Journal of Geometry and Physics最新文献

英文中文

Relating Hamiltonian systems with multiple invariants to generalized Hamiltonian mechanics via multisymplectic geometry

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-30 DOI: 10.1016/j.geomphys.2025.105438

Nathan Duignan , Naoki Sato

Classical Hamiltonian mechanics, characterized by a single conserved Hamiltonian (energy) and symplectic geometry, ‘hides’ other invariants into symmetries of the Hamiltonian or into the kernel of the Poisson tensor. Nambu mechanics aims to generalize classical Hamiltonian mechanics to ideal dynamical systems bearing two Hamiltonians, but its connection to a suitable geometric framework has remained elusive. This work establishes a novel correspondence between generalized Hamiltonian mechanics, defined for systems with a phase space conservation law (invariance of a closed form) and a matter conservation law (invariance of multiple Hamiltonians), and classical Hamiltonian mechanics via multisymplectic geometry. The key lies in the invertibility of differential forms of degree higher than 2. We demonstrate that the cornerstone theorems of classical Hamiltonian mechanics (Lie-Darboux and Liouville) require reinterpretation within this new framework, reflecting the unique properties of invertibility in multisymplectic geometry. Furthermore, we present two key theorems that solidify the connection: i) any classical Hamiltonian system with two or more invariants is also a generalized Hamiltonian system and ii) given a generalized Hamiltonian system with two or more invariants, there exists a corresponding classical Hamiltonian system on the level set of all but one invariant, with the remaining invariant playing the role of the Hamiltonian function.

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引用次数: 0

The three-point Gaudin model and branched coverings of the Riemann sphere

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-27 DOI: 10.1016/j.geomphys.2025.105436

Natalia Amburg , Ilya Tolstukhin

We study the three-point quantum

{sl}_{2}

Gaudin model. In this case the compactification of the parameter space is

\overline{M_{0, 4} (C)}

, which is the Riemann sphere. We analyze sphere coverings by the joint spectrum of the Gaudin Hamiltonians treating them as algebraic curves. We write equations for these curves as determinants of tridiagonal matrices and deduce some consequences regarding the geometric structure of the Gaudin coverings.

引用次数: 0

A correspondence between the quantum K theory and quantum cohomology of Grassmannians

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-27 DOI: 10.1016/j.geomphys.2025.105437

Wei Gu , Jirui Guo , Leonardo Mihalcea , Yaoxiong Wen , Xiaohan Yan

We utilize physics arguments, and the nonabelian/abelian correspondence, to relate the Givental and Lee's quantum K theory ring of Grassmannians to a twisted variant of the quantum cohomology ring. Furthermore, the quantum K pairing is related to correlators arising from supersymmetric localization. We state some mathematical conjectures, which we illustrate in several examples.

引用次数: 0

WDVV solutions associated with the genus one holomorphic differential

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-22 DOI: 10.1016/j.geomphys.2025.105432

Chaabane Rejeb

Consider the Hurwitz space

H_{1} (n_{0}, \dots, n_{m})

of genus one ramified coverings of fixed degree with

m + 1

prescribed poles of order

n_{0} + 1, \dots, n_{m} + 1

, respectively. In this paper, we derive an explicit solution to the WDVV equations associated with the Dubrovin-Frobenius manifold structure on

H_{1} (n_{0}, \dots, n_{m})

induced by the normalized holomorphic differential ϕ. The resulting solution is a function of

2 + 2 m + \sum_{j = 0}^{m} n_{j}

variables and is written in terms of Bell polynomials, Eisenstein series as well as Weierstrass functions. In addition, we construct Landau-Ginzburg superpotentials for these Frobenius manifold structures parameterized explicitly by flat coordinates of the Frobenius manifolds.

