Auxiliary Scalars Research Articles

Localisation without supersymmetry: towards exact results from Dirac structures in 3D N = 0 gauge theory

We show, by introducing purely auxiliary gluinos and scalars, that the quantum path integral for a class of 3D interacting non-supersymmetric gauge theories localises. The theories in this class all admit a ‘Manin gauge theory’ formulation, that we introduce; it is obtained by enhancing the gauge algebra of the theory to a Dirac structure inside a Manin pair. This machinery allows us to do localisation computations for every theory in this class at once, including for 3D Yang-Mills theory, and for its Third Way deformation; the latter calculation casts the Third Way path integral into an almost 1-loop exact form.

Observer-based output feedback control for switched T–S fuzzy systems with local nonlinear segments by using past output measurements

This paper researches the observer-based exponential stabilization problem for switched Takagi–Sugeno (T–S) fuzzy systems with unmeasurable premise variables and local nonlinearities. The local nonlinear parts that satisfy the incremental quadratic constraint exist in the systems and observers, which have the merit of being able to contain many common nonlinearities in a unified framework. Because the premise variables are not measurable, the performance of the observers in estimation is critical. Different from existing observer-based control strategies, the strategy of this paper not only depends on the current outputs of the systems but also relates to the past outputs of the systems which can provide a way to improve performance. In order to tackle the difficulties caused by introducing the past outputs and design a set of non-fragile fuzzy controllers to stabilize the systems, a new augmented state vector is constructed. Then, the conditions for designing the observer-based controller are derived by using auxiliary scalars, projection Lemma and S-procedure. Finally, the effectiveness of the control strategy obtained in this paper is proved by two examples.

Metric perturbations of Kerr spacetime in Lorenz gauge: circular equatorial orbits

We construct the metric perturbation in Lorenz gauge for a compact body on a circular equatorial orbit of a rotating black hole (Kerr) spacetime, using a newly-developed method of separation of variables. The metric perturbation is formed from a linear sum of differential operators acting on Teukolsky mode functions, and certain auxiliary scalars, which are solutions to ordinary differential equations in the frequency domain. For radiative modes, the solution is uniquely determined by the s=±2 Weyl scalars, the s = 0 trace, and s=0,1 gauge scalars whose amplitudes are determined by imposing continuity conditions on the metric perturbation at the orbital radius. The static (zero-frequency) part of the metric perturbation, which is handled separately, also includes mass and angular momentum completion pieces. The metric perturbation is validated against the independent results of a 2+1D time domain code, and we demonstrate agreement at the expected level in all components, and the absence of gauge discontinuities. In principle, the new method can be used to determine the Lorenz-gauge metric perturbation at a sufficiently high precision to enable accurate second-order self-force calculations on Kerr spacetime in future. We conclude with a discussion of extensions of the method to eccentric and non-equatorial orbits.

Supersymmetric localization of (higher-spin) JT gravity: a bulk perspective

We study two-dimensional Jackiw-Teitelboim gravity on the disk topology by using a BF gauge theory in the presence of a boundary term. The system can be equivalently written in a supersymmetric way by introducing auxiliary gauginos and scalars with suitable boundary conditions on the hemisphere. We compute the exact partition function thanks to supersymmetric localization and we recover the result obtained from the Schwarzian theory by accurately identifying the physical scales. The calculation is then easily extended to the higher-spin generalization of Jackiw-Teitelboim gravity, finding perfect agreement with previous results. We argue that our procedure can also be applied to boundary-anchored Wilson lines correlators.

On unfolded approach to off-shell supersymmetric models

We construct and analyze unfolded off-shell systems for chiral and vector supermultiplets using multispinor formalism and external currents. We find that auxiliary variables of multispinor formalism allow for the interesting reorganization of the unfolded off-shell modules: one can unify dynamical and auxiliary scalars in the Wess-Zumino model, as well as an auxiliary pseudoscalar and a zero-helicity subsector in the vector supermultiplet, so that the resulting unfolded equations get simplified, while various constraints like chirality, electric current conservation and P-oddness of the pseudoscalar can be formulated entirely in terms of these auxiliary variables.

