Supersymmetry Transformations Research Articles

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 Supergravity with Local Scaling Symmetry in Four Dimensions

We construct the most general four-dimensional 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 supergravity coupled to an arbitrary number n of vector multiplets in which the global scaling symmetry is gauged, in addition to a subgroup of SL(2, ℝ) × SO(6, n). The various gaugings are parametrized by an embedding tensor built out of 2n+63+4n+6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 2\\left(\\begin{array}{c}n+6\\\\ {}3\\end{array}\\right)+4\\left(n+6\\right) $$\\end{document} parameters that satisfy a specific set of quadratic consistency constraints, to which we provide explicit solutions. We also derive the local supersymmetry transformation rules and the equations of motion for the four-dimensional 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 matter-coupled supergravity with local scaling symmetry. Such supergravity theories do not possess an action, since the scaling symmetry is only an on-shell symmetry of the corresponding ungauged theories.

Fermionic shift symmetries in (anti) de Sitter space

We study extended shift symmetries that arise for fermionic fields on anti-de Sitter (AdS) space and de Sitter (dS) space for particular values of the mass relative to the curvature scale. We classify these symmetries for general mixed-symmetry fermionic fields in arbitrary dimension and describe how fields with these symmetries arise as the decoupled longitudinal modes of massive fermions as they approach partially massless points. For the particular case of AdS4, we look for non-trivial Lie superalgebras that can underly interacting theories that involve these fields. We study from this perspective the minimal such theory, the Akulov-Volkov theory on AdS4, which is a non-linear theory of a spin-1/2 Goldstino field that describes the spontaneous breaking of \\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 supersymmetry on AdS4 down to the isometries of AdS4. We show how to write the nonlinear supersymmetry transformation for this theory using the fermionic ambient space formalism. We also study the Lie superalgebras of candidate multi-field examples and rule out the existence of a supersymmetric special galileon on AdS4.

Inertia I: The global MSp-SUSY induced uniform motion

In this communication our main emphasis is on the review of the foundations of standard Lorentz code (SLC) of a particle motion. To this aim, we develop the theory of global, so-called, `double space´- or master space (MSp)-supersymmetry, subject to certain rules, wherein the superspace is a 14D-extension of a direct sum of background spaces M4⊕ MSp by the inclusion of additional 8D fermionic coordinates. The latter is induced by the spinors θ and ¯θ referred to MSp. While all the particles are living on M4, their superpartners can be viewed as living on MSp. This is a main ground for introducing MSp, which is unmanifested individual companion to the particle of interest. Supersymmetry transformation is defined as a translation in superspace, specified by the group element with corresponding anticommuting parameters. The multiplication of two successive transformations induce the motion. As a corollary, we derive SLC in a new perspective of global double MSp-SUSY transformations in terms of Lorentz spinors (θ, ¯θ). This calls for a complete reconsideration of our ideas of Lorentz motion code, to be now referred to as the individual code of a particle, defined as its intrinsic property. In MSp-SUSY theory, obviously as in standard unbroken SUSY theory, the vacuum zero point energy problem, standing before any quantum field theory in M4, is solved. The particles in M4 themselves can be considered as excited states above the underlying quantum vacuum of background double spaces M4⊕ MSp, where the zero point cancellation occurs at ground-state energy, provided that the natural frequencies are set equal (q 2 0 ≡ νb = νf ), because the fermion field has a negative zero point energy while the boson field has a positive zero point energy. On these premises, we derive the two postulates on which the Special Relativity (SR) is based.

Simplifying (super-)BMS algebras

We show that the non-linear BMS5 symmetry algebra of asymptotically flat Einstein gravity in five dimensions, as well as the super-BMS4 superalgebra of asymptotically flat supergravity, can be redefined so as to take a direct sum structure. In the new presentation of the (super-)algebra, angle-dependent translations and angle-dependent supersymmetry transformations commute with the (super-)Poincaré generators. We also explain in detail the structure and charge-integrability of asymptotic symmetries with symmetry parameters depending on the fields (through the charges themselves), a topic relevant for nonlinear asymptotic symmetry algebras.

