New Quivers and
Good Old Geometry
Amihay Hanany
1
Thursday, September 2, 2010
Motivation
2
Thursday, September 2, 2010
Motivation
• Given a N=1 SUSY theory in 3+1d
2
Thursday, September 2, 2010
Motivation
• Given a N=1 SUSY theory in 3+1d
• get new information - old geometric tools
2
Thursday, September 2, 2010
Motivation
• Given a N=1 SUSY theory in 3+1d
• get new information - old geometric tools
• Chiral ring - a set of chiral operators
2
Thursday, September 2, 2010
Motivation
• Given a N=1 SUSY theory in 3+1d
• get new information - old geometric tools
• Chiral ring - a set of chiral operators
• Combinatorial question:
2
Thursday, September 2, 2010
Motivation
• Given a N=1 SUSY theory in 3+1d
• get new information - old geometric tools
• Chiral ring - a set of chiral operators
• Combinatorial question:
• How many chiral operators?
2
Thursday, September 2, 2010
Free Chiral Field
3
Thursday, September 2, 2010
Free Chiral Field
• chiral field Φ
3
Thursday, September 2, 2010
Free Chiral Field
• chiral field Φ
• all chiral operators are Φ , k=0, 1, 2,…
k
3
Thursday, September 2, 2010
Free Chiral Field
• chiral field Φ
• all chiral operators are Φ , k=0, 1, 2,…
• Theory has a conserved U(1) symmetry
k
3
Thursday, September 2, 2010
Free Chiral Field
• chiral field Φ
• all chiral operators are Φ , k=0, 1, 2,…
• Theory has a conserved U(1) symmetry
• Φ has charge 1; Φ has charge k
k
k
3
Thursday, September 2, 2010
Free Chiral Field
• chiral field Φ
• all chiral operators are Φ , k=0, 1, 2,…
• Theory has a conserved U(1) symmetry
• Φ has charge 1; Φ has charge k
• introduce a chemical potential µ
k
k
3
Thursday, September 2, 2010
Free Chiral Field
• chiral field Φ
• all chiral operators are Φ , k=0, 1, 2,…
• Theory has a conserved U(1) symmetry
• Φ has charge 1; Φ has charge k
• introduce a chemical potential µ
• Fugacity t = exp(-µ)
k
k
3
Thursday, September 2, 2010
Hilbert Series
∞
!
k=0
4
Thursday, September 2, 2010
1
t =
1−t
k
Hilbert Series
∞
!
• Write a function
k=0
4
Thursday, September 2, 2010
1
t =
1−t
k
Hilbert Series
∞
!
1
t =
1−t
k
• Write a function
• Theory is free - moduli space of vacua
k=0
4
Thursday, September 2, 2010
Hilbert Series
∞
!
1
t =
1−t
k
• Write a function
• Theory is free - moduli space of vacua
• parametrized by the value of Φ
k=0
4
Thursday, September 2, 2010
Hilbert Series
∞
!
1
t =
1−t
k
• Write a function
• Theory is free - moduli space of vacua
• parametrized by the value of Φ
• 1 complex dimensional
k=0
4
Thursday, September 2, 2010
Hilbert Series
∞
!
1
t =
1−t
k
• Write a function
• Theory is free - moduli space of vacua
• parametrized by the value of Φ
• 1 complex dimensional
• indeed for a complex variable z, all
k=0
holomorphic functions zk,
4
Thursday, September 2, 2010
One function
represents infinitely
many operators
5
Thursday, September 2, 2010
2 Free Chiral Fields
φ1 , φ2
k1 k2
φ1 φ2 ,
k1 , k2 = 0, 1, . . .
t1 , t2
1
HS(t1 , t2 ) =
(1 − t1 )(1 − t2 )
6
Thursday, September 2, 2010
2 Free Chiral Fields
•
2 chiral fields
φ1 , φ2
k1 k2
φ1 φ2 ,
k1 , k2 = 0, 1, . . .
t1 , t2
1
HS(t1 , t2 ) =
(1 − t1 )(1 − t2 )
6
Thursday, September 2, 2010
2 Free Chiral Fields
•
• all chiral operators
2 chiral fields
φ1 , φ2
k1 k2
φ1 φ2 ,
k1 , k2 = 0, 1, . . .
t1 , t2
1
HS(t1 , t2 ) =
(1 − t1 )(1 − t2 )
6
Thursday, September 2, 2010
2 Free Chiral Fields
•
φ
φ
all
chiral
operators
•
• 2 conserved U(1) charges
2 chiral fields
k1 k2
1
2 ,
φ1 , φ2
k1 , k2 = 0, 1, . . .