引用次数: 0

Goto's deformation theory of geometric structures, a Lie-theoretical description

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-21 DOI: 10.1016/j.geomphys.2025.105434

Grigory Papayanov

Ryushi Goto has constructed the deformation space for a manifold equipped with a collection of closed differential forms and showed that in some important cases (Calabi-Yau,

G_{2}

- and

S p i n (7)

-structures) this deformation space is smooth. This result unifies the classical Bogomolov-Tian-Todorov and Joyce theorems about unobstructedness of deformations. We show that this deformation space could be obtained as the deformation space associated to a certain dg Lie algebra. We also show that for Calabi-Yau,

G_{2}

- and

Spin (7)

-structures this dg Lie algebra is homotopy abelian. This gives a new proof of Goto's theorem.

引用次数: 0

Generalized Harish-Chandra morphism on Reflection Equation algebras

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-21 DOI: 10.1016/j.geomphys.2025.105435

Dimitry Gurevich , Pavel Saponov

We consider the so-called generalized Harish-Chandra morphism, taking the center of the enveloping algebra

U (g l (N))

to the commutative algebra generated by eigenvalues of the generating matrix of this algebra, and generalize this construction to Reflection Equation algebras. To this end we introduce the eigenvalues of the generating matrix of the Reflection Equation algebra (modified or not), corresponding to a skew-invertible Hecke symmetry and define the generalized Harish-Chandra morphism in a similar way. We use this map in order to introduce quantum analogs of the so-called weight systems.

引用次数: 0

Berry connections for 2d (2,2) theories, monopole spectral data & (generalised) cohomology theories

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-16 DOI: 10.1016/j.geomphys.2025.105425

Andrea E.V. Ferrari , Daniel Zhang

We study Berry connections for supersymmetric ground states of 2d

N = (2, 2)

GLSMs quantised on a circle, which are generalised periodic monopoles. Periodic monopole solutions may be encoded into difference modules, as shown by Mochizuki, or into an alternative algebraic construction given in terms of vector bundles endowed with filtrations. By studying the ground states in terms of a one–parameter family of supercharges, we relate these two different kinds of spectral data to the physics of the GLSMs. From the difference modules we derive novel difference equations for brane amplitudes, which in the conformal limit yield novel difference equations for hemisphere or vortex partition functions. When the GLSM flows to a nonlinear sigma model with Kähler target X, we show that the two kinds of spectral data are related to different (generalised) cohomology theories: the difference modules are related to the equivariant quantum cohomology of X, whereas the vector bundles with filtrations are related to its equivariant K–theory.

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引用次数: 0

Squared distance function on the configuration space of a planar spider with applications to Hooke energy and Voronoi distance

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-15 DOI: 10.1016/j.geomphys.2025.105424

Maciej Denkowski , Gaiane Panina , Dirk Siersma

Spider mechanisms are the simplest examples of arachnoid mechanisms, they are one step more complicated than polygonal linkages. Their configuration spaces have been studied intensively, but are yet not completely understood. In the paper we study them using the Morse theory of the squared distance function from the “body” of the spider to some fixed point in the plane. Generically, it is a Morse-Bott function. We list its critical manifolds, describe them as products of polygon spaces, and derive a formula for their Morse-Bott indices. We apply the obtained results to Hooke energy and Voronoi distance.

引用次数: 0

Polynomial graph invariants induced from the gl-weight system

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-10 DOI: 10.1016/j.geomphys.2025.105421

N. Kodaneva , S. Lando

引用次数: 0

Higher-dimensional integrable systems arising from the principal representation of toroidal Lie algebra so2ntor

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics

Pub Date : 2025-01-10 DOI: 10.1016/j.geomphys.2025.105420

Yi Yang

Based on the principal representation of toroidal Lie algebra

{so}_{2 n}^{tor}

, we construct a high–dimensional integrable hierarchy of Hirota bilinear equation. This hierarchy is in fact the

(2 + 1)

D extension of

D_{n}

type Drinfeld–Sokolov hierarchy. Furthermore, we also study the Darboux transformations of

D_{n}

type Drinfeld–Sokolov and its extension by virtue of two-component neutral free fermions.

引用次数: 0

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