Instanton partition function in N=2 4-dim...
Transcript of Instanton partition function in N=2 4-dim...
KEK理論研究会 1
吉田豊
Y Yoshida arXiv11010872[hep-th]
KEK理論研究会 2
Moore Nekrasov amp Shatashivli (1998) Nekrasov(2002)
Instanton partition function in N=2 4-dim SYM
k-Instanton partition function by Localization formula ex) G=U(N) vector multiplet
Instanton number
KEK理論研究会 3
Instanton partition function with surface operator in N=2 SYM
Alday et al(2009) Alday amp Tachikawa Bruzzo et al(2010)
Instanton number The first Chern number
Dimofte Gukov amp Hollands (2010)
Vortex partition function in N=(22) 2dim SQED
KEK理論研究会 4
The moduli space of abelian vortex with vortex number k in two dimension is isomorphic to
Jaffe amp Taubes(1980)
Equivariant character
k-vortex partition function for N=(22) SQED with single chiral multiplet
contour integral representation
KEK理論研究会 5
Contribution from a vector multiplet
Contribution from a chiral multiplet
vortex partition function of N=(22) SQED with chiral multiplet
twisted mass
KEK理論研究会 6
5d Nekrasov partition(K-theoretic instanton counting)
Introduction of Surface operator Introduction of A-brane
Closed A-model on toric CY
G=U(1) 4-dim pure N=2 SYM
ex)
Theory induced on the surface operator is N=(22) U(1) SQED with single chiral mutiplet
string side gauge theory side
Kozcaz Pasquetti amp Wyllard(2010)
1 Introduction
2Vortices in 2d super Yang-Mills theories
3 Localization of vortex in N=(22) SYM
4 Vortex partition and equivariant character
5 Relation to geometric indices
6 Summary
KEK理論研究会 7
KEK理論研究会 8
Vortex equation (Bogomolrsquonyi equation) with G=U(N)
1This equation preserves half of the supersymmetry 2 On-shell action
Vortex number is defined by the first Chern number
complexified FI-parameter
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 2
Moore Nekrasov amp Shatashivli (1998) Nekrasov(2002)
Instanton partition function in N=2 4-dim SYM
k-Instanton partition function by Localization formula ex) G=U(N) vector multiplet
Instanton number
KEK理論研究会 3
Instanton partition function with surface operator in N=2 SYM
Alday et al(2009) Alday amp Tachikawa Bruzzo et al(2010)
Instanton number The first Chern number
Dimofte Gukov amp Hollands (2010)
Vortex partition function in N=(22) 2dim SQED
KEK理論研究会 4
The moduli space of abelian vortex with vortex number k in two dimension is isomorphic to
Jaffe amp Taubes(1980)
Equivariant character
k-vortex partition function for N=(22) SQED with single chiral multiplet
contour integral representation
KEK理論研究会 5
Contribution from a vector multiplet
Contribution from a chiral multiplet
vortex partition function of N=(22) SQED with chiral multiplet
twisted mass
KEK理論研究会 6
5d Nekrasov partition(K-theoretic instanton counting)
Introduction of Surface operator Introduction of A-brane
Closed A-model on toric CY
G=U(1) 4-dim pure N=2 SYM
ex)
Theory induced on the surface operator is N=(22) U(1) SQED with single chiral mutiplet
string side gauge theory side
Kozcaz Pasquetti amp Wyllard(2010)
1 Introduction
2Vortices in 2d super Yang-Mills theories
3 Localization of vortex in N=(22) SYM
4 Vortex partition and equivariant character
5 Relation to geometric indices
6 Summary
KEK理論研究会 7
KEK理論研究会 8
Vortex equation (Bogomolrsquonyi equation) with G=U(N)
1This equation preserves half of the supersymmetry 2 On-shell action
Vortex number is defined by the first Chern number
complexified FI-parameter
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 3
Instanton partition function with surface operator in N=2 SYM
Alday et al(2009) Alday amp Tachikawa Bruzzo et al(2010)
Instanton number The first Chern number
Dimofte Gukov amp Hollands (2010)
Vortex partition function in N=(22) 2dim SQED
KEK理論研究会 4
The moduli space of abelian vortex with vortex number k in two dimension is isomorphic to
Jaffe amp Taubes(1980)
Equivariant character
k-vortex partition function for N=(22) SQED with single chiral multiplet
contour integral representation
KEK理論研究会 5
Contribution from a vector multiplet
Contribution from a chiral multiplet
vortex partition function of N=(22) SQED with chiral multiplet
twisted mass
KEK理論研究会 6