t1 , t2
1
HS(t1 , t2 ) =
(1 − t1 )(1 − t2 )
6
Thursday, September 2, 2010
2 Free Chiral Fields
•
φ
φ
all
chiral
operators
•
• 2 conserved U(1) charges
• 2 fugacities
2 chiral fields
k1 k2
1
2 ,
φ1 , φ2
k1 , k2 = 0, 1, . . .
t1 , t2
1
HS(t1 , t2 ) =
(1 − t1 )(1 − t2 )
6
Thursday, September 2, 2010
2 Free Chiral Fields
•
φ
φ
,
k
,
k
=
0,
1,
.
.
.
all
chiral
operators
•
• 2 conserved U(1) charges
t ,t
• 2 fugacities
1
HS(t , t ) =
•
(1 − t )(1 − t )
2 chiral fields
φ1 , φ2
k1 k2
1
2
1
2
1
1
6
Thursday, September 2, 2010
2
1
2
2
U(2) global symmetry
t
t2 =
x
t1 = tx;
7
Thursday, September 2, 2010
U(2) global symmetry
• In fact the global symmetry is U(2)
t
t2 =
x
t1 = tx;
7
Thursday, September 2, 2010
U(2) global symmetry
• In fact the global symmetry is U(2)
• U(1)xU(1) is a maximal Abelian subgroup
t
t2 =
x
t1 = tx;
7
Thursday, September 2, 2010
U(2) global symmetry
• In fact the global symmetry is U(2)
• U(1)xU(1) is a maximal Abelian subgroup
• should in fact arrange all operators into
irreps of SU(2)
t
t2 =
x
t1 = tx;
7
Thursday, September 2, 2010
U(2) global symmetry
• In fact the global symmetry is U(2)
• U(1)xU(1) is a maximal Abelian subgroup
• should in fact arrange all operators into
irreps of SU(2)
•
t
t2 =
x
t1 = tx;
7
Thursday, September 2, 2010
Non Abelian Fugacity
1
HS(t, x) =
(1 − tx)(1 − t/x)
8
Thursday, September 2, 2010
Non Abelian Fugacity
• t counts operators without taking into
account their index
1
HS(t, x) =
(1 − tx)(1 − t/x)
8
Thursday, September 2, 2010
Non Abelian Fugacity
• t counts operators without taking into
account their index
• x counts the label 1 or 2
1
HS(t, x) =
(1 − tx)(1 − t/x)
8
Thursday, September 2, 2010
Non Abelian Fugacity
• t counts operators without taking into
account their index
• x counts the label 1 or 2
• SU(2) fugacity
1
HS(t, x) =
(1 − tx)(1 − t/x)
8
Thursday, September 2, 2010
Expansion in characters
of SU(2)
∞
!
1
n
HS(t, x) =
=
χn (x)t
(1 − tx)(1 − t/x) n=0
n
2
χn (x) =
!
m=− n
2
9
Thursday, September 2, 2010
2m
x
Expansion in characters
of SU(2)
∞
!
1
n
HS(t, x) =
=
χn (x)t
(1 − tx)(1 − t/x) n=0
• the operators at dimension n transform in
the spin j=n/2 representation of SU(2).
n
2
χn (x) =
!
