Darboux Coordinates Research Articles

Description of Generalized Isomonodromic Deformations of Rank-Two Linear Differential Equations Using Apparent Singularities

In this paper, we consider the generalized isomonodromic deformations of rank-two irregular connections on the Riemann sphere. We introduce Darboux coordinates on the parameter space of a family of rank-two irregular connections by apparent singularities. Using the Darboux coordinates, we describe the generalized isomonodromic deformations as Hamiltonian systems.

Isomonodromic and isospectral deformations of meromorphic connections: the sl2(C) case

Abstract We consider non-twisted meromorphic connections in sl 2 ( C ) and the associated symplectic Hamiltonian structure. In particular, we provide explicit expressions of the Lax pair in the geometric gauge supplementing results of Marchal et al (2022 arXiv:2212.04833) where explicit formulas have been obtained in the oper gauge. Expressing the geometric Lax matrices requires the introduction of specific Darboux coordinates for which we provide the explicit Hamiltonian evolutions. These expressions allow to build bridges between the isomonodromic deformations and the isospectral ones. More specifically, we propose an explicit change of Darboux coordinates to obtain isospectral coordinates for which Hamiltonians match the spectral invariants. This result solves the issue left opened in Bertola et al (2023 J. Math. Phys. 64 083502) in the case of sl 2 ( C ) .

Darboux’s Theorem, Lie series and the standardization of the Salerno and Ablowitz–Ladik models

In the framework of nonlinear Hamiltonian lattices, we revisit the proof of Moser-Darboux’s Theorem, in order to present a general scheme for its constructive applicability to Hamiltonian models with non-standard symplectic structures. We take as a guiding example the Salerno and Ablowitz–Ladik (AL) models: we justify the form of a well-known change of coordinates which is adapted to the Gauge symmetry, by showing that it comes out in a natural way within the general strategy outlined in the proof. Moreover, the full or truncated Lie-series technique in the extended phase-space is used to transform the Salerno model, at leading orders in the Darboux coordinates: thus the dNLS Hamiltonian turns out to be a normal form of the Salerno and AL models; as a byproduct we also get estimates of the dynamics of these models by means of dNLS one. We also stress that, once it is cast into the perturbative approach, the method allows to deal with the cases where the explicit trasformation is not known, or even worse it is not writable in terms of elementary functions.

Coadjoint orbits of vortex sheets in ideal fluids

We describe coadjoint orbits associated to the motion of codimension one singular vorticities in ideal fluids, e.g. vortex sheets in 3D. We show that these coadjoint orbits can be identified with a certain class of decorated nonlinear Grassmannians, that consist of codimension one submanifolds belonging to an isodrast (i.e., enclosing a given volume, provided their homology class vanishes), endowed with a closed 1-form of a given type. Such isodrasts are defined via an integrable distribution associated to a manifold endowed with a volume form, similar to the isodrastic distribution [35] in symplectic setting. With the choice of a Riemannian metric, the expression of the orbit symplectic form is shown to take a particularly simple expression reminiscent of Darboux coordinates. In general we get coadjoint orbits of the universal central extension of the group of exact volume preserving diffeomorphisms and we give a necessary and sufficient condition under which the central extension is not needed. We also focus on the case in which we get coadjoint orbits of the group of volume preserving diffeomorphisms, which is relevant for the ideal fluid, and discuss how our results extend the class of coadjoint orbits of exact vortex sheets considered in [25].

Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R P}^2$-structures in genus one

Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R P}^2$-structures in genus one

't Hooft lines of ADE-type and topological quivers

We investigate 4D Chern-Simons theory with ADE gauge symmetries in the presence of interacting Wilson and ‘t Hooft line defects. We analyse the intrinsic properties of these lines’ coupling and explicate the building of oscillator-type Lax matrices verifying the RLL integrability equation. We propose gauge quiver diagrams Q_{G}^{\mu}Gμ encoding the topological data carried by the Lax operators and give several examples where Darboux coordinates are interpreted in terms of topological bi-fundamental matter. We exploit this graphical description (i) to give new results regarding solutions in representations beyond the fundamentals of sl_{N}slN, $ so_{2N}$ and e_{6,7}e6,7, and (ii) to classify the Lax operators for simply laced symmetries in a unified E_{7}7 CS theory. For quick access, a summary list of the leading topological quivers Q$ _{ADE}^{}$ is given in the conclusion section [Figures 29.(a-e), 30.(a-d) and 31.(a-d)].

Geometry of Inhomogeneous Poisson Brackets, Multicomponent Harry Dym Hierarchies, and Multicomponent Hunter–Saxton Equations

We introduce a natural class of multicomponent local Poisson structures $\mathcal P_k + \mathcal P_1$, where $\mathcal P_1$ is a local Poisson bracket of order one and $\mathcal P_k$ is a homogeneous Poisson bracket of odd order $k$ under assumption that is has Darboux coordinates (Darboux-Poisson bracket) and non-degenerate. For such brackets we obtain the general formulas in arbitrary coordinates, find normal forms (related to Frobenius triples) and provide the description of the Casimirs, using purely algebraic procedure. In two-component case we completely classify such brackets up to the point transformation. From the description of Casimirs we derive new Harry Dym (HD) hierarchies and new Hunter-Saxton (HS) equations for arbitrary number of components. In two component case our HS equation differs from the well-known HS2 equation.

