Tristan Hübsch Department of Physics and Astronomy Howard University, Washington DC

Transcription

1 Tristan Hübsch Department of Physics and Astronomy Howard University, Washington DC In collaborations with: S.J. Gates, Jr., C. Doran, M. Faux, K. Iga, G. Landweber, R. Miller, G. Katona, K. StifBler, J. Hallet

2 Error-Corrected Off-Shell Supermultiplets! Program!! Off- Shell Supermultiplets & ClassiBication! Off- Shell Supermultiplets: Worldline Perspective! Not Your Father s Lie Algebra Representations! Supersymmetry is highly degenerate! Pictures > 10,000 Equations! The Adjoint/Fundamental Representation! Some Simple Examples: Adinkras! Chromotopology and Chromotopography! Supersymmetry, Error- Correction & More! Projections and Their Binary Encryption! Constrained and Quotiented Supermultiplets! and Many Other Supermultiplets 2

3 Off-Shell Supermultiplets & Classification! Off-Shell Supermultiplets!! Nature is quantum; we need partition functionals! where the Bields must not be subject to any spacetime differential equation that could be derived as an equation of motion! Non- differential constraints are OK: they do not propagate! Off- Shell Supermultiplets! Off- shell component Bields! that form a complete orbit of the supersymmetry algebra! φ, Q 1 (φ), Q 2 (φ),, Q 1 (Q 2 (φ)), Q 1 (Q 3 (φ)), Q 1 (Q 2 ( Q N (φ))),! for every φ: (2 N 1, 2 N 1 ) component Bields! E.g.: 4d spacetime è N = 4, (8,8) component Bields! but, chiral super4ields are only half as large 3

4 Off-Shell Supermultiplets & Classification! The Worldline Perspective!! Restrict (dimensionally reduce) to the worldline! Must be included in any (d > 1 spacetime) physical theory! Is present in the Hilbert space of any Bield theory! May well be the underlying M- theory! extends to the worldsheet [ bow- ties obstruction; SJG.Jr. & TH]! Worldline supersymmetry w/o central charges! { Q I, Q J } = 2 δ IJ H and [ H, Q I ] = 0, for all I, J = 1, 2, 3 N.! Lorentz symmetry: Spin(1, d 1) è Spin(1,0) = Z 2 (boson/fermion)! No rotations, boosts, component Bield mixing! Full Spin(1, d 1) etc. may be reconstructed afterwards! by mixing component Bields, dimension- by- dimension! (Q I ) 2 = i d/dt, for each I = 1, 2, 3,, N.! In supermultiplets, (Q I ) 2 1 (Bields as Taylor series/towers) 4

5 Off-Shell Supermultiplets & Classification! Not Your Father s Lie Algebra Representations!! But but irreps of Lie algebras are well known!! As well as super- algebras and their on- shell representations!! But, NOT off- shell representations.! Consider V j := { j,m>, m j } eigenspace of J 2 in su(2),! built from eigenspaces of J 3, mutually commuting generators.! So, in { Q I, Q J } = 2δ IJ H, [ Q I, H ] = 0,! The generator that commutes with everyone is H! but, eigenstates of H satisfy an ODE, EoM, are classical.! We could Biber them over the energy- momentum space! (quantum/free Bields sheaves of classical Bields)! representations Biltered Clifford supermodules! Happily, there is a more user- friendly approach 5

6 Off-Shell Supermultiplets & Classification! Not Your Father s Lie Algebra Representations!! For the Lie algebra congnoscenti:! Standard: Lie algebra = {H α, E α, E α+β, }! [ H α, E α ] = α E α and [ E α, E β ] = N α+β E α+β for α β! [ H, Q I ] = 0 E α and { Q I, Q J } = 0 Q I+J for I J! [ E α, E α ] = 2H α for each α, vs. { Q I, Q I } = 2H, for all I = 1,2,3,! Supersymmetry is a highly degenerate graded algebra! Just in case you are not yet convinced:! Write [ X a, X b ] = i f ab c X c.! The Killing metric g ab := f ac d f bd c! Write H = X 0, and Q I = X I ; f IJ 0 = 2i 0, all other f ab c = 0,! so the Killing metric is identically zero.! Even Tr[ X a X b ] = ( )H, è ODE, EoM 6

