arXiv subjects
Publications and source records attributed to Simonetta Abenda.
We prove that each totally positive Schubert cell is a real component of Jacobian of the maximal Mumford ($\mathtt{MM}$) curve corresponding to the Le-graph of such cell. At this aim we construct a bijection between the points of this positroid cell and Dubrovin-Natanzon Cartier divisors on the $\mathtt{MM}$-curve.
The Kadomtsev-Petviashvili (KP) equation is the cornerstone of integrable systems, whose solutions reflect deep connections in algebraic geometry. Banana curves are reducible rational curves obtained as a degeneration of hyperelliptic curves. In this work, we relate the family of KP multi-solitons arising from banana curves together with non-special divisors of fixed degree to the combinatorics of the tropical theta divisor of the curve. We describe the Voronoi and Delaunay polytopes and show that the latter are combinatorially equivalent to uniform matroid polytopes. As a consequence, the combinatorics of the tropical theta divisor canonically encodes the matroid and Grassmannian structures underlying the associated KP multi-soliton solutions. We define the Hirota variety of a banana graph, which parametrizes all tau functions arising from such a graph. Starting from the matroid arising from Delaunay polytopes and the periods in the tropical limit, we construct an explicit parametrization of this variety which realizes the tau function as a multi-soliton. Our framework specializes naturally to real and positive settings.
Amalgamation in the totally non-negative part of positroid varieties is equivalent to gluing copies of $Gr^{TP}(1,3)$ and $Gr^{TP}(2,3)$. Lam has proposed to represent amalgamation in positroid varieties by equivalence classes of relations on bipartite graphs and identify total non-negativity via edge signatures. Here we provide an explicit characterization of such signatures on the planar bicolored trivalent directed perfect networks in the disk parametrizing positroid cells $S_M^{TNN}$.To a graph $G$ representing $S_M^{TNN}$, we associate a geometric signature satisfying full rank condition and total non--negativity. Such signature is uniquely identified by geometric indices ruled by orientation and gauge ray direction. The image of this map coincides with that of Postnikov boundary measurement map. We solve the system of geometric relations generalizing Postnikov's and Talaska's results for the boundary edges to the internal edges of the graphs: the edge vector components are rational in the weights with subtraction--free denominators, and have explicit expressions in terms of conservative and edge flows. At boundary sources the edge vectors give the boundary measurement matrix. If $G$ is acyclically orientable, all components are subtraction-free rational in the weights w.r.t. a convenient basis. We provide explicit formulas for the transformation rules w.r.t. changes the orientation, the several gauges of the given network, moves and reductions of networks. We show that the image of the boundary measurement map and the dimer partition functions do not coincide if the graph is not bipartite.
The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms of discrete path integration on planar bicolored (plabic) graphs in the disk. An alternative parametrization was proposed by T. Lam [38] introducing systems of relations on vectors on such graphs, depending on some signatures defined on edges. The problem of characterizing the signatures corresponding to the totally non-negative cells, was left open in [38]. In our paper we provide an explicit construction of such signatures, satisfying both the full rank condition and the total non-negativity property on the full positroid cell. If the graph $\mathcal G$ satisfies the following natural constraint: each edge belongs to some oriented path from the boundary to the boundary, then such signature is unique up to a vertex gauge transformation. Such signature is uniquely identified by geometric indices (local winding and intersection number) ruled by the orientation $\mathcal O$ and gauge ray direction $\mathfrak l$ on $\mathcal G$. Moreover, we provide a combinatorial representation of geometric signatures by showing that the total signature of every finite face just depends on the number of white vertices on it. The latter characterization is a Kasteleyn-type property [7,1] and we conjecture a mechanical-statistical interpretation of such relations. An explicit connection between the solution of Lam system of relations and the value of Postnikov boundary measurement map is established using the generalization of Talaska formula [50] obtained in [6]. In particular, the components of the edge vectors are rational in the edge weights with subtraction-free denominators. Finally, we provide explicit formulas for transformations of signatures under Postnikov moves and reductions, and amalgamations of networks.
