One-parameter Deformation Research Articles

N≥ 𝟐 symmetric superpolynomials

The theory of symmetric functions has been extended to the case where each variable is paired with an anticommuting one. The resulting expressions, dubbed superpolynomials, provide the natural N=1 supersymmetric version of the classical bases of symmetric functions. Here we consider the case where two independent anticommuting variables are attached to each ordinary variable. The N=2 super-version of the monomial, elementary, homogeneous symmetric functions, as well as the power sums, are then constructed systematically (using an exterior-differential formalism for the multiplicative bases), these functions being now indexed by a novel type of superpartitions. Moreover, the scalar product of power sums turns out to have a natural N=2 generalization which preserves the duality between the monomial and homogeneous bases. All these results are then generalized to an arbitrary value of N. Finally, for N=2, the scalar product and the homogenous functions are shown to have a one-parameter deformation, a result that prepares the ground for the yet-to-be-defined N=2 Jack superpolynomials.

The para-HK/QK correspondence

We generalise the hyper-Kähler/quaternionic Kähler (HK/QK) correspondence to include para-geometries, and present a new concise proof that the target manifold of the HK/QK correspondence is quaternionic Kähler. As an application, we construct one-parameter deformations of the temporal and Euclidean supergravity c -map metrics and show that they are para-quaternionic Kähler.

Nonlocal symmetries, spectral parameter and minimal surfaces in AdS/CFT

We give a general account of nonlocal symmetries in symmetric space models and their relation to the AdS/CFT correspondence. In particular, we study a master symmetry which generates the spectral parameter and acts as a level-raising operator on the classical Yangian generators. The master symmetry extends to an infinite tower of symmetries with nonlocal Casimir elements as associated conserved charges. We discuss the algebraic properties of these symmetries and establish their role in explaining the recently observed one-parameter deformation of holographic Wilson loops. Finally, we provide a numerical framework, in which discretized minimal surfaces and their master symmetry deformation can be calculated.

$\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields

AbstractIt is shown that the description of certain class of representations of the holonomy Lie algebra $\mathfrak g_{\Delta}$ associated with hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated with $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.

Gerstenhaber–Schack cohomology for Hom-bialgebras and deformations

ABSTRACTHom-bialgebras and Hom-Hopf algebras are generalizations of bialgebra and Hopf algebra structures, where associativity and coassociativity conditions are twisted by a homomorphism. The purpose of this paper is to define a Gerstenhaber–Schack cohomology complex for Hom-bialgebras and then study one-parameter formal deformations.

Conformal geometry of marginally trapped surfaces in $$\mathbb {S}^4_1$$ S 1 4

A spacelike surface S immersed in \(\mathbb {S}^4_1\) is marginally trapped if its mean curvature vector is everywhere lightlike. On any oriented spacelike surface S immersed in \(\mathbb {S}^4_1\) we show that a choice of orientation of the normal bundle \(\nu (S)\) determines a smooth map \(G: S \rightarrow \mathbb {S}^3\) which we call the null Gauss map of S. If S is marginally trapped we show that G is a conformal immersion away from the zeros of certain quadratic Hopf-differential of S and so the surface G(S) is uniquely determined up to conformal transformations of \(\mathbb {S}^3\) by two invariants: the normal Hopf differential \(\kappa \) and the schwartzian derivative s. These invariants plus an additional quadratic differential \(\delta \) are related by a differential equation and determine the geometry of S up to ambient isometries of \(\mathbb {S}^4_1\). This allows us to obtain a characterization of marginally trapped surfaces S whose null Gauss image is a constrained Willmore surface in \(\mathbb {S}^3\) in the sense of Bohle et al. (Calc Var Partial Differ Equ 32:263–277, 2008). As an application of these results we construct and study integrable non-trivial one-parameter deformations of marginally trapped surfaces with non-zero parallel mean curvature vector and those with flat normal bundle.

One-parameter deformations of the diassociative and dendriform operads

Livernet and Loday constructed a polarization of the nonsymmetric associative operad with one operation into a symmetric operad with two operations (the Lie bracket and Jordan product), and defined a one-parameter deformation of , which includes Poisson algebras. We combine this with the dendriform splitting of an associative operation into the sum of two nonassociative operations, and use Koszul duality for quadratic operads, to construct one-parameter deformations of the nonsymmetric dendriform and diassociative operads into the category of symmetric operads.