7 Pictures > 10,000 Equations! The Adjoint/Fundamental Supermultiplets!! Since, { Q I, Q J } = 2δ IJ H, avoid (Q I ) 2 and use! φ, Q 1 (φ), Q 2 (φ),, Q 1 (Q 2 (φ)), Q 1 (Q 3 (φ)), Q 1 (Q 2 ( Q N (φ)))! All of the form Q b (φ), where b is an N- digit binary number! All have the structure (chromotopography) of an N- cube height = mass- dimension N =1 N =2 N =3 N =4 7

8 Pictures > 10,000 Equations! In case you were not convinced!! Even just for N = 2: Q 1 = i 1, F Q 2 = i 2, Q 1 1 =., Q 2 1 = F, Q 1 2 = F, Q 2 2 =., Q 1 F = i. 2, 1 2 Q 2 F = i. 1,! To be precise, N 2 N equations! which is >10,000 when N > 10.! and is 2,048 for N = 8 (double supersymmetry in 4d)! No need to write them all out. There will be no quiz. 8

9 Pictures > 10,000 Equations! Not Your Father s Lie Algebra Representations!! By comparison, the familiar 3 of su(3) would have 3 nodes: raising Cartan lowering 9 In Supersymmetry Q I = Q I (QI) 2 = H (same) raising = lowering degeneracy

10 Pictures > 10,000 Equations! Oh, by the way!! Graphical depiction of supersymmetry transformations! (just as many other methods in physics and mathematics)! has been done routinely, and all the time! However, the practice has not been formalized until Pietro Fré: Introduction to harmonic expansions on coset manifolds and in particular on coset manifolds with Killing spinors, in Supersymmetry and supergravity 1984 (Trieste, 1984), p , (World Sci. Pub., Singapore, 1984). C.F. Doran, M.G. Faux, S.J. Gates, Jr., T. Hübsch, K.M. Iga and G.D. Landweber: On Graph- Theoretic Identi4ications of Adinkras, Supersymmetry Representations and Super4ields, Int. J. Mod. Phys. A22 (2007) , arxiv:math-ph/ ! 10

11 Pictures > 10,000 Equations! Some Simple Examples: Adinkras!! 4d spacetime simple supersymmetry: N = 4! The vector superbield! B4 Wess- Zumino gauge! Real, unconstrained, unreduced, ungauged, unrestricted, un ed = intact supermultiplet! What can we do with it?! Reduce using the D I s! { D I, D J } = 2 δ IJ H, [ H, D I ] = 0,! { Q I, D J } = 0.! Write D I - equations! which are not d/dt- equations. 11

12 Pictures > 10,000 Equations! Chromotopology and Chromotopography!! For example, DeBine quasi- projectors P IJKL ± := D I D J ± ½ ε IJ KL D K D L (P IJKL± ) 2 = H 2 P IJKL ± DeBine S ± := { P IJKL± (SM) = 0 } S + 12

13 Pictures > 10,000 Equations! Chromotopology and Chromotopography!! Example, cont d: chiral S + S + F 1! f 2 := Z dtf twisted chiral S 13

14 Supersymmetry, Error-Correction & More! Projections and Their Binary Encryption!! What about more supersymmetries?! Impose P IJKL ± := D I D J ± ½ ε IJ KL D K D L for some Bixed I, J, K, L! these are quasi- projectors only if {IJ } = 0 (mod 4)! and [ P IJKL ±, P MNPQ ± ] 0, if {IJKL} {MNPQ} = 0 (mod 2)! Write D b := D I D J D K D L ; b has 1 s at I, J, K, L positions, 0 otherwise! E.g.: [111100] = D 1 D 2 D 3 D 4 P 1234±, [110011] P 1256±,! [ P 1234 ±, P 1256 ± ] = H 2 P 3456 ± 0.! That is, [111100]+[110011] = [221111] [001111]! Doubly even linear block code d 6 : error- correcting encryption (?!)! Classify these codes = classify minimal supermultiplets! up to φ è (dφ/dt) and inverse component Bield redebinitions! How hard could that be? 14

15 Supersymmetry, Error-Correction & More! Projections and Their Binary Encryption!! Let s start: Note the non- uniqueness (& non- linearity) of the sub- code embeddings. 15 Algebraically distinct ways to construct the same supermultiplet

16 Supersymmetry, Error-Correction & More! Projections and Their Binary Encryption!! For N 8, this becomes even richer: The (well- known) only two even, unimodular lattices 16 Lie algebras: SO(32) & E8 E8#

