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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
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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
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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
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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?
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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:
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
Component Expansion:
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CM Component Action: Auxiliary Fields
Integrate out q action over d4x :
symmetry:
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
Super- differential relations: a symmetry of the action: Satisfy:
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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
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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
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
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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
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
LI=(RI)T =(RI)
-1 LI=
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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
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
Closure relation: becomes
GR(d,N) or Garden Algebra
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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
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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
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The same (4|4) irreducible adinkra as VM
Tensor multiplet (TM) = Real Linear SF 4 4 2 ,S d xd G G G
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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
Adinkras: Graphical Reps of SUSY
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Two irreducible 1D, N=4 adinkras
“Reflection” about “orange” axis
chiral superfield:
Real linear SF = Gauge fixed VM = TM
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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
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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
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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
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Real Scalar Superfield (RSS): (8|8)
(nc = 1, nt = 0) + (nc = 0, nt = 1)
= (nc = 1, nt = 1)
4 4 2S d xd V
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V V G
containscomponents | | , , |a aV K M N U
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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
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
containscomponents , | , | , , , |a a aK L M N U V
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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 )
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
Reduced to 1D, N=4 (temporal gauge )
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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
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
Reduced to 1D, N=4
Define a seed linear combination:
0: 1c tcis n n 0: 1c ttrans n n
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
Reduced to 1D, N=4
Defines:
0: 1c tcis n n 0: 1c ttrans n n
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
Reduced to 1D, N=4
Defines:
0: 1c tcis n n 0: 1c ttrans n n
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
Reduced to 1D, N=4
Defines:
0: 1c tcis n n 0: 1c ttrans n n
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
Reduced to 1D, N=4
Constraints:
0: 1c tcis n n 0: 1c ttrans n n
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
Reduced to 1D, N=4
Constraints:
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Adinkra for mSG (12|12) 4D, N=1 Transformation Laws
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
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Adinkra for mSG (12|12) nc=1 , nt=2
chiral superfield + conformal supergravity = old-minimal supergravity (mSG)
?
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= complex linear SF + conformal SG (20|20)
4D, N=1 Component Lagrangian:
4D, N=1 Transformation Laws:
Adinkra (20|20)
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1D, N=4 Lagrangian: 1D, N=4 Transformation Laws:
= complex linear SF + conformal SG (20|20) Adinkra (20|20)
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Apply the Adinkranization Procedure: 1D, N=4 Transformation Laws:
0: 1c tcis n n 0: 1c ttrans n n
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
= complex linear SF + conformal SG (20|20) Adinkra (20|20)
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Apply the Adinkranization Procedure:
0: 1c tcis n n 0: 1c ttrans n n
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
= complex linear SF + conformal SG (20|20) Adinkra (20|20)
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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)
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Adinkra (20|20)
Four trans-submultiplets parameterized by the four sets of four parameters:
= complex linear SF + conformal SG (nc=1, nt=4)
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Adinkra (20|20)
Four trans-submultiplets parameterized by the four sets of four parameters, consider the choice:
= complex linear SF + conformal SG (nc=1, nt=4)
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Adinkra (20|20) = complex linear SF + conformal SG (nc=1, nt=4)
complex linear superfield + conformal supergravity
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Adinkra for cSG
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Show that:
is indeed the adinkra for conformal supergravity
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Adinkra for cSG
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
This is trivial: conformal supergravity has component fields: , i.e. mSG- S - P ( , | )ah A
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Adinkra for cSG
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
This is trivial: conformal supergravity has component fields: , i.e. mSG- S - P ( , | )ah A
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Adinkra for cSG
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
This is trivial: conformal supergravity has component fields: , i.e. mSG- S - P ( , | )ah A
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Synthesis
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
Consider the Spin(3,1) character:
Define an analogous Spin(4)R ‘twisted’ character: For Adinkraic Reps. Defined by L and R :
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Synthesis
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
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Synthesis
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
These characters satisfy the Superfield identites:
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Synthesis
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
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Conclusion
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318
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!
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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!
Kory Stiffler - Updates for Off-Shell Supersymmetric Representation Theory Using Adinkras 1212.3318