Bions in Large-N Sigma Models
Toshiaki Fujimori (Keio University)
based onarXiv:1607.04205, arXiv:1702.00589
arXiv:1705.10483, arXiv:1810.03768···

In collaboration with
Syo Kamata Tatsuhiro Misumi
Muneto Nitta Norisuke Sakai

Potential Toolkit to Attack Nonperturbative Aspects of QFT
September 24, 2020

Path Integral and Trans-series
path integral

Z = 𝒟φ exp (−S[φ])∫Path Integral and Trans-series
path integral

Z = 𝒟φ exp (−S[φ])∫Path Integral and Trans-series
path integral

Z = 𝒟φ exp (−S[φ])∫· integral over all field configurations ··· (usually) impossible

Path Integral and Trans-series
path integral

Z = 𝒟φ exp (−S[φ])∫· integral over all field configurations ··· (usually) impossible· perturbative expansion ··· divergent, Borel non-summablePath Integral and Trans-series
path integral

Z = 𝒟φ exp (−S[φ])∫· integral over all field configurations ··· (usually) impossible· perturbative expansion ··· divergent, Borel non-summablePath Integral and Trans-series
path integral

Z = 𝒟φ exp (−S[φ])∫· integral over all field configurations ··· (usually) impossible· perturbative expansion ··· divergent, Borel non-summablenon-perturbative saddle points

Zσ(g) = e −Sσ(g)g ασ (cσ,0 + cσ,1 g + cσ,2 g 2 + ⋯)Z(g) =

∑

σ∈𝔖

nσ Zσ

··· contribution of saddle point σSaddle Points of Action· saddle points : solutions of equation of motion of S

Saddle Points of Action· saddle points : solutions of equation of motion of S
· not only real solutions ··· complex saddle points

Saddle Points of Action· saddle points : solutions of equation of motion of S
· not only real solutions ··· complex saddle points· analytic continuation of action

S[φ] = S[φR + iφI]

Saddle Points of Action· saddle points : solutions of equation of motion of S
· not only real solutions ··· complex saddle points· analytic continuation of action

S[φ] = S[φR + iφI]

Euler-Lagrange equation

δS
=0
δφ

Saddle Points of Action· saddle points : solutions of equation of motion of S
· not only real solutions ··· complex saddle points· analytic continuation of action

S[φ] = S[φR + iφI]

Euler-Lagrange equation

δS
=0
δφ

→ “perturbative vacuum”, “instanton”, etc

Lefschetz Thimble
Z(g) =∑σ∈𝔖
Which saddle points?
Which contour?

nσ Zσ

Lefschetz Thimble
Z(g) =∑σ∈𝔖
Which saddle points?
Which contour?

nσ Zσ
Lefschetz thimbleLefschetz Thimble
Z(g) =∑σ∈𝔖
Which saddle points?
Which contour?

thimblenσ Zσ
Lefschetz thimble

𝒥σ and dual thimble 𝒦σLefschetz Thimble
Z(g) =∑σ∈𝔖
Which saddle points?
Which contour?

thimbleZσ =∫𝒥

Lefschetz thimble

𝒥σ and dual thimble 𝒦σ

𝒟φ exp(−S[φ])
σ

nσ Zσ

Lefschetz Thimble
Z(g) =∑σ∈𝔖
Which saddle points?
Which contour?

thimbleZσ =∫𝒥

Lefschetz thimble

𝒥σ and dual thimble 𝒦σ

𝒟φ exp(−S[φ])
σ

nσ Zσ

nσ = ⟨𝒦σ , 𝒞⟩

Renormalon

Renormalon· 4d QCD, 2d NLSM, etc : diagrams with factorially large contribution

ambiguity ± ie −Sσ with
•••

Borel

1
Sσ ∼ 2g N

Renormalon· 4d QCD, 2d NLSM, etc : diagrams with factorially large contribution

ambiguity ± ie −Sσ with

1
Sσ ∼ 2
g N

•••

Borel

≠ instanton

1
Sinst ∼ 2g··· vanishes for N ! 1, g 2 N = fixed<latexit sha1_base64="hbrwF0WC9oUrAm04DTfmr1l2RBo=">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</latexit>

Renormalon· 4d QCD, 2d NLSM, etc : diagrams with factorially large contribution

ambiguity ± ie −Sσ with

1
Sσ ∼ 2
g N

•••

Borel

≠ instanton

1
Sinst ∼ 2g··· vanishes for N ! 1, g 2 N = fixed

other saddle points?

