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Orthogonal or symplectic Yangians are defined by the Yang–Baxter RLL relation involving the fundamental R-matrix with the corresponding so(n) or sp(2m) symmetry. We investigate the second-order solution conditions, where the expansion of L(u) in u −1 is truncated at the second power, and we derive the relations for the two nontrivial terms in L(u).

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages

In this paper we investigate oscillations of conjoined bases of linear Hamiltonian differential systems related via symplectic transformations. Both systems are considered without controllability (or normality) assumptions and under the Legendre condition for their Hamiltonians. The main result of the paper presents new explicit relations connecting the multiplicities of proper focal points of Y ( t ) and the transformed conjoined basis Y ( t ) = R − 1 ( t ) Y ( t ) , where the symplectic transformation matrix R ( t ) obeys some additional assumptions on the rank of its components. As consequences of the main result we formulate the generalized reciprocity principle for the Hamiltonian systems without normality. The main tool of the paper is the comparative index theory for discrete symplectic systems generalized to the continuous case.

We assume that a symplectic real-analytic map has an invaria nt nor- mally hyperbolic cylinder and an associated transverse hom oclinic cylinder. We prove that generically in the real-analytic category the bo undaries of the invariant cylinder are connected by trajectories of the map.

We prove a formula for twisted Duistermaat-Heckman distributions associated to a Hamiltonian $LG$-space. The terms of the formula are localized at the critical points of the norm-square of the moment map, and can be computed in cross-sections. Our main tools are the theory of quasi-Hamiltonian $G$-spaces, as well as the Hamiltonian cobordism approach to norm-square localization introduced recently by Harada and Karshon., 40 pages, some small additions and corrections

This article considers the Dirac operator on a Riemann surface coupled to a symplectic holomorphic vector bundle W. Each spinor in the null-space (which is a holomorphic section of W⊗K1/2) generates through the moment map a Higgs field Φ, and varying W one obtains a holomorphic Lagrangian subvariety in the moduli space of Higgs bundles. Applying this to the irreducible symplectic representations of SL(2,C) we obtain Lagrangian submanifolds of the rank 2 Higgs bundle moduli space which link up with m-period points on the Prym variety of the spectral curve as well as Brill-Noether loci on the moduli space of semistable bundles. The case of genus 2 is investigated in some detail.

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kahler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kahler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kahler–Einstein metrics.

A 2 n × 2 n complex matrix A is symplectic if A ⊤ [ 0 I − I 0 ] A = [ 0 I − I 0 ] . If A is symplectic and rank ( A − I ) = 1 , then it is called a J-symmetry. For each n, we prove that every 2 n × 2 n symplectic matrix M is a product of n + 1 commutators of J-symmetries and this number cannot be smaller for some M.

Let J 2 n = [ 0 I n − I n 0 ] . An A ∈ M 2 n ( C ) is called symplectic if A T J 2 n A = J 2 n . If n = 1 , then we show that every matrix in M 2 n ( C ) is a sum of two symplectic matrices. If n > 1 , then we show that every matrix in M 2 n ( C ) is a sum of three symplectic matrices; moreover, we show that some matrices cannot be written with less than three symplectic matrices. We also show that for every A ∈ M 2 n ( C ) , there exist symplectic P, Q ∈ M 2 n ( C ) and B, C, D ∈ M n ( C ) such that P A Q = [ B C 0 D ] . If A is skew Hamiltonian ( J 2 n − 1 A T J 2 n = A ), then we show that A is a sum of two symplectic matrices.

On a manifold with an almost contact metric structure we introduce the notions of intrinsic connection, N-extended connection and N-connection. It is shown that the Tanaka–Webster and Schouten–van Kampen connections are special cases of N-connection. We define new classes of N-connections, namely, the Vagner connection and canonical metric N-connection. We also define N-extended symplectic connection. It is proved that the N-extended symplectic connection exists on any manifold with a contact metric structure.

It seems to be of recurring interest in the literature to give alternative proofs for the fact that the determinant of a symplectic matrix is one. We state four short and elementary proofs for symplectic matrices over general fields. Two of them seem to be new.

