Nonlinear Sigma Model Research Articles

New construction of a vacuum doubly rotating black ring by the Ehlers transformation

Using the Ehlers transformation, we derive an exact solution for a doubly rotating black ring in five-dimensional vacuum Einstein theory. It is well known that the vacuum Einstein theory with three commuting Killing vector fields can be reduced to a nonlinear sigma model with SL(3,R) target space symmetry. As shown previously by Giusto and Saxena, the SO(2,1) subgroup in the SL(3,R) can generate a rotating solution from a static solution while preserving asymptotic flatness. This so-called Ehlers transformation actually transforms the five-dimensional Schwarzschild black hole into the five-dimensional Myers-Perry black hole. However, unlike the case with the black hole, applying this method directly to the static black ring or the Emparan-Reall black ring, does not yield a regular rotating black ring due to the emergence of a Dirac-Misner string singularity. To solve this undesirable issue, we use a singular vacuum solution of a rotating black ring/lens that already possesses a Dirac-Misner string singularity as the seed solution for the Ehlers transformation. The resulting solution is regular, indicating the absence of curvature singularities, conical singularities, orbifold singularities, Dirac-Misner string singularities, and closed timelike curves both on and outside the horizon. We show that this solution obtained by the Ehlers transformation coincides precisely with the Pomeransky-Sen’kov solution. We expect that applying this method to other theories may lead to the finding of new exact solutions, such as solutions for black lenses and capped black holes, as well as black ring configurations. Published by the American Physical Society 2024

Isospinning CP2 solitons

We study stationary rotating topological solitons in a (2+1)-dimensional CP2 nonlinear sigma model with a stabilizing potential term. We find families of U(1)×U(1) symmetric solutions with topological degrees equal or larger than 1, which have two angular frequencies and are labeled by two (one topological and the other nontopological) winding numbers k1≥k2. We discuss properties of these solitons and investigate the domains of their existence. Published by the American Physical Society 2024

Covariant quantisation of tensor multiplet models

The Batalin-Vilkovisky formalism is applied to quantise the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetric generalisation of the Freedman-Townsend (FT) model, which was proposed by Lindström and Roček in 1983 in Minkowski superspace and is lifted to a supergravity background in this paper. This super FT theory describes a non-Abelian tensor multiplet and is known to be classically equivalent to a supersymmetric nonlinear sigma model. Using path integral considerations, we demonstrate that this equivalence holds at the quantum level in the sense that the quantum supercurrents in the two theories coincide. A modified Faddeev-Popov procedure is employed to quantise models for the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 tensor multiplet in harmonic superspace. The obtained results agree with those derived by applying the Batalin-Vilkovisky scheme within the harmonic superspace setting.

Nonlinear Sigma model amplitudes to all loop orders are contained in the Tr(Φ3) theory

Scattering amplitudes for the simplest theory of colored scalar particles—the Tr(Φ3) theory—have recently been the subject of active investigations. In this work we describe an unanticipated wider implication of this work: the Tr(Φ3) theory secretly contains nonlinear sigma model (NLSM) amplitudes to all loop orders. The NLSM amplitudes are obtained from Tr(Φ3) amplitudes by a unique shift of kinematic variables. We show that this shifted kinematics produces amplitudes for a cubic theory with a linear term in the potential, with extrema spontaneously breaking U(N)→U(N−k)×U(k). The Goldstone amplitudes for this theory coincide with those of pions in the U(N)×U(N)→U(N) chiral Lagrangian to all orders in the planar limit. We also give a purely on-shell understanding of this correspondence, showing that integrands defined by the kinematic shifts have the correct residues on poles and appropriately produce the Adler zero. Finally, we discuss how similar kinematic shifts produce certain infinite classes of mixed amplitudes of pions and Tr(Φ3) scalars, most of which are not interpretable from the Lagrangian description. Published by the American Physical Society 2024

