A particular dispersive generalization of long water wave equation in 1 + 1 dimensions, which is important in the study of matrix models without scaling limit, known as two–Boson (TB) equation, as well as the associated hierarchy has been derived from the zero curvature condition on the gauge group SL(2, R) ⊗ U(1). The supersymmetric extension of the two–Boson (sTB) hierarchy has similarly been...

New integrable variant of the one-dimensional Hubbard model with variable-range correlated hopping is studied. The Hamiltonian is constructed by applying the quantum inverse scattering method on the infinite interval at zero density to the one-parameter deformation of the L-matrix of the Hubbard model. By construction, this model has Y(su(2))⊕Y(su(2)) symmetry in the infinite chain limit. Multi...

This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. Associated with the two hierarchies, a new Neumann-like and a new Bargmann-like N -dimensional integrable system are presented. The two CH hierarchies possess...

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete and continuum classical integrable models is introduced. Using this framework the associated classical integrals of motion and the corresponding Lax pair ar...

In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevich naturally appear as τ -functions of integrable hierarchies related to topological minimal model...

In this paper we conjecture that the generating function of the intersection numbers on the moduli spaces of Riemann surfaces with boundary, constructed recently by R. Pandharipande, J. Solomon and R. Tessler and extended by A. Buryak, is a taufunction of the KP integrable hierarchy. Moreover, it is given by a simple modification of the Kontsevich matrix integral so that the generating function...

For a real or p-adic connected reductive group G, Harish-Chandra introduced the Plancherel measure on the tempered dual Ĝtemp and founded the Plancherel formula [7, 18] relating functions on G to functions on Ĝtemp. While the Plancherel measure is equal to the formal degree for square-integrable representations, there has been no similar interpretation for tempered but nonsquare-integrable repr...

We present a method for computing the non-perturbative mass-gap in the theory of Bosonic membranes in flat background spacetimes. The analysis is extended to the study of membranes coupled to background fluxes as well. The computation of mass-gaps is carried out using a matrix regularization of the membrane Hamiltonians. The mass gap is shown to be naturally organized as an expansion in a ’hidd...

We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL(2, Z) to its preimage in the universal cover of SL(2, R). With this method we recover the classification of two dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give ...

Abstract We study the SYK 2 model of N Majorana fermions with random quadratic interactions through a detailed spectral analysis and by coupling to two- four-point sources. In particular, we define generalized form factor (SFF) level spacing distribution function generalizing from partition generating function. For = 2, obtain an exact solution SFF. It exhibits qualitatively similar behavior hi...