One-parameter Deformation Research Articles

Phase diagram of a generalized Stephanov model for finite-density QCD

We solve a random matrix model for QCD at finite chemical potential, obtained by generalizing the Stephanov model by modifying the random-matrix integration measure with a one-parameter trace deformation. This allows one to check how important the integration measure is for the qualitative features of random matrix models, as well as to test the robustness and universality of the qualitative picture of the original model. While for a small trace deformation the phase diagram is identical to that of the Stephanov model, for a large deformation an exotic phase with spontaneous charge-conjugation breaking appears. Published by the American Physical Society 2024

Higher-rank sectors in the hexagon formalism and marginal deformations

Abstract The hexagon approach provides an integrability framework for the computation of structure constants in N = 4 super Yang-Mills theory in four dimensions. Three-point functions are cut into two hexagonal patches, on which the excitations of the long-range Bethe ansatz of the spectrum problem scatter. To this end, the Bethe states representing the operators also need to be cut into two parts. In rank-one sectors such entangled states are fairly straightforward to construct so that most applications of the method have so far been restricted to this simplest case. In this article we construct entangled states for operators in psu(1, 1|2) sectors, importing a minimum of information from the nested Bethe ansatz. The idea is successfully tested against free field theory for a sample set of correlators with up to three higher-rank operators.
Further, we take a look at the same correlators in the presence of marginal deformations of the theory. While a systematic modification of the hexagon procedure remains out of reach for now, in practical applications the undeformed amplitudes are surprisingly efficient, especially for a certain one-parameter deformation.

Bridging trails in reflectionless potential deformation: Two paths and one horizon

In contrast to the conventional one-parameter class of isospectral deformation using translation, we calculate the two- and three-parameter classes of isospectral deformation of the well-known reflectionless potential by utilizing a far different approach of scaling methodology. Subsequently, using these results, we find that this more general class of deformations is not unique but instead subsume in the same class of conventional one-parameter translational deformation. We also provide a theoretical foundation for how these two incredibly different approaches converge at the same destination. Finally, we show that the most generic class of potentials, obtained by scaling deformation, are solutions of the nonlinear KdV equation.

Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge

Sensitivity of an eigenvalue λ i to the perturbation of matrix elements is controlled by the eigenvalue condition number defined as κi=⟨Li|Li⟩⟨Ri|Ri⟩ , where ⟨Li| and |Ri⟩ are left and right eigenvectors to the eigenvalue λ i . In random matrix theory the squared eigenvalue condition number is also known as the eigenvector self-overlap. In this work we calculate the asymptotics of the joint probability density function of the real eigenvalue and the square of the corresponding eigenvalue condition number for the real elliptic ensemble in the double scaling regime of almost Hermiticity and close to the edge of the spectrum. As a byproduct, we also calculate the one-parameter deformation of the Scorer’s function.

Cohomology of modified Rota–Baxter Leibniz algebra of weight λ

Rota–Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota–Baxter operators on Leibniz algebras. We investigate modified Rota–Baxter Leibniz algebras from the cohomological point of view. We study a one-parameter formal deformation theory of modified Rota–Baxter Leibniz algebras and define the associated deformation cohomology that controls the deformation. Finally, as an application, we characterize equivalence classes of abelian extensions in terms of second cohomology groups.

On compatible Leibniz algebras

In this paper, we study compatible Leibniz algebras. We explore the classification of complex Leibniz algebras in dimensions 2 and 3 to provide compatible pairs. We characterize compatible Leibniz algebras in terms of Maurer–Cartan elements of a suitable differential graded Lie algebra. Moreover, we define a cohomology theory of compatible Leibniz algebras, which in particular controls one-parameter formal deformations of this algebraic structure. Motivated by a classical application of cohomology, we moreover study abelian extensions of compatible Leibniz algebras.

Nijenhuis operators on Leibniz algebras

In this paper, we study Nijenhuis operators on Leibniz algebras. We discuss the relationship of Nijenhuis operators with Rota-Baxter operators and modified Rota-Baxter operators on Leibniz algebras. We define a representation theory of Nijenhuis Leibniz algebras and construct a cohomology theory. Next, we define a one-parameter formal deformation theory of Nijenhuis Leibniz algebras and study infinitesimals, rigidity, and equivalences along the line of Gerstenhaber deformation theory. As applications of our cohomology theory, we show that our cohomology is deformation cohomology and study abelian extensions of such algebras.

The generalized Born oscillator and the Berry-Keating Hamiltonian

In this study, we introduce and investigate a family of quantum mechanical models in 0 + 1 dimensions, known as generalized Born quantum oscillators. These models represent a one-parameter deformation of a specific system obtained by reducing the Nambu-Goto theory to 0 + 1 dimensions. Despite these systems showing significant similarities with TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extrm{T}\\overline{\ extrm{T}} $$\\end{document}-type perturbations of two-dimensional relativistic models, our analysis reveals their potential as interesting regularizations of the Berry-Keating theory. We quantize these models using the Weyl quantization scheme up to very high orders in ħ. By examining a specific scaling limit, we observe an intriguing connection between the generalized Born quantum oscillators and the Riemann-Siegel θ function.