On universal splittings of tree-level particle and string scattering amplitudes

In this paper, we study the newly discovered universal splitting behavior for tree-level scattering amplitudes of particles and strings [1]: when a set of Mandelstam variables (and Lorentz products involving polarizations for gluons/gravitons) vanish, the n-point amplitude factorizes as the product of two lower-point currents with n+3 external legs in total. We refer to any such subspace of the kinematic space of n massless momenta as “2-split kinematics”, where the scattering potential for string amplitudes and the corresponding scattering equations for particle amplitudes nicely split into two parts. Based on these, we provide a systematic and detailed study of the splitting behavior for essentially all ingredients which appear as integrands for open- and closed-string amplitudes as well as Cachazo-He-Yuan (CHY) formulas, including Parke-Taylor factors, correlators in superstring and bosonic string theories, and CHY integrands for a variety of amplitudes of scalars, gluons and gravitons. These results then immediately lead to the splitting behavior of string and particle amplitudes in a wide range of theories, including bi-adjoint ϕ3 (with string extension known as Z and J integrals), non-linear sigma model, Dirac-Born-Infeld, the special Galileon, etc., as well as Yang-Mills and Einstein gravity (with bosonic and superstring extensions). Our results imply and extend some other factorization behavior of tree amplitudes considered recently, including smooth splittings [2] and factorizations near zeros [3], to all these theories. A special case of splitting also yields soft theorems for gluons/gravitons as well as analogous soft behavior for Goldstone particles near their Adler zeros.

Integrability and renormalizability for the fully anisotropic SU(2) principal chiral field and its deformations

For the class of 1 + 1 dimensional field theories referred to as the non-linear sigma models, there is known to be a deep connection between classical integrability and one-loop renormalizability. In this work, the phenomenon is reviewed on the example of the so-called fully anisotropic SU(2) Principal Chiral Field (PCF). Along the way, we discover a new classically integrable four parameter family of sigma models, which is obtained from the fully anisotropic SU(2) PCF by means of the Poisson-Lie deformation. The theory turns out to be one-loop renormalizable and the system of ODEs describing the flow of the four couplings is derived. Also provided are explicit analytical expressions for the full set of functionally independent first integrals (renormalization group invariants).

All-loop soft theorem for pions

In this paper we discuss a generalization of the Adler zero to loop integrands in the planar limit of the SU(N) nonlinear sigma model (NLSM). The Adler zero for integrands is violated starting at the two-loop order and is only recovered after integration. Here we propose a soft theorem satisfied by loop integrands with any number of loops and legs. This requires a generalization of NLSM integrands to an off shell framework with certain deformed kinematics. Defining an , we identify a simple nonvanishing soft behavior of integrands, which we call the . We find that the proposed soft theorem is satisfied by the “surface” integrand of Arkani-Hamed and Figueiredo [], which is obtained from the shifted Trφ3 surfacehedron integrand. Finally, we derive an on shell version of the algebraic soft theorem that takes an interesting form in terms of self-energy factors and lower-loop integrands in a mixed theory of pions and scalars. Published by the American Physical Society 2024

The Balaban Variational Problem in the Non-Linear Sigma Model

The Balaban Variational Problem in the Non-Linear Sigma Model

Pions from higher-dimensional gluons: general realizations and stringy models

In this paper we revisit the general phenomenon that scattering amplitudes of pions can be obtained from “dimensional reduction” of gluons in higher dimensions in a more general context. We show that such “dimensional reduction” operations universally turn gluons into pions regardless of details of interactions: under such operations any amplitude that is gauge invariant and contains only local simple poles becomes one that satisfies Adler zero in the soft limit. As two such examples, we show that starting from gluon amplitudes in both superstring and bosonic string theories, the operations produce “stringy” completion of pion scattering amplitudes to all orders in α′, with leading order given by non-linear sigma model amplitudes. Via Kawai-Lewellen-Tye relations, they give closed-stringy completion for Born-Infeld theory and the special Galileon theory, which are directly related to gravity amplitudes in closed-string theories. We also discuss how they naturally produce stringy models for mixed amplitudes of pions and colored scalars.