Following [42], positroid cells ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ in totally non-negative Grassmannians ${Gr^{\mbox{TNN}} (k,n)}$ admit parametrizations by positive weights on planar bicolored directed perfect networks in the disk. An explicit formula for elements of matrices representing the points in ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ was obtained in [49] in terms of flows on such networks. The formulas from [42,49] are defined on the boundary edge vectors. In this paper we propose an extension of these formulas for vectors on internal edges defined as summations over paths on the given directed network gauged by the choice of a ray direction. This gauge choice does not affect the boundary edge vectors, which generate the Postnikov boundary measurement map. The systems of internal edge vectors corresponding to different choices of gauge ray directions coincide up to sign, the sign rule admits a simple explicit description. We prove that the components of these edge vectors are rational in the weights with subtraction--free denominators. Moreover, these components are expressed in terms of internal edge flows; these formulas extend the original Talaska ones to the internal edges. These vectors also solve the system of geometric relations associated to the corresponding network. These relations are full rank and respect the total non--negativity property on the full positroid cell. We also provide explicit formulas both for the transformation rules of the edge vectors with respect to the orientation, and for their transformations due to moves and reductions of networks.
In this paper we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell $S$ in the totally non-negative Grassmannian $Gr^{\mbox{TNN}}(k,n)$ to the spectral data for the relevant class of real regular Kadomtsev-Petviashvili II (KP) solutions, thus completing search of real algebraic-geometric data for the KP equation started in [4,6]. The spectral curve is modeled on Krichever construction for degenerate finite-gap solutions, and is a rationally degenerate $M$-curve, $Γ$, dual to the graph. The divisors are real regular KP divisors in the ovals of $Γ$, i.e. they fulfill the conditions for selecting real regular finite--gap solutions KPII solutions in [25]. Since the soliton data are described by points in $S$, we establish a bridge between real regular finite-gap KP solutions [25] and real regular multi-line KP solitons which are known to be parameterized by points in $Gr^{\mbox{TNN}}(k,n)$ [18,43]. We use the geometric characterization of spaces of relations on plabic networks introduced in [7] to prove the invariance of this construction with respect to the many gauge freedoms on the network. Such systems of relations were proposed in [53] for the computation of scattering amplitudes on on--shell diagrams $N=4$ SYM \cite{AGP1} and govern the totally non--negative amalgamation of the little positive Grassmannians, $Gr^{\mbox{TP}}(1,3)$ and $Gr^{\mbox{TP}}(2,3)$, into any given positroid cell $S$. In our setting they rule the reality and regularity properties of the KP divisor. Finally, we explain the transformation of the curve and the divisor both under Postnikov moves and reductions and under amalgamation of positroid cells, and apply our construction to some examples.
Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non--negative part of real Grassmannians (see Refs. [54,43,44,58,7]). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Ref. [5]. We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Refs. [4,6] for the present class of graphs.
We complete the program of Ref. [3,5] of connecting totally non-negative Grassmannians to the reality problem in KP finite-gap theory via the assignment of real regular divisors on rational degenerations of M-curves for the class of real regular multi-line soliton solutions of KP II equation whose asymptotic behavior has been combinatorially characterized by Chakravarthy, Kodama and Williams. We use the plabic networks in the disk introduced by Postnikov to parametrize positroid cells in totally nonnegative Grassmannians. The boundary of the disk corresponds to the rational curve associated to the soliton data in the direct spectral problem, and the bicolored graph is the dual of a reducible curve G which is the rational degeneration of a regular M-curve whose genus g equals the number of faces of the network diminished by one. We assign systems of edge vectors to the planar bicolored networks. The system of relations satisfied by them has maximal rank and may be reformulated in the form of edge signatures.. We prove that the components of the edge vectors are rational in the edge weights with subtraction free denominators and provide their explicit expressions in terms of conservative and edge flows. The edge vectors rule the value of the KP wave function at the double points of G, whereas the signatures at the vertices rule the position of the divisor in the ovals. We prove that the divisor satisfies the conditions of Dubrovin-Natanzon for real finite-gap solutions: there is exactly one divisor point in each oval except for the one containing the essential singularity of the wave function. The divisor may be explicitly computed using the linear relations at the vertices of the network. We explain the role of moves and reductions in the transformation of both the curve and the divisor for given soliton data, and apply our construction to some examples.
We associate real and regular algebraic--geometric data to each multi--line soliton solution of Kadomtsev-Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non--negative part of real Grassmannians $Gr^{TNN}(k,n)$. In Ref.[3] we were able to construct real algebraic-geometric data for soliton data in the main cell $Gr^{TP} (k,n)$ only. Here we do not just extend that construction to all points in $Gr^{TNN}(k,n)$, but we also considerably simplify it, since both the reducible rational $M$-curve $Γ$ and the real regular KP divisor on $Γ$ are directly related to the parametrization of positroid cells in $Gr^{TNN}(k,n)$ via the Le-networks introduced by A. Postnikov in Ref [62]. In particular, the direct relation of our construction to the Le--networks guarantees that the genus of the underlying smooth $M$-curve is minimal and it coincides with the dimension of the positroid cell in $Gr^{TNN}(k,n)$ to which the soliton data belong to. Finally, we apply our construction to soliton data in $Gr^{TP}(2,4)$ and we compare it with that in Ref [3].
