Group OSp Research Articles

$${\hat{Z}}$$-Invariant for SO(3) and OSp(1|2) Groups

Three-manifold invariants $${\hat{Z}}$$ (“Z-hat”), also known as homological blocks, are q-series with integer coefficients. Explicit q-series form for $${\hat{Z}}$$ is known for SU(2) group, supergroup SU(2|1) and orthosymplectic supergroup OSp(2|2). We focus on $${\hat{Z}}$$ for SO(3) group and orthosymplectic supergroup OSp(1|2) in this paper. Particularly, the change of variable relating SU(2) link invariants to the SO(3) and OSp(1|2) link invariants plays a crucial role in explicitly writing the q-series.

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.

Remarks on a D(2,1;a) super-Schwarzian derivative

It was recently demonstrated that N=1,2,3,4 super-Schwarzian derivatives can be constructed by applying the method of nonlinear realisations to finite-dimensional superconformal groups OSp(1|2), SU(1,1|1), OSp(3|2), SU(1,1|2), respectively, thus avoiding the use of superconformal field theory techniques. In this work, a similar construction is applied to the exceptional supergroup D(2,1;a), which describes the most general N=4 supersymmetric extension of SL(2,R), with the aim to study possible candidates for a D(2,1;a) super-Schwarzian derivative.

Formula omitted] super–Schwarzian from OSp(3|2) invariants

It was recently demonstrated that the N=0,1,2,4 super–Schwarzian derivatives can be constructed by applying the method of nonlinear realizations to the finite–dimensional (super)conformal groups SL(2,R), OSp(1|2), SU(1,1|1), and SU(1,1|2), respectively. In this work, a similar scheme is realised for OSp(3|2). It is shown that the N=3 case exhibits a surprisingly richer structure of invariants, the N=3 super–Schwarzian being a particular member. We suggest that the extra invariants may prove useful in building an N=3 supersymmetric extension of the Sachdev–Ye–Kitaev model.

Rankin–Cohen brackets on supermodular forms

Let ℂ1|1 denote the supercomplex plane with local coordinate (z,θ) where θ2=0, and with the canonical supersymmetric (SUSY) structure given by the maximal non-integrable structure distribution of rank 0|1 generated by the odd vector field D=∂∂θ+θ∂∂z. It admits a super action of the super Lie group OSP(1|2,R). This action induces a one parameter action of OSP(1|2,R) on the super-ring, R of C∞-superfunctions on ℂ1|1. A supermodular form is a superfunction F∈R invariant by the action of OSP(1|2,R) on the super-ring R (see Cohen et al., 1997).In this paper, we construct a whole family of operators transforming a given pair of supermodular forms into another supermodular form and apply these operators to define a deformed associative product.

Abelian Yang–Baxter deformations and TsT transformations

We prove that abelian Yang–Baxter deformations of superstring coset σ models are equivalent to sequences of commuting TsT transformations, meaning T dualities and coordinate shifts. Our results extend also to fermionic deformations and fermionic T duality, and naturally lead to a TsT subgroup of the superduality group OSp(db,db|2df). In cases like AdS5×S5, fermionic deformations necessarily lead to complex models. As an illustration of inequivalent deformations, we give all six abelian deformations of AdS3. We comment on the possible dual field theory interpretation of these (super-)TsT models.

Role of ERK signaling pathway in inhibition of OGD/R-induced apoptosis in rat cortical neurons by sevoflurane: the relationship with mPTP