m=− n
2
9
Thursday, September 2, 2010
2m
x
moduli space
1
(1 − t)2
1
(1 − t)n
10
Thursday, September 2, 2010
moduli space
• At x=1 the HS looks like
1
(1 − t)n
10
Thursday, September 2, 2010
1
(1 − t)2
moduli space
• At x=1 the HS looks like
• dimension of moduli space is 2
1
(1 − t)n
10
Thursday, September 2, 2010
1
(1 − t)2
moduli space
• At x=1 the HS looks like
• dimension of moduli space is 2
• n free fields
1
(1 − t)n
10
Thursday, September 2, 2010
1
(1 − t)2
Hilbert Series
11
Thursday, September 2, 2010
Hilbert Series
• Counts Chiral Operators in a
supersymmetric gauge theory
11
Thursday, September 2, 2010
Hilbert Series
• Counts Chiral Operators in a
supersymmetric gauge theory
• a single function encodes information on
infinitely many operators
11
Thursday, September 2, 2010
Hilbert Series
• Counts Chiral Operators in a
supersymmetric gauge theory
• a single function encodes information on
infinitely many operators
• gives information about the moduli space of
vacua of the theory
11
Thursday, September 2, 2010
Hilbert Series
• Counts Chiral Operators in a
supersymmetric gauge theory
• a single function encodes information on
infinitely many operators
• gives information about the moduli space of
vacua of the theory
• representations under global symmetry
11
Thursday, September 2, 2010
Non-trivial use of this
physical quantity
12
Thursday, September 2, 2010
An infinite class of
N=2 SCFT’s
13
Thursday, September 2, 2010
An infinite class of
N=2 SCFT’s
• Write down Lagrangians using a graphical
representation (Quivers)
13
Thursday, September 2, 2010
An infinite class of
N=2 SCFT’s
• Write down Lagrangians using a graphical
representation (Quivers)
• All have N=2 supersymmetry
13
Thursday, September 2, 2010
An infinite class of
N=2 SCFT’s
• Write down Lagrangians using a graphical
representation (Quivers)
• All have N=2 supersymmetry
• 2 multiplets:
13
Thursday, September 2, 2010
An infinite class of
N=2 SCFT’s
• Write down Lagrangians using a graphical
representation (Quivers)
• All have N=2 supersymmetry
• 2 multiplets:
• Vector multiplet: gauge field, 2 Weyl
fermions, 1 complex scalar field
13
Thursday, September 2, 2010
An infinite class of
N=2 SCFT’s
• Write down Lagrangians using a graphical
representation (Quivers)
• All have N=2 supersymmetry
• 2 multiplets:
• Vector multiplet: gauge field, 2 Weyl
fermions, 1 complex scalar field
• 1/2 hyper: 1 complex scalar, 1 Weyl fermion
13
Thursday, September 2, 2010
Graphical rules
14
Thursday, September 2, 2010
Graphical rules
• Consider a diagram made of 2 objects
14
Thursday, September 2, 2010
Graphical rules
• Consider a diagram made of 2 objects
• lines & vertices
14
Thursday, September 2, 2010
Graphical rules
• Consider a diagram made of 2 objects
• lines & vertices
• vertices are 3-valent
14
Thursday, September 2, 2010
Lines
15
Thursday, September 2, 2010
Lines
• Each line of length L is an SU(2) V-plet with
gauge coupling g2 ~ 1/L
15
Thursday, September 2, 2010
Lines
• Each line of length L is an SU(2) V-plet with
gauge coupling g2 ~ 1/L
• a finite line - SU(2) gauge group
15
Thursday, September 2, 2010
Lines
• Each line of length L is an SU(2) V-plet with
gauge coupling g2 ~ 1/L
• a finite line - SU(2) gauge group
• an infinite line - global SU(2) symmetry
15
Thursday, September 2, 2010
Vertices
16
Thursday, September 2, 2010
Vertices
• Each vertex is a matter field charged as
(1/2, 1/2, 1/2) representation of SU(2)3
16
Thursday, September 2, 2010
Vertices
• Each vertex is a matter field charged as
(1/2, 1/2, 1/2) representation of SU(2)3
• each vertex has 8 1/2 hypermultiplets
16
Thursday, September 2, 2010
An infinite class
17
Thursday, September 2, 2010
An infinite class
• These rules define a unique N=2
supersymmetric Lagrangian
17
Thursday, September 2, 2010
An infinite class
• These rules define a unique N=2
supersymmetric Lagrangian
• a product of SU(2) gauge groups with
matter fields that have a global symmetry
that is a product of SU(2)’s.
17
Thursday, September 2, 2010
Example: 8 free 1/2
hypers
18
Thursday, September 2, 2010
Example: 8 free 1/2
hypers
• No lines of finite length - no gauge group
18
Thursday, September 2, 2010
Example: 8 free 1/2
hypers
• No lines of finite length - no gauge group
• 3 infinite lines - global symmetry SU(2)
3
18
Thursday, September 2, 2010
Example: 8 free 1/2
hypers
• No lines of finite length - no gauge group
• 3 infinite lines - global symmetry SU(2)
• 1 vertex - 8 1/2 hypers in (1/2,1/2,1/2) of
3
SU(2)3 - free
18
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
19
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
• 1 finite line - SU(2) gauge theory
19
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
• 1 finite line - SU(2) gauge theory
• 2 vertices - each gives 4 1/2 hyper which
are doublets of this SU(2). Together, have 8
1/2 hypers in the fundamental of SU(2).
19
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
• 1 finite line - SU(2) gauge theory
• 2 vertices - each gives 4 1/2 hyper which
are doublets of this SU(2). Together, have 8
1/2 hypers in the fundamental of SU(2).