We apply the general construction developed in references [4,5] to the first non-trivial case of Gr^{TP}(2,4). In particular, we construct finite-gap KP-II real quasiperiodic solutions in the form of soliton lattice corresponding to a smooth genus 4 M-curve, which is a desingularization of a reducible rational M-curve for soliton data in Gr^{TP}(2,4).
We establish a new connection between the theory of totally positive Grassmannians and the theory of $\mathtt M$-curves using the finite--gap theory for solitons of the KP equation. Here and in the following KP equation denotes the Kadomtsev-Petviashvili 2 equation, which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian $Gr^{TP} (N,M)$ a reducible curve which is a rational degeneration of an $\mathtt M$--curve of minimal genus $g=N(M-N)$, and we reconstruct the real algebraic-geometric data á la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth $ M$-curves. In our approach we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection $Gr^{TP} (r+1,M-N+r+1)\mapsto Gr^{TP} (r,M-N+r)$.
We classify the soliton data in the totally non--negative part of Gr(k,n) which may be associated to algebraic-geometric data on certain rational degenerations of regular hyperelliptic M-curves. Such degenerate rational curve G is a desingularization of that constructed in Abenda-Grinevich (arXiv:1506.00563) for soliton data in the totally positive part of Gr(n-1,n) and the KP wavefunctions are the same in such case. G is also the curve constructed in the paper for soliton data in the totally positive part of Gr(1,n). For any such G and for any fixed k between 2 and (n-2), we show that k-compatible soliton data correspond to a family of KP multi-line solitons (T-hyperelliptic) which parametrize soliton data in an (n-1)--dimensional variety of the totally positive part of the Grassmannian Gr(k,n). We explicitly characterize T-hyperelliptic solitons from the algebraic-point of view. T-hyperelliptic solitons are also connected to the solutions of the finite non-periodic Toda hierarchy because the tau function is the same for both systems. We investigate such relation from the algebraic point of view and compare the two spectral problems. In particular, the vacuum KP divisor and the Toda divisor coincide, while k--compatible divisors may be recursively constructed using known recursions for the Toda system. Finally, we also explain how KP divisors change under the space--time transformation which induces a duality tranformation from soliton data in Gr(k,n) to soliton data in Gr(n-k,n) and compare the action of such transformation both for the KP and the Toda systems.
In our paper we study periodic geodesic motion on multidimensional ellipsoids with elastic impacts along confocal quadrics. We show that the method of isoperiodic deformation is applicable.
We start from a hyperbolic DN hydrodynamic type system of dimension $n$ which possesses Riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that, after such a transformation of the independent variables, one of the metrics associated to the initial system be flat. We prove the following statement: let $n\ge 3$ in the case of reciprocal transformations of a single independent variable or $n\ge 5$ in the case of transformations of both the independent variable; then the reciprocal metric may be flat only if the conservation laws in the transformation are linear combinations of the canonical densities of conservation laws, {\it i.e} the Casimirs, the momentum and the Hamiltonian densities associated to the Hamiltonian operator for the initial metric. Then, we restrict ourselves to the case in which the initial metric is either flat or of constant curvature and we classify the reciprocal transformations of one or both the independent variables so that the reciprocal metric is flat. Such characterization has an interesting geometric interpretation: the hypersurfaces of two diagonalizable DN systems of dimension $n\ge 5$ are Lie equivalent if and only if the corresponding local hamiltonian structures are related by a canonical reciprocal transformation.
The closedness condition for real geodesics on n-dimensional ellipsoids is in general transcendental in the parameters (semiaxes of the ellipsoid and constants of motion). We show that it is algebraic in the parameters if and only if both the real and the imaginary geodesics are closed and we characterize such double--periodicity condition via real hyperelliptic tangential coverings. We prove the density of algebraically closed geodesics on n-dimensional ellipsoids with respect to the natural topology in the (2n)-dimensional real parameter space. In particular, the approximating sequence of algebraic closed geodesics on the approximated ellipsoids may be chosen so to share the same values of the length and of the real period vector as the limiting closed geodesic on the limiting ellipsoid. Finally, for real doubly-periodic geodesics on triaxial ellipsoids, we show how to evaluate algebraically the period mapping and we present some explicit examples of families of algebraically closed geodesics.
We derive the modulation equations or Whitham equations for the Camassa--Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.
We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is known. We give explicit sufficient conditions both for closed geodesics and periodic billiard orbits on Q and discuss their relation with the elliptic KdV solutions and elliptic Calogero system