Mathematical Surprises from off- shell SUSY Representation Theory

Mathematical Surprises from off- 

shell SUSY Representation Theory 

Kory Stiffler 

Miami 2011 

Based on 4D, N = 1 Supersymmetry Genomics (II), 1112.2147 [hep-th] 

S. James Gates, Jr.†, Jared Hallett‡, James Parker†, 

Vincent G. J. Rodgers∗, and Kory Stiffler† 

and 

The Real Anatomy of Complex Linear Superfields (manuscript in progress) 

S.James Gates, Jr. †, Jared Hallett‡, Tristan Hübsch**, and Kory Stiffler† 

†Center for String and Particle Theory, Department of Physics, University of Maryland, College Park MD 

‡Department of Mathematics, Williams College, Williamstown, MA 

∗Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 

** Dept. of Physics, University of Central Florida, Orlando, FL and 

Dept. of Physics & Astronomy, Howard University, Washington, DC

Kory Stiffler - Mathematical Surprises from off-shell SUSY Representation Theory 1112.2147 

Review: Representation Theory for 

Compact Lie Groups 

• SU(2) representation size: d= 2 j + 1, 

j = 0,½,1,… 

• j = 0, trivial 

• j= ½, d=2 fundamental representation, e.g. 

acted on by 2x2 matrices generated by Pauli 

spin matrices 

• j=1, d=3 Adjoint representation 

• ……more


Review: Representation Theory for 

Compact Lie Groups 

• SU(3) representation size: 

d = (p+1)(q+1)(p+q+2)/2, p,q =0,1,2,…. 

• p,q = 0: d=1 trivial 

• p=1, q =0: d=3 fundamental: e.g. acted on by 

3x3 matrices generated by Gell-Mann 

matrices 

• p=0, q =1: d=3 anti-fundamental 

• p=1,q=1: d=8 adjoint 

• p=2,q=0 or p=0, q=2: d=6 sextet,…..more


Representation Theory and the 

Standard Model 

SU(3) color 

Representation Fundamental (matter) Adjoint (force carriers) 

Particles Quarks gluons 

SU(2) Left 

Representation Fundamental (l. h. matter) Adjoint (force carriers) 

Particles Quarks, leptons weak bosons: W ± , Z 

Generally speaking, matter transforms in the fundamental representation, force 

carriers in the adjoint.


Supersymmetric (SUSY) 

Extension of Standard Model (SM)? 

• sleptons, squarks, gluinos, winos, binos, 

Higgsinos, more? 

• No off-shell representation theory of 

supersymmetry exists, SUSY extensions of SM 

would benefit from its construction


SUSY Representation Theory, 

Currently Under Construction 

F: fundamental, A: Anti-fundamental, O: Octet(adjoint), D: Decuplet 

C: chiral multiplet, V:vector multiplet, T: tensor multiplet, RSS: real scalar superfield 

multiplet, CLS: complex linear superfield multiplet 

d: size of representation 

p, q: 0,1,2,…. 

The topic of the rest of the talk are n c and n t: the number of cis- and trans-Adinkras, 

respectively, in the SUSY representation.


Adinkras: Chiral Multiplet (CM) 

Lagrangian is supersymmetric 

with respect to the transformation laws: 

which satisfy:


Adinkras: CM One Dimensional 

Reduction 

• Consider only time dependent fields (prime (‘) denotes time derivative) 

• SUSY transformation laws become


Adinkras (West Africa) : 

“symbols with hidden meaning” 




Adinkras (West Africa) : 

“symbols with hidden meaning” 




Build your own Adinkra in 6 easy steps! 

e.g. CM 

• Color encodes transformation identity: D 1, D 2, D 3, D 4



e.g. CM 

• Color encodes transformation identity: D 1, D 2, D 3, D 4 

• D-operator on lower node yields upper node



e.g. CM 


• D-operator on lower node yields upper node 

• D-operator on upper node yields derivative of lower node



e.g. CM 



• D-operator on upper node yields derivative of lower node 

• So higher node has ½ more engineering dimensions than lower node



e.g. CM 




• So higher node has ½ more engineering dimensions than lower node 

• Dashed line encodes an overall minus sign in both transformations



e.g. CM 




• So higher node has ½ more engineering dimensions than lower node 

• Dashed line encodes an overall minus sign in both transformations 

• Transformations from fermion to boson have an additional factor of i 

(imaginary number) multiplying boson.


