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Andrei Grigorev

Publications and source records attributed to Andrei Grigorev.

5 recordsLinked to original sources
arXiv2026

(1,k) CFT and RH problem with the c=-2 case

Following approach of Iorgov--Lisovyy--Teschner, we construct solutions of the (modified) Riemann--Hilbert problem using conformal blocks of $(1,k)$ Virasoro models. For $k>1$ case, the solution of this Riemann--Hilbert problem is not unique due to more singular behavior at punctures. On the CFT side the dimension of the space of conformal blocks also increases. We specifically study the $k=2$ case, which corresponds to the central charge $c=-2$ and symplectic fermions. We explicitly construct a corresponding solution of the modified Riemann--Hilbert problem in the case of 3 punctures and prove its uniqueness under suitable initial data conditions. We also obtain new bilinear relations for $c=-2$ tau functions.

math-ph↗
arXiv2026

Complex dynamics perspective for birational maps of the plane arising from cluster algebra mutations

Using the methods of holomorphic dynamics we investigate planar birational mappings that arise from the theory of cluster algebras and integrable systems. Computing dynamical degrees of these mappings, many of which are greater than one, allows us to show that many of the mappings do not have a conserved quantity (nor an invariant fibration). In most of the examples, invariant fibrations can also be ruled out by finding superattracting periodic points. This answers a question posted by Machacek and Ovenhouse 2024 and by Chen and Li 2024. Moreover, having found a good algebraically stable model for the mappings and having computed the dynamical degree, we can then apply results from the ergodic theory of birational maps to produce invariant measures with positive entropy and positive Lyapunov exponents.

math.DS↗
arXiv2026

Monodromy free Schr\"odinger operators and affine $\mathfrak{sl}_2$ master functions

Given a non-zero polynomial $P(x)$, we study Fuchsian differential operators of the form $L=\partial_x^2-u(x)$ such that for all $\lambda\in\mathbb{C}$ the operator $L+\lambda P(x)$ is monodromy free. We prove that all such operators are obtained from populations of critical points of ${\widehat{\mathfrak{sl}}_2}$ master functions. Moreover, we show that the reproduction procedure of critical points corresponds to a Darboux transformation of operator $P^{-1}(x)L$. As a result, we obtain a classification of all operators $L$ with such properties in the case of $P(x)=x^k$.

math.QA↗
arXiv2022

Hamiltonian reductions in Matrix Painlev\'e systems

For certain finite groups $G$ of B\"acklund transformations we show that the dynamics of $G$-invariant configurations of $n|G|$ Calogero--Painlev\'e particles is equivalent to certain $n$-particle Calogero--Painlev\'e system. We also show that the reduction of dynamics on $G$-invariant subset of $n|G|\times n|G|$ matrix Painlev\'e system is equivalent to certain $n\times n$ matrix Painlev\'e system. The groups $G$ correspond to folding transformations of Painlev\'e equations. The proofs are based on the Hamiltonian reductions.

nlin.SI↗