Tri-Vertices & SU(2)’s
Amihay Hanany
Noppadol Mekareeya
1
Monday, May 9, 2011
Noppadol Mekareeya
Monday, May 9, 2011
Introduction
3
Monday, May 9, 2011
Introduction
• N=2 gauge theories in 3+1 dimensions
3
Monday, May 9, 2011
Introduction
• N=2 gauge theories in 3+1 dimensions
• Vector Multiplet Moduli Space
3
Monday, May 9, 2011
Introduction
• N=2 gauge theories in 3+1 dimensions
• Vector Multiplet Moduli Space
• Hypermultiplet Moduli Space
3
Monday, May 9, 2011
Introduction
• N=2 gauge theories in 3+1 dimensions
• Vector Multiplet Moduli Space
• Hypermultiplet Moduli Space
• revisit - study a class made out of SU(2)’s
3
Monday, May 9, 2011
Introduction
• N=2 gauge theories in 3+1 dimensions
• Vector Multiplet Moduli Space
• Hypermultiplet Moduli Space
• revisit - study a class made out of SU(2)’s
• interesting class of HyperKahler manifolds
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Monday, May 9, 2011
Hypermultiplet
moduli space
4
Monday, May 9, 2011
Hypermultiplet
moduli space
• Most attention in the literature - V plet
moduli space (Coulomb branch)
4
Monday, May 9, 2011
Hypermultiplet
moduli space
• Most attention in the literature - V plet
moduli space (Coulomb branch)
• Receives quantum corrections
4
Monday, May 9, 2011
Hypermultiplet
moduli space
• Most attention in the literature - V plet
moduli space (Coulomb branch)
• Receives quantum corrections
• H plet moduli space is classical
4
Monday, May 9, 2011
Hypermultiplet
moduli space
• Most attention in the literature - V plet
moduli space (Coulomb branch)
• Receives quantum corrections
• H plet moduli space is classical
• argue it has many interesting features
4
Monday, May 9, 2011
From Quivers to
Skeleton Diagrams
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Monday, May 9, 2011
From Quivers to
Skeleton Diagrams
• N=2 Quiver - nodes connected by lines
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Monday, May 9, 2011
From Quivers to
Skeleton Diagrams
• N=2 Quiver - nodes connected by lines
• each node - V plet
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Monday, May 9, 2011
From Quivers to
Skeleton Diagrams
• N=2 Quiver - nodes connected by lines
• each node - V plet
• each line - H plet, bifundamental
5
Monday, May 9, 2011
From Quivers to
Skeleton Diagrams
• N=2 Quiver - nodes connected by lines
• each node - V plet
• each line - H plet, bifundamental
• extend to other matter?
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Monday, May 9, 2011
Tri-Vertices
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Monday, May 9, 2011
Tri-Vertices
• First - change notation
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Monday, May 9, 2011
Tri-Vertices
• First - change notation
• Inspired by brane configurations, use lines
for V plets and nodes for H plets
6
Monday, May 9, 2011
Tri-Vertices
• First - change notation
• Inspired by brane configurations, use lines
for V plets and nodes for H plets
• restrict to 3-valent vertices and lines
representing SU(2) gauge group
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Monday, May 9, 2011
Skeleton Diagrams
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Monday, May 9, 2011
Skeleton Diagrams
• Lines & 3-valent Vertices
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Monday, May 9, 2011
Skeleton Diagrams
• Lines & 3-valent Vertices
• Each Line - V plet with SU(2) gauge group
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Monday, May 9, 2011
Skeleton Diagrams
• Lines & 3-valent Vertices
• Each Line - V plet with SU(2) gauge group
• Each vertex - a tri-fundamental of SU(2)
3
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Monday, May 9, 2011
Skeleton Diagrams
• Lines & 3-valent Vertices
• Each Line - V plet with SU(2) gauge group
• Each vertex - a tri-fundamental of SU(2)
• 8 half hypermultiplets in (2,2,2) of SU(2)
3
3
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Monday, May 9, 2011
Skeleton Diagrams
• Lines & 3-valent Vertices
• Each Line - V plet with SU(2) gauge group
• Each vertex - a tri-fundamental of SU(2)
• 8 half hypermultiplets in (2,2,2) of SU(2)
• Length of the line L ~ 1/g
3
3
2
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Monday, May 9, 2011
Skeleton Diagrams
• Lines & 3-valent Vertices
• Each Line - V plet with SU(2) gauge group
• Each vertex - a tri-fundamental of SU(2)
• 8 half hypermultiplets in (2,2,2) of SU(2)
• Length of the line L ~ 1/g
• infinite line - global SU(2) symmetry
3
3
2
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Monday, May 9, 2011
Example: 8 free 1/2
Hypers
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Monday, May 9, 2011
Example: 8 free 1/2
Hypers
•
global symmetry SU(2)3
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Monday, May 9, 2011
Example: 8 free 1/2
Hypers
•
• No gauge group
global symmetry SU(2)3
