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Exact calculation for AB-phase effective potential via supersymmetric localization
A.T, A. Tomiya, T. Shimotani
Exact calculation for AB-phase effective potential via supersymmetric localization
A.T, A. Tomiya, T. Shimotani
work in progress
Todayʼs concern ispurely theoretical...
Do these plots represent different vacua?
Introduction
Introduction
Phenomena Tools
Introduction
...SSB
Higgs mechanism
Hosotani mechanism
Phenomena Tools
Introduction
...SSB
Higgs mechanism
Hosotani mechanism
Phenomena Tools
Introduction
...SSB
Higgs mechanism
Hosotani mechanism
...Quantum correctionperturbationLattice
Phenomena Tools
Introduction
...SSB
Higgs mechanism
Hosotani mechanism
...Quantum correctionperturbationLattice
Phenomena Tools
Introduction
Hosotani mechanism (1983)
Introduction
Hosotani mechanism
×M S1(1983)
Introduction
Hosotani mechanism
×M S1(1983)
Introduction
Hosotani mechanism
×M S1FMN = 0(1983)
Introduction
Hosotani mechanism
×M S1FMN = 0
Aµ = 0
AS1 = θ
(1983)
Introduction
Hosotani mechanism
×M S1
Aµ = 0
AS1 = θ
�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
(1983)
=�
dθe−Γ(θ)
Introduction
Hosotani mechanism
How to perform?
(1983)�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
=�
dθe−Γ(θ)
Introduction
Hosotani mechanism
1. perturbation
(1983)�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
=�
dθe−Γ(θ)
Veff = V treeeff + V 1−loop
eff + V 2−loopeff + ...
Introduction
Hosotani mechanism
1. perturbation
0 Finite !
(1983)
arXiv:hep-ph/0504272
SU(3)
�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
=�
dθe−Γ(θ)
Veff = V treeeff + V 1−loop
eff + V 2−loopeff + ...
Introduction
Hosotani mechanism
2. Lattice
(1983)�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
=�
dθe−Γ(θ)
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
PRPolyakov loop via lattice action
=�
dθe−Γ(θ)
∝ �PR�
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
Polyakov loop via lattice action
arXiv:0904.1353PR=
�dθe−Γ(θ)
∝ �PR�
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
Polyakov loop via lattice action
arXiv:0904.1353PR=
�dθe−Γ(θ)
∝ �PR�
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
Polyakov loop via lattice action
arXiv:0904.1353PR=
�dθe−Γ(θ)
∝ �PR�
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
Polyakov loop via lattice action
arXiv:0904.1353PR=
�dθe−Γ(θ)
∝ �PR�
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
Polyakov loop via lattice action
arXiv:0904.1353PR=
�dθe−Γ(θ)
∝ �PR�
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
Polyakov loop via lattice action
arXiv:0904.1353PR=
�dθe−Γ(θ)
∝ �PR�
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
Polyakov loop via lattice action
arXiv:0904.1353PR=
�dθe−Γ(θ)
∝ �PR�
Introduction
Hosotani mechanism�D(A + θ)e−S(A+θ)
=�
dθ
�DAe−S(A+θ)
2. Lattice
(1983)
PR
PR
Polyakov loop via lattice action
arXiv:0904.1353PR=
�dθe−Γ(θ)
∝ �PR�
Introduction
...SSB
Higgs mechanism
Hosotani mechanism
...Quantum correctionperturbationLattice
Phenomena Tools
Introduction
...SSB
Higgs mechanism
Hosotani mechanism
...Quantum correctionperturbationLattice SUSY localization
Phenomena Tools
Hosotani mechanismSUSY localization
1. Geometry setup
×2. SUSY on ×
3. Localizationd
dtZ(t) = 0
λ
σ
D
λ†
Aµ
Mass:k
Hosotani mechanismSUSY localization
1. Geometry setup
×2. SUSY on ×
3. Localizationd
dtZ(t) = 0
4. Results on Veff�D(fields)e−S =
�dθe−Veff (θ)
?
