arXiv subjects

Noureddine Chair

Publications and source records attributed to Noureddine Chair.

14 recordsLinked to original sources

arXiv2014

Trigonometrical sums connected with the chiral Potts model, Verlinde dimension formula, two-dimensional resistor network, and number theory

\ \ We have recently developed methods for obtaining exact two-point resistance of the complete graph minus $N$ edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving the Scott's conjecture on permanent of Cauchy matrix, and in the perturbative chiral Potts model. The generalized trigonometrical sums of the chiral Potts model are shown to satisfy recursion formulas that are transparent and direct, and differ from those of Gervois and Mehta. By making a change of variables in these recursion formulas, the dimension of the space of conformal blocks of $SU(2)$ and $SO(3)$ WZW models may be computed recursively. Our methods are then extended to compute the corner-to-corner resistance, and the Kirchhoff index of the first non-trivial two-dimensional resistor network, $2\times N$. Finally, we obtain new closed formulas for variant of trigonometrical sums, some of which appear in connection with number theory.

math-ph↗

arXiv2014

The effective resistance of the $N$-cycle graph with four nearest neighbors

The exact expression for the effective resistance between any two vertices of the $N$-cycle graph with four nearest neighbors $C_{N}(1,2)$, is given. It turns out that this expression is written in terms of the effective resistance of the $N$-cycle graph $C_{N}$, the square of the Fibonacci numbers, and the bisected Fibonacci numbers. As a consequence closed form formulas for the total effective resistance, the first passage time (FPT), and the mean first passage time (MFPT) for the simple random walk on the the $N$-cycle graph with four nearest neighbors are obtained. Finally, a closed form formula for the effective resistance of $C_{N}(1,2)$ with all first neighbors removed is obtained.

math-ph↗

arXiv2014

Generalized Penner model and the Gaussian beta ensemble

In this paper, a new expression for the partition function of the generalized Penner model given by Goulden, Harer and Jackson is derived. The Penner and the orthogonal Penner partition functions are special cases of this formula. The parametrized Euler characteristic $ξ^s_g(γ)$ deduced from our expression of the partition function is shown to exhibit a contribution from the orbifold Euler characteristic of the moduli space of Riemann surfaces of genus $g$, with $s$ punctures, for all parameters $γ$ and $g$ odd. The other contributions for $g$ even are linear combinations of the Bernoulli polynomials at rational arguments. It turns out that the free energy coefficients of the generalized Penner model in the continuum limit, are identical to those coefficients in the large $N$ expansion of the Gaussian $β$-ensemble. Moreover, the duality enjoyed by the generalized Penner model, is also the duality symmetry of the Gaussian $β$-ensemble. Finally, a shift in the 't Hooft coupling constant required by the refined topological string, would leave the Gaussian $β$-ensemble duality intact. This duality is identified with the remarkable duality of the $c=1$ string at radius $R=β$.

hep-th↗

arXiv2013

Exact two-point resistance, and the simple random walk on the complete graph minus $ N$ edges

An analytical approach is developed to obtain the exact expressions for the two-point resistance, and the total effective resistance of the complete graph minus $N$ edges of the opposite vertices. These expressions are written in terms of certain numbers that we introduced which we call the Bejaia and the Pisa numbers, these numbers are the natural generalizations of the bisected Fibonacci and Lucas numbers. The correspondence between random walks and the resistor networks is then used to obtain the exact expressions for the the first passage and mean first passage times on this graph.

math-ph↗

arXiv2013

The Euler-Riemann Gases, and Partition Identities

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice ($s$-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order $s$ of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions. If the parity of the order $s$ is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of $s$ is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function $θ_{4}$, and generate sequences in partition theory. Our partition functions coincide with the overpartitions, and jagged partitions in conformal field theory. Also, The partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore-Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi Theta function is $\frac{ζ(2t)}{ζ(t)^2}$.

math-ph↗

arXiv2012

The Goulden-Harer-Jackson matrix model

An alternative formula for the partition function of the Goulden-Harer-Jackson matrix model is derived, in which the Penner and the orthogonal Penner partition functions are special cases of this formula. Then the free energy that computes the parametrized Euler characteristic $ξ^s_g(γ)$ of the moduli spaces as yet an unidentified, for $g$ is odd, shows that the expression for $ξ^s_g(γ)$ contains the orbifold Euler characteristic of the moduli space of Riemann surfaces of genus $g$, with $s$ punctures for all parameters $γ$. The other contributions are written as a linear combinations of Bernoulli polynomials at rational arguments . It is also shown that in the continuum limit, both the Goulden-Harer-Jackson matrix model and the Penner model have the same critical points.

