Updates for Off-Shell Supersymmetric Representation · PDF fileUpdates for Off-Shell Supersymmetric Representation Theory Using Adinkras Kory Stiffler Based on: 4D, N = 1 Supergravity

Updates for Off-Shell Supersymmetric Representation

Theory Using Adinkras

Kory Stiffler

Based on:

4D, N = 1 Supergravity Genomics, 1212.3318 [hep-th]

Isaac Chappell, S. James Gates, Jr., William D. Linch III, James Parker,

Stephen Randall, Alexander Ridgway, and Kory Stiffler

Center for String and Particle Theory,

Department of Physics, University of Maryland,

College Park, MD 20742 USA

Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318

Outline

Why SUSY? Why Off-Shell? Adinkras: an off-shell representations theory? 4D, N =1 Supergravity and compensating fields Main Result: Adinkras for

•old-minimal supergravity (mSG) •non-minimal supergravity ( ) •Conformal supergravity (cSG)

Adinkras as characters of Spin(4)R

Conclusions

Why SUSY? Elegance •Do math, find the physics it describes – P.A.M. Dirac Answer to the hierarchy problem? •Maybe naturalness isn’t a guide •Split SUSY String theory


Why off-shell?

Don’t impose equations of motion Actions utilizing superfields have manifest supersymmetry Off-shell formulations not known for

•10 D SUGRA, string theory •11 D SUGRA, M-theory

Adinkras: graphical representations of supersymmetry

•Categorize off-shell representations •A way to study unknown off-shell systems?


Example: Off-shell Chiral Multiplet (CM)

2

2 2 2

( ) ( )

( )

a a b

a ab

a b

ab

A iB G iF i A iB

i A iB

4 4S d xd

511 0

2

b

a baD D Chiral Superfield (4|4):

Super- differential relations: a symmetry of the action: Satisfy:


Component Expansion:

CM Component Action: Auxiliary Fields

Integrate out q action over d4x :

symmetry:


Super- differential relations: a symmetry of the action: Satisfy:

Adinkras: Graphical Reps of SUSY

Dimensionally reduce a 4D, N =1 to 1D, N=4

Adinkras: graphical representations of 1D, N=4 Chiral Superfield: a (4|4) irreducible adinkra










LI=(RI)T =(RI)

-1 LI=





Closure relation: becomes

GR(d,N) or Garden Algebra

A distinct (4|4) irreducible adinkra

Gauge fixed vector multiplet(VM) = Real Linear SF:

4 4 2 ,a

aS d xd V D D D V V V G

0

2, 0G G D G


A distinct (4|4) irreducible adinkra

Gauge fixed vector multiplet(VM) = Real Linear SF:

4 4 2 ,a

aS d xd V D D D V V V G

0

2, 0G G D G


The same (4|4) irreducible adinkra as VM

Tensor multiplet (TM) = Real Linear SF 4 4 2 ,S d xd G G G


Real Linear Superfield = TM=gauge-fixed VM The same (4|4) irreducible adinkras

Is there a way to distinguish them? Possibly: related to a “hodge duality” between permutation subsets of the adinkras. See 1208.5999, 1210.0478, outside the realm of this talk



Two irreducible 1D, N=4 adinkras

“Reflection” about “orange” axis

chiral superfield:

Real linear SF = Gauge fixed VM = TM


Draw an analogy: Enantiomers

“Reflection” about

“orange” axis

Draw an analogy from chemistry: molecules that are identical up to a spatial reflection are called cis- and trans-enantiomers (mirror images)

Spatial Reflections

Chiral superfield Real linear superfield


Define: ‘SUSY enantiomer’ numbers

“Reflection” about

“orange” axis

Draw an analogy from chemistry: molecules that are identical up to a spatial reflection are called cis- and trans-enantiomers (mirror images)

Spatial Reflections

(nc = 1, nt = 0) (nc = 0, nt = 1) cis-Adinkra trans-Adinkra


Larger Off-shell Representations Denote these by (4k|4k) Built out of the irreducible adinkras k= nc + nt = # irred. Adinkras composing the rep.