5. Results on �P3�
6. Preliminary results
λ
σ
D
λ†
Aµ
Mass:k
Hosotani mechanismSUSY localization
1. Geometry setup
×
1. Geometry setup
×
1. Geometry setup
×
×
1. Geometry setup
×
×
β
1. Geometry setup
×ClaimIf CFT,
×β
limβ→0
= ×
β
1. Geometry setup
×ClaimIf CFT,
×β
limβ→0
= ×
1. Geometry setup
×ClaimIf CFT,
×β
limβ→0
= ×
1. Geometry setup
ClaimIf CFT,
βlimβ→0
=
××
1. Geometry setup
ClaimIf CFT,
βlimβ→0
=
××
Hosotani mechanismSUSY localization
1. Geometry setup
2. SUSY on
3. Localizationd
dtZ(t) = 0
4. Results on Veff�D(fields)e−S =
�dθe−Veff (θ)
?
5. Results on �P3�
6. Preliminary results
λ
σ
D
λ†
Aµ
×
Mass:k
Hosotani mechanismSUSY localization
2. SUSY on
λ
σ
D
λ†
Aµ
×
Mass:k
2. SUSY on ×
Spin 1
Spin 1/2
Spin 0
λ
σ
D
λ†
Aµ
2. SUSY on
λ
σ
D
λ†
Aµ
×
2. SUSY on
λ
σ
D
λ†
Aµ
×
∇µ� =12γµγ3�
curved effect
2. SUSY on
λ
σ
D
λ†
Aµ
×
δ�SSCS = 0 δη†SSCS = 0
SUSY invariance
SSCS =14π
�d3x
√gTr
� 1√
g�µνλ(Aµ∂νAλ +
2i
3AµAνAλ)− λ†λ + 2Dσ
�
2. SUSY on
λ
σ
D
λ†
Aµ
×
δ�SSCS = 0 δη†SSCS = 0
SUSY invariance
SSCS =14π
�d3x
√gTr
� 1√
g�µνλ(Aµ∂νAλ +
2i
3AµAνAλ)− λ†λ + 2Dσ
�
“Mass”
2. SUSY on
λ
σ
D
λ†
Aµ
×
SUSY invarianceδ�SSY M = 0 δη†SSY M = 0
2. SUSY on
λ
σ
D
λ†
Aµ
×
SUSY invarianceδ�SSY M = 0 δη†SSY M = 0
SUSY exactnessSSY M = δ�V
2. SUSY on ×
Hosotani mechanismSUSY localization
1. Geometry setup
×
3. Localizationd
dtZ(t) = 0
4. Results on Veff�D(fields)e−S =
�dθe−Veff (θ)
?
5. Results on �P3�
6. Preliminary results
λ
σ
D
λ†
Aµ
Mass:k
Hosotani mechanismSUSY localization
3. Localizationd
dtZ(t) = 0
3. Localization
d
dtZ(t) = 0
Why?
Z(t) :=�D(fields)eikSSCS−tSSY M
fields : (Aµ,λ†,λ,σ, D)
δ�V
=
d
dtZ(t) =
�D(fields)
d
dteikSSCS−tδ�V
=�D(fields)(−δ�V )eikSSCS−tδ�V
=�D(fields)δ�
�− V eikSSCS−tδ�V
�
=�D(fields)(total derivative)
=0
3. Localization
d
dtZ(t) = 0
Z(1) =Z(∞) ←Steepest decent method is exact
Z(t) :=�D(fields)eikSSCS−tSSY M
fields : (Aµ,λ†,λ,σ, D)
δ�V
=
3. Localization
d
dtZ(t) = 0
Z(1) =Z(∞) ←Steepest decent method is exact
Z(t) :=�D(fields)eikSSCS−tSSY M
fields : (Aµ,λ†,λ,σ, D)
δ�V
=
3. Localization
d
dtZ(t) = 0
FMN = 0
Aµ = 0
AS1 = θ
Z(1) =Z(∞) ←Steepest decent method is exact
Z(t) :=�D(fields)eikSSCS−tSSY M
fields : (Aµ,λ†,λ,σ, D)
δ�V
=
3. Localization
d
dtZ(t) = 0
AS1 = θ
Aµ = a(m)
Z(1) =Z(∞) ←Steepest decent method is exact
Z(t) :=�D(fields)eikSSCS−tSSY M
fields : (Aµ,λ†,λ,σ, D)
δ�V
=
3. Localization
d
dtZ(t) = 0
AS1 = θ
Aµ = a(m)
Z(1) =Z(∞) ←Steepest decent method is exact
Integers
Z(t) :=�D(fields)eikSSCS−tSSY M
fields : (Aµ,λ†,λ,σ, D)