Objective To evaluate the role of extracellular signal-regulated kinase(ERK)signaling pathway in inhibition of oxygen-glucose deprivation and restoration(OGD/R)-induced apoptosis in rat cortical neurons by sevoflurane and the relationship with mitochondrial permeability transition pore(mPTP). Methods The rat cortical neurons were cultured in vitro and seeded in 6-well or 12-well culture plates.The neurons were randomly divided into 4 groups(n=18 each)using a random number table: control group(group C); OGD/R group(group O); sevoflurane group(group OS); sevoflurane+ ERK1/2 inhibitor PD98059 group(group OSP). The neurons were subjected to O2-glucose deprivation for 90 min followed by restoration of O2-glucose supply for 24 h in group O. The neurons were incubated with 2% sevoflurane for 2 h after OGD/R in group OS.OGD/R was performed at 1 h after ERK1/2 inhibitor PD98059 30 μmol/L was added, and the neurons were incubated with 2% sevoflurane for 2 h after OGD/R in group OSP.At 24 h of restoration of O2-glucose supply, the expression of phosphorylated ERK1/2(p-ERK1/2)in neurons was measured by Western blot, the neuronal apoptosis was detected using Annexin V-FITC/PI double staining combined with flow cytometry, and the opening of mPTP was determined through measuring the optical density at 540 nm.The apoptosis rate was calculated. Results Compared with group C, the expression of p-ERK1/2 in neurons was significantly up-regulated, and the apoptosis rate and mPTP opening were significantly increased in group O(P<0.05). Compared with group O, the expression of p-ERK1/2 in neurons was significantly up-regulated, and the apoptosis rate and mPTP opening were significantly decreased in group OS(P<0.05). Compared with group OS, the expression of p-ERK1/2 in neurons was significantly down-regulated, and the apoptosis rate and mPTP opening were significantly increased in group OSP(P<0.05). Conclusion The mechanism by which sevoflurane inhibits OGD/R-induced apoptosis in rat cortical neurons is related to inhibition of mPTP opening after activation of ERK signaling pathway. Key words: Extracellular signal-regulated MAP kinases; Mitochondrial membrane transport proteins; Anesthetics, inhalation; Neurons; Apoptosis

Superconformal structures on the three-sphere

With the motivation to develop superconformal field theory on S^3, we introduce a 2n-extended supersphere S^{3|4n}, with n=1,2,..., as a homogeneous space of the three-dimensional Euclidean superconformal group OSp(2n|2,2) such that its bosonic body is S^3. Supertwistor and bi-supertwistor realizations of S^{3|4n} are derived. We study in detail the n=1 case, which is unique in the sense that the R-symmetry subgroup SO^*(2n) of the superconformal group is compact only for n=1. In particular, we show that the OSp(2|2,2) transformations preserve the chiral subspace of S^{3|4}. Several supercoset realizations of S^{3|4n} are presented. Harmonic/projective extensions of the supersphere by auxiliary bosonic fibre directions are sketched.

Minimal unitary representation of [formula omitted] and its [formula omitted] deformations and [formula omitted], [formula omitted] superconformal models

Quantization of the geometric quasiconformal realizations of noncompact groups and supergroups leads directly to their minimal unitary representations (minreps). Using quasiconformal methods massless unitary supermultiplets of superconformal groups SU(2,2|N) and OSp(8⁎|2n) in four and six dimensions were constructed as minreps and their U(1) and SU(2) deformations, respectively. In this paper we extend these results to SU(2) deformations of the minrep of N=4 superconformal algebra D(2,1;λ) in one dimension. We find that SU(2) deformations can be achieved using n pair of bosons and m pairs of fermions simultaneously. The generators of deformed minimal representations of D(2,1;λ) commute with the generators of a dual superalgebra OSp(2n⁎|2m) realized in terms of these bosons and fermions. We show that there exists a precise mapping between symmetry generators of N=4 superconformal models in harmonic superspace studied recently and minimal unitary supermultiplets of D(2,1;λ) deformed by a pair of bosons. This can be understood as a particular case of a general mapping between the spectra of quantum mechanical quaternionic Kähler sigma models with eight super symmetries and minreps of their isometry groups that descends from the precise mapping established between the 4d, N=2 sigma models coupled to supergravity and minreps of their isometry groups.

Superconformal Yang-Mills quantum mechanics and Calogero model with $$ {\text{OSp}}\left( {\mathcal{N}|{2},\mathbb{R}} \right) $$ symmetry

In spacetime dimension two, pure Yang-Mills possesses no physical degrees of freedom, and consequently it admits a supersymmetric extension to couple to an arbitrary number, N say, of Majorana-Weyl gauginos. This results in (N,0) super Yang-Mills. Further, its dimensional reduction to mechanics doubles the number of supersymmetries, from N to N+N, to include conformal supercharges, and leads to a superconformal Yang-Mills quantum mechanics with symmetry group OSp(N|2,R). We comment on its connection to AdS_2 \times S^{N-1} and reduction to a supersymmetric Calogero model.

