Supersymmetry Algebra Research Articles

Supersymmetric celestial OPEs and soft algebras from the ambitwistor string worldsheet

Using the ambitwistor string, we complete the list of celestial operator product expansion (OPE) coefficients for supersymmetric theories. This uses the ambitwistor string worldsheet conformal field theory to dynamically generate the OPE coefficients for maximally supersymmetric gauge theory, as well as gravity and Einstein-Yang-Mills theories, including all helicity and orientation configurations. This extends previous purely bosonic results [T. Adamo et al., arXiv:2111.02279] to include supersymmetry and provides explicit formulas which are, to the best of our knowledge, not in the literature. We also examine how the supersymmetric infinite dimensional soft algebras behave compared to the purely bosonic cases.

Supersymmetry and Hodge theory on Sasakian and Vaisman manifolds

Sasakian manifolds are odd-dimensional counterpart to Kähler manifolds. They can be defined as contact manifolds equipped with an invariant Kähler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kähler supersymmetry algebra is associated to a Kähler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly.

Supersymmetry algebras in arbitrary signature and their R-symmetry groups

String theory, specifically type-II superstring theory, can be formulated in any ten-dimensional signature. To facilitate the study of supergravity and superstring theories in this setting, we present a uniform construction of supersymmetry algebras in arbitrary dimension and signature, which generalizes the ideas underlying symplectic Majorana spinors. In our formalism R-symmetry acts on an auxiliary multiplicity space which makes its action manifest. This allows us to provide extensive tables which list the R-symmetry groups of extended supersymmetry algebras for all signatures together with other useful information. Twisted (‘type-*’) supersymmetry algebras in Lorentz signature with non-compact R-symmetry groups are shown to be part of a general pattern resulting from the interplay between complex superbrackets and reality conditions. As an application we show how the relations between type-II string theories in ten and nine dimensions can be extracted from their supersymmetry algebras. We also use our results to determine the special geometry of vector and hypermultiplet scalar manifolds of four-dimensional mathcal{N} = 2 and three-dimensional mathcal{N} = 4 supergravity theories for all signatures.

SL(5) Supersymmetry

AbstractWe consider supersymmetry in five dimensions, where the fermionic parameters are a 2‐form under . Supermultiplets are investigated using the pure spinor superfield formalism, and are found to be closely related to infinite‐dimensional extensions of the supersymmetry algebra: the Borcherds superalgebra , the tensor hierarchy algebra and the exceptional superalgebra E(5, 10). A theorem relating and E(5, 10) to all levels is given.

Supersymmetry, T-duality and heterotic \u03b1\u2032-corrections

Higher-derivative interactions and transformation rules of the fields in the effective field theories of the massless string states are strongly constrained by space-time symmetries and dualities. Here we use an exact formulation of ten dimensional mathcal{N} = 1 supergravity coupled to Yang-Mills with manifest T-duality symmetry to construct the first order α′-corrections of the heterotic string effective action. The theory contains a supersymmetric and T-duality covariant generalization of the Green-Schwarz mechanism that determines the modifications to the leading order supersymmetry transformation rules of the fields. We compute the resulting field-dependent deformations of the coefficients in the supersymmetry algebra and construct the invariant action, with up to and including four-derivative terms of all the massless bosonic and fermionic fields of the heterotic string spectrum.

Supersymmetric structures of Dirac oscillators in commutative and noncommutative spaces* *Project supported by the National Natural Science Foundation of China (Grant No. 11465006).

The supersymmetric properties of a charged planar Dirac oscillator coupling to a uniform perpendicular magnetic field are studied. We find that there is an N = 2 supersymmetric structure in both commutative and noncommutative cases. We construct the generators of the supersymmetric algebras explicitly and show that the generators of the supersymmetric algebras can be mapped onto ones which only contain the left or right-handed chiral phonons by unitary transformations.

Left-covariant first order differential calculus on quantum Hopf supersymmetry algebra

We introduce a Hopf algebra structure on the N = 2 quantum supersymmetry algebra and formulate a first order quantum differential calculus on it. Then, it is enhanced to three *-calculi by defining three appropriate involution maps on the quantum super-algebra. Two of the *-structures correspond to quantum complex super-algebra and the other correspond to a quantum real one. An appropriate quantum super-Hopf algebra including two even and two odd generators and also its corresponding quantum super-group are introduced. Compared to the quantum super-algebra, the quantum super-group also has three different *-structures. It is shown that the differential calculus over the quantum super-algebra is left-covariant with respect to the quantum super-group. Besides, it is shown that the graded differential algebra for the case q = 1 is a bicovariant bimodule over the undeformed Hopf supersymmetry algebra.

