Large Representation Recurrences in Large N Random Unitary Matrix Models

In a random unitary matrix model at large N, we study the properties of the expectation value of the character of the unitary matrix in the rank k symmetric tensor representation. We address the problem of whether the standard semiclassical technique for solving the model in the large N limit can be applied when the representation is very large, with k of order N. We find that the eigenvalues do indeed localize on an extremum of the effective potential; however, for finite but sufficiently large k/N, it is not possible to replace the discrete eigenvalue density with a continuous one. Nonetheless, the expectation value of the character has a well-defined large N limit, and when the discreteness of the eigenvalues is properly accounted for, it shows an intriguing approximate periodicity as a function of k/N.Comment: 24 pages, 11 figure

Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions

We give a concise summary of the impressive recent development unifying a number of different fundamental subjects. The quiver Nekrasov functions (generalized hypergeometric series) form a full basis for all conformal blocks of the Virasoro algebra and are sufficient to provide the same for some (special) conformal blocks of W-algebras. They can be described in terms of Seiberg-Witten theory, with the SW differential given by the 1-point resolvent in the DV phase of the quiver (discrete or conformal) matrix model (\beta-ensemble), dS = ydz + O(\epsilon^2) = \sum_p \epsilon^{2p} \rho_\beta^{(p|1)}(z), where \epsilon and \beta are related to the LNS parameters \epsilon_1 and \epsilon_2. This provides explicit formulas for conformal blocks in terms of analytically continued contour integrals and resolves the old puzzle of the free-field description of generic conformal blocks through the Dotsenko-Fateev integrals. Most important, this completes the GKMMM description of SW theory in terms of integrability theory with the help of exact BS integrals, and provides an extended manifestation of the basic principle which states that the effective actions are the tau-functions of integrable hierarchies.Comment: 14 page

The scalar gluonium correlator: large-beta_0 and beyond

The investigation of the scalar gluonium correlator is interesting because it carries the quantum numbers of the vacuum and the relevant hadronic current is related to the anomalous trace of the QCD energy-momentum tensor in the chiral limit. After reviewing the purely perturbative corrections known up to next-next-to-leading order, the behaviour of the correlator is studied to all orders by means of the large-beta_0 approximation. Similar to the QCD Adler function, the large-order behaviour is governed by the leading ultraviolet renormalon pole. The structure of infrared renormalon poles, being related to the operator product expansion are also discussed, as well as a low-energy theorem for the correlator that provides a relation to the renormalisation group invariant gluon condensate, and the vacuum matrix element of the trace of the QCD energy-momentum tensor.Comment: 14 pages, references added, discussion of IR renormalon pole at u=3 extended, similar version to appear in JHE

Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals

The AGT relation identifies the Nekrasov functions for various N=2 SUSY gauge theories with the 2d conformal blocks, which possess explicit Dotsenko-Fateev matrix model (beta-ensemble) representations the latter being polylinear combinations of Selberg integrals. The "pure gauge" limit of these matrix models is, however, a non-trivial multiscaling large-N limit, which requires a separate investigation. We show that in this pure gauge limit the Selberg integrals turn into averages in a Brezin-Gross-Witten (BGW) model. Thus, the Nekrasov function for pure SU(2) theory acquires a form very much reminiscent of the AMM decomposition formula for some model X into a pair of the BGW models. At the same time, X, which still has to be found, is the pure gauge limit of the elliptic Selberg integral. Presumably, it is again a BGW model, only in the Dijkgraaf-Vafa double cut phase.Comment: 21 page

The matrix model version of AGT conjecture and CIV-DV prepotential

Recently exact formulas were provided for partition function of conformal (multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and analytically continued in the number of screening insertions, represents generic Virasoro conformal blocks. Actually these formulas describe the lowest terms of the q_a-expansion, where q_a parameterize the shape of the Penner potential, and are exact in the filling numbers N_a. At the same time, the older theory of CIV-DV prepotential, straightforwardly extended to arbitrary beta and to non-polynomial potentials, provides an alternative expansion: in powers of N_a and exact in q_a. We check that the two expansions coincide in the overlapping region, i.e. for the lowest terms of expansions in both q_a and N_a. This coincidence is somewhat non-trivial, since the two methods use different integration contours: integrals in one case are of the B-function (Euler-Selberg) type, while in the other case they are Gaussian integrals.Comment: 27 pages, 1 figur

Classical conformal blocks from TBA for the elliptic Calogero-Moser system

The so-called Poghossian identities connecting the toric and spherical blocks, the AGT relation on the torus and the Nekrasov-Shatashvili formula for the elliptic Calogero-Moser Yang's (eCMY) functional are used to derive certain expressions for the classical 4-point block on the sphere. The main motivation for this line of research is the longstanding open problem of uniformization of the 4-punctured Riemann sphere, where the 4-point classical block plays a crucial role. It is found that the obtained representation for certain 4-point classical blocks implies the relation between the accessory parameter of the Fuchsian uniformization of the 4-punctured sphere and the eCMY functional. Additionally, a relation between the 4-point classical block and the $N_f=4$, ${\sf SU(2)}$ twisted superpotential is found and further used to re-derive the instanton sector of the Seiberg-Witten prepotential of the $N_f=4$, ${\sf SU(2)}$ supersymmetric gauge theory from the classical block.Comment: 25 pages, no figures, latex+JHEP3, published versio

Generalized matrix models and AGT correspondence at all genera

We study generalized matrix models corresponding to n-point Virasoro conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT correspondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge theories with generalized quiver diagrams. We obtain the generalized matrix models from the perturbative evaluation of the Liouville correlation functions and verify the consistency of the description with respect to degenerations of the Riemann surface. Moreover, we derive the Seiberg-Witten curve for the N=2 gauge theory as the spectral curve of the generalized matrix model, thus providing a check of AGT correspondence at all genera.Comment: 19 pages; v2: version to appear in JHE

GLOBAL STABILITY AND BIFURCATIONS ANALYSIS OF AN EPIDEMIC MODEL WITH CONSTANT REMOVAL RATE OF THE INFECTIVE

In this thesis we consider an epidemic model with a constant removal rate of infective individuals is proposed to understand the effect of limited resources for treatment of infective on the disease spread. It is found that it is unnecessary to take such a large treatment capacity that endemic equilibria disappear to eradicate the disease. It is shown that the outcome of disease spread may depend on the position of the initial states for certain range of parameters. It is also shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, subcritical Hopf bifurcation. Keyword: Epidemic model, nonlinear incidence rate, basic reproduction number, local and global stabilit