An infinite supermultiplet of massive higher-spin fields

The representation theory underlying the infinite-component relativistic wave equation written by Majorana is revisited from a modern perspective. On the one hand, the massless solutions of this equation are shown to form a supermultiplet of the superPoincare algebra with tensorial central charges; it can also be obtained as the infinite spin limit of massive solutions. On the other hand, the Majorana equation is generalized for any space-time dimension and for arbitrary Regge trajectories. Inspired from these results, an infinite supermultiplet of massive fields of all spins and of equal mass is constructed in four dimensions and proved to carry an irreducible representation of the orthosymplectic group OSp(1|4) and of the superPoincare group with tensorial charges.

Collective states of the odd-mass nuclei within the framework of the interacting vector boson model

A supersymmetric extension of the dynamical symmetry group SpB(12, R) of the interacting vector boson model (IVBM) to the orthosymplectic group OSp(2Ω/12, R) is developed in order to incorporate fermion degrees of freedom into the nuclear dynamics and to encompass the treatment of odd-mass nuclei. The bosonic sector of the supergroup is used to describe the complex collective spectra of the neighboring even–even nuclei and is considered as a core structure of the odd nucleus. The fermionic sector is represented by the fermion spin group SOF(2Ω) ⊃ SUF(2). The so obtained, new exactly solvable limiting case is applied for the description of the nuclear collective spectra of odd-mass nuclei. The theoretical predictions for different collective bands in three odd-mass nuclei, namely 157Gd, 173Yb and 163Dy from the rare earth region are compared with the experiment. The B(E2) transition probabilities for 157Gd and 163Dy between the states of the ground band are also studied. The important role of the symplectic structure of the model for the proper reproduction of the B(E2) behavior is revealed. The obtained results reveal the applicability of the model's extension.

NewN=8nonlinear supermultiplet

We construct a new off-shell $\mathcal{N}=8$, $d=1$ nonlinear supermultiplet $(\mathbf{4},\mathbf{8},\mathbf{4})$ proceeding from the nonlinear realization of the $\mathcal{N}=8$, $d=1$ superconformal group $OSp({4}^{\ensuremath{\star}}|4)$ in its supercoset $\frac{OSp({4}^{\ensuremath{\star}}|4)}{SU(2{)}_{\mathcal{R}}\ensuremath{\bigotimes}{D,K}\ensuremath{\bigotimes}SO(4)}$. The irreducibility constraints for the superfields automatically follow from appropriate covariant conditions on the $osp({4}^{\ensuremath{\star}}|4)$-valued Cartan superforms. We present the most general sigma-model type action for $(\mathbf{4},\mathbf{8},\mathbf{4})$ supermultiplet. The relations between linear and nonlinear $(\mathbf{4},\mathbf{8},\mathbf{4})$ supermultiplets and linear $\mathcal{N}=8$ $(\mathbf{5},\mathbf{8},\mathbf{3})$ vector supermultiplet are discussed.

Nonlinear (4,8,4) multiplet of N=8, d=1 supersymmetry

Abstract We construct a nonlinear version of the d = 1 off-shell N = 8 multiplet ( 4 , 8 , 4 ) , proceeding from a nonlinear realization of the superconformal group OSp ( 4 ⋆ | 4 ) in the N = 8 , d = 1 analytic bi-harmonic superspace. The new multiplet is described by a double-charged analytic Goldstone superfield q 1 , 1 subjected to some nonlinear OSp ( 4 ⋆ | 4 ) covariant harmonic constraints. We construct the most general N = 8 supersymmetric actions of q 1 , 1 and find the general expression for the bosonic target metric in terms of the original analytic Lagrangian superfield density which is thus the target geometry prepotential. It also completely specifies the scalar potential. The metric is conformally flat and, in the SO ( 4 ) invariant case, is a deformation of the metric of S 4 .

N=8 superconformal mechanics

We construct new models of N=8 superconformal mechanics associated with the off-shell N=8, d=1 supermultiplets ( 3,8,5 ) and ( 5,8,3 ) . These two multiplets are derived as N=8 Goldstone superfields and correspond to nonlinear realizations of the N=8, d=1 superconformal group OSp(4 ★|4) in its supercosets OSp(4 ★|4) U(1) R⊗SO(5) and OSp(4 ★|4) SU(2) R⊗SO(4) , respectively. The irreducibility constraints for these superfields automatically follow from appropriate superconformal covariant conditions on the Cartan superforms. The N=8 superconformal transformations of the superspace coordinates and the Goldstone superfields are explicitly given. Interestingly, each N=8 supermultiplet admits two different off-shell N=4 decompositions, with different N=4 superconformal subgroups SU(1,1|2) and OSp(4 ★|2) of OSp(4 ★|4) being manifest as superconformal symmetries of the corresponding N=4, d=1 superspaces. We present the actions for all such N=4 splittings of the N=8 multiplets considered.

