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Marta Mazzocco

Publications and source records attributed to Marta Mazzocco.

31 records · Page 2Linked to original sources
arXiv2013

Quantum ordering for quantum geodesic functions of orbifold Riemann surfaces

We determine the explicit quantum ordering for a special class of quantum geodesic functions corresponding to geodesics joining exactly two orbifold points or holes on a non-compact Riemann surface. We discuss some special cases in which these quantum geodesic functions form sub--algebras of some abstract algebras defined by the reflection equation and we extend our results to the quantisation of matrix elements of the Fuchsian group associated to the Riemann surface in Poincaré uniformization. In particular we explore an interesting relation between the deformed $U_q(\mathfrak{sl}_2)$ and the Zhedanov algebra AW(3).

math.QA↗
arXiv2012

Confluence on the Painlevé Monodromy Manifolds, their Poisson Structure and Quantisation

In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlevé equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that quantisation and confluence commute. We also investigate the underlying cluster algebra structure and the relation to the versal deformations of singularities of type $D_4,A_3,A_2$, and $A_1$.

math-ph↗
arXiv2011

Poisson algebras of block-upper-triangular bilinear forms and braid group action

In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on $C^{N}$ with the property that for any $n,m\in N$ such that $n m =N$, the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size $m\times m$ is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case $m=1$ the quantum affine algebra is the twisted $q$-Yangian for ${o}_n$ and for $m=2$ is the twisted $q$-Yangian for ${sp}_{2n}$. We describe the quantum braid group action in these two examples and conjecture the form of this action for any $m>2$.

math-ph↗
arXiv2011

Cubic and quartic transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence

A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painleve VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard's solutions of Painleve VI.

nlin.SI↗
arXiv2011

Teichmüller spaces as degenerated symplectic leaves in Dubrovin--Ugaglia Poisson manifolds

In this paper we study the Goldman bracket between geodesic length functions both on a Riemann surface $Σ_{g,s,0}$ of genus $g$ with $s=1,2$ holes and on a Riemann sphere $Σ_{0,1,n}$ with one hole and $n$ orbifold points of order two. We show that the corresponding Teichmüller spaces $\mathcal T_{g,s,0}$ and $\mathcal T_{0,1,n}$ are realised as real slices of degenerated symplectic leaves in the Dubrovin--Ugaglia Poisson algebra of upper--triangular matrices $S$ with 1 on the diagonal.

math-ph↗
arXiv2010

Shear coordinate description of the quantised versal unfolding of D_4 singularity

In this paper by using Teichmuller theory of a sphere with four holes/orbifold points, we obtain a system of flat coordinates on the general affine cubic surface having a D_4 singularity at the origin. We show that the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere coincides with the Etingof-Ginzburg Poisson bracket on the affine D_4 cubic. We prove that this bracket is the image under the Riemann-Hilbert map of the Poisson Lie bracket on the direct sum of three copies of sl_2. We realise the action of the mapping class group by the action of the braid group on the geodesic functions . This action coincides with the procedure of analytic continuation of solutions of the sixth Painlevé equation. Finally, we produce the explicit quantisation of the Goldman bracket on the geodesic functions on the four-holed/orbifold sphere and of the braid group action.

math-ph↗
arXiv2009

Isomonodromic deformations and twisted Yangians arising in Teichmüller theory

In this paper we build a link between the Teichmuller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincare' uniformization. In the case of a one-sheeted hyperboloid with n orbifold points we show that the Poisson algebra D_n of geodesic length functions is the semiclassical limit of the twisted q-Yangian for the orthogonal Lie algebra defined by Molev, Ragoucy and Sorba. We give a representation of the braid group action on this algebra in terms of an adjoint matrix action. We characterize two types of finite-dimensional Poissonian reductions and give an explicit expression for the generating function of their central elements. Finally, we interpret the algebra D_n as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semisimple point.

math.AG↗
arXiv2007

A remark on the Hankel determinant formula for solutions of the Toda equation

We consider the Hankel determinant formula of the $τ$ functions of the Toda equation. We present a relationship between the determinant formula and the auxiliary linear problem, which is characterized by a compact formula for the $τ$ functions in the framework of the KP theory. Similar phenomena that have been observed for the Painlevé II and IV equations are recovered. The case of finite lattice is also discussed.

nlin.SI↗
arXiv2007

Canonical structure and symmetries of the Schlesinger equations

The Schlesinger equations $S_{(n,m)}$ describe monodromy preserving deformations of order $m$ Fuchsian systems with $n+1$ poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of $n$ copies of $m\times m$ matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations $S_{(n,m)}$ for all $n$, $m$ and we compute the action of the symmetries of the Schlesinger equations in these coordinates.

math.DG↗
arXiv2006

The Hamiltonian Structure of the Second Painleve Hierarchy

In this paper we study the Hamiltonian structure of the second Painleve hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The n-th element of the hierarchy is a non linear ODE of order 2n in the independent variable $z$ depending on n parameters denoted by ${t}_1,...,{t}_{n-1}$ and $α_n$. We introduce new canonical coordinates and obtain Hamiltonians for the $z$ and $t_1,...,t_{n-1}$ evolutions. We give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.

nlin.SI↗
arXiv2001

The geometry of the classical solutions of the Garnier systems

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in $n$-variables, ${\cal G}_n$, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system ${\cal G}_n$. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system ${\cal G}_{n}$ is such that the associated Fuchsian system has $l$ monodromy matrices equal to $\pm\ID$, then it can be reduced classically to a solution of a the Garnier system with $n-l$ variables ${\cal G}_{n-l}$. When $n$ monodromy matrices are equal to $\pm\ID$, we have classical solutions of ${\cal G}_n$. We give also another mechanism to produce classical solutions: we show that the solutions of the Garnier systems having reducible monodromy groups can be reduced to the classical solutions found by Okamoto and Kimura in terms of Lauricella hypergeometric functions. In the case of the Garnier system in 1-variables, i.e. for the Painlevé VI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least one monodromy matrix equal to $\pm\ID$.

math.CA↗