Surprise 1: Can Succinctly Represent 

the CM as an Adinkra picture


Where: 

Adinkra Matrices 

We can write these transformation laws as:


Where: 


We can write these transformation laws as:


Where: 


We can write these transformation laws as: 

Important Result: L I=(R I) T =(R I) -1 

without which, the representation would not be Adinkraic:


Where: 

We also have: 

Adinkra Matrices and the 

Garden Algebra 

We can write these transformation laws as: 

Important Result: L I=(R I) T =(R I) -1 

without which, the representation would not be Adinkraic 

We call this a GR(d,N) Algebra: General Real d x d matrices encoding N SUSY’s. 

• AKA: A Garden Algebra which is essentially a real representation of a Clifford algebra.


One Basis is as Good as the Next 

We would like to preserve the basic features



We would like to preserve the basic features 

• Transformation Laws: 

• Adinkraic: L I=(R I) T =(R I) -1 

• Garden Algebra:



We would like to preserve the basic features 

• Transformation Laws: 

• Adinkraic: L I=(R I) T =(R I) -1 

• Garden Algebra: 

Can accomplish this with simultaneous orthogonal transformations on: 

• bosons: 

• fermions: 

• Adinkra matrices: 

T 

O , O O I 

B B B 

T T 

O , O O I 

F F F 

L OL O , R ORO 

T T 

I B I F I F I B


e.g.: the Valise Adinkra for CM 

Valise Adinkra: An adinkra with one row each of bosons and fermions, drawn fermions 

over bosons. 

Via Integrating to lower F and G and applying Orthogonal Transformations: 

T 

T T 

O , O O I O , O O I 

B B B 

L OL O , R ORO 

T T 

I B I F I F I B 

F F F


Adinkras: Vector Multiplet (VM) 

Lagrangian is supersymmetric 

with respect to the transformation laws:



1-D reduced SUSY transformation laws 

can be succinctly written canonically as the Adinkra




vector multiplet 

valise Adinkra:


Compare with the Adinkra for the CM 

(flip back and forth between previous slide, watch the colors, are these the same Adinkra?) 


chiral multiplet 

valise Adinkra:



valise Adinkra : 


valise Adinkra: 

Surprise 2: 

“Reflection” about 

“orange” axis



valise Adinkra : 


valise Adinkra: 

But are these truly distinct? 

? 

Are they related by O B 

and/or O F 

transformations ?


Chromocharacters: Basis Independent 

Comparison Tools 

Invariant w.r.t. 

orthogonal 

transformations 

O B and O F



P=1: (VM) 



orthogonal 


O B and O F 

(CM)



P=1: (VM) 

P=2: 

(VM) 

(CM) 



orthogonal 


O B and O F 

(CM)



P=1: 

P=2: 

(VM) 

(CM) 


(VM) 


orthogonal 


O B and O F 

(CM) 

The sign in front of the epsilon shows that the valise Adinkras for the 

vector and chiral multiplets are indeed distinct: no O B and O F 

transformations exist between them: they are indeed distinct!


Draw an analogy: 

Enantiomers 

chiral multiplet valise Adinkra 

“Reflection” 

about 

“orange” axis 

vector multiplet valise Adinkra 

Draw an analogy from chemistry: molecules that are identical up to a spatial 

reflection are called cis- and trans-enantiomers (mirror images) 

Spatial Reflections


Define: 

‘SUSY enantiomer’ numbers 


about 


(nc = 1, nt = 0) (nc = 0, nt = 1) 

cis-Adinkra trans-Adinkra 

Draw an analogy from chemistry: molecules that are identical up to a spatial 

reflection are called cis- and trans-enantiomers (mirror images) 

Spatial Reflections


Surprise 3: A conjecture: 

All 4D N=1 SUSY off-shell multiplets can be represented by their SUSY enantiomer numbers: 

the numbers of its n c cis- and n t trans-valise Adinkras 


about 


(n c = 1, n t = 0) (n c = 0, n t = 1) 


Conjecture also: all chromocharacters for 4D N=1 SUSY off-shell Adinkras


Use Tensor Multiplet to test Conjecture: 




about 


(n c = 1, n t = 0) (n c = 0, n t = 1) 


The tensor multiplet:






about 


(n c = 1, n t = 0) (n c = 0, n t = 1) 


The tensor multiplet: 1-D reduction






about 


(n c = 1, n t = 0) (n c = 0, n t = 1) 


The tensor multiplet can be represented by the trans-Adinkra and has (n c = 0, n t = 1)


Conjecture works for Tensor Multiplet: 




about 


(n c = 1, n t = 0) (n c = 0, n t = 1) 