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Monday, May 9, 2011
Example: N=4 SU(2)
gauge theory
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Monday, May 9, 2011
Example: N=4 SU(2)
gauge theory
• 1 gauge group
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Monday, May 9, 2011
Example: N=4 SU(2)
gauge theory
• 1 gauge group
• 2 adjoints - an N=2 H plet
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Monday, May 9, 2011
Example: N=4 SU(2)
gauge theory
• 1 gauge group
• 2 adjoints - an N=2 H plet
• 2 singlets - decouple
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Monday, May 9, 2011
Example: N=4 SU(2)
gauge theory
• 1 gauge group
• 2 adjoints - an N=2 H plet
• 2 singlets - decouple
• infinite line - SU(2) global symmetry
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Monday, May 9, 2011
Example: N=4 SU(2)
gauge theory
• 1 gauge group
• 2 adjoints - an N=2 H plet
• 2 singlets - decouple
• infinite line - SU(2) global symmetry
• possible N=2 breaking mass term
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Monday, May 9, 2011
Example: SU(2) with 4
flavors
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Monday, May 9, 2011
Example: SU(2) with 4
flavors
• 1 gauge group
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Monday, May 9, 2011
Example: SU(2) with 4
flavors
• 1 gauge group
• SU(2) global symmetry
4
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Monday, May 9, 2011
Infinite class of SU(2)
theories
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Monday, May 9, 2011
Infinite class of SU(2)
theories
• For each such diagram write a unique
Lagrangian in 3+1 dimensions with N=2
supersymmetry
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Monday, May 9, 2011
Infinite class of SU(2)
theories
• For each such diagram write a unique
Lagrangian in 3+1 dimensions with N=2
supersymmetry
• Feynmann diagram in phi cubed field theory
but each diagram represents a unique
Lagrangian
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Monday, May 9, 2011
Conformal Invariance
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Monday, May 9, 2011
Conformal Invariance
• Each finite line has 2 ends
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Monday, May 9, 2011
Conformal Invariance
• Each finite line has 2 ends
• giving 8 fundamental half hypers charged
under it
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Monday, May 9, 2011
Conformal Invariance
• Each finite line has 2 ends
• giving 8 fundamental half hypers charged
under it
• beta function is 0
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Monday, May 9, 2011
An infinite class of N=2
SCFT’s
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Monday, May 9, 2011
Topology of skeleton
diagrams
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Monday, May 9, 2011
Topology of skeleton
diagrams
• each diagram has g loops & e external legs
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Monday, May 9, 2011
Topology of skeleton
diagrams
• each diagram has g loops & e external legs
• Number of gauge groups 3g-3+e
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Monday, May 9, 2011
Topology of skeleton
diagrams
• each diagram has g loops & e external legs
• Number of gauge groups 3g-3+e
• Number of matter fields 2g-2+e
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Monday, May 9, 2011
Topology of skeleton
diagrams
• each diagram has g loops & e external legs
• Number of gauge groups 3g-3+e
• Number of matter fields 2g-2+e
• global symmetry SU(2)
e
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Monday, May 9, 2011
Hypermultiplet
Moduli Space
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Monday, May 9, 2011
Hypermultiplet
Moduli Space
• at generic point gauge group is broken
down to
g
U(1)
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Monday, May 9, 2011
Hypermultiplet
Moduli Space
• at generic point gauge group is broken
down to
g
U(1)
• dimension of moduli space (quaternionic)
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Monday, May 9, 2011
Hypermultiplet
Moduli Space
• at generic point gauge group is broken
down to
g
U(1)
• dimension of moduli space (quaternionic)
• (2g-2+e) x 8 x 1/2 - [(3g-3+e)x3 - g] = e+1
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Monday, May 9, 2011
Hypermultiplet
Moduli Space
• at generic point gauge group is broken
down to
g
U(1)
• dimension of moduli space (quaternionic)
• (2g-2+e) x 8 x 1/2 - [(3g-3+e)x3 - g] = e+1
• No dependence on g
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Monday, May 9, 2011
Hypermultiplet
Moduli Space
• at generic point gauge group is broken
down to
g
U(1)
• dimension of moduli space (quaternionic)
• (2g-2+e) x 8 x 1/2 - [(3g-3+e)x3 - g] = e+1
• No dependence on g
• Kibble Branch
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Monday, May 9, 2011