<latexit sha1_base64="hbrwF0WC9oUrAm04DTfmr1l2RBo=">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</latexit>

renormalon

Bion Saddle Points· bion : pair of fractional instanton - instanton

Bion Saddle Points· bion : pair of fractional instanton - instanton

1
Sbion ∼ 2
g N··· survives in the large N limit

Bion Saddle Points· bion : pair of fractional instanton - instanton

1
Sbion ∼ 2
g N··· survives in the large N limit

relation between bion and renormalon?
[Argyres-Ünsal, 2012]

Bion Saddle Points· bion : pair of fractional instanton - instanton

1
Sbion ∼ 2
g N··· survives in the large N limit

relation between bion and renormalon?
[Argyres-Ünsal, 2012]· 2d CPN model is a good testing ground2d analog of 4d gauge theory
2d CPN model: toy model 4d QCD-like models

common properties :asymptotic freedom,
instanton, large-N limit …

2d analog of 4d gauge theory
2d CPN model: toy model 4d QCD-like models

common properties :asymptotic freedom,
instanton, large-N limit …

similar resurgence structure as 4d QCD

2d analog of 4d gauge theory
2d CPN model: toy model 4d QCD-like models

common properties :asymptotic freedom,
instanton, large-N limit …

similar resurgence structure as 4d QCD

bion：non-trivial resurgence structure?

Fractional Instanton· Euclidean action density of instanton in CP1 model on

high

y

x

· instanton(caloron)0[Eto-TF-Isozumi-NittaOhashi-Ohta-Sakai, 2006]
N fractional instantons in CPN-1 modelFractional Instanton· Euclidean action density of instanton in CP1 model on

high

y

x

· instanton(caloron)0[Eto-TF-Isozumi-NittaOhashi-Ohta-Sakai, 2006]
N fractional instantons in CPN-1 modelCPN-1 Model on Cylinder· cylinder : Euclidean time τ ∈ ℝ and spacial S 1x ∼ x + 2πR

R : compactification radius

1S=dτdx | 𝒟iϕA | + iD( | ϕA | − 2 )
∑∫
[
]
g
N
A=1
N

2

2

ϕA (A = 1,⋯, N) : charged scalar fields (qA = 1)
𝒟i : covariant derivative 𝒟iϕA = (∂i + iAi)ϕA
Ai : auxiliary U(1) gauge field, D : Lagrange multiplierTwisted Boundary Condition
ϕA(x + 2πR) = exp(2πiRmA) ϕA(x)Background Holonomy
for SU(N) Global Symmetry(m1, ⋯, mN ) ∈ 𝔲(1)N−1 ⊂ 𝔰𝔲(N)Twisted Boundary Condition
ϕA(x + 2πR) = exp(2πiRmA) ϕA(x)Background Holonomy
for SU(N) Global Symmetry(m1, ⋯, mN ) ∈ 𝔲(1)N−1 ⊂ 𝔰𝔲(N)A 1: enhancement of ZN-symmetry
mA =
NR
1
ix
Ax → Ax −ϕA → e NR ϕA+1
NR
adiabatic continuity

Instantons in CPN-1 Model· single instanton (in the A-th classical vacuum)ϕBu − u01 − 1 ψ −m R=e 2 u B ×{gaB
1Aτ = ∂xψ,2