Five-brane current algebras in type II string theories

We study the current algebras of the NS5-branes, the Kaluza-Klein (KK) five-branes and the exotic {5}_2^2 -branes in type IIA/IIB superstring theories. Their worldvolume theories are governed by the six-dimensional mathcal{N} = (2, 0) tensor and the mathcal{N} = (1, 1) vector multiplets. We show that the current algebras are determined through the S- and T-dualities. The algebras of the mathcal{N} = (2, 0) theories are characterized by the Dirac bracket caused by the self-dual gauge field in the five-brane worldvolumes, while those of the mathcal{N} = (1, 1) theories are given by the Poisson bracket. By the use of these algebras, we examine extended spaces in terms of tensor coordinates which are the representation of ten-dimensional supersymmetry. We also examine the transition rules of the currents in the type IIA/IIB supersymmetry algebras in ten dimensions. Based on the algebras, we write down the section conditions in the extended spaces and gauge transformations of the supergravity fields.

Supersymmetric Bi-Hamiltonian Systems

We construct super Hamiltonian integrable systems within the theory of supersymmetric Poisson vertex algebras (SUSY PVAs). We provide a powerful tool for the understanding of SUSY PVAs called the super master formula. We attach some Lie superalgebraic data to a generalized SUSY W-algebra and show that it is equipped with two compatible SUSY PVA brackets. We reformulate these brackets in terms of odd differential operators and obtain super bi-Hamiltonian hierarchies after performing a supersymmetric analog of the Drinfeld–Sokolov reduction on these operators. As an example, an integrable system is constructed from \({\mathfrak {g}}=\mathfrak {osp}(2|2)\).

Supersymmetric W-algebras

We develop a general theory of $W$-algebras in the context of supersymmetric vertex algebras. We describe the structure of $W$-algebras associated with odd nilpotent elements of Lie superalgebras in terms of their free generating sets. As an application, we produce explicit free generators of the $W$-algebra associated with the odd principal nilpotent element of the Lie superalgebra $\mathfrak{gl}(n+1|n).$

Projective oscillator representations of and

Motivated from understanding higher-dimensional supersymmetric conformal field theory, we study various inhomogeneous oscillator representations of and on supersymmetric polynomial algebras and on spaces of supersymmetric exponential–polynomial functions, analogous to the projective oscillator representations of and . In particular, we find the explicit structures of certain Harish-Chandra modules and Whittaker-type modules of these Lie superalgebras.

Fluxes, twisted tori, monodromy and U(1) supermembranes

We show that the D = 11 supermembrane theory (M2-brane) compactified on a M9× T2 target space, with constant fluxes C± naturally incorporates the geometrical structure of a twisted torus. We extend the M2-brane theory to a formulation on a twisted torus bundle. It is consistently fibered over the world volume of the M2-brane. It can also be interpreted as a torus bundle with a nontrivial U(1) connection associated to the fluxes. The structure group G is the area preserving diffeomorphisms. The torus bundle is defined in terms of the monodromy associated to the isotopy classes of symplectomorphisms with π0(G) = SL(2, Z), and classified by the coinvariants of the subgroups of SL(2, Z). The spectrum of the theory is purely discrete since the constant flux induces a central charge on the supersymmetric algebra and a modification on the Hamiltonian which renders the spectrum discrete with finite multiplicity. The theory is invariant under symplectomorphisms connected and non connected to the identity, a result relevant to guarantee the U-dual invariance of the theory. The Hamiltonian of the theory exhibits interesting new U(1) gauge and global symmetries on the worldvolume induced by the symplectomorphim transformations. We construct explicitly the supersymmetric algebra with nontrivial central charges. We show that the zero modes decouple from the nonzero ones. The nonzero mode algebra corresponds to a massive superalgebra that preserves either 1/2 or 1/4 of the original supersymmetry depending on the state considered.

Four-dimensional vector multiplets in arbitrary signature (I)

We derive a necessary and sufficient condition for Poincaré Lie superalgebras in any dimension and signature to be isomorphic. This reduces the classification problem, up to certain discrete operations, to classifying the orbits of the Schur group on the vector space of superbrackets. We then classify four-dimensional [Formula: see text] supersymmetry algebras, which are found to be unique in Euclidean and in neutral signature, while in Lorentz signature there exist two algebras with R-symmetry groups [Formula: see text] and [Formula: see text], respectively.

Massless Infinite Spin (Super)particles and Fields

A new twistorial field formulation of a massless infinite spin particle is derived. We find a twistorial infinite spin field and derive its helicity decomposition. The twistorial equations of motion for infinite spin fields in the cases of integer and half-integer helicities are derived. We show that the infinite integer-spin field and infinite half-integer-spin field form the $\mathcal{N}{=}\,1$ infinite spin supermultiplet. The corresponding supersymmetry transformations are presented. We prove that the supersymmetry algebra is closed on-shell.