For the quantum symplectic group SP q (2n), we describe the C ∗-algebra of continuous functions on the quotient space S P q (2n)/S P q (2n−2) as an universal C ∗-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the K-groups of this C ∗-algebra in terms of generators of the C ∗-algebra.

Let $M$ be a closed Fano symplectic manifold with a semifree Hamiltonian circle action with isolated maximum. We compute the Gromov width and the Hofer-Zehnder capacity of $M$ using a moment map., 14 pages, weakened the assumption to Fano case, added a result on the Hofer-Zehnder capacity, changed title

We show that, for a quadratic extension of p-adic fields, no cuspidal representation of the quasi-split unitary group admits a non-trivial linear form invariant by the symplectic subgroup. Our proof is purely local.

For the general integrability case of M. Adler and P. van Moerbeke, invariant relations are obtained in which the rank of the momentum map is 1. Thereby, special periodic solutions generating the edges of the bifurcation diagram are defined. All phase variables are expressed in terms of a set of constants and one auxiliary variable, for which a differential equation integrable in elliptic functions time is given. An explicit expression for the characteristic exponent determining the type of special periodic solutions is presented, which makes it possible to study the character of stability of the obtained solution.

A strong from of invariance under a group G is manifested in a family over the classifying space BG. We advocate a differential-geometric avatar of BG when G is a Lie group. Applied to G-equivariant connections on smooth principal or vector bundles, the equivariance-->families principle converts the G-equivariant extensions of curvature and Chern-Weil forms to the standard nonequivariant versions. An application of this technique yields the moment map of the determinant line of a G-equivariant Dirac operator, which in turn sheds light on some anomaly formulas in quantum field theory.

We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

Let X be a symplectic homotopy K3 surface and G = S 5 act on X symplectically. In this paper, we give a weak classification of the G action on X by discussing the fixed-point set structure. Besides, we analyse the exoticness of smooth structures of X under the action of G.

We give a scheme-theoretic description of μ − 1 ( 0 ) and μ − 1 ( 0 ) / / Sp ( 2 ) , where μ is the moment map defining the space of ADHM data associated to framed SO ( 3 , R ) -instantons with instanton number 1 . The irreducible component in μ − 1 ( 0 ) / / Sp ( 2 ) , other than the Zariski closure of the instanton moduli space, is non-reduced, which is a corrigendum to [Choy (2016), Theorem 4.3]. We further describe this component using the factorization property Braverman et al. (2006).

We describe the Cartan-Münzner polynomials of degree four on an $N$-dimensional Euclidean space by using the square norm of the moment map of certain group actions. In the case of OT-FKM type, the action is a spin action which is not necessarily transitive on isoprametric hypersurfaces.

A general-covariant statistical framework capable of describing classical fluctuations of the gravitational field is a thorny open problem in theoretical physics, yet ultimately necessary to understand the nature of the gravitational interaction and a key to quantum gravity. Inspired by Souriau's symplectic generalization of the Maxwell-Boltzmann-Gibbs equilibrium in Lie group thermodynamics, we investigate a spacetime-covariant formulation of statistical mechanics for parametrized first-order field theories, as a simplified model sharing essential general covariant features with canonical general relativity. Starting from a covariant multi-symplectic phase space formulation, we define a general-covariant notion of Gibbs state in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space. We show how such a covariant notion of equilibrium encodes the whole information about symmetry, gauge and dynamics carried by the theory, associated to a canonical spacetime foliation, where the covariant choice of a reference frame reflects in a Lie algebra-valued notion of local temperature. We investigate how physical equilibrium, hence time evolution, emerges from such a state and the role of the gauge symmetry in the thermodynamic description., 46 pages, 1 figure

This paper puts the theory of quasi-Hamiltonian reduction in the framework of shifted symplectic structures developed by Pantev, To\"{e}n, Vaqui\'{e} and Vezzosi. We compute the symplectic structures on mapping stacks and show how the AKSZ topological field theory defined by Calaque allows one to neatly package the constructions used in quasi-Hamiltonian reduction. Finally, we explain how a prequantization of character stacks can be obtained purely locally., Comment: 33 pages