ψ = log( | u |

−2mAR

1
Ax = − ∂τ ψ,22| u − u0 | +

∑

B≠A

u=e

(B = A)(B ≠ A)i 2
D = ∂i ψ
2−2mB R|u|| aB | )2

τ + ixR

Instantons in CPN-1 Model· single instanton (in the A-th classical vacuum)ϕBu − u01 − 1 ψ −m R=e 2 u B ×{gaB
1Aτ = ∂xψ,2

ψ = log( | u |

−2mAR

1
Ax = − ∂τ ψ,22| u − u0 | +

∑

B≠A

u=e

(B = A)(B ≠ A)i 2
D = ∂i ψ
2−2mB R|u|| aB | )2

τ + ixR

Instantons in CPN-1 Model· single instanton (in the A-th classical vacuum)ϕBu − u01 − 1 ψ −m R=e 2 u B ×{gaB
1Aτ = ∂xψ,2

ψ = log( | u |

−2mAR

1
Ax = − ∂τ ψ,22| u − u0 | +

∑

B≠A

u=e

(B = A)(B ≠ A)i 2
D = ∂i ψ
2−2mB R|u|| aB | )2

τ + ixR

Instantons in CPN-1 Model· single instanton (in the A-th classical vacuum)ϕBu − u01 − 1 ψ −m R=e 2 u B ×{gaB
1Aτ = ∂xψ,2

ψ = log( | u |

−2mAR

1
Ax = − ∂τ ψ,22| u − u0 | +∑u=e(B = A)(B ≠ A)i 2
D = ∂i ψ
2−2mB R|u|

B≠A| aB | )· value of action for single instanton

2π
S = 2 ≠ IR renormalong2

τ + ixR

Fractional Instantons· fractional instanton solutions (N solutions A=1,···,N)

ϕA = e− 12 ψ

u

−mAR

×a

ϕA+1 = eϕB = 0 (B ≠ A, A + 1)

− 12 ψ

u −mA+1R

Fractional Instantons· fractional instanton solutions (N solutions A=1,···,N)

ϕA = e− 12 ψu−mAR

ϕA+1 = e×aϕB = 0 (B ≠ A, A + 1)
mA+1R
Σ = g2N∑A

mAR

mA | ϕA |2

τ− 12 ψ

u −mA+1R

Fractional Instantons· fractional instanton solutions (N solutions A=1,···,N)

ϕA = e− 12 ψu−mAR

ϕA+1 = e×a− 12 ψ

u −mA+1R

ϕB = 0 (B ≠ A, A + 1)
mA+1R
Σ = g2N∑A

mAR· fractional instanton = kinkmA | ϕA |2τFractional Instantons· fractional instanton solutions (N solutions A=1,···,N)

ϕA = e− 12 ψu−mAR

ϕA+1 = e×a− 12 ψ

u −mA+1R

ϕB = 0 (B ≠ A, A + 1)
mA+1R
Σ = g2N∑A

mAR· fractional instanton = kinkmA | ϕA |2τ· value of action for single kinkS=

2π(mA+1 − mA)R
g2

Fractional Instantons· fractional instanton solutions (N solutions A=1,···,N)

ϕA = e− 12 ψu−mAR

ϕA+1 = e×a− 12 ψ

u −mA+1R

ϕB = 0 (B ≠ A, A + 1)
mA+1R
Σ = g2N∑A· fractional instanton = kinkmA | ϕA |2τ· value of action for single kinkmAR

in ZN symmetric case

2π
S= 2
g N

S=

2π(mA+1 − mA)R
g2

(1/N × single instanton)

Bion· fractional instantons ··· no contribution to partition function

Z = tr e−βH : periodic boundary condition τ ∼ τ + β

Bion· fractional instantons ··· no contribution to partition function

Z = tr e−βH : periodic boundary condition τ ∼ τ + β

fractional instantons are not periodic but

Bion· fractional instantons ··· no contribution to partition function

Z = tr e−βH : periodic boundary condition τ ∼ τ + β

fractional instantons are not periodic but
kink

bion
can be periodic

anti-kinkτBion· fractional instantons ··· no contribution to partition function

Z = tr e−βH : periodic boundary condition τ ∼ τ + β

fractional instantons are not periodic but
kink

bion

anti-kinkτ

can be periodic → non-perturbative contribution to ZBion
· fractional instantons ··· no contribution to partition function

Z = tr e−βH : periodic boundary condition τ ∼ τ + β

fractional instantons are not periodic but
kink

bion

anti-kinkτ

can be periodic → non-perturbative contribution to Z

Question : relation between bion and renormalon ?

