For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a tau function attached to an algebraic curve. These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e. the (p,q) minimal models of conformal field theory. As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV tau-function.
In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's hypergeometric solutions of the 2-component BKP hierarchy. We derive the closed algebraic formulas for the correlation functions associated with these tau-functions, and under reasonable analytical assumptions we prove the loop equations (the blobbed topological recursion). Finally, we prove a version of topological recursion for the spin Hurwitz numbers with the spin …
In this work we study and exactly solve the Dirac oscillator with three different topological defects, namely the cosmic string spacetime ($Λ_\mp$), the magnetic cosmic string spacetime ($Θ_\mp$) and the cosmic dislocation spacetime ($Π_\mp$). Moreover, we show that the radial part of this problem possess an $SU(1,1)$ symmetry. From this, we obtain the wave functions and their respective energy spectrum by means of the Schrödinger factorization. Also, we compute the radial coherent states of the eigenfunctions of each problem and their time evolution.
We prove that the BGW and HCIZ integrals admit large N topological expansions for complex coupling and complex external fields, provided the coupling is sufficiently strong. The expansion coefficients are holomorphic functions which are genus-specific generating functions for the monotone single and double Hurwitz numbers, respectively.
We construct Lewis-Riesenfeld invariants from two dimensional point transformations for two oscillators that are coupled to each other in space in a PT-symmetrical and time-dependent fashion. The non-Hermitian Hamiltonian of the model is conveniently expressed in terms of generators of the symplectic $sp(4)$ Lie algebra. This allows for an alternative systematic approach to find Lewis-Riesenfeld invariants leading to a set of coupled differential equations that we solve by using time-ordered exponentials. We also demonstrate that point transformations may be utilized to directly construct time…
We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.
Cluster expansions of particle system state with topological nearest-neighbor interaction
Institute of Mathematics of NAS of Ukraïne（乌克兰国家科学院数学研究所）；Taras Shevchenko National University of Kyïv, Department of Mechanics and Mathematics（基辅塔拉斯舍甫琴科国立大学力学与数学系）