Look for generic results
depending on g & e
not on Lagrangian
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Monday, May 9, 2011
Chiral Operators on
Kibble Branch
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Monday, May 9, 2011
Chiral Operators on
Kibble Branch
• Write the theory in N=1 language
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Monday, May 9, 2011
Chiral Operators on
Kibble Branch
• Write the theory in N=1 language
• Many chiral operators
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Monday, May 9, 2011
Chiral Operators on
Kibble Branch
• Write the theory in N=1 language
• Many chiral operators
• Count Chiral Operators on the Kibble
branch
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Monday, May 9, 2011
Chiral Operators on
Kibble Branch
• Write the theory in N=1 language
• Many chiral operators
• Count Chiral Operators on the Kibble
branch
• Hilbert Series
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Monday, May 9, 2011
What is a
Hilbert Series?
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Monday, May 9, 2011
What is a
Hilbert Series?
g({ti }) =
!
ii
di1 ,...,ik t1
i1 ,...,ik
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Monday, May 9, 2011
ik
· · · tk
What is a
Hilbert Series?
g({ti }) =
!
ii
di1 ,...,ik t1
i1 ,...,ik
i1 , . . . , ik
di1 ,...,ik
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Monday, May 9, 2011
ik
· · · tk
Fugacities
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Monday, May 9, 2011
Fugacities
• t - keeps track of the dimension of
operator
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Monday, May 9, 2011
Fugacities
• t - keeps track of the dimension of
operator
• x - SU(2)
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Monday, May 9, 2011
Fugacities
• t - keeps track of the dimension of
operator
• x - SU(2)
• for e SU(2)’s there are e x’s
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Monday, May 9, 2011
Fugacities
• t - keeps track of the dimension of
operator
• x - SU(2)
• for e SU(2)’s there are e x’s
• HS(t, x , x , ..., x )
1
2
e
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Monday, May 9, 2011
Example: g=0, e=3
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Monday, May 9, 2011
Plethystics
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Monday, May 9, 2011
Example: g=0, e=4
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Monday, May 9, 2011
Example: g=0, e=4
• To compute the HS observe
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Monday, May 9, 2011
Example: g=0, e=4
• To compute the HS observe
• Higgs branch is the moduli space of 1
SO(8) instanton on R4
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Monday, May 9, 2011
Example: g=0, e=4
• To compute the HS observe
• Higgs branch is the moduli space of 1
SO(8) instanton on R4
• HS was computed and gives
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Monday, May 9, 2011
Example: g=0, e=4
• To compute the HS observe
• Higgs branch is the moduli space of 1
SO(8) instanton on R4
• HS was computed and gives
•
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Monday, May 9, 2011
HS: g=0, e=4
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Monday, May 9, 2011
HS: g=0, e=4
•
Next decompose irreps of SO(8) to SU(2)4
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Monday, May 9, 2011
HS: g=0, e=4
•
• Use fugacity map
Next decompose irreps of SO(8) to SU(2)4
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Monday, May 9, 2011
HS: g=0, e=4
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Monday, May 9, 2011
Gluing Theories
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Monday, May 9, 2011
Gluing Theories
• 2 theories with (g1, e1) & (g2, e2)
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Monday, May 9, 2011
Gluing Theories
• 2 theories with (g1, e1) & (g2, e2)
• glue by identifying an infinite leg from each
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Monday, May 9, 2011
Gluing Theories
• 2 theories with (g1, e1) & (g2, e2)
• glue by identifying an infinite leg from each
• turn it into finite line
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Monday, May 9, 2011
Gluing Theories
• 2 theories with (g1, e1) & (g2, e2)
• glue by identifying an infinite leg from each
• turn it into finite line
• gauging an SU(2) global symmetry
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Monday, May 9, 2011
Gluing Theories
• 2 theories with (g1, e1) & (g2, e2)
• glue by identifying an infinite leg from each
• turn it into finite line
• gauging an SU(2) global symmetry
• get the theory with (g1+g2, e1+e2-2)
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Monday, May 9, 2011
Gluing 2 theories with
(0,3) to form (0,4)
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Monday, May 9, 2011
