Effective Scalar Products for D-finite Symmetric Functions

Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as sub-series and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are D-finite. We extend Gessel's work by providing algorithms that compute differential equations these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel's class. Examples of applications to k-regular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself.Comment: 51 pages, full paper version of FPSAC 02 extended abstract; v2: corrections from original submission, improved clarity; now formatted for journal + bibliograph

Kronecker product identities from D-finite symmetric functions

Using an algorithm for computing the symmetric function Kronecker product of D-finite symmetric functions we find some new Kronecker product identities. The identities give closed form formulas for trace-like values of the Kronecker product.Comment: 6 page

Beta functions and anomalous dimensions up to three loops

We derive an algorithm for automatic calculation of perturbative $\beta$-functions and anomalous dimensions in any local quantum field theory with canonical kinetic terms. The infrared rearrangement is performed by introducing a common mass parameter in all the propagator denominators. We provide a set of explicit formulae for all the necessary scalar integrals up to three loops.Comment: 22 pages, 4 figures, uses epsf.st

Higher Spin de Sitter Hilbert Space

We propose a complete microscopic definition of the Hilbert space of minimal higher spin de Sitter quantum gravity and its Hartle-Hawking vacuum state. The fundamental degrees of freedom are $2N$ bosonic fields living on the future conformal boundary, where $N$ is proportional to the de Sitter horizon entropy. The vacuum state is normalizable. The model agrees in perturbation theory with expectations from a previously proposed dS-CFT description in terms of a fermionic Sp(N) model, but it goes beyond this, both in its conceptual scope and in its computational power. In particular it resolves the apparent pathologies affecting the Sp(N) model, and it provides an exact formula for late time vacuum correlation functions. We illustrate this by computing probabilities for arbitrarily large field excursions, and by giving fully explicit examples of vacuum 3- and 4-point functions. We discuss bulk reconstruction and show the perturbative bulk QFT canonical commutations relations can be reproduced from the fundamental operator algebra, but only up to a minimal error term $\sim e^{-\mathcal{O}(N)}$, and only if the operators are coarse grained in such a way that the number of accessible "pixels" is less than $\mathcal{O}(N)$. Independent of this, we show that upon gauging the higher spin symmetry group, one is left with $2N$ physical degrees of freedom, and that all gauge invariant quantities can be computed by a $2N \times 2N$ matrix model. This suggests a concrete realization of the idea of cosmological complementarity

Spinor gravity and diffeomorphism invariance on the lattice

The key ingredient for lattice regularized quantum gravity is diffeomorphism symmetry. We formulate a lattice functional integral for quantum gravity in terms of fermions. This allows for a diffeomorphism invariant functional measure and avoids problems of boundedness of the action. We discuss the concept of lattice diffeomorphism invariance. This is realized if the action does not depend on the positioning of abstract lattice points on a continuous manifold. Our formulation of lattice spinor gravity also realizes local Lorentz symmetry. Furthermore, the Lorentz transformations are generalized such that the functional integral describes simultaneously euclidean and Minkowski signature. The difference between space and time arises as a dynamical effect due to the expectation value of a collective metric field. The quantum effective action for the metric is diffeomorphism invariant. Realistic gravity can be obtained if this effective action admits a derivative expansion for long wavelengths.Comment: 13 pages, proceedings 6th Aegean Summer School, Naxos 201

BMN operators with vector impurities, Z_2 symmetry and pp-waves

We calculate the coefficients of three-point functions of BMN operators with two vector impurities. We find that these coefficients can be obtained from those of the three-point functions of scalar BMN operators by interchanging the coefficient for the symmetric-traceless representation with the coefficient for the singlet. We conclude that the Z_2 symmetry of the pp-wave string theory is not manifest at the level of field theory three-point correlators.Comment: 25 pages, 7 figures. v1: A reference and a footnote added; v2: New contributions found, Z_2 symmetry lost in 3-point function

Non-commutative Gross-Neveu model at large N

The non-commutative O(N) Gross-Neveu model is solved in the large N limit in two and three space-time dimensions. The commutative version of the two dimensional model is a renormalizable quantum field theory, both in a coupling constant expansion and an expansion in 1/N. The non-commutative version has a renormalizable coupling constant expansion where ultraviolet divergences can be removed by adjusting counterterms to each order. On the other hand, in a previous work, we showed that the non-commutative theory is not renormalizable in the large N expansion. This is argued to be due to a combined effect of asymptotic freedom and the ultraviolet/infrared mixing that occurs in a non-commutative field theory. In the present paper we will elaborate on this result and extend it to study the large N limit of the three dimensional Gross-Neveu model. We shall see that the large N limit of the three dimensional theory is also trivial when the ultraviolet cutoff is removed.Comment: 23 page