Gauged D = 4 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 supergravity

We present the full Lagrangian and supersymmetry transformation rules for the gauged D = 4, 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 (half-maximal) supergravity coupled to an arbitrary number of vector multiplets. Using the embedding tensor formulation, the final results are universal and valid in arbitrary symplectic frames. We also analyze the conditions for the critical points of the scalar potential and specify the full spectrum of the quadratic fluctuations about Minkowski vacua. This allows us also to exclude the appearance of quadratic divergences in the 1-loop corrections to the scalar potential for any Minkowski vacuum fully breaking supersymmetry. We also provide some interesting byproducts of our analysis, like the field equations and the quadratic constraints for the fermion shifts characterizing the gauging (also known as T-tensor identities).

Edge states and SUSY in (2+1)-dimensional Maxwell Chern–Simons theory

In a [Formula: see text]-dimensional Maxwell–Chern–Simons theory coupled with a fermion and a scalar, which has [Formula: see text] SUSY in the absence of the boundary, supersymmetry is broken on the insertion of a spatial boundary. We show that only a subset of the boundary conditions allowed by the self-adjointness of the Hamiltonian can preserve partial ([Formula: see text]) supersymmetry, while for the remaining boundary conditions SUSY is completely broken. In the latter case, we demonstrate two distinct SUSY-breaking mechanisms. In one scenario, the SUSY-breaking boundary conditions are not consistent with the supersymmetry transformations. In another scenario, despite the boundary conditions being consistent with the SUSY transformations, unpaired fermionic edge states in the domain of the Hamiltonian leads to the breaking of the supersymmetry.

Engineering isospectrality in multidimensional photonic systems

Abstract Selective manipulation of energy levels plays an essential role in realizing multichannel wave devices. One of the representative examples is to utilize the concept of quasi-isospectrality: a family of wave systems with an almost identical spectrum except for a part of energy levels. Most approaches toward quasi-isospectrality have employed analytical methods based on symmetry or tridiagonalization, such as supersymmetry, Householder, or Lanczos transformations. Although such analytical approaches provide deterministic and stable designs based on operator factorizations, the mathematical strictness in the factorizations, at the same time, hinders isospectral engineering in a given multidimension. Here we develop the semi-analytical method for engineering isospectrality in multidimensional photonic systems. The method provides the systematic perturbation for the target energy level shifts by decomposing the allowed form of system changes into the perturbation basis. We demonstrate the isospectrality of lower-, higher-, and random-order states while imposing the designed shifts on the other states. The stability analysis shows that the accuracy of the method is determined by the ranges of isospectral state numbers and perturbation strength. The systematic, free-form, and multidimensional natures of the proposed method show great potential for the platform-transparent design of multichannel devices.

Type II double field theory in superspace

We explore type II supersymmetric double field theory in superspace. The double supervielbein is an element of the orthosymplectic group OSp(10, 10|64), which also governs the structure of generalized superdiffeomorphisms. Unlike bosonic double field theory, the local tangent space must be enhanced from the double Lorentz group in order to eliminate unphysical components of the supervielbein and to define covariant torsion and curvature tensors. This leads to an infinite hierarchy of local tangent space symmetries, which are connected to the super-Maxwell∞ algebra. A novel feature of type II is the Ramond-Ramond sector, which can be encoded as an orthosymplectic spinor (encoding the complex of super p-forms in conventional superspace). Its covariant field strength bispinor itself appears as a piece of the supervielbein. We provide a concise discussion of the superspace Bianchi identities through dimension two and show how to recover the component supersymmetry transformations of type II DFT. In addition, we show how the democratic formulation of type II superspace may be recovered by gauge-fixing.

Supersymmetric gradient flow in 4d $${{\mathcal {N}}}=1$$ SQCD

A supersymmetric gradient flow for four-dimensional \({{\mathcal {N}}}=1\) supersymmetric QCD (SQCD) is proposed. The flow equation is given in both the superfield and component fields of the Wess–Zumino gauge. The superfield flow equation is defined for each of the gauge and matter multiplets individually. Adding a gauge fixing, the component-field flow equation is defined in the Wess–Zumino gauge in a gauge covariant manner. We find that the latter equation is supersymmetric in a sense that the commutator of the flow time derivative and the supersymmetry transformation vanishes up to a gauge transformation. We also discuss a simplified flow using the gradient of supersymmetric Yang–Mills (SYM) action instead of using SQCD action to define a gauge multiplet flow.

SUSY and Tri-Vector Deformations

We analyze conditions for a tri-vector deformation of a supergravity background to preserve some supersymmetry. Working in the formalism of the SL(5) exceptional field theory, we present its supersymmetry transformations and introduce an additional USp(4) transformation to stay in the supergravity frame. This transformation acts on local indices and deforms BPS equations of exceptional field theory. The requirement for the deformation to vanish is the desired condition. The condition is shown to be consistent with previous results on bi-vector deformations.