Slow-roll inflation and growth of perturbations in Kaniadakis modification of Friedmann cosmology

Kaniadakis entropy is a one-parameter deformation of the classical Boltzmann–Gibbs–Shannon entropy, arising from a self-consistent relativistic statistical theory. Assuming a Kaniadakis-type generalization of the entropy associated with the apparent horizon of Friedmann–Robertson–Walker (FRW) Universe and using the gravity-thermodynamics conjecture, a new cosmological scenario is obtained based on the modified Friedmann equations. By employing such modified equations, we analyze the slow-roll inflation, driven by a scalar field with power-law potential, at the early stages of the Universe. We explore the phenomenological consistency of this model by computation of the scalar spectral index and tensor-to-scalar ratio. Comparison with the latest BICEP and Planck data allows us to constrain Kaniadakis parameter to κ≲O(10-12÷10-11)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa \\lesssim \\mathcal {O}(10^{-12}\\div 10^{-11})$$\\end{document}, which is discussed in relation to other observational bounds in the literature. We also disclose the effects of Kaniadakis correction term on the growth of perturbations at the early stages of the Universe by employing the spherically symmetric collapse formalism in the linear regime of density perturbations. We find out that the profile of density contrast is non-trivially affected in this scenario. Interestingly enough, we observe that increasing Kaniadakis parameter κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document} corresponds to a faster growth of perturbations in a Universe governed by the corrected Friedmann equations. Finally, we comment on the consistency of the primordial power spectrum for scalar perturbations with the best data-fit provided by Planck.

Cohomology and formal deformations of n -Hom–Lie color algebras

UDC 512.5 The aim of this paper is to provide a cohomology of n -Hom–Lie color algebras, in particular, a cohomology governing one-parameter formal deformations. Then we also study formal deformations of the n -Hom–Lie color algebras and introduce the notion of Nijenhuis operator on a n -Hom–Lie color algebra, which may give rise to infinitesimally trivial ( n - 1 ) -order deformations. Furthermore, in connection with Nijenhuis operators, we introduce and discuss the notion of product structure on n -Hom–Lie color algebras.

Diassociative family algebras and averaging family operators

Algebraic structures relative to a semigroup Ω (also called family algebraic structures) first appeared in the work of renormalization in quantum field theory. In this paper, we first consider Ω-diassociative algebras and diassociative family algebras as family analogues of diassociative algebras. We define the cohomologies of these algebras generalizing the cohomology of diassociative algebras introduced by Frabetti. We also introduce (relative) averaging family operators as the family analogue of averaging operators and show that they induce diassociative family algebras and Ω-diassociative algebras. Next, we define the cohomology of a (relative) averaging family operator as the cohomology of the induced Ω-diassociative algebra with coefficients in a suitable representation. Finally, as an application of our cohomology, we study formal one-parameter deformations of relative averaging family operators.

On unitarity of the Coon amplitude

The Coon amplitude is a one-parameter deformation of the Veneziano amplitude. We explore the unitarity of the Coon amplitude through its partial wave expansion using tools from q-calculus. Our analysis establishes manifest positivity on the leading and sub-leading Regge trajectories in arbitrary spacetime dimensions D, while revealing a violation of unitarity in a certain region of (q, D) parameter space starting at the sub-sub-leading Regge order. A combination of numerical studies and analytic arguments allows us to argue for the manifest positivity of the partial wave coefficients in fixed spin and Regge asymptotics.

Precision test of gauge/gravity duality in D0-brane matrix model at low temperature

We test the gauge/gravity duality between the matrix model and type IIA string theory at low temperatures with unprecedented accuracy. To this end, we perform lattice Monte Carlo simulations of the Berenstein-Maldacena-Nastase (BMN) matrix model, which is the one-parameter deformation of the Banks-Fischler-Shenker-Susskind (BFSS) matrix model, taking both the large N and continuum limits. We leverage the fact that sufficiently small flux parameters in the BMN matrix model have a negligible impact on the energy of the system while stabilizing the flat directions so that simulations at smaller N than in the BFSS matrix model are possible. Hence, we can perform a precision measurement of the large N continuum energy at the lowest temperatures to date. The energy is in perfect agreement with supergravity predictions including estimations of α′-corrections from previous simulations. At the lowest temperature where we can simulate efficiently (T = 0.25λ1/3, where λ is the ’t Hooft coupling), the difference in energy to the pure supergravity prediction is less than 10%. Furthermore, we can extract the coefficient of the 1/N4 corrections at a fixed temperature with good accuracy, which was previously unknown.

Hopf Structure for the q-Lévy-Meixner Oscillator Algebra

The main purpose of this paper is to investigate a generalized oscillator algebra, naturally associated with the [Formula: see text]-Lévy-Meixner polynomials. We solve the problem of the Hopf algebraic structure for the [Formula: see text]-deformed Lévy-Meixner oscillator algebra based on the one-parameter deformation of canonical commutation relations.

