Dimensional Nonlinear Sigma Model Research Articles

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.

Scale symmetry breaking, quantum anomalous energy and proton mass decomposition

We study the anomalous scale symmetry breaking effects on the proton mass in QCD due to quantum fluctuations at ultraviolet scales. We confirm that a novel contribution naturally arises as a part of the proton mass, which we call the quantum anomalous energy (QAE). We discuss the QAE origins in both lattice and dimensional regularizations and demonstrate its role as a scheme-and-scale independent component in the mass decomposition. We further argue that QAE role in the proton mass resembles a dynamical Higgs mechanism, in which the anomalous scale symmetry breaking field generates mass scales through its vacuum condensate, as well as its static and dynamical responses to the valence quarks. We demonstrate some of our points in two simpler but closely related quantum field theories, namely the 1+1 dimensional non-linear sigma model in which QAE is non-perturbative and scheme-independent, and QED where the anomalous energy effect is perturbative calculable.

Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders

We propose a new type of quantum liquids, dubbed Stiefel liquids, based on $2+1$ dimensional nonlinear sigma models on target space $SO(N)/SO(4)$, supplemented with Wess-Zumino-Witten terms. We argue that the Stiefel liquids form a class of critical quantum liquids with extraordinary properties, such as large emergent symmetries, a cascade structure, and nontrivial quantum anomalies. We show that the well known deconfined quantum critical point and $U(1)$ Dirac spin liquid are unified as two special examples of Stiefel liquids, with $N=5$ and $N=6$, respectively. Furthermore, we conjecture that Stiefel liquids with $N>6$ are non-Lagrangian, in the sense that under renormalization group they flow to infrared (conformally invariant) fixed points that cannot be described by any renormalizable continuum Lagrangian. Such non-Lagrangian states are beyond the paradigm of parton gauge mean-field theory familiar in the study of exotic quantum liquids in condensed matter physics. The intrinsic absence of (conventional or parton-like) mean-field construction also means that, within the traditional approaches, it will be difficult to decide whether a non-Lagrangian state can actually emerge from a specific UV system (such as a lattice spin system). For this purpose we hypothesize that a quantum state is emergible from a lattice system if its quantum anomalies match with the constraints from the (generalized) Lieb-Schultz-Mattis theorems. Based on this hypothesis, we find that some of the non-Lagrangian Stiefel liquids can indeed be realized in frustrated quantum spin systems, for example, on triangular or Kagome lattice, through the intertwinement between non-coplanar magnetic orders and valence-bond-solid orders.

Quantum anomalous energy effects on the nucleon mass

Apart from the quark and gluon kinetic and potential energies, the nucleon mass includes a novel energy of pure quantum origin resulting from anomalous breaking of scale symmetry. We demonstrate the effects of this quantum anomalous energy (QAE) in QED, as well as in a toy 1 + 1 dimensional non-linear sigma model where it contributes non-perturbatively, in a way resembling the Higgs mechanism for the masses of matter particles in electro-weak theory. The QAE contribution to the nucleon mass can be explained using a similar mechanism, in terms of a dynamical response of the gluonic scalar field through Higgs-like couplings between the nucleon and scalar resonances. In addition, the QAE sets the scale for other energies in the nucleon through a relativistic virial theorem, and contributes a negative pressure to confine the colored quarks.

Sigma models on flags

We study (1+1)-dimensional non-linear sigma models whose target space is the flag manifold U(N)\over U(N_1)\times U(N_2)\cdots U(N_m), with a specific focus on the special case U(N)/U(1)^{N}U(N)/U(1)N. These generalize the well-known \mathbb{CP}^{N-1}ℂℙN−1 model. The general flag model exhibits several new elements that are not present in the special case of the \mathbb{CP}^{N-1}ℂℙN−1 model. It depends on more parameters, its global symmetry can be larger, and its ’t Hooft anomalies can be more subtle. Our discussion based on symmetry and anomaly suggests that for certain choices of the integers N_INI and for specific values of the parameters the model is gapless in the IR and is described by an SU(N)_1SU(N)1 WZW model. Some of the techniques we present can also be applied to other cases.

Towards the gravity/CYBE correspondence beyond integrability — Yang-Baxter deformations of T1,1 —

Yang-Baxter sigma models, proposed by Klimcik and Delduc-Magro-Vicedo, have been recognized as a powerful framework for studying integrable deformations of two dimensional non-linear sigma models. In this short article, as an important generalization, we review a non-integrable sigma model in the Yang-Baxter sigma model approach based on [arXiv:1406.2249]. In particular, we discuss a family of deformations of the 5D Sasaki-Einstein manifold T1,1, instead of the standard deformations of the 5-sphere S5. For this purpose, we first describe a novel construction of T1,1 as a supercoset, and provide a physical interpretation of this construction from viewpoint of the dual Klebanov-Witten field theory. Secondly, we consider a 3-parameter deformation of T1,1 by using classical r-matrices satisfying the classical Yang-Baxter equation (CYBE). The resulting metric and NS-NS two-form completely agree with the ones previously obtained via TsT (T-dual - shift - T-dual) transformations, and contain the Lunin-Maldacena background as a special case. Our result indicates that what we refer to as the gravity/CYBE(Classical Yang-Baxter Equation) correspondence can be extended beyond integrable cosets.