Magnetic charges in supergravity

In this paper we study the dual charges of mathcal{N} = 1 supergravity in asymptotically flat space-time. The action considered is the usual supergravity action with a topological contribution. This is the Nieh-Yan term and the magnetic term of the free Rarita-Schwinger field. Through methods of the covariant phase space formalism we construct the charges conjugate to supersymmetry, diffeomorphism and Lorentz transformations. The additional term in the action will lead to new, non-vanishing contributions to these charges. The magnetic diffeomorphism charges are equivalent to the ones previously found for gravity, while the dual supersymmetric charges are new and do not appear for the free Rarita-Schwinger field. The dual Lorentz charges serve to regularize the previous two. We find that the asymptotic symmetry group for supergravity can only include globally well-defined super-rotations.

On commutator-based linearization of vector-spinor nonlinear supersymmetry and Rarita–Schwinger fields

In this paper, we discuss the linearization of vector-spinor (spin-3/2) nonlinear supersymmetry (vsNLSUSY) transformations for both [Formula: see text] and [Formula: see text]-extended SUSY in flat spacetime based on the commutator algebra by using functionals (composites) of spin-3/2 Nambu–Goldstone (NG) fermions, which are expressed as simple products of powers of the spin-3/2 NG fermions and a fundamental determinant in the vsNLSUSY theory. We define basic component fields by means of those functionals in a linearized vsSUSY theory including spin-3/2 fields, general auxiliary ones and a [Formula: see text]-term. The general forms of linear (rigid) vsSUSY transformations for the component fields are determined uniquely from the commutator-based linearization procedure. By considering appropriate recombinations of the functionals of the spin-3/2 NG fermions for [Formula: see text] SUSY, we find that variations of the recombinations under the vsNLSUSY transformations include linear spin-1/2 SUSY ones of spin-[Formula: see text] fields with [Formula: see text] gauge invariance. The spinorial gauge invariance of the Rarita–Schwinger action in the linearization process is also discussed together with the [Formula: see text] gauge invariance.

Irreducible representations of Z22-graded N=2 supersymmetry algebra and Z22-graded supermechanics

Irreducible representations (irreps) of Z22-graded supersymmetry algebra of N=2 are obtained by the method of induced representation, and they are used to derive Z22-graded supersymmetric classical actions. The irreps are four-dimensional for λ = 0, where λ is an eigenvalue of the Casimir element, and eight-dimensional for λ ≠ 0. The eight-dimensional irreps reduce to four-dimensional ones only when λ and an eigenvalue of Hamiltonian satisfy a particular relation. The reduced four-dimensional irreps are used to define Z22-graded supersymmetry transformations, and two types of classical actions invariant under the transformations are presented. It is shown that one of the Noether charges vanishes if all the variables of specific Z22-degree are auxiliary.

High-efficiency topological pumping with discrete supersymmetry transformations.

Making use of the isospectrality of Supersymmetry transformations, we propose a general and high-fidelity method to prepare gapped topological modes in discrete systems from a single-site excitation. The method consists of adiabatically connecting two superpartner structures, deforming the input state into the desired mode. We demonstrate the method by pumping topological states of the Su-Schrieffer-Heeger model in an optical waveguide array, where the adiabatic deformation is performed along the propagation direction. We obtain fidelities above F = 0.99 for a wide range of coupling strengths when pumping edge and interface states.

Supersymmetric gradient flow in mathcal{N}=1 SYM

The gradient flow equation is derived in four-dimensional mathcal{N}=1 supersymmetric Yang–Mills theory in terms of the component field of the Wess–Zumino gauge. We show that the flow-time derivative and supersymmetry transformation that is naively extended to 4+1 dimensions by replacing the four-dimensional fields with the corresponding flowed fields commute with each other up to a gauge transformation. In this sense, the obtained flow is supersymmetric in the Wess–Zumino gauge. We also discuss more about the symmetry of the flow equation.