From the emergence of cosmic space to horizon thermodynamics in Barrow entropy-based Cosmology

Padmanabhan's paradigm states that the spatial expansion of our Universe can be understood as the consequence of the emergence of space with the progress of cosmic time. Based on this argument, here we extract the first Friedmann equation for a curved (n+1)-dimensional Friedmann-Robertson-Walker Universe and analyze the consistency of Padmanabhan's proposal with horizon entropy maximization in the framework of Barrow entropy. The latter is a one-parameter deformation of Bekenstein-Hawking entropy induced by quantum-gravitational effects. We show that the viability of Padmanabhan's paradigm and its relationship with horizon thermodynamics are well supported in Barrow model, providing preliminary indications on how the emergent gravity perspective should appear in a quantum gravity-like scenario.

The limit cycles in a generalized Rayleigh-Liénard oscillator

We compute the cyclicity of open period annuli of the following generalized Rayleigh-Liénard equation \begin{document}$ \ddot{x}+ax+bx^3-(\lambda_1+\lambda_2 x^2+\lambda_3\dot{x}^2+\lambda_4 x^4+\lambda_5\dot{x}^4+\lambda_6 x^6)\dot{x} = 0 $\end{document} and the equivalent planar system $ X_\lambda $, where the coefficients of the perturbation $ \lambda_j $ are independent small parameters and $ a, b $ are fixed nonzero constants. Our main tool is the machinery of the so called higher-order Poincaré-Pontryagin-Melnikov functions (Melnikov functions $ M_n $ for short), combined with the explicit computation of center conditions and the corresponding Bautin ideal.We consider first arbitrary analytic arcs $ \varepsilon \to \lambda(\varepsilon) $ and explicitly compute all possible Melnikov functions $ M_n $ related to the deformation $ X_{ \lambda(\varepsilon)} $. At a second step we obtain exact bounds for the number of the zeros of the Melnikov functions (complete elliptic integrals depending on parameter) in an appropriate complex domain, using a modification of Petrov's method.To deal with the general case of six-parameter deformations $ \lambda \to X_\lambda $, we compute first the related Bautin ideal. To do this we carefully study the Melnikov functions up to order three, and then use Nakayama lemma from Algebraic geometry. The principalization of the Bautin ideal (achieved after a blow up) reduces finally the study of general deformations $ X_{ \lambda } $ to the study of one-parameter deformations $ X_{ \lambda(\varepsilon)} $.

Root-TT[over ¯] Deformations in Two-Dimensional Quantum Field Theories.

In this Letter, we introduce a one-parameter deformation of two-dimensional quantum field theories generated by a nonanalytic operator that we call Root-TT[over ¯]. For a conformal field theory, the operator coincides with the square root of the TT[over ¯] operator. More generally, the operator is defined so that classically it is marginal and generates a flow that commutes with the TT[over ¯] flow. Intriguingly, the Root-TT[over ¯] flow is closely related to the ModMax theory recently constructed by Bandos, Lechner, Sorokin, and Townsend.

Non-orientable branched coverings, b-Hurwitz numbers, and positivity for multiparametric Jack expansions

We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter b with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate b-weighting that “measures” in some sense their non-orientability.Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so far towards the Matching-Jack conjecture and the “b-conjecture” of Goulden and Jackson from 1996, expansions of the β-ensemble matrix model, deformations of the HCIZ integral, and b-Hurwitz numbers that we introduce here and that are b-deformations of classical (single or double) Hurwitz numbers obtained for b=0.A key role in our proof is played by a combinatorial model of non-orientable constellations equipped with a suitable b-weighting, whose partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax equations, the other one following an explicit combinatorial decomposition.

On equivariant Lie–Yamaguti algebras and related structures

In this paper, we first discuss cohomology and a one-parameter formal deformation theory of Lie–Yamaguti algebras. Next, we study finite group actions on Lie–Yamaguti algebras and introduce equivariant cohomology for Lie–Yamaguti algebras equipped with group actions. Finally, we study an equivariant one-parameter formal deformation theory and show that our equivariant cohomology is the suitable deformation cohomology.

Confinement/deconfinement transition in the D0-brane matrix model — A signature of M-theory?

We study the confinement/deconfinement transition in the D0-brane matrix model (often called the BFSS matrix model) and its one-parameter deformation (the BMN matrix model) numerically by lattice Monte Carlo simulations. Our results confirm general expectations from the dual string/M-theory picture for strong coupling. In particular, we observe the confined phase in the BFSS matrix model, which is a nontrivial consequence of the M-theory picture. We suggest that these models provide us with an ideal framework to study the Schwarzschild black hole, M-theory, and furthermore, the parameter region of the phase transition between type IIA superstring theory and M-theory. A detailed study of M-theory via lattice Monte Carlo simulations of the D0-brane matrix model might be doable with much smaller computational resources than previously expected.