Supersymmetry and non-commutative geometry

&
SUPERSYMMETRY
NONCOMMUTATIVE GEOMETRY

Thijs van den Broek Workshop Bayrischzell
Radboud Univ. Nijmegen / NIKHEF May 22nd, 2011
Wednesday, May 30, 2012

Intro
INTRODUCTION

The project Approach ApplicaNon Preliminary results Outlook

The research project

Joint work with Walter van Suijlekom and Wim Beenakker

Try to extend the Standard Model from NCG with supersymmetry
(Everywhere: N=1 supersymmetry , i.e. MSSM)

Thijs van den Broek (RU Nijmegen) NoncommutaNve geometry & supersymmetry





How supersymmetric is the resulNng acNon?
(So: no superﬁelds or anything...)







Does it share the merits of ‘ordinary’ supersymmetry?
(E.g. hierarchy problem)








Can we predict anything from this?
(E.g. scalar masses, c.f Higgs mass)








Can we predict anything from this?
(E.g. scalar masses, c.f Higgs mass)
Why want this?






Why want this?






Why want this?

Promising BSM candidate.






Why want this?


To see what NCG might have in store for us.






Why want this?


To see what NCG might have in store for us.

UniﬁcaNon of coupling constants:

vs



MoNvaNng example: super-‐QCD [1] (1/2)

Take:

tensored with

where

parametrizing a 3-‐tuple and its conjugate.

1TvdB, W. D. van Suijlekom, Physics Letters B 699 (2011), 119–122




Take:

tensored with

where

‘quark’ ‘anNquark’
‘gluino’

parametrizing a 3-‐tuple and its conjugate.





Inner ﬂuctuaNons

parametrize (anN)squark





Inner ﬂuctuaNons


Gauge group : superpartners





Inner ﬂuctuaNons



Spectral acNon , extra terms:

Inner product:





Inner ﬂuctuaNons



Spectral acNon , extra terms:

Inner product:

SUSY automaNcally broken: (minus) mass terms for squarks.


APP
APPROACH


The approach

Problem: More realisNc situaNons: calculaNons get out of hand

More systemaNcal approach needed (cf. superﬁelds)



The approach

Problem: More realisNc situaNons: calculaNons get out of hand

More systemaNcal approach needed (cf. superﬁelds)

Plan: 1) Deﬁne ‘supersymmetric spectral triple‘

2) Prove ‘susy spectral triple’ supersymmetric acNon

spectral acNon

3) MSSM as a special case



Intermezzo: Krajewski diagrams
Finite spectral triple:

Krajewski diagram:




Krajewski diagram: ... ...
...
...




Grading

...
...




Grading

Dirac operator

...
...




Grading

Dirac operator ‘KO-‐dimension’

...
...



Superpartners (1/2)

General scheme as in super-‐QCD:

ParIcle Superpartner

fermions: sfermions:
Hilbert space ﬁnite Dirac operator

gauge bosons: gauginos:
Dirac operator on Hilbert space (adjoint reps.)

Higgs: Higgsinos:
ﬁnite Dirac operator Hilbert space



Superpartners (2/2)
Gauge group:

:

ParIcle Superpartner

fermions: sfermions:
Hilbert space ﬁnite Dirac operator

gauge bosons: gauginos:
Dirac operator on Hilbert space (adjoint reps.)



R-‐parity & KO-‐dimension (1/2)

Problem the gaugino-‐sector (adjoint elements of )
incompaNble with




incompaNble with

In fact parts of ﬁnite spectral triple possibly of diﬀerent KO-‐
dimensions




incompaNble with

In fact parts of ﬁnite spectral triple possibly of diﬀerent KO-‐
dimensions

SoluNon given:
two spectral triples
of KO-‐dimension (say)

an operator with:

Direct sum:




Direct sum:

Use to ‘even out’ the KO dimensions:

three new signs (‘super-‐KO-‐dimension’?)




Direct sum:

Use to ‘even out’ the KO dimensions:

three new signs (‘super-‐KO-‐dimension’?)

Example KO-‐dimensions 6 (SM) and 0 (gauginos) has:

i.e.