The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

We develop a non-commutative polynomial version of the invariant theory of the quantum general linear supergroup Uq(glm|n). A non-commutative Uq(glm|n)-module superalgebra Pr|sk|l is constructed, which is the quantum analogue of the supersymmetric algebra over Ck|l⊗Cm|n⊕Cr|s⊗(Cm|n)⁎. We analyse the structure of the subalgebra of Uq(glm|n)-invariants in Pr|sk|l by using a quantum super analogue of Howe duality.The subalgebra of Uq(glm|n)-invariants in Pr|sk|l is shown to be finitely generated. We determine its generators and establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto it. This establishes the first fundamental theorem of invariant theory for Uq(glm|n).We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if m⩾min⁡{k,r} and n⩾min⁡{l,s}, and obtain a PBW basis for the subalgebra of invariants in this case. When the homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel. This way we obtain the relations obeyed by the generators of the subalgebra of invariants, producing the second fundamental theorem of invariant theory for Uq(glm|n).We consider the special case n=0 in greater detail, obtaining a complete treatment of the non-commutative polynomial version of the invariant theory of Uq(glm). In particular, the explicit SFT proved here is believed to be new. We also recover the FFT and SFT of invariant theory for the general linear superalgebra from the classical limit (i.e. q→1) of our results.

Lorentz symmetry breaking in supersymmetric quantum electrodynamics

Lorentz symmetry is one of the fundamental symmetries of nature; however, it can be broken by several proposals such as quantum gravity effects, low energy approximations in string theory and dark matter. In this paper, Lorentz symmetry is broken in supersymmetric quantum electrodynamics using aether superspace formalism without breaking any supersymmetry. To break the Lorentz symmetry in three-dimensional quantum electrodynamics, we must use the [Formula: see text] aether superspace. A new constant vector field is introduced and used to deform the deformed generator of supersymmetry. This formalism is required to fix the unphysical degrees of freedom that arise from the quantum gauge transformation required to quantize this theory. By using Yokoyama’s gaugeon formalism, it is possible to study these gaugeon transformations. As a result of the quantum gauge transformation, the supersymmetric algebra gets modified and the theory is invariant under BRST symmetry. These results could aid in the construction of the Gravity’s Rainbow theory and in the study of superconformal field theory. Furthermore, it is demonstrated that different gauges in this deformed supersymmetric quantum electrodynamics can be related to each other using the gaugeon formalism.

Quantum collapse-revival effect in a supersymmetric Jaynes–Cummings model and its possible application in supersymmetric qubits

In the photon-atom interaction, one of the most interesting phenomena can be collapse-revival effect in population inversion of atomic levels. Since a single-photon-transition Jaynes–Cummings model can be generalized to a multiphoton-transition Jaynes–Cummings model, of which the Hamiltonian can exhibit a supersymmetric Lie algebra structure, in this paper, quantum collapse and revival in atomic population inversion, which is a collaborative effect of a variety of quantum Rabi oscillations with various quantum Rabi frequencies, is studied for the supersymmetric multiphoton-transition Jaynes–Cummings model. In the literature, it has been pointed out that quantum collapse and revival might find an application in qubit design. We expect that a supersymmetric two-level system might serve as such a kind of qubits. Compared with ordinary two-level qubits, more information could be stored in the basis vectors of supersymmetric qubits, and this could increase the degrees of freedom of tensor product (multi-component extension) of basis vectors of this multiphoton-atom interacting system. For this reason, some characteristics of the collapse-revival effect driven by ordinary coherent states and Yurke–Stoler coherent states will be addressed in this paper. The influence of various characteristic parameters of this supersymmetric Jaynes–Cummings model and the quantized optical fields on the quantum collapse and revival in the population inversion will be indicated in some numerical illustrative examples.

Quantum entanglement, supersymmetry, and the generalized Yang-Baxter equation

Entangled states, such as the Bell and GHZ states, are generated from separable states using matrices known to satisfy the Yang-Baxter equation and its generalization. This remarkable fact hints at the possibility of using braiding operators as quantum entanglers, and is part of a larger speculated connection between topological and quantum entanglement. We push the analysis of this connection forward, by showing that supersymmetry algebras can be used to construct large families of solutions of the spectral parameter-dependent generalized Yang-Baxter equation. We present a number of explicit examples and outline a general algorithm for arbitrary numbers of qubits. The operators we obtain produce, in turn, all the entangled states in a multi-qubit system classified by the Stochastic Local Operations and Classical Communication protocol introduced in quantum information theory.

Extended supersymmetry with central charges in Dirac action with curved extra dimensions

We discuss a new realization of $\mathcal{N}$-extended quantum-mechanical supersymmetry (QM SUSY) with central charges hidden in the four-dimensional (4D) mass spectrum of higher dimensional Dirac action with curved extra dimensions. We show that this $\mathcal{N}$-extended QM SUSY results from symmetries in extra dimensions, and the supermultiplets in this supersymmetry algebra correspond to the Bogomol'nyi--Prasad--Sommerfield states. Furthermore, we examine the model of the $S^2$ extra dimension with a magnetic monopole background and confirm that the $\mathcal{N}$-extended QM SUSY explains the degeneracy of the 4D mass spectrum.

An introduction to a supersymmetric graph algebra

In this paper, we propose a graph superalgebra which is the supersymmetric analogue of Leavitt path algebras. We find a basis for these superalgebras and characterize when they have polynomial growth.