Robust guaranteed cost control of networked Takagi–Sugeno fuzzy systems with local nonlinear parts and multiple quantizations

This paper researches the robust guaranteed cost (GC) control problem for a class of networked Takagi–Sugeno (T-S) fuzzy systems with local nonlinear segments and multiple quantizations, where the multiple quantizations mean that both the control input and the state are quantized through diverse dynamic quantizers. Three aspects are taken into account to perform the present study in depth. Firstly, the multiple quantizations case is investigated. Second, a novel cost function is exploited to perform the GC analysis issue, which consists of the quantized state and quantized control input. At last, a new control scheme is developed to fulfill the design of GC controller, where an additional nonlinear term is also employed besides the quantized state. By introducing auxiliary scalars and matrices, new conditions with regard to designing the dynamic quantizers and robust GC controllers are derived, which are expressed as a set of linear matrix inequalities (LMIs). Finally, a practical example is provided to verify the effectiveness of the obtained results.

Nonfragile H ∞ Control for Uncertain Takagi–Sugeno Fuzzy Systems Under Digital Communication Channels and Its Application

This work addresses the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> controller design problem for a class of discrete-time uncertain nonlinear systems. The norm-bounded uncertainty is contained in the nonlinear plant, which is described by the well-known Takagi–Sugeno (T–S) fuzzy model. The controller gain perturbation is also considered. When the input signal is transmitted from the controller to the system through the digital communication channels, it will be quantized by a static quantizer. The main attention is to design the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> state-feedback controller for the closed-loop quantized uncertain T–S fuzzy system. By introducing some auxiliary scalars and the fuzzy basis-dependent Lyapunov function approach, sufficient conditions are established in the form of linear matrix inequalities (LMIs). The construction for the nonfragile <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math></inline-formula> controller can be completed by solving these LMIs. In the end, the applicability and effectiveness of the proposed method have been illustrated by two simulation examples.

Admissibility analysis of Takagi–Sugeno fuzzy descriptor systems with time-varying delay via nonorthogonal free-matrix-based inequality

This paper studies the admissibility of Takagi–Sugeno (T-S) fuzzy descriptor systems with time-varying delay. To analyze the admissibility, various free-matrix-based inequalities (FMIs) are used in the existing works, since they directly produce bounds for integral terms without time-varying delay in denominator. However, the slack matrices in these inequalities are incomplete with some zero components, which may lead to incomplete coupling system information. In this case, conservatism is unavoidable. To remove the issue, a nonorthogonal FMI (NFMI) is proposed by constructing nonorthogonal polynomials with auxiliary scalars and auxiliary vectors, which includes some existing FMIs as its special cases. Based on the NFMI, a less conservative admissibility criterion is obtained for T-S fuzzy descriptor system. Two examples are illustrated to demonstrate the effectiveness of the obtained results.

Quantized Guaranteed Cost Output Feedback Control for Nonlinear Networked Control Systems and Its Applications

The quantized guaranteed cost static output feedback control problem is investigated for a class of discrete-time nonlinear networked control systems in this article. In this article, the Takagi–Sugeno fuzzy model is put to use for the representation of considered nonlinear networked control systems, where local nonlinear models instead of local linear models are used in the Takagi–Sugeno fuzzy model. Two different dynamic quantizers are applied to quantize the control input and measurement output, respectively. Different from some previous work, a novel guaranteed cost performance function including the quantized control input is used in this article. Through using the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$S$</tex-math></inline-formula> -procedure and introducing some auxiliary scalars, sufficient conditions for the design of guaranteed cost static output feedback controller and dynamic quantizers are obtained in the form of linear matrix inequalities. Finally, the applicability of the proposed method is illustrated through the application in nonlinear mass-spring-damper mechanical system.

