International Journal of Modern Physics A

A review of the appearance of integrable structures in the matrix model description of 2D gravity is presented. Most of the ideas are demonstrated with technically simple but ideologically important examples. Matrix models are considered as a sort of “effective” description of continuum 2D field theory formulation. The main physical role in such a description is played by the Virasoro-W conditions, which can be interpreted as certain unitarity or factorization constraints. Both discrete and continuum (generalized Kontsevich) models are formulated as the solutions to those discrete (continuous) Virasoro-W constraints. Their integrability properties are proved, using mostly the determinant technique highly related to the representation in terms of free fields. The paper also contains some new observations connected with formulation of more-general-than-GKM solutions and deeper understanding of their relation to 2D gravity.

We put forward the hypothesis that the weak W boson be a compound of two 2-component Lorentz spinors. The resulting novel γWW vertex is no gauge field structure. Nevertheless, the Born amplitude of γγ→W_LW_L respects partial-wave unitarity. As in the Yang-Mills case, the amplitude consists of a direct term, a crossed term, and a sea-gull term, and no unobserved particles are to be involved to get the “good” high-energy behavior. This is due to an imaginary pseudoscalar γWW interaction term. Significant differences between angular distributions and total cross sections of the non-Abelian case and the case of the composite bosons are displayed. The unitarity constraint applied to the reaction γγ→W_TW_T leads to the prediction of the existence of a composite charged weak scalar Φ^±. It constitutes the spin 0 state of the constituents forming W^±. Furthermore, the existence of a second and heavy scalar-vector pair ω-X is predicted. These weak boson states are found to exclude the presence of a seagull graph. In the threshold region, the total cross section of γγ→WW in the compositeness case is smaller than in the non-Abelian case. In a broad intermediate energy region it can be larger. Upper unitarity mass-bounds are estimated. They suggest m_Φ≈m_w so that Φ^± might be discovered by forthcoming experiments. The structure of the γΦW, γXW and γωW transition vertices can be inferred without making recourse to unitarity. However, unitarity requires that the mass relation m_Φ/m_W=m_ω/m_X be valid.

We study the Turaev-Viro (TV) invariant as the Euclidean Chern-Simons-Witten gravity partition function with positive cosmological constant. After explaining why it can be identified as the partition function of three-dimensional gravity, we show that the initial data of the TV invariant can be constructed from the duality data of a certain class of rational conformal field theories, and that, in particular, the original TV initial data are associated with the A_k+1 modular invariant SU(2) WZW model. As a corollary we then show that the partition function Z(M) is bounded from above by , where g is the smallest genus of handlebodies with which M can be presented by Hegaard splitting. Z(M) is generically very large near Λ~+0 if M is neither S³ nor a lens space, and many-wormhole configurations dominate near Λ~+0 in the sense that Z(M) generically tends to diverge faster as the “number of wormholes” g becomes larger.

According to some of the recent analyses, the cosmological constant Λ may be extremely small but finite. The decaying cosmological constant model can naturally explain the smallness of Λ from the oldness of our universe, provided that some of the parameters in the Lagrangian are fine-tuned so that the asymptotic value of Λ is zero. In this report we show that Coleman’s wormhole mechanism can naturally explain why the parameters are fine-tuned in this model.

We investigate the renormalization of N=2 SUSY Landau-Ginsburg models with central charge c=3p/(2+p) perturbed by an almost marginal chiral operator. We calculate the renormalization of the chiral fields up to the gg* order and of the nonchiral fields up to the g(g*) order. We propose a formulation of the nonrenormalization theorem and show that it holds in the lowest nontrivial order. It turns out that, in this approximation, the chiral fields cannot get renormalized: . The β function then remains unchanged: β=∈g.

Following our earlier work we argue in detail for the equivalence of the nonlinear σ model with Hopf term at θ=π/2s and an interacting spin-s theory. We give an ansatz for spin-s operators in the σ model and show the equivalence of the correlation functions. We also show the relation between topological and Noether currents. We obtain the Lorentz and discrete transformation properties of the spin-s operator from the fields of the σ model. We also explore the connection of this model with quantum Hall fluids.

We develop further a new geometrical model of a discretized string, proposed in Ref. 1, and establish its basic physical properties. The model can be considered as the natural extension of the usual Feynman amplitude of the random walks to random surfaces. Both amplitudes coincide in the case, when the surface degenerates into a single particle world line. We extend the model to open surfaces as well. The boundary contribution is proportional to the full length of the boundary, and the coefficient of proportionality can be treated as a hopping parameter of the quarks. In the limit when this parameter tends to infinity, the theory is essentially simplified. We prove that the contribution of a given triangulation to the partition function is finite and have found the explicit form for the upper bound. The question of the convergence of the full partition function remains open. In this model the string tension may vanish at the critical point, if the last one exists, and possesses a nontrivial scaling limit. The model contains hidden fermionic variables and can be considered as an independent model of hadrons.