Topological defects in K3 sigma models

We consider the topological defect lines commuting with the spectral flow and the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = (4, 4) superconformal symmetry in two dimensional non-linear sigma models on K3. By studying their fusion with boundary states, we derive a number of general results for the category of such defects. We argue that while for certain K3 models infinitely many simple defects, and even a continuum, can occur, at generic points in the moduli space the category is actually trivial, i.e. it is generated by the identity defect. Furthermore, we show that if a K3 model is at the attractor point for some BPS configuration of D-branes, then all topological defects have integral quantum dimension. We also conjecture that a continuum of topological defects arises if and only if the K3 model is a (possibly generalized) orbifold of a torus model. Finally, we test our general results in a couple of examples, where we provide a partial classification of the topological defects.

NS5-brane backgrounds and coset CFT partition functions

Worldsheet string theory is solvable for a variety of backgrounds involving Neveu-Schwarz fivebranes, in terms of gauged nonlinear sigma models on group manifolds. We compute the worldsheet torus partition function of these models, and propose gauging of null isometries as a unifying principle and conceptual framework for this large family of string backgrounds. In the process, we explain how partition functions of asymmetrically gauged Wess-Zumino-Witten models can be computed from the path integral, and organize and systematize various results scattered throughout the literature.

Joule–Thomson expansion and images of black hole in SU(N)-non-linear sigma model

We consider recently constructed black holes in the Einstein SU(N)-non-linear sigma model and study the Joule–Thomson expansion and observable characteristics. We calculate Joule–Thomson coefficient, inversion curves and isenthalpic curves to investigate the heating and cooling phases in T-P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T-P$$\\end{document} plane of the considered model. Our findings reveal distinct behavior of these black holes different from those reported in existing studies. Notably, inversion curves vanish in the heating regions, resulting in the black holes consistently remaining in the cooling phase. Additionally, we investigate the impact of the flavor number N on the Joule–Thomson expansion of these black holes. We observe that employing higher values of enthalpy and flavor number N extends the ranges of isenthalpic and inversion curves but the black holes remain in cooling phase. We also study the shadow and visual characteristics of the these black hole focusing on its illumination by two theoretical models of static accretion. We analyze that higher flavor numbers have high peaks of observed intensities and it shifts to smaller impact parameter which leads to smaller images.

SO(5) Deconfined Phase Transition under the Fuzzy-Sphere Microscope: Approximate Conformal Symmetry, Pseudo-Criticality, and Operator Spectrum

The deconfined quantum critical point (DQCP) is an example of phase transitions beyond the Landau symmetry-breaking paradigm that attracts wide interest. However, its nature has not been settled after decades of study. In this paper, we apply the recently proposed fuzzy-sphere regularization to study the SO(5) nonlinear sigma model with a topological Wess-Zumino-Witten term, which serves as a dual description of the DQCP with an exact SO(5) symmetry. We demonstrate that the fuzzy sphere functions as a powerful microscope, magnifying and revealing a wealth of crucial information about the DQCP, ultimately paving the way toward its final answer. In particular, through exact diagonalization, we provide clear evidence that the DQCP exhibits approximate conformal symmetry. The evidence includes the existence of a conserved SO(5) symmetry current, a stress tensor, and integer-spaced levels between conformal primaries and their descendants. Most remarkably, we identify 23 primaries and 76 conformal descendants. Furthermore, by examining the renormalization group flow of the lowest symmetry singlet as well as other primaries, we provide numerical evidence in favor of DQCP being pseudo-critical, with the approximate conformal symmetry plausibly emerging from nearby complex fixed points. The primary spectrum we compute also has important implications, including the conclusion that the SO(5) DQCP cannot describe a direct transition from the Néel to valence bond solid phase on the honeycomb lattice. Published by the American Physical Society 2024

Spectral flow of fermions in the ℂP2 (anti-)instanton, and the sphaleron with vanishing topological charge

The spectral flow is ubiquitous in the physics of soliton-fermion interacting systems. We study the spectral flows related to a continuous deformation of background soliton solutions, which enable us to develop insight into the emergence of fermionic zero modes and the localization mechanism of fermion densities. We investigate a ℂP2 nonlinear sigma model in which there are the (anti-) instantons and also the sphalerons with vanishing topological charge. The standard Yukawa coupling of the fermion successfully generates infinite towers of the spectra and the spectral flow is observed when increasing the size of such solitons. At that moment, the localization of the fermions on the solitons emerges. The avoided crossings are also observed in several stages of the exchange of the flows, they are indicating a manifestation of the fermion exchange of the localizing nature.

