noncommutative geometry and supersymmetry
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Non[com,mut]ative geometry
Supersymmetry&Is | Thijs van den Broek
From | Radboud University Nijmegen / Nikhef
With | Walter van Suijlekom & Wim Beenakker
Supersymmetry
Wednesday, May 30, 2012
Geometry & physics
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Classical physics: flat space, time
Wednesday, May 30, 2012
Geometry & physics
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Classical physics: flat space, time
General relativity (Einstein, 1916):
Wednesday, May 30, 2012
Geometry & physics
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Classical physics: flat space, time
General relativity (Einstein, 1916):Curved spacetime
Wednesday, May 30, 2012
Geometry & physics
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Classical physics: flat space, time
General relativity (Einstein, 1916):Curved spacetime
Metric
Wednesday, May 30, 2012
Geometry & physics
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Classical physics: flat space, time
General relativity (Einstein, 1916):Curved spacetime
Metric
Einstein equations:
Wednesday, May 30, 2012
Geometry & physics
Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Classical physics: flat space, time
General relativity (Einstein, 1916):Curved spacetime
Metric
Einstein equations:
Or: Einstein - Hilbert action
Wednesday, May 30, 2012
?Does this set-up allow for generalisations to noncommutative spaces
3Wednesday, May 30, 2012
Noncommutative geometry | Basics
QM:
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Noncommutative geometry | Basics
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Noncommutative geometry | Basics
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Extend geometry to noncommutative geometry:
Noncommutative geometry | Basics
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Extend geometry to noncommutative geometry:
Noncommutative geometry | Basics
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Extend geometry to noncommutative geometry:
Noncommutative geometry | Basics
Noncommutative space (...)
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Extend geometry to noncommutative geometry:
Noncommutative geometry | Basics
Noncommutative space (...)
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Extend geometry to noncommutative geometry:
Noncommutative geometry | Basics
Noncommutative space (...)Determines metric
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Extend geometry to noncommutative geometry:
Noncommutative geometry | Basics
Noncommutative space (...)Determines metric
Fermions / spinors
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Extend geometry to noncommutative geometry:
Noncommutative geometry | Basics
Noncommutative space (...)Determines metric
Fermions / spinors
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
Extend geometry to noncommutative geometry:
Noncommutative geometry | Basics
Noncommutative space (...)Determines metric
Fermions / spinors
is a special example.GRT
QM: acting on wave functions.
4Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Analogy:
Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
To be a bit more precise, we’re working with in the case of GRT:
NCG | Details
5Wednesday, May 30, 2012
Wish:
NCG | ActionAn action
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Wish:
NCG | ActionAn action
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
‘Spectral action’:
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Analogy:
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
‘Spectral action’:
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Analogy:
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
‘Spectral action’:
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Analogy:
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
‘Spectral action’:
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Analogy:
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
‘Spectral action’:
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Analogy:
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
‘Spectral action’:
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Analogy:
Wish:
NCG | ActionAn actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
‘Spectral action’:
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Analogy:
Wish:
NCG | Action
GRT: gives Einstein - Hilbert action + more ( )
An actionThat is fixed by the noncommutative geometry
Ac#on / Lagrangian
Feynman rules
Crosssec#ons
NCG
‘Spectral action’:
6Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
?How different are the internal and external degrees of freedom of a particle?
Wednesday, May 30, 2012
Standard Model | Set upThe geometry:
8Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Standard Model | Set upThe geometry:
8Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Standard Model | Set upThe geometry:
8Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Curved spacetime(external degrees of freedom)
“Gauge group”(internal degrees of freedom)
Standard Model | Set upThe geometry:
8Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Curved spacetime(external degrees of freedom)
“Gauge group”(internal degrees of freedom)
Standard Model | Set up
Choice for F automatically gives fermions (spinors)‘inner structure’ (i.e. color etc).
The geometry:
8Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Curved spacetime(external degrees of freedom)
“Gauge group”(internal degrees of freedom)
Standard Model | Set up
Choice for F automatically gives fermions (spinors)‘inner structure’ (i.e. color etc).
The geometry:
8Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Can naturally coincide with the SM particle content.
Wednesday, May 30, 2012
Curved spacetime(external degrees of freedom)
“Gauge group”(internal degrees of freedom)
Standard Model | Set up
Choice for F automatically gives fermions (spinors)‘inner structure’ (i.e. color etc).
The geometry:
8Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Can naturally coincide with the SM particle content.
Spectral action gives:
General Relativity + Standard Model + more ( )
Wednesday, May 30, 2012
Curved spacetime(external degrees of freedom)
“Gauge group”(internal degrees of freedom)
Standard Model | Set up
Choice for F automatically gives fermions (spinors)‘inner structure’ (i.e. color etc).
The geometry:
A geometrical understanding of the Standard Model!8Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Can naturally coincide with the SM particle content.