5d Nekrasov partition(K-theoretic instanton counting)
Introduction of Surface operator Introduction of A-brane
Closed A-model on toric CY
G=U(1) 4-dim pure N=2 SYM
ex)
Theory induced on the surface operator is N=(22) U(1) SQED with single chiral mutiplet
string side gauge theory side
Kozcaz Pasquetti amp Wyllard(2010)
1 Introduction
2Vortices in 2d super Yang-Mills theories
3 Localization of vortex in N=(22) SYM
4 Vortex partition and equivariant character
5 Relation to geometric indices
6 Summary
KEK理論研究会 7
KEK理論研究会 8
Vortex equation (Bogomolrsquonyi equation) with G=U(N)
1This equation preserves half of the supersymmetry 2 On-shell action
Vortex number is defined by the first Chern number
complexified FI-parameter
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 4
The moduli space of abelian vortex with vortex number k in two dimension is isomorphic to
Jaffe amp Taubes(1980)
Equivariant character
k-vortex partition function for N=(22) SQED with single chiral multiplet
contour integral representation
KEK理論研究会 5
Contribution from a vector multiplet
Contribution from a chiral multiplet
vortex partition function of N=(22) SQED with chiral multiplet
twisted mass
KEK理論研究会 6
5d Nekrasov partition(K-theoretic instanton counting)
Introduction of Surface operator Introduction of A-brane
Closed A-model on toric CY
G=U(1) 4-dim pure N=2 SYM
ex)
Theory induced on the surface operator is N=(22) U(1) SQED with single chiral mutiplet
string side gauge theory side
Kozcaz Pasquetti amp Wyllard(2010)
1 Introduction
2Vortices in 2d super Yang-Mills theories
3 Localization of vortex in N=(22) SYM
4 Vortex partition and equivariant character
5 Relation to geometric indices
6 Summary
KEK理論研究会 7
KEK理論研究会 8
Vortex equation (Bogomolrsquonyi equation) with G=U(N)
1This equation preserves half of the supersymmetry 2 On-shell action
Vortex number is defined by the first Chern number
complexified FI-parameter
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 5
Contribution from a vector multiplet
Contribution from a chiral multiplet
vortex partition function of N=(22) SQED with chiral multiplet
twisted mass
KEK理論研究会 6
5d Nekrasov partition(K-theoretic instanton counting)
Introduction of Surface operator Introduction of A-brane
Closed A-model on toric CY
G=U(1) 4-dim pure N=2 SYM
ex)
Theory induced on the surface operator is N=(22) U(1) SQED with single chiral mutiplet
string side gauge theory side
Kozcaz Pasquetti amp Wyllard(2010)
1 Introduction
2Vortices in 2d super Yang-Mills theories
3 Localization of vortex in N=(22) SYM
4 Vortex partition and equivariant character
5 Relation to geometric indices
6 Summary
KEK理論研究会 7
KEK理論研究会 8
Vortex equation (Bogomolrsquonyi equation) with G=U(N)
1This equation preserves half of the supersymmetry 2 On-shell action
Vortex number is defined by the first Chern number
complexified FI-parameter
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 6
5d Nekrasov partition(K-theoretic instanton counting)
Introduction of Surface operator Introduction of A-brane
Closed A-model on toric CY
G=U(1) 4-dim pure N=2 SYM
ex)
Theory induced on the surface operator is N=(22) U(1) SQED with single chiral mutiplet
string side gauge theory side
Kozcaz Pasquetti amp Wyllard(2010)
1 Introduction
2Vortices in 2d super Yang-Mills theories
3 Localization of vortex in N=(22) SYM
4 Vortex partition and equivariant character
5 Relation to geometric indices
6 Summary
KEK理論研究会 7
KEK理論研究会 8
Vortex equation (Bogomolrsquonyi equation) with G=U(N)
1This equation preserves half of the supersymmetry 2 On-shell action
Vortex number is defined by the first Chern number
complexified FI-parameter
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
1 Introduction
2Vortices in 2d super Yang-Mills theories
3 Localization of vortex in N=(22) SYM
4 Vortex partition and equivariant character
5 Relation to geometric indices
6 Summary
KEK理論研究会 7
KEK理論研究会 8
Vortex equation (Bogomolrsquonyi equation) with G=U(N)
1This equation preserves half of the supersymmetry 2 On-shell action
Vortex number is defined by the first Chern number
complexified FI-parameter
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 8