•nc : number of cis-adinkras •nt : number of trans-adinkras

Off-shell Reps. Completed: •Complex Linear Superfield (CLS) •Real Scalar Superfield (RSS) •Supergravities: mSG, , cSG •Gravitino-matter multiplet


Real Scalar Superfield (RSS): (8|8)

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

= (nc = 1, nt = 1)

4 4 2S d xd V


V V G

containscomponents | | , , |a aV K M N U

Complex Linear Superfield (CLS) (12|12)

= (nc = 1, nt = 2)

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

2 0 ,D complex 4 4S d xd


containscomponents , | , | , , , |a a aK L M N U V

4D, N=1 Linear Supergravity (4k|4k)

cis-adinkra trans-adinkra

Representation = Base + Compensator k adinkras

cSG cSG N/A 2

mSG cSG Chiral SF 3

cSG Complex Linear SF 5

How many cis- (nc) and trans- (nt) adinkras compose each Representation? (k = nc + nt )


Adinkra for mSG (12|12) 4D, N=1 Transformation Laws



Reduced to 1D, N=4 (temporal gauge )


Adinkra for mSG (12|12) Want to fit into either the cis- or the trans- adinkra

0: 1c tcis n n 0: 1c ttrans n n



Reduced to 1D, N=4

Define a seed linear combination:




Reduced to 1D, N=4

Defines:




Reduced to 1D, N=4

Defines:




Reduced to 1D, N=4

Defines:




Reduced to 1D, N=4

Constraints:




Reduced to 1D, N=4

Constraints:



Reduced to 1D, N=4

or

One free parameter u1 encodes an overall scaling symmetry of all the fields nc=1

Two free parameters two possible linearly independent solutions nt=2


Adinkra for mSG (12|12) nc=1 , nt=2

chiral superfield + conformal supergravity = old-minimal supergravity (mSG)

?


= complex linear SF + conformal SG (20|20)

4D, N=1 Component Lagrangian:

4D, N=1 Transformation Laws:

Adinkra (20|20)


1D, N=4 Lagrangian: 1D, N=4 Transformation Laws:

= complex linear SF + conformal SG (20|20) Adinkra (20|20)


Apply the Adinkranization Procedure: 1D, N=4 Transformation Laws:


Force linear combos to fit into:

Pick seed linear combination:

Iterate with four colors:

0: 1c tcis n n 0: 1c ttrans n n or



Apply the Adinkranization Procedure:


Force linear combos to fit into:

Pick seed linear combination:

Iterate with four colors:

0: 1c tcis n n 0: 1c ttrans n n or

Leaves us with:

or



Adinkranization gives us:

or

= complex linear SF + conformal SG (nc=1, nt=4)

unique up to overall scaling symmetry of all the fields nc=1

Four free parameters four possible linearly independent solutions nt=4

Adinkra (20|20)


Adinkra (20|20)

Four trans-submultiplets parameterized by the four sets of four parameters:



Adinkra (20|20)

Four trans-submultiplets parameterized by the four sets of four parameters, consider the choice:



Adinkra (20|20) = complex linear SF + conformal SG (nc=1, nt=4)

complex linear superfield + conformal supergravity


Adinkra for cSG


Show that:

is indeed the adinkra for conformal supergravity

Adinkra for cSG


This is trivial: conformal supergravity has component fields: , i.e. mSG- S - P ( , | )ah A

Adinkra for cSG



Adinkra for cSG



Synthesis


Consider the Spin(3,1) character:

Define an analogous Spin(4)R ‘twisted’ character: For Adinkraic Reps. Defined by L and R :

Synthesis


Synthesis


These characters satisfy the Superfield identites:

Synthesis


Conclusion


Adinkras: graphical reps of Spin(4)R characters Classify 4D, N=1 off-shell SUSY New results: adinkras for mSG, , cSG Add like characters as expected for these reps:

•old-minimal SG = chiral +conformal SG •non-minimal SG = complex linear+ conformal SG •gauge fixed vector = real scalar – (chiral + anti-chiral) = real linear •Complex unconstrained = complex linear + chiral

cis and trans numbers reported so far in this talk: •nc=1 or 0 , nt=0,1,2,3,4 •Does this pattern continue? •Preliminary results: Gravitino-matter (20|20): nc=4 , nt= 1 •Roles or cis and trans seem reversed from the others!

Thank You

• You the audience • Co-authors: Profs. Gates and Linch, S. Randall, I. Chappell, J.

Parker, A. Ridgway • 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

• K. Burghardt, K. Koutrolikos, M. Calkins, T. Rimlinger • Greg Landweber, creator of Adinkramat ©2008 • Fort Lauderdale Airport where this all began 1 year ago!


Documents

Updates for Off-Shell Supersymmetric Representation · PDF fileUpdates for Off-Shell Supersymmetric Representation Theory Using Adinkras Kory Stiffler Based on: 4D, N = 1 Supergravity