Spectral action gives:
General Relativity + Standard Model + more ( )
Wednesday, May 30, 2012
Standard Model | PredictionsBonus: spectral action leads to
9Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Standard Model | PredictionsBonus: spectral action leads to
9Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Standard Model | PredictionsBonus: spectral action leads to
(Theory ‘lives’ here)Value of
9Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Standard Model | PredictionsBonus: spectral action leads to
(Theory ‘lives’ here)Value of
Values of coupling constants
9Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Standard Model | PredictionsBonus: spectral action leads to
(Theory ‘lives’ here)Value of
Values of coupling constants
9Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Standard Model | PredictionsBonus: spectral action leads to
Higgs mass predic#on: 158 -‐ 173 GeV
(Theory ‘lives’ here)Value of
Values of coupling constants
9Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Supersymmetry & MSSMN = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2...
10Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Supersymmetry & MSSMN = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2...
... such that action is invariant under supersymmetry transformations.
10Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Supersymmetry & MSSMN = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2...
MSSM: N = 1 supersymmetric version of SM:
... such that action is invariant under supersymmetry transformations.
10Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Supersymmetry & MSSMN = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2...
Chiral fermions (quarks, leptons)
Gauge bosons (gluons, ...)
MSSM: N = 1 supersymmetric version of SM:
... such that action is invariant under supersymmetry transformations.
10Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12Wednesday, May 30, 2012
Supersymmetry & MSSMN = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2...
Chiral fermions (quarks, leptons)
Gauge bosons (gluons, ...)
MSSM: N = 1 supersymmetric version of SM:
... such that action is invariant under supersymmetry transformations.
10Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Sfermions (squarks, sleptons)
Gauginos (gluinos, ...)
Wednesday, May 30, 2012
Supersymmetry & MSSMN = 1 supersymmetry: each particle in the theory has a superpartner, differing in spin by 1/2...
Chiral fermions (quarks, leptons)
Gauge bosons (gluons, ...)
Possible solution for various (theoretical) problems(e.g. provides dark matter candidate).
MSSM: N = 1 supersymmetric version of SM:
... such that action is invariant under supersymmetry transformations.
10Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Sfermions (squarks, sleptons)
Gauginos (gluinos, ...)
Wednesday, May 30, 2012
NCG & Supersymmetry
11Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Complication:Different origins ofparticles
Wednesday, May 30, 2012
NCG & SupersymmetrySUSY NCG
Chiral fermions
Sfermions (scalar)
Gauge bosons
Gauginos
11Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Complication:Different origins ofparticles
Wednesday, May 30, 2012
NCG & Supersymmetry
Moreover: different questions:
SUSY NCG
Chiral fermions
Sfermions (scalar)
Gauge bosons
Gauginos
11Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Complication:Different origins ofparticles
Wednesday, May 30, 2012
NCG & Supersymmetry
Moreover: different questions:
SUSY NCG
Chiral fermions
Sfermions (scalar)
Gauge bosons
Gauginos
11Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Complication:Different origins ofparticles
Which actions are supersymmetric?SUSY:
Wednesday, May 30, 2012
NCG & Supersymmetry
Moreover: different questions:
SUSY NCG
Chiral fermions
Sfermions (scalar)
Gauge bosons
Gauginos
11Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Complication:Different origins ofparticles
For which noncommutative geometries is the spectral action supersymmetric?
NCG:
Which actions are supersymmetric?SUSY:
Wednesday, May 30, 2012
NCG & Supersymmetry
Moreover: different questions:
SUSY NCG
Chiral fermions
Sfermions (scalar)
Gauge bosons
Gauginos
11Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Complication:Different origins ofparticles
For which noncommutative geometries is the spectral action supersymmetric?
NCG:
Which actions are supersymmetric?SUSY:
Status: SYM ✓super-QCD✓
Wednesday, May 30, 2012
NCG & Supersymmetry
Moreover: different questions:
SUSY NCG
Chiral fermions
Sfermions (scalar)
Gauge bosons
Gauginos
11Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Complication:Different origins ofparticles
For which noncommutative geometries is the spectral action supersymmetric?
NCG:
Which actions are supersymmetric?SUSY:
General: soon to be finished.
Status: SYM ✓super-QCD✓
Wednesday, May 30, 2012
NCG & Supersymmetry
Moreover: different questions:
SUSY NCG
Chiral fermions
Sfermions (scalar)
Gauge bosons
Gauginos
11Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
Complication:Different origins ofparticles
For which noncommutative geometries is the spectral action supersymmetric?
NCG:
Which actions are supersymmetric?SUSY:
General: soon to be finished.
Status: SYM ✓super-QCD✓
Up next:
? The Higgs mass.
Wednesday, May 30, 2012
12Thijs van den Broek (RU) | FOM Veldhoven | 17/1/12
I Thank you.
Wednesday, May 30, 2012
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