• Called SU(2) with 4 flavors.
19
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
• 1 finite line - SU(2) gauge theory
• 2 vertices - each gives 4 1/2 hyper which
are doublets of this SU(2). Together, have 8
1/2 hypers in the fundamental of SU(2).
• Called SU(2) with 4 flavors.
• Global symmetry is SU(2) - subgroup of
4
SO(8)
19
Thursday, September 2, 2010
Example: loop adjoint
representation
20
Thursday, September 2, 2010
Example: loop adjoint
representation
• 1 finite line - SU(2) gauge group
20
Thursday, September 2, 2010
Example: loop adjoint
representation
• 1 finite line - SU(2) gauge group
• 1 vertex - 2 copies of (1/2,1/2) of this
SU(2); decomposes as 1+0 of SU(2)
20
Thursday, September 2, 2010
Example: loop adjoint
representation
• 1 finite line - SU(2) gauge group
• 1 vertex - 2 copies of (1/2,1/2) of this
SU(2); decomposes as 1+0 of SU(2)
• 1 infinite line - SU(2) global symmetry
f
20
Thursday, September 2, 2010
Example: loop adjoint
representation
• 1 finite line - SU(2) gauge group
• 1 vertex - 2 copies of (1/2,1/2) of this
SU(2); decomposes as 1+0 of SU(2)
• 1 infinite line - SU(2) global symmetry
• 2 adjoints, 2 singlets transform as 1/2 of
f
SU(2)f
20
Thursday, September 2, 2010
N=2 SCFT
21
Thursday, September 2, 2010
N=2 SCFT
• each finite line is an SU(2), the 2 vertices
contribute 4 flavors - beta function = 0.
21
Thursday, September 2, 2010
N=2 SCFT
• each finite line is an SU(2), the 2 vertices
contribute 4 flavors - beta function = 0.
• all these theories are SCFT’s
21
Thursday, September 2, 2010
An infinite class of N=2
SCFT’s
22
Thursday, September 2, 2010
An infinite class of N=2
SCFT’s
• See pdf file
22
Thursday, September 2, 2010
An infinite class of N=2
SCFT’s
• See pdf file
• Parametrized by
22
Thursday, September 2, 2010
An infinite class of N=2
SCFT’s
• See pdf file
• Parametrized by
• number of vertices, n
22
Thursday, September 2, 2010
An infinite class of N=2
SCFT’s
• See pdf file
• Parametrized by
• number of vertices, n
• number of loops, g
22
Thursday, September 2, 2010
An infinite class of N=2
SCFT’s
• See pdf file
• Parametrized by
• number of vertices, n
• number of loops, g
• call number of external legs e
22
Thursday, September 2, 2010
An infinite class of N=2
SCFT’s
• See pdf file
• Parametrized by
• number of vertices, n
• number of loops, g
• call number of external legs e
• n = 2g-2+e
22
Thursday, September 2, 2010
Tri-vertices & SU(2)’s
(g, e)
23
Thursday, September 2, 2010
Tri-vertices & SU(2)’s
(g, e)
• Number of gauge groups: 3g-3+e
23
Thursday, September 2, 2010
Tri-vertices & SU(2)’s
(g, e)
• Number of gauge groups: 3g-3+e
• number of matter fields (1/2)x8x(2g-2+e)
23
Thursday, September 2, 2010
Tri-vertices & SU(2)’s
(g, e)
• Number of gauge groups: 3g-3+e
• number of matter fields (1/2)x8x(2g-2+e)
• global symmetry SU(2)
e
23
Thursday, September 2, 2010
Tri-vertices & SU(2)’s
(g, e)
• Number of gauge groups: 3g-3+e
• number of matter fields (1/2)x8x(2g-2+e)
• global symmetry SU(2)
• Low energy spectrum
e
23
Thursday, September 2, 2010
Tri-vertices & SU(2)’s
(g, e)
• Number of gauge groups: 3g-3+e
• number of matter fields (1/2)x8x(2g-2+e)
• global symmetry SU(2)
• Low energy spectrum
• give VEV’s to H-plets - g massless photons
e
23
Thursday, September 2, 2010
Tri-vertices & SU(2)’s
(g, e)
• Number of gauge groups: 3g-3+e
• number of matter fields (1/2)x8x(2g-2+e)
• global symmetry SU(2)
• Low energy spectrum
• give VEV’s to H-plets - g massless photons
• moduli space has cplx dimension 2(e+1)
e
23
Thursday, September 2, 2010
Many Lagrangians
24
Thursday, September 2, 2010
Many Lagrangians
• For given (g,e) there are more than one
lagrangian
24
Thursday, September 2, 2010
Example:
g=2, e=0
25
Thursday, September 2, 2010
Example:
g=2, e=0
•
SU(2)3
25
Thursday, September 2, 2010
Example:
g=2, e=0
•
• One theory with adjoints
SU(2)3
25
Thursday, September 2, 2010
Example:
g=2, e=0
•
• One theory with adjoints
• One with fundamentals
SU(2)3
25
Thursday, September 2, 2010
Example:
g=2, e=0
•
• One theory with adjoints
• One with fundamentals
• Different Lagrangians, different matter
SU(2)3
contect
25
Thursday, September 2, 2010
Duality conjecture
26
Thursday, September 2, 2010
Duality conjecture
• The low energy dynamics depends on g & e
only and not on the choice of the
Lagrangian
26
Thursday, September 2, 2010
Duality conjecture
• The low energy dynamics depends on g & e
only and not on the choice of the
Lagrangian
• Many to one dualities!!