Large-N Analysis
large-N bion contributions : two routes
1. g expansion ··· trans-series
2. large-N limit

large-N limit’t Hooft coupling expansion[ Ishikawa, Morikawa, Nakayama, Shibata, Suzuki, Takaura, 2019]ambiguity of ⟨F 2⟩ ∼ Λ3 on R×S1 with TBC (route 2)≠ n-bion contribution Λ2n (route 1)

Question
Can we correctly capture
the bion contributions in the large-N analysis?

Large-N Analysis
large-N bion contributions : two routes
1. g expansion ··· trans-series
2. large-N limit

large-N limit’t Hooft coupling expansion[ Ishikawa, Morikawa, Nakayama, Shibata, Suzuki, Takaura, 2019]ambiguity of ⟨F 2⟩ ∼ Λ3 on R×S1 with TBC (route 2)≠ n-bion contribution Λ2n (route 1)

Question
Can we correctly capture
the bion contributions in the large-N analysis?

Plan of Talk
Intoduction

Route 1 : Semi-classical (small g) Expansion

Route 2 : Large-N Analysis

Setup2d 𝒩 = (2,2) SUSY CPN-1 sigma model1| 𝒟 ϕ | + | σϕA | + iD( | ϕA | − 2 ) + fermions + ℒsourceℒ=∑[ i A
]
g
N
A=1
N

2

2

2

· model on torus (τ, x) ∼ (τ + β, x) ∼ (τ, x + 2πR)
· periodic boundary condition for τ ∼ τ + β(β → ∞)

Setup2d 𝒩 = (2,2) SUSY CPN-1 sigma model1| 𝒟 ϕ | + | σϕA | + iD( | ϕA | − 2 ) + fermions + ℒsourceℒ=∑[ i A
]
g
N
A=1
N

2

2

2

· model on torus (τ, x) ∼ (τ + β, x) ∼ (τ, x + 2πR)
· periodic boundary condition for τ ∼ τ + β(β → ∞)

twisted boundary condition
AAA

ϕ (x + 2πR) = exp(2πim R) ϕ (x)AAψl,r(x + 2πR) = exp(2πim AR) ψl,r(x)

Source Term
ℒsource = ϵ ΣΣ ≡ g 2N∑A

mA | ϕA |2 : moment map for U(1) ⊂ SU(N)Source Term
ℒsource = ϵ ΣΣ ≡ g 2N∑A

mA | ϕA |2 : moment map for U(1) ⊂ SU(N)Z(g, ϵ) =∫𝒟φ exp (−S) : generating function

Source Term
ℒsource = ϵ ΣΣ ≡ g 2N∑A

mA | ϕA |2 : moment map for U(1) ⊂ SU(N)Z(g, ϵ) =∫𝒟φ exp (−S) : generating function

n
1
∂
1
(n)
− lim log Zn-point correlation function E =()]
n! ∂ϵ [ β→∞ β
ϵ=0

E (1) = ⟨Σ⟩E (2) =∫dτ ⟨Σ(0) Σ(τ)⟩···

Route 1
Semi-classical Expansion(Small Coupling Expansion)

