180,875 research outputs found

    Topological Graph Polynomials in Colored Group Field Theory

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    In this paper we analyze the open Feynman graphs of the Colored Group Field Theory introduced in [arXiv:0907.2582]. We define the boundary graph \cG_{\partial} of an open graph \cG and prove it is a cellular complex. Using this structure we generalize the topological (Bollobas-Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension

    From Jack polynomials to minimal model spectra

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    In this note, a deep connection between free field realisations of conformal field theories and symmetric polynomials is presented. We give a brief introduction into the necessary prerequisites of both free field realisations and symmetric polynomials, in particular Jack symmetric polynomials. Then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods. While these results on the representation theory of the minimal models are all known, this note exploits the full power of Jack polynomials to present significant simplifications of the original proofs in the literature.Comment: 14 pages, corrected typos and added comment on connections to the AGT conjecture in introduction, version to appear in J. Phys.

    Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime

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    In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and Kohler, and by Brunetti and Fredenhagen, but they did not impose any ``locality'' or ``covariance'' condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an ``extended Wick polynomial algebra''-large enough to contain the Wick polynomials and their time ordered products. We then define the notion of a {\it local, covariant quantum field}, and seek a definition of {\it local} Wick polynomials and their time ordered products as local, covariant quantum fields. We impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements-such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigationComment: 45 pages; no figures; latex fil

    Exact Multi-Restricted Schur Polynomial Correlators

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    We derive a product rule satisfied by restricted Schur polynomials. We focus mostly on the case that the restricted Schur polynomial is built using two matrices, although our analysis easily extends to more than two matrices. This product rule allows us to compute exact multi-point correlation functions of restricted Schur polynomials, in the free field theory limit. As an example of the use of our formulas, we compute two point functions of certain single trace operators built using two matrices and three point functions of certain restricted Schur polynomials, exactly, in the free field theory limit. Our results suggest that gravitons become strongly coupled at sufficiently high energy, while the restricted Schur polynomials for totally antisymmetric representations remain weakly interacting at these energies. This is in perfect accord with the half-BPS (single matrix) results of hep-th/0512312. Finally, by studying the interaction of two restricted Schur polynomials we suggest a physical interpretation for the labels of the restricted Schur polynomial: the composite operator χR,(rn,rm)(Z,X)\chi_{R,(r_n,r_m)}(Z,X) is constructed from the half BPS ``partons'' χrn(Z)\chi_{r_n}(Z) and χrm(X)\chi_{r_m}(X).Comment: 42 page