Robust guaranteed cost control for uncertain discrete-time systems with state and input quantizations

The robust guaranteed cost control problem for uncertain discrete-time systems with state and input quantizations has been studied in this paper. The polytope type uncertainties are considered in the plants. Different from previous related works, a novel guaranteed cost control strategy has been put forward in this paper. The novelty and challenge lie in that the quantized state and quantized input are included in the guaranteed cost function. Through introducing some auxiliary scalars and combining with the S-procedure, new criteria are developed for the robust stability and guaranteed cost performance for discrete-time systems with state and input quantizations. Based on a new two-step design strategy, the controller and dynamic quantizers can be easily obtained by means of linear matrix inequalities. In the end, two examples are given to demonstrate the effectiveness and applicability of the proposed method.

Stochastic daily hydrothermal scheduling based on decomposition and parallelization

This paper studies the stochastic short term hydrothermal scheduling of systems that have a significant contribution from controllable reservoirs, focusing on the reservoirs’ operation policy when non-linearity and deployment of supplementary reserves at the real operation stage are modeled. Wind power and forced outages of power plants and transmission lines are treated as stochastic variables, and the total reserve is an endogenous variable to the model. The problem includes cascaded hydro systems with head-sensitive plants, a DC power flow representation with nonlinear transmission losses, and four types of operating reserves; and it aims at pre-positioning generation and reserves schedules in the light of the expected deployment of operating reserves throughout the day. To solve the optimization problem, a hybrid scheme elaborated from outer approximation and Benders decomposition is applied. The proposed solution method is tested in a real-size system, where its effectiveness, in terms of computational speed and accuracy, is demonstrated.The main nomenclature used is presented next. Other auxiliary variables, coefficients, dual variables and scalars are detailed within the section they are mentioned.

Robust Guaranteed Cost Control Under Digital Communication Channels

This paper investigates the feedback guaranteed cost control problem for a class of uncertain discrete-time systems with input quantization. The plant with polytope type uncertainties is considered. The input signal will be quantized by a dynamic quantizer before it is transmitted from the controller to the system under digital communication channels. First, a design method of the feedback control for the quantized closed-loop system is given to attain the same guaranteed cost performance as the one without signal quantization. Then, by introducing some useful auxiliary scalars, the dynamic parameter of the quantizer is also designed effectively so that the desired performance level can be achieved. A simulation example is provided to show the effectiveness of the proposed design method.

Complex three-form supergravity and membranes

There exist two variants of the old minimal formulation for mathcal{N} = 1 supergravity in four dimensions, in which one or each of the two auxiliary scalars is replaced by the field strength of a gauge three-form. These theories are known as three-form supergravity and complex three-form supergravity, respectively. For each of them, we present a super-Weyl invariant coupling of supergravity to the supermembrane and prove kappa-invariance of the resulting action. In the case of three-form supergravity, we demonstrate that the action constructed reduces to that given by Ovrut and Waldram twenty years ago upon imposing a super-Weyl gauge in which the compensating three-form superfield is set to a constant.

Particle actions and brane tensions from double and exceptional geometry

Massless particles in n + 1 dimensions lead to massive particles in n dimensions on Kaluza-Klein reduction. In string theory, wrapped branes lead to multiplets of massive particles in n dimensions, in representations of a duality group G. By encoding the masses of these particles in auxiliary worldline scalars, also transforming under G, we write an action which resembles that for a massless particle on an extended spacetime. We associate this extended spacetime with that appearing in double field theory and exceptional field theory, and formulate a version of the action which is invariant under the generalised diffeomorphism symmetry of these theories. This provides a higher-dimensional perspective on the origin of mass and tension in string theory and M-theory. Finally, we consider the reduction of exceptional field theory on a twisted torus, which is known to give the massive IIA theory of Romans. In this case, our particle action leads naturally to the action for a D0 brane in massive IIA. Here an extra vector field is present on the worldline, whose origin in exceptional field theory is a vector field introduced to ensure invariance under generalised diffeomorphisms.