δ�V
=
3. Localization
d
dtZ(t) = 0
Z(1) =Z(∞) ←Steepest decent method is exact
Integers
= ∞�
m1,m2,...=−∞ψ(m)
� π/β
0dθ1dθ2...e
2kiP
miθiβ�
i<j
�coshβ(mi −mj)− cos 2β(θi − θj)
�
Z(t) :=�D(fields)eikSSCS−tSSY M
fields : (Aµ,λ†,λ,σ, D)
δ�V
=
3. Localization
d
dtZ(t) = 0
Z(1) =Z(∞) ←Steepest decent method is exact
Integers
=cannot determine phases
∞�
m1,m2,...=−∞ψ(m)
� π/β
0dθ1dθ2...e
2kiP
miθiβ�
i<j
�coshβ(mi −mj)− cos 2β(θi − θj)
�
Z(t) :=�D(fields)eikSSCS−tSSY M
fields : (Aµ,λ†,λ,σ, D)
δ�V
=
Hosotani mechanismSUSY localization
1. Geometry setup
×2. SUSY on ×
3. Localizationd
dtZ(t) = 0
4. Results on Veff�D(fields)e−S =
�dθe−Veff (θ)
?
5. Results on �P3�
6. Preliminary results
λ
σ
D
λ†
Aµ
Mass:k
Hosotani mechanismSUSY localization
4. Results on Veff�D(fields)e−S =
�dθe−Veff (θ)
?
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
ClaimIf CFT,
×β
limβ→0
= ×
∞�
m1,m2,...=−∞ψ(m)
� π/β
0dθ1dθ2...e
2kiP
miθiβ�
i<j
�coshβ(mi −mj)− cos 2β(θi − θj)
�
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
ClaimIf CFT,
×β
limβ→0
= ×
θ̃ := βθ
∞�
m1,m2,...=−∞ψ(m)
� π/β
0dθ1dθ2...e
2kiP
miθiβ�
i<j
�coshβ(mi −mj)− cos 2β(θi − θj)
�
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
ClaimIf CFT,
×β
limβ→0
= ×
βN
� π
0dθ̃1dθ̃2...
θ̃ := βθ
e2kiP
miθ̃i
∞�
m1,m2,...=−∞ψ(m)
� π/β
0dθ1dθ2...e
2kiP
miθiβ�
i<j
�coshβ(mi −mj)− cos 2β(θi − θj)
�
cos 2(θ̃i − θ̃j)
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
ClaimIf CFT,
×β
limβ→0
= ×
θ̃ := βθ
cosh 0 = 1βN
� π
0dθ̃1dθ̃2...e
2kiP
miθ̃i
∞�
m1,m2,...=−∞ψ(m)
� π/β
0dθ1dθ2...e
2kiP
miθiβ�
i<j
�coshβ(mi −mj)− cos 2β(θi − θj)
�
cos 2(θ̃i − θ̃j)
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
∞�
m1,m2,...=−∞ψ̃(m)
� π
0dθ̃1dθ̃2...e
2kiP
miθ̃i�
i<j
sin2(θ̃i − θ̃j)
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
Choice of “wave function” ψ̃(m)
∞�
m1,m2,...=−∞ψ̃(m)
� π
0dθ̃1dθ̃2...e
2kiP
miθ̃i�
i<j
sin2(θ̃i − θ̃j)
Example with SU(3) 1:
Veff (θ̃1, θ̃2)
ψ̃(m,n,−m− n) = (δm,0 + δm,0)(δn,0 + δn,0)
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
Choice of “wave function” ψ̃(m)
∞�
m1,m2,...=−∞ψ̃(m)
� π
0dθ̃1dθ̃2...e
2kiP
miθ̃i�
i<j
sin2(θ̃i − θ̃j)
Example with SU(3) 2: ψ̃(m,n,−m− n) = (δm,1 + δm,−1)(δn,1 + δn,−1)
Veff (θ̃1, θ̃2)
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
Choice of “wave function” ψ̃(m)
∞�
m1,m2,...=−∞ψ̃(m)
� π
0dθ̃1dθ̃2...e
2kiP
miθ̃i�
i<j
sin2(θ̃i − θ̃j)
Example with SU(3) 3:
Veff (θ̃1, θ̃2)
ψ̃(m,n,−m− n) = 1
=?, but a little bit interesting.