17 Supersymmetry, Error-Correction & More! Projections and Their Binary Encryption!! And then it becomes really, Really, REALLY hard: 17

18 Supersymmetry, Error-Correction & More! Projections and Their Binary Encryption!! So, can we classify these codes? w/robert Miller N = # # # # # # # # # # # # # # # # # # # # # # # # # # # # # 2 # # # # # # # # # # # # # # # # # # # # # # # # # # # 3 # # # # # # # # # # # # # # # # # # # # # # # # # # 4 # # # # # # # # # # # # # # # # # # # # # # # # + 5 # # # # # # # # # # # # # # # # # # # + + E8 6 # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # k 8 # # # # # # # # # # # # # E8 E8, E codes# 9 # # # # # # # # # # # # # # # # # # # # # # # # # # # codes# 13 # # # # # + 15 # # 16 # codes#

19 Supersymmetry, Error-Correction & More! Constrained and Quotiented Supermultiplets!! More generally,! With two off- shell supermultiplets, D b : X è Y, debine:! Kernel, A := { X: D b (X) = 0 }, (e.g., chiral supermultiplet)! Cokernel, B := { Y (mod D b (X)) }, (e.g., vector sm. in WZ gauge)! Already for N = 3, 0 This debines new (and ever bigger) supermultiplets: 1, C A = the several steps in the procedure (14) (21), level by level and starting from th! depicts (3 Y)/(D I X) = a new (5 8 3)- component supermultiplet! Would you have preferred the 3 (3 2 3 ) = 72 equations? 19

20 Supersymmetry, Error-Correction & More! Constrained and Quotiented Supermultiplets!! And, an N = 4 (in fact, 4d simple) supersymmetry example:! The complex linear supermultiplet/superbield [SJG.Jr, JH, TH & KS]: X 24 X 23 X 21 =X 43 =X 41 =Y 41 =Y 43 Y 21 Y 23 Y K Figure 2! Bigger supermultiplets = networks of Adinkras, connected by one- way (BRST- like) supersymmetry transformations 20 L

21 Supersymmetry, Error-Correction & More! Constrained and Quotiented Supermultiplets!! And, an N = 4 (in fact, 4d simple) supersymmetry example:! The complex linear supermultiplet/superbield [SJG.Jr, JH, TH & KS]: X 24 X 23 X 21 =X 43 =X 41 =Y 41 =Y 43 Y 21 Y 23 Y K! Bigger supermultiplets = networks of Adinkras, connected by one- way (BRST- like) supersymmetry transformations 21 L

22 Supersymmetry, Error-Correction & More! and Many Other Supermultiplets!! Weyl Construction 0 Lie algebras: R 1 R 2 = R 3 R 4 R k and R 1 R 2 unique.! Not so in supersymmetry: 1, C A = the several! Thus, steps Y in the procedure (14) (21), level by level and starting from th 1 Y 2 Y 3 reduces also differently; it is not unique.! In turn, already for N = 1, (" # apple 0 apple 00 >== apple (2i $ apple 00 ( 0 00 ) ( ) ( ) ( ) 0 00 ( 0 00 ) The parametrization is this simple only in the simplest, N = 1 case 22

23 Supersymmetry, Error-Correction & More! and Many Other Supermultiplets!! Weyl- esque Construction:! Form Y 1 Y 2, and reduce as D b - constraining/gauging [GK & TH]: Lie algebra irrreps (Weyl construction) Off-shell supermultiplets Starting object(s) and their depiction Combining operator fundamental irrep ( ) Adinkras ( & & ) Reduction methods Resulting objects and their depiction Young symmetrization, traces w/inv. tensors arbitrarily large irreps, & their Young tableaux (1-quadrant graphs) Construction 3.1: ker(µ) and cok(µ) of supersymmetric maps networks of otherwise proper Adinkras, connected by one-way Q-action edges! IndeBinite, but unlike Lie algebras, no proof of completeness 23

24 Adinkras, off- shell supermultiplets/superbields & classibication: arxiv : math- ph/ , hep- th/ , , , , , From worldline to worldsheets and beyond: arxiv : , , Constructing actions using Adinkras arxiv : hep- th/ , , Symmetries and structures in Adinkras: arxiv : , , , Reducibility and new supermultiplets: arxiv : , , Rigorous mathematical foundation: arxiv : math- ph/

25 Thanks!! Tristan Hubsch Department of Physics and Astronomy Howard University, Washington DC