Dynamics of interaction of domain walls in (2+1)-dimensional non-linear sigma-model

By numerical simulation the dynamics of interactions of 180‑degree Neel type domain walls in (2+1) dimensional O (3) vectorial nonlinear sigma model is investigated. Are obtained numerically: new solutions in the form of domain walls with the rotation of the vector of A3 field in isotopic space; long-range model of interaction of the domain walls; oscillating (bion) model of the bound states of the domain walls.

Численное моделирование взаимодействия топологических вихрей с доменной стенкой в (2+1)-мерной нелинейной сигма-модели

Численное моделирование взаимодействия топологических вихрей с доменной стенкой в (2+1)-мерной нелинейной сигма-модели

Correlation functions of theSU(∞)principal chiral model

We obtain exact matrix elements of physical operators of the ($1+1$)-dimensional nonlinear sigma model of an $\mathrm{SU}(N)$-valued bare field, in the 't Hooft limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Specifically, all the form factors of the Noether current and the stress-energy-momentum tensor are found with an integrable bootstrap method. These form factors are used to find vacuum expectation values of products of these operators.

Resurgence and trans-series in Quantum Field Theory: the $\mathbb{C}{{\mathbb{P}}^{N-1 }}$ model

This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle resummation. The ideas from QFT use continuity on R^1 x S^1_L, i.e, the absence of any phase transition as N \to infinity, or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) in the semiclassical expansion. We find a set of "confluence equations" that encode the exact cancellation of the two different type of ambiguities. We show that a new notion of "graded resurgence triangle" is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Theta angle dependence of vacuum energy are calculated from first principles, and are in accord with large-N and lattice results.

Four-dimensional description of five-dimensionalN-brane models

We propose a new method for obtaining the four-dimensional effective gravitational theory for five-dimensional braneworld models with arbitrary numbers of branes in a low energy regime, based on a two-lengthscale expansion. The method is straightforward and computationally efficient, and is applicable to both compactified and uncompactified models. Particular emphasis is placed on the behavior of the radion modes of the model, while the massive effective fields are automatically truncated. Generally, the radion modes are found to form a (N-1)-dimensional nonlinear sigma model. We illustrate the method for a N-brane model in an uncompactified extra dimension.

Conformal invariance and quantum integrability of sigma models on symmetric superspaces

We consider two dimensional nonlinear sigma models on few symmetric superspaces, which are supergroup manifolds of coset type. For those spaces where one loop beta function vanishes, two loop beta function is calculated and is shown to be zero. Vanishing of beta function in all orders of perturbation theory is shown for the principal chiral models on group supermanifolds with zero Killing form. Sigma models on symmetric (super) spaces on supergroup manifold G/H are known to be classically integrable. We investigate a possibility to extend an argument of absence of quantum anomalies in nonlocal current conservation from nonsuper case to the case of supergroup manifolds which are asymptotically free in one loop.

BRST Invariant Theory of a Generalized 1 + 1 Dimensional Nonlinear Sigma Model with Topological Term

We give a generalized Lagrangian density of 1+1 Dimensional O(3) nonlinear sigma model with subsidiary constraints, different Lagrange multiplier fields and topological term, find a lost intrinsic constraint condition, convert the subsidiary constraints into inner constraints in the nonlinear sigma model, give the example of not introducing the lost constraint, by comparing the example with the case of introducing the lost constraint, we obtain that when not introducing the lost constraint, one has to obtain a lot of various non-intrinsic constraints. We further deduce the gauge generator, give general BRST transformation of the model under the general conditions. It is discovered that there exists a gauge parameter originating from the freedom degree of BRST transformation in a general O(3) nonlinear sigma model, and we gain the general commutation relations of ghost field.

Three Dimensional Nonlinear Sigma Models in the Wilsonian Renormalization Method

The three dimensional nonlinear sigma model is unrenormalizable in perturbative method. By using the $\beta$ function in the nonperturbative Wilsonian renormalization group method, we argue that ${\cal N}=2$ supersymmetric nonlinear $\sigma$ models are renormalizable in three dimensions. When the target space is an Einstein-K\"{a}hler manifold with positive scalar curvature, such as C$P^N$ or $Q^N$, there are nontrivial ultraviolet (UV) fixed point, which can be used to define the nontrivial continuum theory. If the target space has a negative scalar curvature, however, the theory has only the infrared Gaussian fixed point, and the sensible continuum theory cannot be defined. We also construct a model which interpolates between the C$P^N$ and $Q^N$ models with two coupling constants. This model has two non-trivial UV fixed points which can be used to define the continuum theory. Finally, we construct a class of conformal field theories with ${\bf SU}(N)$ symmetry, defined at the fixed point of the nonperturbative $\beta$ function. These conformal field theories have a free parameter corresponding to the anomalous dimension of the scalar fields. If we choose a specific value of the parameter, we recover the conformal field theory defined at the UV fixed point of C$P^N$ model and the symmetry is enhanced to ${\bf SU}(N+1)$.