Poisson resummation:∞�
n=−∞e2πixn =
∞�
p=−∞δ(x− p)
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
Choice of “wave function” ψ̃(m)
∞�
m1,m2,...=−∞ψ̃(m)
� π
0dθ̃1dθ̃2...e
2kiP
miθ̃i�
i<j
sin2(θ̃i − θ̃j)
Example with SU(3) 3:
Veff (θ̃1, θ̃2)
ψ̃(m,n,−m− n) = 1
=?, but a little bit interesting.
2θ̃1 + θ̃2 =2π
kp, θ̃1 + 2θ̃2 =
2π
kq, p, q ∈ Z
4. Results on Veff
�D(fields)e−S =
�dθe−Veff (θ)
?
Choice of “wave function” ψ̃(m)
∞�
m1,m2,...=−∞ψ̃(m)
� π
0dθ̃1dθ̃2...e
2kiP
miθ̃i�
i<j
sin2(θ̃i − θ̃j)
Example with SU(3) 3:
Veff (θ̃1, θ̃2)
ψ̃(m,n,−m− n) = 1
=?, but a little bit interesting.
2θ̃1 + θ̃2 =2π
kp, θ̃1 + 2θ̃2 =
2π
kq, p, q ∈ Z
Hosotani mechanismSUSY localization
1. Geometry setup
×2. SUSY on ×
3. Localizationd
dtZ(t) = 0
4. Results on Veff�D(fields)e−S =
�dθe−Veff (θ)
?
5. Results on �P3�
6. Preliminary results
λ
σ
D
λ†
Aµ
Mass:k
Hosotani mechanismSUSY localization
5. Results on �P3�
5. Results on �P3�
×
5. Results on �P3�
×
5. Results on �P3�
×
5. Results on �P3�
×
5. Results on �P3�
5. Results on �P3�
5. Results on �P3�
Calculable if δ�P3 = 0
2 possibilities
5. Results on �P3�
Calculable if δ�P3 = 0
2 possibilities
5. Results on �P3�
Calculable if δ�P3 = 0
2 possibilities
We consider this with SU(3).
5. Results on �P3�
�P3�=
∞�
m1,m2,...=−∞ψ(m)
� π/β
0dθ1dθ2...e
2kiP
miθiβ�
i<j
�coshβ(mi −mj)− cos 2β(θi − θj)
�
× TrRn(2βiθ + βm)
Hosotani mechanismSUSY localization
1. Geometry setup
×2. SUSY on ×
3. Localizationd
dtZ(t) = 0
4. Results on Veff�D(fields)e−S =
�dθe−Veff (θ)
?
5. Results on �P3�
6. Preliminary results
λ
σ
D
λ†
Aµ
Mass:k
Hosotani mechanismSUSY localization
6. Preliminary results
6. Preliminary results
2θ̃1 + θ̃2 =2π
kp,
θ̃1 + 2θ̃2 =2π
kq,
p, q ∈ Z
�P3� via various vacua?
k = 4
ψ̃(m,n,−m− n) = (δm,1 + δm,−1)(δn,1 + δn,−1)
ψ̃(m,n,−m− n) = 1
Running p,q
Importance sampling
6. Preliminary results �P3� via various vacua?
6. Preliminary results �P3� via various vacua?
arXiv:0904.1353
6. Preliminary results
2θ̃1 + θ̃2 =2π
kp,
θ̃1 + 2θ̃2 =2π
kq,
p, q ∈ Z
�P3� via various vacua?
k = 4
ψ̃(m,n,−m− n) = (δm,1 + δm,−1)(δn,1 + δn,−1)
ψ̃(m,n,−m− n) = 1
Running p,q
Importance sampling
6. Preliminary results �P3� via various vacua?
6. Preliminary results �P3� via various vacua?
6. Preliminary results
2θ̃1 + θ̃2 =2π
kp,
θ̃1 + 2θ̃2 =2π
kq,
p, q ∈ Z
�P3� via various vacua?
k = 4
ψ̃(m,n,−m− n) = (δm,1 + δm,−1)(δn,1 + δn,−1)
ψ̃(m,n,−m− n) = 1
Running p,q
Importance sampling
Thank you.