Symmetries of string, M-and F-theories

The d = 10 type II string theories, d = 11 M-theory and d = 12 F-theory have the same symmetry group. It can be viewed either as asubgroup of a conformal group OSp(1|64) or as a contraction ofOSp(1|32). The theories are related by different identifications of theirsymmetry operators as generators of OSp(1|32). T- and S-dualities arerecognized as redefinitions of generators. Some (s,t) signatures ofspacetime allow reality conditions on the generators. All of those that allowa real structure are related again by redefinitions within the algebra, dueto the fact that the algebra OSp(1|32) has only one real realization. Theredefinitions include space/space, time/time and space/time dualities. Afurther distinction between the theories is made by the identification ofthe translation generator. This distinguishes various versions of type IIstring theories, in particular the so-called *-theories, characterized bythe fact that the P0 generator is not the (unique) positive-definiteenergy operator in the algebra.

Supersymmetric two-time physics

We construct an Sp(2,R) gauge invariant particle action which possesses manifest space-time SO(d,2) symmetry, global supersymmetry and kappa supersymmetry. The global and local supersymmetries are non-abelian generalizations of Poincare type supersymmetries and are consistent with the presence of two timelike dimensions. In particular, this action provides a unified and explicit superparticle representation of the superconformal groups OSp(N/4), SU(2,2/N) and OSp(8*/N) which underlie various AdS/CFT dualities in M/string theory. By making diverse Sp(2,R) gauge choices our action reduces to diverse one-time physics systems, one of which is the ordinary (one-time) massless superparticle with superconformal symmetry that we discuss explicitly. We show how to generalize our approach to the case of superalgebras, such as OSp(1/32), which do not have direct space-time interpretations in terms of only zero branes, but may be realizable in the presence of p-branes.

Superconformal symmetry in six dimensions and its reduction to four

Superconformal symmetry in six dimensions is analyzed in terms of coordinate transformations on superspace. A superconformal Killing equation is derived and its solutions are identified in terms of supertranslations, dilations, Lorentz transformations, R-symmetry transformations and special superconformal transformations. The full superconformal symmetry, which is shown to form the group OSp(2, 6| N), is possible only if the supersymmetry algebra has N spinorial generators of the same chirality, corresponding to ( N,0) supersymmetry. The R-symmetry group is then Sp( N) and the corresponding superspace is R 6/vb8n . We define superinversion as a map to the associated superspace of opposite chirality. General formulae for two- and three-point correlation functions of quasi-primary superfields are exhibited. The superconformal group in six dimensions is reduced to a corresponding extended superconformal group in four dimensions. Superconformally covariant differential operators are also discussed.

Supersymmetry and supercoherent states of a nonrelativistic free particle

Coordinate atypical representation of the orthosymplectic superalgebra osp (2/2) in a Hilbert superspace of square integrable functions constructed in a special way is given. The quantum nonrelativistic free particle Hamiltonian is an element of this superalgebra which turns out to be a dynamical superalgebra for this system. The supercoherent states, defined by means of a supergroup displacement operator, are explicitly constructed. These are the coordinate representation of the known atypical abstract super group OSp (2/2) coherent states. We interpret obtained results from the classical mechanics viewpoint as a model of classical particle which is immovable in the even sector of the phase superspace and is in rectilinear movement (in the appropriate coordinate system) in its odd sector.

Osp(1|2) Chern-Simons gauge theory as 2DN=1induced supergravity

We demonstrate the close relationship between Chern-Simons gauge theory with gauge group Osp(1|2) and N=1 induced supergravity in two dimensions. More precisely, the inner product of the physical states in the former yields the partition function of the latter evaluated in the Wess-Zumino supergauge. It is also shown that the moduli space of flat Osp(1|2) connections naturally includes super Teichm\"uller space of super Riemann surfaces.