Superembedding methods for 4d N=1 SCFTs

We extend SO(4,2) covariant lightcone embedding methods of four-dimensional CFTs to N=1 superconformal field theory (SCFT). Manifest superconformal SU(2,2|1) invariance is achieved by realizing 4D superconformal space as a surface embedded in the projective superspace spanned by certain complex chiral supermatrices. Because SU(2,2|1) acts linearly on the ambient space, the constraints on correlators implied by superconformal Ward identities are automatically solved in this formalism. Applications include new, compact expressions for correlation functions containing one anti-chiral superfield and arbitrary chiral superfield insertions, and manifestly invariant expressions for the superconformal cross-ratios that parametrize the four-point function of two chiral and two anti-chiral fields. Superconformal expressions for the leading singularities in the OPE of chiral and anti-chiral operators are also given. Because of covariance, our expressions are valid in any superconformally flat background, e.g., AdS_4 or R times S^3.Comment: 33 pages, clarification of constraints, version to appear in PR

On the biparametric quantum deformation of GL(2) x GL(1)

We study the biparametric quantum deformation of GL(2) x GL(1) and exhibit its cross-product structure. We derive explictly the associated dual algebra, i.e., the quantised universal enveloping algebra employing the R-matrix procedure. This facilitates construction of a bicovariant differential calculus which is also shown to have a cross-product structure. Finally, a Jordanian analogue of the deformation is presented as a cross-product algebra.Comment: 16 pages LaTeX, published in JM

Duality and Representations for New Exotic Bialgebras

We find the exotic matrix bialgebras which correspond to the two non-triangular nonsingular 4x4 R-matrices in the classification of Hietarinta, namely, R_{S0,3} and R_{S1,4}. We find two new exotic bialgebras S03 and S14 which are not deformations of the of the classical algebras of functions on GL(2) or GL(1|1). With this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements. We also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail. We also discuss in detail a special case of R_{S1,4} in which the corresponding algebra turns out to be a special case of the two-parameter quantum group deformation GL_{p,q}(2).Comment: 33 pages, LaTeX2e, using packages: cite,amsfonts,amsmath,subeqn; reference updated; v3: corrections in subsection 3.

RTT relations, a modified braid equation and noncommutative planes

With the known group relations for the elements $(a,b,c,d)$ of a quantum matrix $T$ as input a general solution of the $RTT$ relations is sought without imposing the Yang - Baxter constraint for $R$ or the braid equation for $\hat{R} = PR$. For three biparametric deformatios, $GL_{(p,q)}(2), GL_{(g,h)}(2)$ and $GL_{(q,h)}(1/1)$, the standard,the nonstandard and the hybrid one respectively, $R$ or $\hat{R}$ is found to depend, apart from the two parameters defining the deformation in question, on an extra free parameter $K$,such that only for two values of $K$, given explicitly for each case, one has the braid equation. Arbitray $K$ corresponds to a class (conserving the group relations independent of $K$) of the MQYBE or modified quantum YB equations studied by Gerstenhaber, Giaquinto and Schak. Various properties of the triparametric $\hat{R}(K;p,q)$, $\hat{R}(K;g,h)$ and $\hat{R}(K;q,h)$ are studied. In the larger space of the modified braid equation (MBE) even $\hat{R}(K;p,q)$ can satisfy $\hat{R}^2 = 1$ outside braid equation (BE) subspace. A generalized, $K$- dependent, Hecke condition is satisfied by each 3-parameter $\hat{R}$. The role of $K$ in noncommutative geometries of the $(K;p,q)$,$(K;g,h)$ and $(K;q,h)$ deformed planes is studied. K is found to introduce a "soft symmetry breaking", preserving most interesting properties and leading to new interesting ones. Further aspects to be explored are indicated.Comment: Latex, 17 pages, minor change

Applications of Two-Body Dirac Equations to the Meson Spectrum with Three versus Two Covariant Interactions, SU(3) Mixing, and Comparison to a Quasipotential Approach