Pauli–Villars Regularization of Non-Abelian Gauge Theories

As an extension of earlier work on QED, we construct a BRST-invariant Lagrangian for SU(N) Yang-Mills theory with fundamental matter, regulated by the inclusion of massive Pauli-Villars (PV) gluons and PV quarks. The underlying gauge symmetry for massless PV gluons is generalized to accommodate the PV-index-changing currents that are required by the regularization. Auxiliary adjoint scalars are used, in a mechanism due to Stueckelberg, to attribute mass to the PV gluons and the PV quarks. The addition of Faddeev--Popov ghosts then establishes a residual BRST symmetry. Although there are drawbacks to the approach, in particular the computational load of a large number of PV fields and a nonlocal interaction of the ghost fields, this formulation could provide a foundation for renormalizable nonperturbative solutions of light-front QCD in an arbitrary covariant gauge.

Exact Path Integral for 3D Quantum Gravity.

Three-dimensional Euclidean pure gravity with a negative cosmological constant can be formulated in terms of the Chern-Simons theory, classically. This theory can be written in a supersymmetric way by introducing auxiliary gauginos and scalars. We calculate the exact partition function of this Chern-Simons theory by using the localization technique. Thus, we obtain the quantum gravity partition function, assuming that it can be obtained nonperturbatively by summing over partition functions of the Chern-Simons theory on topologically different manifolds. The resultant partition function is modular invariant, and, in the case in which the central charge is expected to be 24, it is the J function, predicted by Witten.

Ghost-free [formula omitted] bigravity and accelerating cosmology

We propose a bigravity analogue of the F(R) gravity. Our construction is based on recent ghost-free massive bigravity where additional scalar fields are added and the corresponding conformal transformation is implemented. It turns out that F(R) bigravity is easier to formulate in terms of the auxiliary scalars as the explicit presentation in terms of F(R) is quite cumbersome. The consistent cosmological reconstruction scheme of F(R) bigravity is developed in detail, showing the possibility to realize nearly arbitrary physical universe evolution with consistent solution for second metric. The examples of accelerating universe which includes phantom, quintessence and ΛCDM acceleration are worked out in detail and their physical properties are briefly discussed.

Stability analysis of f(R)-AdS black holes

We study the stability of the f(R)-AdS (Schwarzschild–AdS) black hole obtained from f(R) gravity. In order to resolve the difficulty of solving fourth-order linearized equations, we transform f(R) gravity into scalar–tensor theory by introducing two auxiliary scalars. In this case, the linearized curvature scalar becomes a dynamical scalaron, showing that all linearized equations are second order. Using the positivity of gravitational potentials and S-deformed technique allows us to guarantee the stability of f(R)-AdS black hole if the scalaron mass squared satisfies the Breitenlohner–Freedman bound. This is confirmed by computing quasinormal frequencies of the scalaron for the f(R)-AdS black hole.

Instability of a rotating black hole in a limited form off(R)gravity

We investigate the stability of $f(R)$-rotating (Kerr) black hole obtained from a limited form of $f(R)$ gravity. In order to avoid the difficulty of handling fourth order linearized equations, we transform this form of $f(R)$ gravity into the scalar-tensor theory by introducing two auxiliary scalars. In this case, the linearized curvature scalar equation leads to a massive scalaron equation. It turns out that the $f(R)$-rotating black hole is unstable because of the superradiant instability known as the black-hole bomb idea.

Stability off(R)black holes

We investigate the stability of $f(R)$ (Schwarzschild) black hole obtained from the $f(R)$ gravity. It is difficult to carry out the perturbation analysis around the black hole because the linearized Einstein equation is fourth order in $f(R)$ gravity. In order to resolve this difficulty, we transform $f(R)$ gravity into the scalar-tensor theory by introducing two auxiliary scalars. In this case, the linearized curvature scalar becomes a scalaron, showing that all linearized equations are second order, which are the same equations for the massive Brans-Dicke theory. It turns out that the $f(R)$ black hole is stable against the external perturbations if the scalaron does not have a tachyonic mass.