Vortex equation (Bogomolrsquonyi equation) with G=U(N)
1This equation preserves half of the supersymmetry 2 On-shell action
Vortex number is defined by the first Chern number
complexified FI-parameter
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 9
Super YM theory with 8 SUSY (2-dim N=(44) SYM)
The vector multiplet in N=(44) SYM consists of
Hypermultiplets in N=(44) theory consists of
matter content of N=(44) theory
N=(22) vector multiplet
N=(22) adjoint chiral multiplet
N=(22) fundametnal chiral multiplet
N=(22) anti-fundametnal chiral multiplet
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 10
Vacuum (Higgs branch)
rFI-parameter
Symmetry group of Vacuum
Bosonic part of Lagrangian
Global gauge group
Flavor group
twisted mass
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 11
k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSY by k Dp- N D(p+2) brane construction(Hanany amp Tong 2002)
0 1 2 3 4 5 6 7 8 9 NS5 o o o o o o D2 o o o D0 o
vortex partition function(zero mode theory) in N=(44) SYM
from brane system
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 12
D0-D0
D0-D2
I orientational moduli
B translational moduli
DRED of vector with gauge group
DRED of adjoint chiral multiplet
DRED of chiral malutiplet
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 13
k-vortex partition functions Chen and Tong (2006)
Mass deformation
D-term condition
The moduli space of k-vortex Eto et al(2005) Hanany amp Tong(2002)
We consider mass deformation N=(44) theory Taking large mass limit we obtain N=(22) SYM with N chiral multiplets
Edalati amp Tong (2007)
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 14
DRED of 2d (02) chiral multipet
DRED of 2d (02) fermi multipet
In the presence of the mass term vortex partition function is deformed
multiplets decouple from the vortex theory
heavy mass limit
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 15
k-vortex partition function for N=(22) U(N) SYM with N-fundamental matter
with
This action is expressed in Q-exact form
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 16
SUSY transformation generates the following vector field on
Nekrasov (2002) Bruzzo et al (2002)
Superdeterminant
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 17
k-vortex parition function in G=U(N) N=(22) SYM N-flavor
Vortex partition function in G=U(1) N=(22) SQED
This agree with the result from the equivariant character
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 18
We introduce the following torus action
Vortex moduli space
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 19
At the fixed points we can decompose the representation space as
Gauge transformation
Restriction map
Fixed point condition
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 20
2d partition (Young diagram)
1d partition
In the case of 4-dim instantonhellip
In the case of 2-dim vortex
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 21
bullcharacter of each spaces
Infinitesimal gauge transformation
Tangent space of k-vortex moduli space
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 22
equivariant character
3d vortex partition function
Replacement
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 23
-genus of complex manifold M
Equivariant case
The fixed points
The weight at the point
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
KEK理論研究会 24
3d vortex partition function
This corresponds to geometric genus
This corresponds to Euler number
N=(22) case
N=(44) case
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25
We have obtained N=(22) vortex partition function from the mass deformation of N=(44) vortex partition function
N=(22) vortex partition function can be written with Q-exact form
rArr We can apply Localization formula especially we reproduce abelian vortex from open BPS state counting or
equivariant character of Vortex parition function is expressed by 1d partition Cf) Nekrasov partition is expressed by 2d partition(Young diagram) 3d vortex partition is related to certain geometric indices of the k-vortex moduli space Future direction Relation to integrable structure( KP hierarchy spin chain) etchellip
KEK理論研究会 25