Development of Algorithm for Off-shell Completion of 1D Supermultiplets
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1 Development of Algorithm for Off-shell Completion of 1D Supermultiplets Delilah Gates University of Maryland, College Park Center for String and Particle Theory
2 The Problem: SUSY Auxiliary Off-shell SUSY Off-shell completion: Field Problem Adding auxiliary fields to a supersymmetric theory such that the supersymmetric algebra can be satisfied SUSY Auxiliary Off-shell Field Problem: There is not well defined method for off-shell completion of a general theory for which the supersymmetric algebra is not satisfied Seeming contradiction: Prime Example: 4D, N = 4 & 10D, N=1 Maxwell vector supermultiplets Superspace argument: can be embedded in within a formulation involving unconstrained super p-forms. Thus there should existence some type of off-shell formulation. Siegel & Rocek: No-Go Theorem Applying a simple counting argument to all supermultiplets, we find that for the N = 4 super Yang-Mills theory the auxiliary field problem cannot have a solution within any previously known framework. We propose alternatives. Solution: Develop new frameworks
3 Method of Study: RADIO Start: on-shell D-dimensional N-extended theory R: reduced to a 1D on-shell theory AD: perform automorphic duality transformations I: integrate additional fields into 1D representations O: oxide completed theory back up to D dimensions
4 Reduction & Valise Supermultplets 0-brane reduction: reduce a linear theory along one time like direction field redefinitions produce representation as valise supermultiplet: Valise formulation of supersymmetric algebra: SO(1,3) replaced by SU(2) SU(2) generators of spacial rotations: generators of extended SU(2) R-symmetry:
5 4D, N=1 Chiral Multiplet Off-shell 4D, N=1 Chiral Multiplet 0-brane reduction Vectors: On-shell
6 4D, N=1 Chiral Multiplet L and R Matrices Off-shell On-shell
7 Adinkra Networks & SUSY Off-shell Auxiliary Field Problem Adinkra Network vector of bosonic fields, and vector of fermionic fields, supercharges, L-matices, R-matrices assumes nothing about Lorentz invariance: L and R matrices need not be created from SU(2) generators Off-shell condition for valise adinkra: GR(d, N ) or Garden Algebra result of observation about valise matrices from 4D, N=1 theories where: Off-shell completion Deform L and R matrices be n x n matrix with n a multiple of 4 Satisfy Graden Algebra
8 Off-shell completion of 4D, N=1 Chiral Multiplet on 0-brane via Adinkra Network Start: On-shell L and R matrices Off-shell Conditions Deform L and R matrices n x n where n is multiple of 4 (deform matrices with real variables) Demanding L and R matrices satisfy Garden Algebra gives with all others zero Thus we recover the Off-shell 4D, N=1 Chiral Multiplet on the 0-brane up to field redefinitions (F -> ± F & G -> ±G )
9 Summary and Moving Forward Summary: We have produced a well defined method by which we reduced the off-shell auxiliary field problem to a 1D problem by taking a D-dimensional N-Extended on-shell theory reduced it to along on dimension (R), made it into a valise supermultiplet (AD), and integrated in new yields that produced an off-shell representation of the 1D theory (I). Success Next: We have applied this method to on-shell 4D, N=1 chiral and vector multiplets (not shown here), recovering the well known off-shell representations of these theories on the 0-brane, thus giving us promise that this method is a viable step in a new method of off-shell completion for general on-shell supersymmetric theories. Have applied this method with an extra constraint add to the Garden Algebra to on-shell 4D, N=2 tensor and vector and recovered recovering the well known off-shell representations of these theories on the 0-brane We want to test other 4D, N=1 & N=2 theories with well known off-shell and on-shell representations Develop method for the oxidation of off-shell theories on 0-brane back into fully realized higher dimensional theories Apply to on-shell theories for which no none off-shell competition exists A glimpse into a larger question: General Cryptographic Problem Analogy to Adinkra Network Auxiliary Field Problem
10 General Cryptographic Problem Analogy to Auxiliary Field Problem Consider more generally the a set of matrices where: I=1,,N p is a fixed integer r1 to r4p can range from 0 to 4p-1, s1 to s4p can range from 0 to 4p-1 Encryption corresponds to setting l and l-hat parameters to 0. Understanding this problem is equivalent to solving the adinkra network auxiliary problem in full generality
11 Thank you for listening References Mathew. Calkins, D. E. A. Gates, S. James Gates,Jr., and William M. Golding. Think Different: Applying the Old Macintosh Mantra to the Computability of the SUSY Auxiliary Field Problem. JHEP04 (2015) 056 S. J. Gates Jr., and S. Vashakidze, On D=10, N=1 Supersymmetry, Superspace Geometry and Superstring Effects, Nucl. Phys. B291 (1987) 172 W. Siegel, and M. Rocek, On Off-shell Supermultiplets, Phys. Lett. B105 (1981) 275. M. Faux, S. J. Gates Jr., Adinkras: A Graphical Technology for Supersymmetric Representation Theory, Phys. Rev. D71 (2005) C. F. Doran, M. G. Faux, S. J. Gates, Jr., T. Hu bsch, K. M. Iga, and G. D. Landweber, On graph-theoretic identifications of Adinkras, supersym- metry representations and superfields, Int. J. Mod. Phys. A22 (2007) , [math-ph/ ]
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