In a previous paper Crater and Van Alstine applied the Two Body Dirac equations of constraint dynamics to the meson quark-antiquark bound states using a relativistic extention of the Adler-Piran potential and compared their spectral results to those from other approaches, ones which also considered meson spectroscopy as a whole and not in parts. In this paper we explore in more detail the differences and similarities in an important subset of those approaches, the quasipotential approach. In the earlier paper, the transformation properties of the quark-antiquark potentials were limited to a scalar and an electromagnetic-like four vector, with the former accounting for the confining aspects of the overall potential, and the latter the short range portion. A part of that work consisted of developing a way in which the static Adler-Piran potential was apportioned between those two different types of potentials in addition to covariantization. Here we make a change in this apportionment that leads to a substantial improvement in the resultant spectroscopy by including a time-like confining vector potential over and above the scalar confining one and the electromagnetic-like vector potential. Our fit includes 19 more mesons than the earlier results and we modify the scalar portion of the potential in such a way that allows this formalism to account for the isoscalar mesons {\eta} and {\eta}' not included in the previous work. Continuing the comparisons made in the previous paper with other approaches to meson spectroscopy we examine in this paper the quasipotential approach of Ebert, Faustov, and Galkin for a comparison with our formalism and spectral results.Comment: Revisions of earlier versio

Tensor Operators for Uh(sl(2))

Tensor operators for the Jordanian quantum algebra Uh(sl(2)) are considered. Some explicit examples of them, which are obtained in the boson or fermion realization, are given and their properties are studied. It is also shown that the Wigner-Eckart's theorem can be extended to Uh(sl(2)).Comment: 11pages, LaTeX, to be published in J. Phys.

Twist Deformation of the rank one Lie Superalgebra

The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra $osp(1|2)$. The twist element is the same as for the $sl(2)$ Lie algebra due to the embedding of the $sl(2)$ into the superalgebra $osp(1|2)$. The R-matrix has the direct sum structure in the irreducible representations of $osp(1|2)$. The dual quantum group is defined using the FRT-formalism. It includes the Jordanian quantum group $SL_\xi(2)$ as subalgebra and Grassmann generators as well.Comment: LaTeX, 9 page

Duality for Exotic Bialgebras

In the classification of Hietarinta, three triangular $4\times 4$ $R$-matrices lead, via the FRT formalism, to matrix bialgebras which are not deformations of the trivial one. In this paper, we find the bialgebras which are in duality with these three exotic matrix bialgebras. We note that the $L-T$ duality of FRT is not sufficient for the construction of the bialgebras in duality. We find also the quantum planes corresponding to these bialgebras both by the Wess-Zumino R-matrix method and by Manin's method.Comment: 25 pages, LaTeX2e, using packages: cite, amsfonts, amsmath, subeq

Duality for the Jordanian Matrix Quantum Group GLg,h(2)GL_{g,h}(2)

We find the Hopf algebra $U_{g,h}$ dual to the Jordanian matrix quantum group $GL_{g,h}(2)$. As an algebra it depends only on the sum of the two parameters and is split in two subalgebras: $U'_{g,h}$ (with three generators) and $U(Z)$ (with one generator). The subalgebra $U(Z)$ is a central Hopf subalgebra of $U_{g,h}$. The subalgebra $U'_{g,h}$ is not a Hopf subalgebra and its coalgebra structure depends on both parameters. We discuss also two one-parameter special cases: $g =h$ and $g=-h$. The subalgebra $U'_{h,h}$ is a Hopf algebra and coincides with the algebra introduced by Ohn as the dual of $SL_h(2)$. The subalgebra $U'_{-h,h}$ is isomorphic to $U(sl(2))$ as an algebra but has a nontrivial coalgebra structure and again is not a Hopf subalgebra of $U_{-h,h}$.Comment: plain TeX with harvmac, 16 pages, added Appendix implementing the ACC nonlinear ma

On Jordanian U_{h,\alpha}(gl(2)) Algebra and Its T Matrices Via a Contraction Method

The $R_h^{j_1;j_2}$ matrices of the Jordanian U$_h$(sl(2)) algebra at arbitrary dimensions may be obtained from the corresponding $R_q^{j_1;j_2}$ matrices of the standard $q$-deformed U$_q$(sl(2)) algebra through a contraction technique. By extending this method, the coloured two-parametric ($h, \alpha$) Jordanian $R_{h,\alpha}^{j_1,z_1;j_2,z_2}$ matrices of the U$_{h,\alpha}$(gl(2)) algebra may be derived from the corresponding coloured $R_{q,\lambda}^{j_1,z_1;j_2,z_2}$ matrices of the standard ($q, \lambda$)-deformed U$_{q,\lambda}$(gl(2)) algebra. Moreover, by using the contraction process as a tool, the coloured $T_{h,\alpha}^{j,z}$ matrices for arbitrary ($j, z$) representations of the Jordanian Fun$_{h,\alpha}$(GL(2)) algebra may be extracted from the corresponding $T_{q,\lambda}^{j,z}$ matrices of the standard Fun$_{q,\lambda}$(GL(2)) algebra.Comment: LaTeX, uses amssym.sty, 24 pages, no figure, to be published in Int. J. Mod. Phys.