permutation symmetry
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Monday, May 9, 2011
permutation symmetry
• The result has S
symmetry - permutation
of external legs (global symmetries)
4
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Monday, May 9, 2011
Example: g=1, e=1
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Monday, May 9, 2011
g=1, e=1 another
method
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Monday, May 9, 2011
g=1, e=1 another
method
• two commuting adjoints
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Monday, May 9, 2011
g=1, e=1 another
method
• two commuting adjoints
• symmetric product of 2 C
2
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Monday, May 9, 2011
s
Example: g=1, e=2
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Monday, May 9, 2011
Example: g=2, e=0
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Monday, May 9, 2011
examples: g=3, e=0
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Monday, May 9, 2011
Duality statement
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Monday, May 9, 2011
Duality statement
• physics depends on g & e and not on the
particular choice of the Lagrangian
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Monday, May 9, 2011
Duality statement
• physics depends on g & e and not on the
particular choice of the Lagrangian
• Many to 1 correspondence
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Monday, May 9, 2011
general case: (g,e)
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Monday, May 9, 2011
any g, e=0
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Monday, May 9, 2011
any g, e=1
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Monday, May 9, 2011
g, e=1
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Monday, May 9, 2011
g, e=1
• complete intersection
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Monday, May 9, 2011
g, e=1
• complete intersection
• generated by 5 operators
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Monday, May 9, 2011
g, e=1
• complete intersection
• generated by 5 operators
• triplet of dimension 2; doublet of dimension
2g-1
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Monday, May 9, 2011
g, e=1
• complete intersection
• generated by 5 operators
• triplet of dimension 2; doublet of dimension
2g-1
• one relation of degree 4g
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Monday, May 9, 2011
generators of the
moduli space
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Monday, May 9, 2011
generators
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Monday, May 9, 2011
generators
• 3e at dimension 2
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Monday, May 9, 2011
generators
• 3e at dimension 2
• 2 at dimension 2g-2+e
e
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Monday, May 9, 2011
generators
• 3e at dimension 2
• 2 at dimension 2g-2+e
• Indication of the complexity at high e
e
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Monday, May 9, 2011
generators
• 3e at dimension 2
• 2 at dimension 2g-2+e
• Indication of the complexity at high e
• HyperKahler moduli space
e
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Monday, May 9, 2011
generators
• 3e at dimension 2
• 2 at dimension 2g-2+e
• Indication of the complexity at high e
• HyperKahler moduli space
• of dimension e+1
e
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Monday, May 9, 2011
generators
• 3e at dimension 2
• 2 at dimension 2g-2+e
• Indication of the complexity at high e
• HyperKahler moduli space
• of dimension e+1
• with SU(2) isometry
e
e
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Monday, May 9, 2011
Summary
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Monday, May 9, 2011
Summary
• Study SU(2)’s & matter in tri-fundamental
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Monday, May 9, 2011
Summary
• Study SU(2)’s & matter in tri-fundamental
• properties of H plet moduli space (Kibble
branch)
40
Monday, May 9, 2011
Summary
• Study SU(2)’s & matter in tri-fundamental
• properties of H plet moduli space (Kibble
branch)
• Special class of HyperKahler manifolds with
SU(2)e isometries
40
Monday, May 9, 2011
Summary
• Study SU(2)’s & matter in tri-fundamental
• properties of H plet moduli space (Kibble
branch)
• Special class of HyperKahler manifolds with
SU(2)e isometries
• Multi-ality: Physics depends on (g,e)
40
Monday, May 9, 2011
Summary
• Study SU(2)’s & matter in tri-fundamental
• properties of H plet moduli space (Kibble
branch)
• Special class of HyperKahler manifolds with
SU(2)e isometries
• Multi-ality: Physics depends on (g,e)
• Symmetry: symmetric group in e elements
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Monday, May 9, 2011
Thank you!
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Monday, May 9, 2011