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Register a new userAdvening to Adynkrafields: Young Tableaux to Component Fields of the 10D, N = 1 Scalar Superfield
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Starting from higher dimensional adinkras constructed with nodes referenced by Dynkin Labels, we define adynkras. These suggest a computationally direct way to describe the component fields contained within supermultiplets in all superspaces. We explicitly discuss the cases of ten dimensional superspaces. We show this is possible by replacing conventional $theta$-expansions by expansions over Young Tableaux and component fields by Dynkin Labels. Without the need to introduce $sigma$-matrices, this permits rapid passages from Adynkras $to$ Young Tableaux $to$ Component Field Index Structures for both bosonic and fermionic fields while increasing computational efficiency compared to the starting point that uses superfields. In order to reach our goal, this work introduces a new graphical method, tying rules, that provides an alternative to Littlewoods 1950 mathematical results which proved branching rules result from using a specific Schur function series. The ultimate point of this line of reasoning is the introduction of mathematical expansions based on Young Tableaux and that are algorithmically superior to superfields. The expansions are given the name of adynkrafields as they combine the concepts of adinkras and Dynkin Labels.
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The first complete and explicit SO(1,9) Lorentz descriptions of all component fields contained in $mathcal{N} = 1$, $mathcal{N} = 2$A, and $mathcal{N} = 2$B unconstrained scalar 10D superfields are presented. These are made possible by the discovery of the relation of the superfield component expansion as a consequence of the branching rules of irreducible representations in one ordinary Lie algebra into one of its Lie subalgebras. Adinkra graphs for ten dimensional superspaces are defined for the first time, whose nodes depict spin bundle representations of SO(1,9). An analog of Breitenlohners approach is implemented to scan for superfields that contain graviton(s) and gravitino(s), which are the candidates for the prepotential superfields of 10D off-shell supergravity theories and separately abelian Yang-Mills theories are similarly treated. Version three contains additional content, both historical and conceptual, which broaden the reach of the scan in the 10D Yang-Mills case.
The details of Lagrangian description of irreducible integer higher-spin representations of the Poincare group with an Young tableaux $Y[hat{s}_1,hat{s}_2]$ having $2$ columns are considered for Bose particles propagated on an arbitrary dimensional Minkowski space-time. The procedure is based, first, on using of an auxiliary Fock space generated by Fermi oscillators (antisymmetric basis), second, on construction of the Verma module and finding auxiliary oscillator realization for $sl(2)oplus sl(2)$ algebra which encodes the second-class operator constraints subsystem in the HS symmetry superalgebra. Application of an BRST-BFV receipt permits to reproduce gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive mixed-antisymmetric bosonic fields of any spin with appropriate number of gauge and Stueckelberg fields. The general prescription possesses by the possibility to derive constrained Lagrangians with only BRST-invariant extended algebraic constraints which describes the Poincare group irreducible representations in terms of mixed-antisymmetric tensor fields with 2 group indices.
We present a four-dimensional (4D) ${cal N}=1$ superfield description of supersymmetric Yang-Mills (SYM) theory in ten-dimensional (10D) spacetime with certain magnetic fluxes in compactified extra dimensions preserving partial ${cal N}=1$ supersymmetry out of full ${cal N}=4$. We derive a 4D effective action in ${cal N}=1$ superspace directly from the 10D superfield action via dimensional reduction, and identify its dependence on dilaton and geometric moduli superfields. A concrete model for three generations of quark and lepton superfields are also shown. Our formulation would be useful for building various phenomenological models based on magnetized SYM theories or D-branes.
The superspace formulation of N=1 conformal supergravity in four dimensions is demonstrated to be equivalent to the conventional component field approach based on the superconformal tensor calculus. The detailed correspondence between two approaches is explicitly given for various quantities; superconformal gauge fields, curvatures and curvature constraints, general conformal multiplets and their transformation laws, and so on. In particular, we carefully analyze the curvature constraints leading to the superconformal algebra and also the superconformal gauge fixing leading to Poincare supergravity since they look rather different between two approaches.
We present a superfield construction of Hamiltonian quantization with N=2 supersymmetry generated by two fermionic charges Q^a. As a byproduct of the analysis we also derive a classically localized path integral from two fermionic objects Sigma^a that can be viewed as ``square roots of the classical bosonic action under the product of a functional Poisson bracket.
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