Topological triviality of linear deformations with constant Lê numbers

Let f(t,z) = f0(z) + tg(z) be a holomorphic function defined in a neighbourhood of the origin in C × Cn. It is well known that if the one-parameter deformation family {ft} defined by the function f is a μ-constant family of isolated singularities, then {ft} is topologically trivial—a result of A. Parusinski. It is also known that Parusinski's result does not extend to families of non-isolated singularities in the sense that the constancy of the Le numbers of ft at 0, as t varies, does not imply the topological triviality of the family ft in general—a result of J. Fernandez de Bobadilla. In this paper, we show that Parusinski's result generalizes all the same to families of non-isolated singularities if the Le numbers of the function f itself are defined and constant along the strata of an analytic stratification of C × (f0−1(0) $\cap$ g−1(0)). Actually, it suffices to consider the strata that contain a critical point of f.

The Lerch Zeta function III. Polylogarithms and special values

This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $\Phi(s, z, c)$ is obtained from the Lerch zeta function $\zeta(s, a, c)$ by the change of variable $z=e^{2 \pi i a}$. We show that it analytically continues to a maximal domain of holomorphy in three complex variables $(s, z, c)$ as a multivalued function defined over the base manifold ${\bf C} \times P^1({\bf C} \smallsetminus \{0, 1, \infty\}) \times ({\bf C}\smallsetminus {\bf Z})$. and compute the monodromy functions defining the multivaluedness. For positive integer values s=m and c=1 this function is closely related to the classical m-th order polylogarithm $Li_m(z)$ We study its behavior as a function of two variables $(z, c)$ for special values where s=m is an integer. For $m \ge 1$ it gives a one-parameter deformation of the polylogarithm, and satisfies a linear ODE with coefficients depending on c, of order m+1 of Fuchsian type. We determine its (m+1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of $Li_m(z).$

Realizations of Galilei algebras

All inequivalent realizations of the Galilei algebras of dimensions not greater than five are constructed using the algebraic approach proposed by Shirokov. The varieties of the deformed Galilei algebras are discussed and families of one-parametric deformations are presented in explicit form. It is also shown that a number of well-known and physically interesting equations and systems are invariant with respect to the considered Galilei algebras or their deformations.

The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.

Which pseudodifferential operators with radial symbols are non-negative?

Operators the Weyl symbols of which are of the special kind \(f(x,\xi )=F(|x|^2+|\xi |^2)\) are in the spectral-theoretic sense functions \(\Psi \left(L^{(n)}\right)\) of the harmonic oscillator \(L^{(n)}\). We give the exact conditions on the real function F which ensure that the function \(\Psi \) (defined on the set \(\{\frac{n}{2}+k,\,k=0,1,\dots \}\)) will be non-negative, non-decreasing or convex. In the one-dimensional case, the first point can also be treated within a one-parameter relativistic deformation of the whole situation, involving in place of Hermite functions the no longer elementary (modified) Mathieu functions.

Geometric auxetics.

We formulate a mathematical theory of auxetic behaviour based on one-parameter deformations of periodic frameworks. Our approach is purely geome- tric, relies on the evolution of the periodicity lattice and works in any dimension. We demonstrate its usefulness by predicting or recognizing, without experiment, computer simulations or numerical approximations, the auxetic capabilities of several well-known structures available in the literature. We propose new principles of auxetic design and rely on the stronger notion of expansive behaviour to provide an infinite supply of planar auxetic mechanisms and several new three-dimensional structures.

Type IIB supergravity solution for the T-dual of the η-deformed AdS 5 × S 5 superstring

We find an exact type IIB supergravity solution that represents a one-parameter deformation of the T-dual of the AdS 5 × S 5 background (with T-duality applied in all 6 abelian bosonic isometric directions). The non-trivial fields are the metric, dilaton and RR 5-form only. The latter has remarkably simple “undeformed” form when written in terms of a “deformation-rotated” vielbein basis. An unusual feature of this solution is that the dilaton contains a linear dependence on the isometric coordinates of the metric precluding a straightforward reversal of T-duality. If we still formally dualize back, we find exactly the metric, B-field and product of dilaton with RR field strengths as recently extracted from the η-deformed AdS 5 × S 5 superstring action in arXiv:1507.04239 . We also discuss similar solutions for deformed AdS n × S n backgrounds with n = 2, 3. In the η → i limit we demonstrate that all these backgrounds can be interpreted as special limits of gauged WZW models and are also related to (a limit of) the Pohlmeyer-reduced models of the AdS n × S n superstrings.

