau) (induction) and of HomH(πH,τ)Hom_H(\pi|_H,\tau) (restriction) by means of the real flag variety G/PG/P, and discover that uniform boundedness property of these multiplicities is independent of real forms and characterized by means of the complex flag variety.Comment: to appear in Advances in Mathematic

    Superposition Formulas for Darboux Integrable Exterior Differential Systems

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    In this paper we present a far-reaching generalization of E. Vessiot's analysis of the Darboux integrable partial differential equations in one dependent and two independent variables. Our approach provides new insights into this classical method, uncovers the fundamental geometric invariants of Darboux integrable systems, and provides for systematic, algorithmic integration of such systems. This work is formulated within the general framework of Pfaffian exterior differential systems and, as such, has applications well beyond those currently found in the literature. In particular, our integration method is applicable to systems of hyperbolic PDE such as the Toda lattice equations, 2 dimensional wave maps and systems of overdetermined PDE.Comment: 80 page report. Updated version with some new sections, and major improvements to other

    Quantum Mechanics On Spaces With Finite Fundamental Group

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    We consider in general terms dynamical systems with finite-dimensional, non-simply connected configuration-spaces. The fundamental group is assumed to be finite. We analyze in full detail those ambiguities in the quantization procedure that arise from the non-simply connectedness of the classical configuration space. We define the quantum theory on the universal cover but restrict the algebra of observables \O to the commutant of the algebra generated by deck-transformations. We apply standard superselection principles and construct the corresponding sectors. We emphasize the relevance of all sectors and not just the abelian ones.Comment: 40 Pages, Plain-TeX, no figure

    Weaving Worldsheet Supermultiplets from the Worldlines Within

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    Using the fact that every worldsheet is ruled by two (light-cone) copies of worldlines, the recent classification of off-shell supermultiplets of N-extended worldline supersymmetry is extended to construct standard off-shell and also unidextrous (on the half-shell) supermultiplets of worldsheet (p,q)-supersymmetry with no central extension. In the process, a new class of error-correcting (even-split doubly-even linear block) codes is introduced and classified for p+q8p+q \leq 8, providing a graphical method for classification of such codes and supermultiplets. This also classifies quotients by such codes, of which many are not tensor products of worldline factors. Also, supermultiplets that admit a complex structure are found to be depictable by graphs that have a hallmark twisted reflection symmetry.Comment: Extended version, with added discussion of complex and quaternionic tensor products demonstrating that certain quotient supermultiplets do not factorize over any ground fiel

    Two-dimensional models as testing ground for principles and concepts of local quantum physics

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    In the past two-dimensional models of QFT have served as theoretical laboratories for testing new concepts under mathematically controllable condition. In more recent times low-dimensional models (e.g. chiral models, factorizing models) often have been treated by special recipes in a way which sometimes led to a loss of unity of QFT. In the present work I try to counteract this apartheid tendency by reviewing past results within the setting of the general principles of QFT. To this I add two new ideas: (1) a modular interpretation of the chiral model Diff(S)-covariance with a close connection to the recently formulated local covariance principle for QFT in curved spacetime and (2) a derivation of the chiral model temperature duality from a suitable operator formulation of the angular Wick rotation (in analogy to the Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational chiral theories. The SL(2,Z) modular Verlinde relation is a special case of this thermal duality and (within the family of rational models) the matrix S appearing in the thermal duality relation becomes identified with the statistics character matrix S. The relevant angular Euclideanization'' is done in the setting of the Tomita-Takesaki modular formalism of operator algebras. I find it appropriate to dedicate this work to the memory of J. A. Swieca with whom I shared the interest in two-dimensional models as a testing ground for QFT for more than one decade. This is a significantly extended version of an ``Encyclopedia of Mathematical Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
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