We identify a family of torus representations such that the corresponding singular symplectic quotients at the $0$-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic., 19 pages. This is a pre-print of an article published in Beitr\"age zur Algebra und Geometrie / Contributions to Algebra and Geometry. The final authenticated version is available online at: https://doi.org/10.1007/s13366-020-00486-8

We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the intersection theory of such Lagrangian orbits, giving many examples., 22 pages. Final version to appear in Mathematische Zeitschrift

The paper clarifies some subtle points surrounding the definition of scalar curvature in generalized K$\ddot{a}$hler (GK) geometry. We have solved an open problem in GK geometry of symplectic type posed by R. Goto \cite{Go1} on relating the scalar curvature defined in terms of generalized pure spinors \emph{directly} to the underlying biHermitian structure. In particular, we apply this solution to toric GK geometry of symplectic type and prove that the scalar curvature suggested in this setting by L. Boulanger \cite{Bou} coincides with Goto's definition., A continuation of the preprints arXiv:1810.08265v1 and arXiv:1811.06848v1. 38 pages

In 1969, Jean-Marie Souriau has introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines thanks to cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau model is more general if we consider another Souriau theorem, that we can associate to a Lie group, an homogeneous symplectic manifold with a KKS 2-form on their coadjoint orbits. Souriau method could then be applied on Lie Groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincare unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant.

If a Lie group acts on a symplectic manifold preserving the symplectic form, it is possible that each fundamental vector field is the Hamiltonian vector field of a Hamiltonian function called the moment map which was defined in Definition 2.9.

In this chapter, we will define Hamiltonian flows, Hamiltonian actions and moment maps. The layout of the chapter is as follows. In Sect. 2.1, we recall the original example of a Hamiltonian flow, namely, Hamilton’s equations. In Sect. 2.2, we will start by understanding what Hamiltonian vector fields and Hamiltonian functions are. In Sect. 2.3, we will introduce a bracket on the set of smooth functions on a symplectic manifold which will satisfy the Jacobi identity and will make the former into a Lie algebra. We will see some examples of such vector fields on \(S^2\) and the 2-torus. In the final section (Sect. 2.4), we will define a moment map and will list some conditions which will guarantee the existence of moment maps and other conditions which guarantee their uniqueness.

We investigate the non-BPS realm of 3d ${\cal N} = 4$ superconformal field theory by uniting the non-perturbative methods of the conformal bootstrap and supersymmetric localization, and utilizing special features of 3d ${\cal N} = 4$ theories such as mirror symmetry and a protected sector described by topological quantum mechanics (TQM). Supersymmetric localization allows for the exact determination of the conformal and flavor central charges, and the latter can be fed into the mini-bootstrap of the TQM to solve for a subset of the OPE data. We examine the implications of the $\mathbb{Z}_2$ mirror action for the SCFT single- and mixed-branch crossing equations for the moment map operators, and apply numerical bootstrap to obtain universal constraints on OPE data for given flavor symmetry groups. A key ingredient in applying the bootstrap analysis is the determination of the mixed-branch superconformal blocks. Among other results, we show that the simplest known self-mirror theory with $SU(2) \times SU(2)$ flavor symmetry saturates our bootstrap bounds, which allows us to extract the non-BPS data and examine the self-mirror $\mathbb{Z}_2$ symmetry thereof., Comment: 50+27 pages, 6 figures, 7 tables. v2: typos corrected, references added. v3: a brief summary added, minor notations changed, SciPost published version

This paper extends the theory of controlled Hamiltonian systems with symmetries due to [9, 10, 6, 7, 11] to the case of non-abelian symmetry groups $G$. The notion of symmetry actuating forces is introduced and it is shown, that Hamiltonian systems subject to such forces permit a conservation law, which arises as a controlled perturbation of the $G$-momentum map. Necessary and sufficient matching conditions are given to relate the closed-loop dynamics, associated to the forced Hamiltonian system, to an unforced Hamiltonian system. These matching conditions are then applied to general Lie-Poisson systems, to the example of ideal charged fluids in the presence of an external magnetic field ([20]), and to the satellite with a rotor example ([9, 10])., Comment: v2: minor changes