A hyperkähler geometry associated to the BPS structure of the resolved conifold

We associate to the resolved conifold an affine special Kähler (ASK) manifold of complex dimension 1, and an instanton corrected hyperkähler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland.

On exceptional 't Hooft lines in 4D-Chern-Simons theory

We study ‘t Hooft lines and the associated L- operators in topological 4D Chern-Simons theory with gauge symmetry given by the exceptional groups E6 and E7. We give their oscillator realisations and propose topological gauge quivers encoding the properties of these topological lines where Darboux coordinates are interpreted in terms of topological fundamental matter. Other related aspects are also described.

Symplectic geometry of Cartan–Hartogs domains

This paper studies the geometry of Cartan–Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan–Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan–Hartogs domains and their dual. Further, we compute their symplectic capacity and show that a Cartan–Hartogs domain admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.

Exact WKB and the quantum Seiberg-Witten curve for 4d N = 2 pure SU(3) Yang-Mills. Abelianization

We investigate the exact WKB method for the quantum Seiberg-Witten curve of 4d N = 2 pure SU(3) Yang-Mills in the language of abelianization. The relevant differential equation is a third-order equation on ℂℙ1 with two irregular singularities. We employ the exact WKB method to study the solutions to such a third-order equation and the associated Stokes phenomena. We also investigate the exact quantization condition for a certain spectral problem. Moreover, exact WKB analysis leads us to consider new Darboux coordinates on a moduli space of flat SL(3,ℂ)-connections. In particular, in the weak coupling region we encounter coordinates of the higher length-twist type generalizing Fenchel-Nielsen coordinates. The Darboux coordinates are conjectured to admit asymptotic expansions given by the formal quantum periods series and we perform numerical analysis supporting this conjecture.

Hitchin Systems on Hyperelliptic Curves

We describe a class of spectral curves and find explicit formulas for the Darboux coordinates of Hitchin systems corresponding to classical simple groups on hyperelliptic curves. We consider in detail the systems with rank $$2$$ groups on genus $$2$$ curves.

Hodge and Prym Tau Functions, Strebel Differentials and Combinatorial Model of $${\mathcal {M}}_{g,n}$$

The goal of the paper is to apply the theory of integrable systems to construct explicit sections of line bundles over the combinatorial model of the moduli space of pointed Riemann surfaces based on Strebel differentials. These line bundles are tensor products of the determinants of the Hodge or Prym vector bundles with the standard tautological line bundles $$\mathcal {L}_j$$ , and the sections are constructed in terms of tau functions. The combinatorial model is interpreted as the real slice of a complex analytic moduli space of quadratic differentials where the phase of each tau-function provides a section of a circle bundle. The phase of the ratio of the Prym and Hodge tau functions gives a section of the $$\kappa _1$$ -circle bundle. By evaluating the increment of the phase around co-dimension 2 sub-complexes, we identify the Poincare dual cycles to the Chern classes of the corresponding line bundles: they are expressed explicitly as combination of Witten’s cycle $$W_{5} $$ and Kontsevich’s boundary. This provides combinatorial analogues of Mumford’s relations on $${\mathcal {M}}_{g,n}$$ and Penner’s relations in the hyperbolic combinatorial model. The free homotopy classes of loops around $$W_{5} $$ are interpreted as pentagon moves while those of loops around Kontsevich’s boundary as combinatorial Dehn twists. Throughout the paper we exploit the classical description of the combinatorial model in terms of Strebel differentials, parametrized in terms of period, or homological coordinates; we show that they provide Darboux coordinates for the symplectic structure introduced by Kontsevich. We also express the latter as the intersection pairing in the odd homology of the canonical double cover.

Faithful realizations of semiclassical truncations

Realizations of algebras in terms of canonical or bosonic variables can often be used to simplify calculations and to exhibit underlying properties. There is a long history of using such methods in order to study symmetry groups related to collective motion, for instance in nuclear shell models. Here, related questions are addressed for algebras obtained by turning the quantum commutator into a Poisson bracket on moments of a quantum state, truncated to a given order. In this application, canonical realizations allow one to express the quantum back-reaction of moments on basic expectation values by means of effective potentials. In order to match degrees of freedom, faithfulness of the realization is important, which requires that, at least locally, the space of moments as a Poisson manifold is realized by a complete set of Casimir–Darboux coordinates in local charts. A systematic method to derive such variables is presented and applied to certain sets of moments which are important for physical applications. If only second-order moments are considered, their Poisson-bracket relations are isomorphic to the Lie bracket of sp(2N,R), providing an interesting link with realizations of nuclear shell models.