On the off-shell superfield Lagrangian formulation of 4D, [formula omitted] supersymmetric infinite spin theory

We develop a complete off-shell Lagrangian description of the free 4D,N=1 supersymmetric theory of infinite spin. Bosonic and fermionic fields are formulated in terms of spin-tensor fields with dotted and undotted indices. The corresponding Lagrangians for bosonic and fermionic infinite spin fields entering into the on-shell supersymmetric model are derived within the BRST method. Lagrangian for this supersymmetric model is written in terms of the complex infinite spin bosonic field and infinite spin fermionic Weyl field subject to supersymmetry transformations. The fields involved into the on-shell supersymmetric Lagrangian can be considered as components of six infinite spin chiral and antichiral multiplets. These multiplets are extended to the corresponding infinite spin chiral and antichiral superfields so that two chiral and antichiral superfields contain among the components the basic fields of an infinite spin supermultiplet and extra four chiral and antichiral superfields containing only the auxiliary fields needed for the Lagrangian formulation. The superfield Lagrangian is constructed in terms of these six chiral and antichiral supefields, and we show that the component form of this superfield Lagrangian exactly coincides with the previously found component supersymmetric Lagrangian after eliminating the component fields added to construct (anti)chiral superfields.

Dimensional reduction of higher derivative heterotic supergravity

Higher derivative couplings of hypermultiplets to 6D, N = (1, 0) supergravity are obtained from dimensional reduction of 10D heterotic supergravity that includes order α′ higher derivative corrections. Reduction on T4 is followed by a consistent truncation. In the resulting action the hyperscalar fields parametrize the coset SO(4, 4)/(SO(4) × SO(4)). While the SO(4, 4) symmetry is ensured by Sen’s construction based on string field theory, its emergence at the field theory level is a nontrivial phenomenon. A number of field redefinitions in the hypermultiplet sector are required to remove several terms that break the SO(4) × SO(4) down to its SO(4) diagonal subgroup in the action and the supersymmetry transformation rules. Working with the Lorentz Chern-Simons term modified 3-form field strength, where the spin connection has the 3-form field strength as torsion, is shown to simplify considerably the dimensional reduction.

Black hole superpotential as a unifying entropy function and BPS thermodynamics

We develop an effective superpotential formalism for the SU(2)×U(1) invariant sector of mathcal{N} = 2 gauged supergravity in five dimensions with a U(1)3 Fayet-Iliopoulos gauging, and determine the exact superpotential that describes all 1/4 BPS solutions in this sector. This includes the Gutowski-Reall black holes, but also a much broader class of solutions with a squashed S3, magnetic flux and vector multiplet sources, as well as complex Euclidean BPS saddles. Some of these solutions are known only numerically, but the exact superpotential allows us to analytically evaluate the on-shell action, holographic one-point functions and conserved charges of all BPS solutions and to study their thermodynamics. In particular, by examining the supersymmetry Ward identities we show that solutions with supersymmetric vector multiplet sources break supersymmetry spontaneously. We also demonstrate the first law for black holes in the SU(2)×U(1) invariant sector and show that the conserved charges of supersymmetric solutions satisfy the generalized BPS relation derived in [1], which includes the supersymmetric Casimir energy as a consequence of the anomalous supersymmetry transformation of the mathcal{N} = 1 supercurrent at the boundary. Finally, we show that the effective superpotential provides a unifying entropy extremization principle, reproducing Sen’s entropy function for near extremal black holes and the Hosseini-Hristov-Zaffaroni functional for complex Euclidean BPS saddles.

On the Wess-Zumino Model: a supersymmetric field theory

We consider the free massless Wess-Zumino Model in 4D which describes a supersymmetric field theory that is invariant under the rigid or global supersymmetry transformations where the transformation parameter ϵ (or ϵ ) is a constant Grassmann spinor. We quantize the theory using the Hamiltonian and path integral formulations.

Off-shell supersymmetric continuous spin gauge theory

We construct, for the first time, an off-shell supersymmetric continuous spin gauge theory in 4-dimensional Minkowski spacetime, in both constrained and unconstrained Lagrangian formulations. As an extension to the on-shell description [1], we employ an auxiliary field to close the algebra of supersymmetry transformations off-shell. The 4d mathcal{N} = 1 massless continuous spin supermultiplet is then denoted by (Φ, H; Ψ), comprised of a dynamical and a non-dynamical complex scalar continuous spin gauge fields Φ and H, as well as a Dirac continuous spin gauge field Ψ. In particular, we demonstrate that the off-shell continuous spin supermultiplet, in a limit, reproduces off-shell supersymmetry transformations of the known scalar supermultiplet (0, 1/2), all integer-spin supermultiplets (s, s + 1/2), s ≥ 1, and all half-integer spin supermultiplets (s − 1/2, s), s ≥ 1.