Constructions and Cohomology of Hom–Lie Color Algebras

The main purpose of this paper is to define representations and a cohomology of Hom–Lie color algebras and to study some key constructions and properties. We describe Hartwig–Larsson–Silvestrov Theorem in the case of Γ-graded algebras, study one-parameter formal deformations, discuss α k -generalized derivations and provide examples.

The equivariant A-twist and gauged linear sigma models on the two-sphere

We study two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on the $\Omega$-deformed sphere, $S^2_\Omega$, which is a one-parameter deformation of the $A$-twisted sphere. We provide an exact formula for the $S^2_\Omega$ supersymmetric correlation functions using supersymmetric localization. The contribution of each instanton sector is given in terms of a Jeffrey-Kirwan residue on the Coulomb branch. In the limit of vanishing $\Omega$-deformation, the localization formula greatly simplifies the computation of $A$-twisted correlation functions, and leads to new results for non-abelian theories. We discuss a number of examples and comment on the $\epsilon_\Omega$-deformation of the quantum cohomology relations. Finally, we present a complementary Higgs branch localization scheme in the special case of abelian gauge groups.

One-parameter supersymmetric Hamiltonians in momentum space

Recent results on the one-parameter supersymmetric deformation in momentum space by Curtright and Zachos (2014 J. Phys. A: Math. Theor. 47 145201) are presented in a more general framework following our own papers. We extend the analysis of Curtright and Zachos by including the supersymmetric partner one-parameter deformation.

One-parameter formal deformations of Hom-Lie-Yamaguti algebras

This paper studies one-parameter formal deformations of Hom-Lie-Yamaguti algebras. The first, second, and third cohomology groups on Hom-Lie-Yamaguti algebras extending ones on Lie-Yamaguti algebras are provided. It is proved that first and second cohomology groups are suitable to the deformation theory involving infinitesimals, equivalent deformations, and rigidity. However, the third cohomology group is not suitable for the obstructions.

Anisotropic scale invariant spacetimes and black holes in Zwei-Dreibein Gravity

We show that Zwei-Dreibein Gravity (ZDG), a bigravity theory recently proposed by Bergshoeff, de Haan, Hohm, Merbis, and Townsend in Phys.Rev.Lett. 111 (2013) 111102, admits exact solutions with anisotropic scale invariance. These type of geometries are the three-dimensional analogues of the spacetimes which were proposed as gravity duals for condensed matter systems. In particular, we find Schr\"odinger invariant spaces as well as Lifshitz spaces with arbitrary dynamical exponent $z$. We also find black holes that are asymptotically Lifshitz with $z=3$, showing that these (non-constant curvature) solutions of New Massive Gravity (NMG) are persistent after the introduction of the infinite tower of higher-curvature terms of ZDG, provided a renormalization of the parameters. Black holes in asymptotically warped Anti-de Sitter spaces are also found. Interestingly, in almost all the geometries studied in this work, the metric associated with the second dreibein turns out to be equivalent, up to a constant global factor, to the first one. This phenomenon has been previously observed in other bigravity theories in asymptotically flat and asymptotically Anti-de Sitter backgrounds. However, for the particular case of the $z=3$ Lifshitz black hole, here we found that the second metric corresponds to a different black hole that coincides with the former only in the asymptotic region. In fact, we find a new family of $z=3$ black holes that corresponds to a one-parameter deformation of the NMG solution.

A one-parameter refinement of the Razumov–Stroganov correspondence

We introduce and prove a one-parameter refinement of the Razumov–Stroganov correspondence. This is achieved for fully-packed loop configurations (FPL) on domains which generalise the square domain, and which are endowed with the gyration operation. We consider one given side of the domain, and FPLs such that the only straight-line tile on this side is black. We show that the enumeration vector associated with such FPLs, weighted according to the position of the straight line and refined according to the link pattern for the black boundary points, is the ground state of the scattering matrix, an integrable one-parameter deformation of the O(1) Dense Loop Model Hamiltonian. We show how the original Razumov–Stroganov correspondence, and a conjecture formulated by Di Francesco in 2004, follows from our results.