Abstract. The Liouville tori, having an integrable metric of the form (U1(q1)− U2(q2))(dq 2 1 + dq 2 ), allow the separation of variables in the study of the Laplacian spectrum. The asymptotic analysis of the resulting Sturm-Liouville problems allows to reduce the count of the eigenvalues of the spectrum to the count of the lattice points inside a plane domain. The present paper is concerned wi...

Making use of the N = 2 Liouville theory and world-sheet techniques, we study the properties of D-branes wrapped around vanishing SUSY cycles of singular Calabi-Yau n-folds (n = 2, 3, 4). After constructing boundary states describing the wrapped branes, we evaluate the disc amplitudes corresponding to the periods of SUSY cycles. We use the old technique of KPZ scaling in Liouville theory and de...

We consider bosonic noncritical strings as QCD string. We present an basic strategy to construct them, in the context of Liouville theory. We show that Dirichlet boundary condition fills important roles in generalized Liouville theory. Specifically, it is used to stabilize the classical configuration of strings, and also utilized to treat the tachyon condensation, in our model. We point out tha...

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ2 beyond the transition phase (i.e. γ2 > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a sim...

The Liouville number, denoted l, is defined by l := 0.100101011101101111100 . . . , where the nth bit is given by 1 2 (1 + λ(n)); here λ is the Liouville function for the parity of prime divisors of n. Presumably the Liouville number is transcendental, though at present, a proof is unattainable. Similarly, define the Gaussian Liouville number by γ := 0.110110011100100111011 . . . where the nth ...

We construct a stochastic process, called the Liouville Brownian motion which we conjecture to be the scaling limit of random walks on large planar maps which are embedded in the euclidean plane or in the sphere in a conformal manner. Our construction works for all universality classes of planar maps satisfying γ < γc = 2. In particular, this includes the interesting case of γ = √ 8/3 which cor...

The quantum theory of the Liouville model with imaginary field is considered using the Quantum Inverse Scattering Method. An integrable structure with nontrivial spectral parameter dependence is developed for lattice Liouville theory by scaling the L-matrix of lattice sine-Gordon theory. This L-matrix yields Bethe Ansatz equations for Liouville theory, by the methods of the algebraic Bethe Ansa...

In this study by modifying some techniques of classical Sturm-Liouville theory and suggesting own approaches we investigate some spectral properties of one Sturm-Liouville type problem on two disjoint intervals.

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009)) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is d...

The origin of the rather mysterious duality symmetry found in quantum Liouville theory is investigated by considering the Liouville theory as the reduction of a WZW-like theory in which the form of the potential for the Cartan field is not fixed a priori. It is shown that in the classical theory conformal invariance places no condition on the form of the potential, but the conformal invariance ...