math-ph↗

arXiv2012

The Symplectic-Orthogonal Penner Models

The generating function for the orbifold Euler characteristic of the moduli space of real algebraic curves of genus $2g$ (locally orientable surfaces) with $n$ marked points $χ^r(\mathfrak{M}_{2g,n})$, is identified with a simple formula. It is shown that the free energy in the continuum limit of both the symplectic and the orthogonal Penner models are almost identical, with the structure $F^{SP/SO}(μ)=1/2F(μ)\mp F^{NO}(μ)$, where $F(μ)$ is the Penner free energy and $F^{NO}(μ)$ is the free energy contributions from the non-orientable surfaces. Both of these models have the same critical point as the Penner model.

math-ph↗

arXiv2012

$SO/Sp$ Chern-Simons Gauge Theories At Large $N$, $SO/Sp$ Penner Models And The Gauge Group Volumes

We construct a deformed $SO/Sp$ Penner generating function responsible for the close connection between $SO/Sp$ Chern-Simons gauge theories at large $N$ and the $SO/Sp$ Penner models. This construction is then shown to follow from a sector of a Chern-Simons gauge theory with coupling constant $λ$. The free energy and its continuum limit of the perturbative Chern-Simons gauge theory are obtained from the Penner model. Finally, asymptotic expansions for the logarithm of the gauge group volumes are given for every genus $g\geq 0$ and shown to be equivalent to the continuum limits of the $SO/Sp$ Chern-Simons gauge theories and the $SO/Sp$ Penner models

hep-th↗

arXiv2007

Perturbative Chern-Simons Theory From The Penner Model

We show explicitly that the perturbative SU(N) Chern-Simons theory arises naturally from two Penner models, with opposite coupling constants. As a result computations in the perturbative Chern-Simons theory are carried out using the Penner model, and it turns out to be simpler and transparent. It is also shown that the connected correlators of the puncture operator in the Penner model, are related to the connected correlators of the operator that gives the Wilson loop operator in the conjugacy class.

hep-th↗

arXiv2007

Comment on "Remark on the renormalization group equation for the Penner model"

We show explicitly that the sum over punctures for the three times derivative for the Penner free energy $F_{0}^{3}$, given by D.A. Johnston, Phys.Rev.D 51 (1995) is not correct. As a consequence, Eq.(21), the differentiated version for the renormalization group (RG) equation, is wrong. Also, his conclusion that the differentiated version of the (RG) equation for the three-times derivative of the free energy can be obtained from the higher genus (RG) equation can not be true. Finally, the differentiated version of the (RG) equation is extended to any $s$ derivative of the free energy $F_{0}$.

hep-th↗

arXiv2004

The Noncommutative Quadratic Stark Effect For The H-Atom

Using both the second order correction of perturbation theory and the exact computation due to Dalgarno-Lewis, we compute the second order noncommutative Stark effect,i.e., shifts in the ground state energy of the hydrogen atom in the noncommutative space in an external electric field. As a side result we also obtain a sum rule for the mean oscillator strength. The energy shift at the lowest order is quadratic in both the electric field and the noncommutative parameter $θ$. As a result of noncommutative effects the total polarizability of the ground state is no longer diagonal.

hep-th↗

arXiv2004

Partition Identities From Partial Supersymmetry

In the quantum theory, using the notion of partial supersymmetry, in which some, but not all, operators have superpartners we derive the Euler theorem in partition theory. The paraferminic partition function gives another identity in partition theory with restrictions. Also an explicit formula for the graded parafermionic partition function is obtained. It turns out that the ratio of the former partition function to the latter is given in terms of the Jacobi Theta function, $θ_{4}$. The inverted graded parafermionic partition function is shown to be a generating function of partitions of numbers with restriction that generalizes the Euler generating function and as a result we obtain new sequences of partitions of numbers with given restrictions.

hep-th↗

arXiv1998

Explicit Computations for the Intersection Numbers on Grassmannians, and on the Space of Holomorphic Maps from CP^1 into G_r(C^n)

We derive some explicit expressions for correlators on Grassmannian G_r(C^n) as well as on the moduli space of holomorphic maps, of a fixed degree d, from sphere into the Grassmannian. Correlators obtained on the Grassmannain are a first step generalization of the Schubert formula for the self-intersection. The intersection numbers on the moduli space for r=2,3 are given explicitly by two closed formulas, when r=2 the intersection numbers, are found to generate the alternate Fibonacci numbers, the Pell numbers and in general a random walk of a particle on a line with absorbing barriers. For r=3 the intersection numbers form a well organized pattern.

hep-th↗

arXiv1997

Grassmannian Cohomolgy Rings and Fusion Rings from Algebraic Equations

The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for $su(N)_K$ is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for $sp(N)_K$ from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers.

hep-th↗