On the direct quantization of Maxwell field

In this paper, we apply the generalized integration constants (GCI) method (Belhadi 2023 https://arxiv.org/abs/2303.08236), in field theory to quantize Maxwell and the Klein–Gordon free fields. The study is performed in both position and momentum spaces, to obtain the equal-time Dirac brackets among the fields and their conjugate momenta. The idea is to compute the brackets near the initial instant using the Taylor polynomial expansion, and then deduce directly their expressions at any later time. In the case of the Maxwell field, the interdependence of the field components (constraints) requires the use of the Helmholtz theorem to separate the transversal and longitudinal parts. Our work finishes with the study of the O(3) nonlinear sigma model using the GCI approach.

One-loop Bern-Carrasco-Johansson numerators on quadratic propagators from the worldsheet

We introduce a novel approach for deriving one-loop Bern-Carrasco-Johansson (BCJ) numerators and reveal the world-sheet origin of the one-loop double copy. Our work shows that expanding Cachazo-He-Yuan half-integrands into generalized Parke-Taylor factors intrinsically generates BCJ numerators on quadratic propagators satisfying Jacobi identities. We validate our methodology by successfully reproducing one-loop BCJ numerators for the nonlinear sigma model as well as those of pure Yang-Mills theory in four dimensions with all-plus or single-minus helicities. Published by the American Physical Society 2024

Hybrid Goldstone Modes from the Double Copy Bootstrap

We perform a systematic classification of scalar field theories whose amplitudes admit a double copy formulation and identify two building blocks at four-point and 13 at five-point. Using the four-point blocks simultaneously as bootstrap seeds, this naturally leads to a single copy theory that is a gauged nonlinear sigma model. Moreover, its double copy includes a novel theory that can be written in terms of Lovelock invariants of an induced metric, and includes Dirac-Born-Infeld and the special Galileon in specific limits. The amplitudes of these Goldstone modes have two distinct soft behavior regimes, corresponding to a hybrid of nonlinear symmetries. Published by the American Physical Society 2024

Point particle E-models

We show that the same algebraic data that permit to construct the Lax pair and the r-matrix of an integrable non-linear σ-model in 1 + 1 dimensions can be also used for the construction of Lax pairs and of r-matrices of several other non-trivial integrable theories in 1 + 0 dimension. We call those new integrable theories the point particle E-models, we describe their structure and give their physical interpretation. We work out in detail the point particle E-modelsassociated to the bi-Yang–Baxter deformation of the SU(N) principal chiral model. In particular, for each complex flag manifold we thus obtain a two-parameter family of integrable models living on it.

Using strain to uncover the interplay between two- and three-dimensional charge density waves in high-temperature superconducting YBa2Cu3Oy

Uniaxial pressure provides an efficient approach to control charge density waves in YBa2Cu3Oy. It can enhance the correlation volume of ubiquitous short-range two-dimensional charge-density-wave correlations, and induces a long-range three-dimensional charge density wave, otherwise only accessible at large magnetic fields. Here, we use x-ray diffraction to study the strain dependence of these charge density waves and uncover direct evidence for a form of competition between them. We show that this interplay is qualitatively described by including strain effects in a nonlinear sigma model of competing superconducting and charge-density-wave orders. Our analysis suggests that strain stabilizes the 3D charge density wave in the regions between disorder-pinned domains of 2D charge density waves, and that the two orders compete at the boundaries of these domains. No signatures of discommensurations nor of pair density waves are observed. From a broader perspective, our results underscore the potential of strain tuning as a powerful tool for probing competing orders in quantum materials.

Geometric construction of non-linear Sigma models with [formula omitted]-ball/[formula omitted]-kink solutions

Non-linear Sigma models involving U(1) symmetry group are studied using a geometrical formalism. In this type of models, Q-balls and Q-Kinks solutions are found. The geometrical framework described in this article allows the identification of the necessary conditions on the metric and the potential to guarantee the existence of these Q-balls and Q-Kinks. Using this procedure, Sigma models where both types of solutions coexist, have been identified. Only the internal rotational frequency distinguishes which one of these defects will arise.