180,875 research outputs found

### Topological Graph Polynomials in Colored Group Field Theory

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

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

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

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 $\chi_{R,(r_n,r_m)}(Z,X)$ is constructed from the half BPS
``partons'' $\chi_{r_n}(Z)$ and $\chi_{r_m}(X)$.Comment: 42 page

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