Non-critical string, Liouville theory and geometric bootstrap hypothesis

The applications of the existing Liouville theories for the description of the longitudinal dynamics of non-critical Nambu-Goto string are analyzed. We show that the recently developed DOZZ solution to the Liouville theory leads to the cut singularities in tree string amplitudes. We propose a new version of the Polyakov geometric approach to Liouville theory and formulate its basic consistency condition - the geometric bootstrap equation. Also in this approach the tree amplitudes develop cut singularieties.Comment: 16 pages; revised versio

Super-Liouville - Double Liouville correspondence

The AGT motivated relation between the tensor product of the N = 1 super-Liouville field theory with the imaginary free fermion (SL) and a certain projected tensor product of the real and the imaginary Liouville field theories (LL) is analyzed. Using conformal field theory techniques we give a complete proof of the equivalence in the NS sector. It is shown that the SL-LL correspondence is based on the equivalence of chiral objects including suitably chosen chiral structure constants of all the three Liouville theories involved.Comment: The Introduction expanded, main points of the paper clarified. Misprints corrected and references added. Published in JHE

Braiding properties of the N=1 super-conformal blocks (Ramond sector)

Using a super scalar field representation of the chiral vertex operators we develop a general method of calculating braiding matrices for all types of N=1 super-conformal 4-point blocks involving Ramond external weights. We give explicit analytic formulae in a number of cases.Comment: LaTeX, 42+1 pages, typo correcte

Elliptic recurrence representation of the N=1 superconformal blocks in the Ramond sector

The structure of the 4-point N=1 super-conformal blocks in the Ramond sector is analyzed. The elliptic recursion relations for these blocks are derived.Comment: 21 pages, no figures. An error in the description of the R-NS block of the Ramond field and all its consequences correcte

Joining-splitting interaction of non-critical string

The joining--splitting interaction of non-critical bosonic string is analyzed in the light-cone formulation. The Mandelstam method of constructing tree string amplitudes is extended to the bosonic massive string models of the discrete series. The general properties of the Liouville longitudinal excitations which are necessary and sufficient for the Lorentz covariance of the light-cone amplitudes are derived. The results suggest that the covariant and the light-cone approach are equivalent also in the non-critical dimensions. Some aspects of unitarity of interacting non-critical massive string theory are discussed.Comment: 38 pages, 4 embedded figures, discussion in the Introduction clarified, Appendix D and some material from Section 5 remove

Singular vector structure of quantum curves

We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.Comment: 33 pages; proceedings of the 2016 AMS von Neumann Symposiu