The tensor multiplet can be represented by the trans-Adinkra and has (n c = 0, n t = 1), and 

of course, it’s chromocharacters match up with these numbers:


Data Supporting Surprise 3: A conjecture: 

All 4D N=1 SUSY off-shell multiplets can be represented by their SUSY enantiomer numbers, the 

numbers of its n c cis- and n t trans-Adinkras 

Conjecture works for the following 4D N=1 off shell multiplets each with 

d = 4 fermion = 4 boson d.o.f: 

Chiral Multiplet 

(n c = 1, n t = 0) 

Vector Multiplet 

(n c = 0, n t = 1) 

Tensor Multiplet 

(n c = 0, n t = 1)


Data Supporting Surprise 3: A conjecture: 

All 4D N=1 SUSY off-shell multiplets can be represented by their SUSY enantiomer numbers, the 

numbers of its n c cis- and n t trans-Adinkras 

Conjecture works for the following 4D N=1 off shell multiplets each with 

d = 4 fermion = 4 boson d.o.f: 

Chiral Multiplet 

(n c = 1, n t = 0) 

Vector Multiplet 

(n c = 0, n t = 1) 

Tensor Multiplet 

(n c = 0, n t = 1) 

What about more complicated multiplets? 

• Real Scalar Superfield Multiplet (d=8 fermion = 8 boson d.o.f.) 

• Complex Linear Superfield Multiplet (d=12 fermion =12 boson d.o.f.)


Real Scalar Superfield (RSS)


Real Scalar Superfield (RSS) 

bosons: fermions:


Real Scalar Superfield (RSS): 

Tesseract Adinkra 

* Notice, we have had to 

combine fields in nodes 

to make this Adinkraic



Valise Adinkra 

(n c = 1, n t = 0) + (n c = 0, n t = 1) 

= (n c = 1, n t = 1)




Check chromocharacters: 

(n c = 1, n t = 1)





(n c = 1, n t = 1)





(n c = 1, n t = 1)





(n c = 1, n t = 1)





(n c = 1, n t = 1)


Complex Linear Superfield (CLS)


Complex Linear Superfield (CLS) 

bosons: fermions:




(n c = 1, n t = 0) + (n c = 0, n t = 1) + (n c = 0, n t = 1) 

= (n c = 1, n t = 2)




*Notice the node content has rather non-trivial linear combinations of the fields, this seems to 

be the main challenge in finding these larger representations 

(n c = 1, n t = 0) + (n c = 0, n t = 1) + (n c = 0, n t = 1) 

= (n c = 1, n t = 2)







(n c = 1, n t = 2)







(n c = 1, n t = 2)







(n c = 1, n t = 2)







(n c = 1, n t = 2)







(n c = 1, n t = 2)


Summary 

• Standard model: matter transforms in the fundamental representation, force carriers in the 

adjoint. 

• No representation theory for off-shell supersymmetry exists 

• SUSY extensions of the Standard Model would benefit from its construction 

F: fundamental, A: Anti-fundamental, O: Octet(adjoint), D: Decuplet 

C: chiral multiplet, V:vector multiplet, T: tensor multiplet, RSS: real scalar superfield 

multiplet, CLS: complex linear superfield multiplet 

d: size of representation 

p, q: 0,1,2,…. 

n c and n t: the number of cis- and trans-Adinkras, respectively, in the SUSY representation.


Summary 

(n c = 1, n t = 0) (n c = 0, n t = 1) (n c = 0, n t = 1) 

(n c = 1, n t = 1) 

(nc = 1, nt = 2) 

+ other 4D N=1 SUSY Multiplets? + extensions to d=10 

YM, SG, Superstring/M-theory? To be continued…


Thank You 

• You the audience 

• Co-authors: Profs. Gates, Rodgers, and Hübsch, and Jared 

Hallett and James Parker 

• Prof. Curtright , the organizers, and the U 

• Lago Mar Resort 

• Funding: Endowment of John S. Toll Professorship, the 

University of Maryland, CSPT, MCFP, NSF Grant PHY- 

0354401, MLK visiting professorship and MIT Center for 

Theoretical Physics 

• Leo Rodgriguez, Abdul Khan, many SSTPRS summer school 

students, and Universities of Iowa and Maryland for hosting 

• Gred Landweber, creator of Adinkramat ©2008


Back-Up Slides


Canonical Adinkra to Valise Adinkra: 

Chiral Multiplet


Canonical Adinkra to Valise Adinkra: 

Vector Multiplet

Mathematical Surprises from off- shell SUSY Representation Theory

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