The space of the Fuchsian systems is the algebraic Poisson manifold, and the equations of the Isomonodromic Deformations are the Hamiltonian equations. The internal symmetry of the problem makes it possible to reduce the dimension of the problem using the symplectic-quotient theory. The phase-space is constructed from the orbits of (co)adjoint representation of the general linear group. The presented parametrisation of the quotientspace is based on the construction of the flag coordinates on the orbits. The simplest non-trivial case that is Painleve VI case is considered as an example.

We construct a Poisson map $\mathbf{M}\colon T^{*}\mathbb{C}^{2} \to \mathfrak{se}(3)^{*}$ with respect to the canonical Poisson bracket on $T^{*}\mathbb{C}^{2} \cong T^{*}\mathbb{R}^{4}$ and the $(-)$-Lie--Poisson bracket on the dual $\mathfrak{se}(3)^{*}$ of the Lie algebra of the special Euclidean group $\mathsf{SE}(3)$. The essential part of this map is the momentum map associated with the cotangent lift of the natural right action of the semidirect product Lie group $\mathsf{SU}(2) \ltimes \mathbb{C}^{2}$ on $\mathbb{C}^{2}$. This Poisson map gives rise to a canonical Hamiltonian system on $T^{*}\mathbb{C}^{2}$ whose solutions are mapped by $\mathbf{M}$ to solutions of the heavy top equations. We show that the Casimirs of the heavy top dynamics and the additional conserved quantity of the Lagrange top correspond to the Noether conserved quantities associated with certain symmetries of the canonical Hamiltonian system. We also construct a Lie--Poisson integrator for the heavy top dynamics by combining the Poisson map $\mathbf{M}$ with a simple symplectic integrator, and demonstrate that the integrator exhibits either exact or near conservation of the conserved quantities of the Kovalevskaya top., Comment: 17 pages, 2 figures

Using the Grosshans Principle, we develop a method for proving lower bounds for the maximal degree of a system of generators of an invariant ring. This method also gives lower bounds for the maximal degree of a set of invariants that define Hilbert's null cone. We consider two actions: The first is the action of SL ( V ) on S 3 ( V ) ⊕ 4 , the space of 4-tuples of cubic forms, and the second is the action of SL ( V ) × SL ( W ) × SL ( Z ) on the tensor space ( V ⊗ W ⊗ Z ) ⊕ 9 . In both these cases, we prove an exponential lower degree bound for a system of invariants that generate the invariant ring or that define the null cone.

We produce for each tropical hypersurface $V(\phi)\subset Q=\mathbb{R}^n$ a Lagrangian $L(\phi)\subset (\mathbb{C}^*)^n$ whose moment map projection is a tropical amoeba of $V(\phi)$. When these Lagrangians are admissible in the Fukaya-Seidel category, we show that they are unobstructed objects of the Fukaya category, and mirror to sheaves supported on complex hypersurfaces in a toric mirror., Comment: 48 Pages. First portion of arXiv:1904.06005 which was split in two. This version has been accepted for publication by the Journal of Topology

We consider T[SU(N)] and its mirror, and we argue that there are two more dual frames, which are obtained by adding flipping fields for the moment maps on the Higgs and Coulomb branch. Turning on a monopole deformation in T[SU(N)], and following its effect on each dual frame, we obtain four new daughter theories dual to each other. We are then able to construct pairs of 3d spectral dual theories by performing simple operations on the four dual frames of T[SU(N)]. Engineering these 3d spectral pairs as codimension-two defect theories coupled to a trivial 5d theory, via Higgsing, we show that our 3d spectral dual theories descends from the 5d spectral duality, or fiber base duality in topological string. We provide further consistency checks about the web of dualities we constructed by matching partition functions on the three sphere, and in the case of spectral duality, matching exactly topological string computations with holomorphic blocks., 74 pages, 15 pictures