Remarks on type IIBppwaves with Ramond-Ramond fluxes and massive two dimensional nonlinear sigma models

We continue the study of supersymmetric type IIB pp wave solutions by Maldacena and Maoz (hep-th/0207284), who showed Ramond-Ramond five-forms can induce potential terms in the light cone string actions which are nonlinear sigma models with special holonomy target spaces. We show nonvanishing Ramond-Ramond three-forms provide extra potential terms involving Killing vectors in the string action and identify the supersymmetry requirements. In particular, in solutions with (1,1) worldsheet supersymmetry, the Killing vectors are required to be self-dual in Spin(7).

Theoretical investigation of the phases of the organic insulator(TMTTF)2PF6

An explanation of some phases of the organic insulator ${(\mathrm{TMTTF})}_{2}{\mathrm{PF}}_{6}$ as a function of temperature, magnetic field, and pressure is given. It is shown that the problem can be mapped onto the (2+1)-dimensional nonlinear sigma model. The physics is described in terms of the coupling constant of this model, which encodes the one-dimensional two-body interactions and phonon effects. A phase diagram is proposed, including a possible quantum critical fixed point.

Existence of a confinement phase in quantum electrodynamics

We show that the four-dimensional U(1) gauge theory in the continuum formulation has a confining phase (exhibiting area law of the Wilson loop) in the strong coupling region above a critical coupling $g_c$. This result is obtained by taking into account topological non-trivial sectors in U(1) gauge theory. The derivation is based on the reformulation of gauge theory as a deformation of a topological quantum field theory and subsequent dimensional reduction of the D-dimensional topological quantum field theory to the (D-2)-dimensional nonlinear sigma model. The topological quantum field theory part of the four-dimensional U(1) gauge theory is exactly equivalent to the two-dimensional O(2) nonlinear sigma model. The confining (resp. Coulomb) phase of U(1) gauge theory corresponds to the high (resp. low) temperature phase of O(2) nonlinear sigma model and the critical point $g_c$ is determined by the Berezinskii-Kosterlitz-Thouless phase transition temperature. The quark (charge) confinement in the strong coupling phase is caused by the vortex condensation. Thus the continuum gauge theory has the direct correspondence to the compact formulation of lattice gauge theory.

A dual description of the four dimensional non-linear sigma model

The dual of the four dimensional non-linear sigma model is constructed using techniques familiar to string theory. This construction necessitates the introduction of a rank two antisymmetric tensor field whose properties are examined. The physics of the dual theory and that of the original model are compared. As an illustration we study in detail the SU(2}) chiral model. We find that the scattering amplitudes of the charged Goldstone bosons in the two theories are in complete agreement at the one loop level.

A gauged nonlinear sigma-model realization of the symplectic blowing up

In this paper a two dimensional non-linear sigma model with a general symplectic manifold with isometry as target space is used to study symplectic blowing up of a point singularity on the zero level set of the moment map associated with a quasi-free Hamiltonian action. We discuss in general the relation between symplectic reduction and gauging of the symplectic isometries of the sigma model action. In the case of singular reduction, gauging has the same effect as blowing up the singular point by a small amount. Using the exponential mapping of the underlying metric, we are able to construct symplectic diffeomorphisms needed to glue the blow-up to the global reduced space which is regular, thus providing a transition from one symplectic sigma model to another one free of singularities.

Gauge-independent analysis of O(3) nonlinear sigma model with Hopf and Chern-Simons terms

The (2 + 1)-dimensional nonlinear sigma model with a Hopf term (Wilczek-Zee model) is generalised to include a Chern-Simons three-form. Contrary to the Wilczek-Zee model, this model turns out to be a local gauge theory. It is quantised in the hamiltonian (Dirac) formalism without gauge fixing. Poincaré covariance of the theory is established, and subtleties related to the structure of different (canonical or symmetric) forms of the energy-momentum tensor are illuminated. The role of constraints and gauge invariance in the analysis is emphasised. Advantages of our gauge-independent formalism compared with usual gauge-fixed approaches are illustrated. The differences, at the conceptual and technical levels, between the usual Wilczek-Zee model and that treated here are high-lighted both in the canonical and path-integral formalisms. The angular-momentum operator is analysed, and the conventional result connecting the topological charge with the “fractional spin” is elaborated.