Recurrence relations for symplectic realization of (quasi)-Poisson structures

It is known that any Poisson manifold can be embedded into a bigger space which admits a description in terms of a global symplectic structure. Such a procedure is known as a symplectic realization and has a number of important applications like the quantization of the original Poisson manifold. In the present paper we extend the above idea to the case of quasi-Poisson structures which should not necessarily satisfy the Jacobi identity. For any given quasi-Poisson structure we provide a closed recursive formula for local embedding functions and Darboux coordinates. Our construction is illustrated for the examples of the constant R-flux algebra, quasi-Poisson structure isomorphic to the commutator algebra of imaginary octonions and the non-geometric M-theory R-flux background. In all cases we derive explicit formulas for the symplectic realization and the corresponding expression for Darboux coordinates.

On the Hamiltonian formulation, integrability and algebraic structures of the Rajeev-Ranken model

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is ‘pseudodual’ to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serve as a toy-model for non-perturbative properties of theories with a Landau pole. Unlike the ‘Euclidean’ current algebra of the PCM, its pseudodual is based on a nilpotent current algebra. Recently, Rajeev and Ranken obtained a mechanical reduction by restricting the nilpotent scalar field theory to a class of constant energy-density classical waves expressible in terms of elliptic functions, whose quantization survives the passage to the strong-coupling limit. We study the Hamiltonian and Lagrangian formulations of this model and its classical integrability from an algebraic perspective, identifying Darboux coordinates, Lax pairs, classical r-matrices and a degenerate Poisson pencil. We identify Casimirs as well as a complete set of conserved quantities in involution and the canonical transformations they generate. They are related to Noether charges of the field theory and are shown to be generically independent, implying Liouville integrability. The singular submanifolds where this independence fails are identified and shown to be related to the static and circular submanifolds of the phase space. We also find an interesting relation between this model and the Neumann model allowing us to discover a new Hamiltonian formulation of the latter.

Separation of variables for linear Lax algebras and classical r-matrices

We consider the problem of separation of variables for the algebraically integrable Hamiltonian systems possessing gl(n)-valued Lax matrices depending on a spectral parameter that satisfy linear Poisson brackets with some gl(n) ⊗ gl(n)-valued classical r-matrices. We formulate, in terms of the corresponding r-matrices, a sufficient condition that guarantees that the “separating polynomials” of Sklyanin [Commun. Math. Phys. 150, 181 (1992)], Scott [J. Math. Phys. 35, 5831 (1994)], Gekhtman [Commun. Math. Phys. 167, 593 (1995)], and Diener and Dubrovin (Algebraic-geometrical Darboux coordinates in R-matrix formalism, SISSA, Preprint Report No. 88-94-FM, 1994) produce a system of canonical variables. We consider two examples of classical r-matrices and separating polynomials. One of these examples, namely, the n-parametric family of non-skew-symmetric non-dynamical classical r-matrices of Skrypnyk [Phys. Lett. A 334, 390 (2005); 347, 266 (2005)] and the corresponding separating polynomials is new. We show that the separating polynomials of Diener and Dubrovin produce in this case a complete set of separated variables for the corresponding generalized Gaudin models with or without external magnetic field.

Congruence method for global Darboux reduction of finite-dimensional Poisson systems

A new procedure for the global construction of the Casimir invariants and Darboux canonical form for finite-dimensional Poisson systems is developed. This approach is based on the concept of matrix congruence and can be applied without the previous determination of the Casimir invariants (recall that their prior knowledge is unavoidable for the standard reduction methods, thus requiring either the integration of a system of PDEs or solving some equivalent problem). Well the opposite, in the new congruence method, both the Darboux coordinates and the Casimir invariants arise simultaneously as the outcome of the reduction algorithm. In fact, the congruence algorithm proceeds only in terms of matrix-algebraic transformations and direct quadratures, thus avoiding the need of previously integrating a system of PDEs and therefore improving previously known approaches. Physical examples illustrating different aspects of the theory are provided.

Multiple D3-Instantons and Mock Modular Forms II

We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an isometric action of S-duality on this moduli space: the generating function of DT invariants in the large volume attractor chamber must be a vector-valued mock modular form with specified modular properties. In this work, we prove that this condition is also sufficient at two-instanton order. This is achieved by producing a holomorphic action of SL(2,Z) on the twistor space which preserves the holomorphic contact structure. The key step is to cancel the anomalous modular variation of the Darboux coordinates by a local holomorphic contact transformation, which is generated by a suitable indefinite theta series. For this purpose we introduce a new family of theta series of signature (2,n-2), find their modular completion, and conjecture sufficient conditions for their convergence, which may be of independent mathematical interest.

Coadjoint orbits in duals of Lie algebras with admissible ideals

We analyze the symplectic structure of the coadjoint orbits of Lie groups with Lie algebras that contain admissible ideals. Such ideals were introduced by Pukanszky to investigate the global symplectic structure of simply connected coadjoint orbits of connected, simply connected, solvable Lie groups. Using the theory of symplectic reduction of cotangent bundles, we identify classes of coadjoint orbits which are vector bundles. This implies Pukanszky's earlier result that such orbits have a symplectic form which is the sum of the canonical form and a magnetic term. This approach also allows us to provide many of the essential details of Pukanszky's result regarding the existence of global Darboux coordinates for the simply connected coadjoint orbits of connected, simply connected solvable Lie groups. Bibliography: 26 titles.