Role ‘R-‐parity’, where



A supersymmetric spectral triple

DeﬁniNon We call an R-‐parity extended spectral triple:

a spectral triple that is extended with a grading
saNsfying:

such that
where

with only

We write:





saNsfying:

such that (...)

DeﬁniNon An R-‐parity extended spectral triple is supersymmetric when:
each element that transforms under the gauge group
comes in both -‐values.
all allowed components of the -‐ part of the Dirac operator
are nonzero.





saNsfying:

such that (...)

DeﬁniNon An R-‐parity extended spectral triple is supersymmetric when:
each element that transforms under the gauge group
comes in both -‐values.
all allowed components of the -‐ part of the Dirac operator
are nonzero.
Hope (sNll) The acNon resulNng from such a spectral triple (via the spectral
acNon principle) is automaNcally supersymmetric.

APP
APPLICATION

The project Approach ApplicaIon Preliminary results Outlook

Why the SM?

A nice way to look at things is provided by Chamseddine & Connes [2]:

Look for irreducible soluNons of a pair :

Chamseddine & Connes, Why the Standard Model, 0706.3688v1 [hep-‐th]



Why the SM?



Either: acNng on
with




Why the SM?



Either: acNng on
with

Or: acNng on

with




Why the SM?



Either: acNng on
with
IncompaNble with

Or: acNng on

with




Why the SM Why the MSSM

ObservaNon:
Given the soluNon for the algebra we we can take not
only but in addiNon to that also the soluNon
for each of the two components of the algebra:

with




ObservaNon:

with

There is an R-‐parity operator:

(From )




ObservaNon:

with

SM parNcles

(From )




ObservaNon:

with

SM parNcles “Gaugino’s”

(From )




ObservaNon:

with

SM parNcles “Gaugino’s”

(From )

(Krajewski diagrams: representaNons have a solid ﬁll.)



The supersymmetric spectral triple for the MSSM’

Three steps to the (MS)SM

IniNal situaNon:
1.




Three steps to the (MS)SM: A vs A^C

1.

2.

As the result of a grading:




Three steps to the (MS)SM:

1.

2.

3.

By adding a Majorana mass
for the right handed neutrino





1.

2.

3.

for the right handed neutrino Bino





1.

2.

3.


Gluino



Three steps to the (MS)SM: Wino/Zino

1.

2.

3.


Gluino



Three steps to the (MS)SM: Higgsinos Wino/Zino

1.

2.

3.


Gluino



Three steps to the (MS)SM: Higgsinos Wino/Zino
+ new parNcles

1.

2.

3.


Gluino

Prel
PRELIMINARY RESULTS


Gauge group | UniﬁcaNon
The gauge group:

is sNll



The gauge group:

is sNll

We sNll have coupling constant uniﬁcaNon:



The gauge group:

is sNll

We sNll have coupling constant uniﬁcaNon:

This happens only because we have more parNcles than the MSSM itself
provides!


Fermion doubling | Chiral anomalies

Copies of fermions exceed those of gaugino’s by a factor of four.
Change inner product in:





Hypercharges:





Hypercharges:

All come in pairs of opposite charges: chiral anomalies cancel


Comments on supersymmetry

NCG treats bosons & fermions diﬀerently

No auxiliary ﬁelds (on-‐shell descripNon)

AutomaNcally broken by sfermion masses







Nonetheless: deﬁnitely susy-‐like properIes








Try to prove susy modulo sfermion potenNal terms:

1. prove susy for both soluNons given by C&C:








Try to prove susy modulo sfermion potenNal terms:

1. prove susy for both soluNons given by C&C:

2. prove that susy stays intact upon breaking

OUT
OUTLOOK


Summary & Outlook

✓ ‘Supersymmetric spectral triple’

? Supersymmetric acNon / explicit susy transformaNons



Summary & Outlook



✓ Applied to SM-‐algebra gives MSSM’

✓ Gauge group intact, anomaly free theory

✓ Coupling constant uniﬁcaNon

? Role & eﬀects extra parNcles?



Summary & Outlook







? PredicNons?



Summary & Outlook







? PredicNons?

For more (conclusive) results: stay tuned!

Supersymmetry and non-commutative geometry

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