26
Thursday, September 2, 2010
Use Hilbert Series to
give a non-trivial check
of the conjecture
27
Thursday, September 2, 2010
Example: 8 free 1/2
hypers
28
Thursday, September 2, 2010
Example: 8 free 1/2
hypers
• No lines of finite length - no gauge group
28
Thursday, September 2, 2010
Example: 8 free 1/2
hypers
• No lines of finite length - no gauge group
• 3 infinite lines - global symmetry SU(2)
3
28
Thursday, September 2, 2010
Example: 8 free 1/2
hypers
• No lines of finite length - no gauge group
• 3 infinite lines - global symmetry SU(2)
• 1 vertex - 8 1/2 hypers in (1/2,1/2,1/2) of
3
SU(2)3 - free
28
Thursday, September 2, 2010
Expansion; first few
lower dimensions
1 + [1; 1; 1]t + ([2; 2; 2] + [2; 0; 0] + [0; 2; 0] + [0; 0; 2])t + . . .
2
29
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
30
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
• 1 finite line - SU(2) gauge theory
30
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
• 1 finite line - SU(2) gauge theory
• 2 vertices - each gives 4 1/2 hyper which
are doublets of this SU(2). Together, have 8
1/2 hypers in the fundamental of SU(2).
30
Thursday, September 2, 2010
Example: SU(2) with 4
flavors
• 1 finite line - SU(2) gauge theory
• 2 vertices - each gives 4 1/2 hyper which
are doublets of this SU(2). Together, have 8
1/2 hypers in the fundamental of SU(2).
•
Global symmetry is SU(2)4 - subgroup of
SO(8)
30
Thursday, September 2, 2010
Example
g=1, e=2
31
Thursday, September 2, 2010
Example
g=1, e=2
• 2 SU(2) gauge groups; 2 global SU(2)’s
31
Thursday, September 2, 2010
general HS:
(g,e)
32
Thursday, September 2, 2010
general HS:
(g,e)
• coefficient of m is 2g-2+e
32
Thursday, September 2, 2010
general HS:
(g,e)
• coefficient of m is 2g-2+e
• constant factor is 2g+e
32
Thursday, September 2, 2010
Summary
33
Thursday, September 2, 2010
Summary
• tri-vertices & lines give an infinite class of
N=2 SCFT’s with SU(2) gauge groups and
global symmetries
33
Thursday, September 2, 2010
Summary
• tri-vertices & lines give an infinite class of
N=2 SCFT’s with SU(2) gauge groups and
global symmetries
• 2 parameter family parametrized by g & e
33
Thursday, September 2, 2010
Summary
• tri-vertices & lines give an infinite class of
N=2 SCFT’s with SU(2) gauge groups and
global symmetries
• 2 parameter family parametrized by g & e
• Low energy dynamics - conjectured to
depend on g & e and not on the choice of
the Lagrangian
33
Thursday, September 2, 2010
Summary
34
Thursday, September 2, 2010
Summary
• The Hilbert Series is a physical quantity
that depends only on g & e
34
Thursday, September 2, 2010
Summary
• The Hilbert Series is a physical quantity
that depends only on g & e
• Hence gives a non-trivial strong support of
the conjecture
34
Thursday, September 2, 2010
Summary
• The Hilbert Series is a physical quantity
that depends only on g & e
• Hence gives a non-trivial strong support of
the conjecture
• An interesting class to study - challenge
problem to young researchers!!
34
Thursday, September 2, 2010