Eliminating Auxiliary Fields· e.o.m. for auxiliary vector multiplet (Ai , σ , D, λ)i ϕ̄A∂iϕA − ϕA∂iϕ̄A
Ai =
+
⋯
2
| ϕA |2σ =0+⋯

etc

Eliminating Auxiliary Fields· e.o.m. for auxiliary vector multiplet (Ai , σ , D, λ)i ϕ̄A∂iϕA − ϕA∂iϕ̄A
Ai =
+
⋯
2
| ϕA |2σ =0+⋯· D-term constraint | ϕA |

2

= 1/g

2

etc

Eliminating Auxiliary Fields· e.o.m. for auxiliary vector multiplet (Ai , σ , D, λ)i ϕ̄A∂iϕA − ϕA∂iϕ̄A
Ai =
+
⋯
2
| ϕA |2σ =0+⋯· D-term constraint | ϕA |

2

= 1/g

etc

2

ϕa
φ =
(a = 1,⋯, N − 1) : inhomogeneous coordinatesϕNaEliminating Auxiliary Fields· e.o.m. for auxiliary vector multiplet (Ai , σ , D, λ)i ϕ̄A∂iϕA − ϕA∂iϕ̄A
Ai =
+
⋯
2
| ϕA |2σ =0+⋯· D-term constraint | ϕA |

2

= 1/g

etc

2

ϕa
φ =
(a = 1,⋯, N − 1) : inhomogeneous coordinates
ϕN
a

2
1
a μ b̄b̄ab̄
aa b̄ c d¯ℒ = 2 Gab̄ (Dμφ D φ̄ − ψ̄ l Dzψl − ψ̄ r Dz̄ ψr ) + Rab̄cd¯ ψl ψ̄ l ψr ψ̄ r + ℒsorce]g [2

2d CP1 Sigma Model
2d N = (2, 2) SUSY CP1 sigma model<latexit sha1_base64="LteuNxskO5184jmScK2t4D44pLg=">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</latexit>ψ̄r ψr ψ̄lψl
1
1
2
¯
|
|
ℒ= 2
∂
ϕ
−
ψ̄
𝒟ψ
−
ψ̄
𝒟ψ
+
i
r
r
l
l
2 2]
g 1 + | φ |2 2 [((1 + | φa | )a )2d CP1 Sigma Model
2d N = (2, 2) SUSY CP1 sigma model<latexit sha1_base64="LteuNxskO5184jmScK2t4D44pLg=">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</latexit>ψ̄r ψr ψ̄lψl
1
1
2
¯
|
|
ℒ= 2
∂
ϕ
−
ψ̄
𝒟ψ
−
ψ̄
𝒟ψ
+
i
r
r
l
l
2 2]
g 1 + | φ |2 2 [((1 + | φa | )a )(τ, x) ∼ (τ, x + 2πR) ∈ ℝ × S 1
twisted boundary condition

φ(x + 2πR) = e 2πimR φ(x)ψl,r(x + 2πR) = e 2πimR ψl,r(x)2d CP1 Sigma Model
2d N = (2, 2) SUSY CP1 sigma model<latexit sha1_base64="LteuNxskO5184jmScK2t4D44pLg=">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</latexit>ψ̄r ψr ψ̄lψl
1
1
2
¯
|
|
ℒ= 2
∂
ϕ
−
ψ̄
𝒟ψ
−
ψ̄
𝒟ψ
+
i
r
r
l
l
2 2]
g 1 + | φ |2 2 [((1 + | φa | )a )(τ, x) ∼ (τ, x + 2πR) ∈ ℝ × S

1twisted boundary condition

φ(x + 2πR) = e 2πimR φ(x)ψl,r(x + 2πR) = e 2πimR ψl,r(x)ℒsorce = ϵ

1 − | φ |2
1 + | φ |2

deformation term in LagrangianZ(ϵ) : generating function for “height”

Equation of Motion

Equation of Motion· complexificatoin

SU(2)SU(2)ℂ
→
U(1)
U(1)ℂ

Equation of Motion· complexificatoin· coordinates (φ, φ̃)SU(2)SU(2)ℂ
→
U(1)
U(1)ℂ

ℂP

1 corresponds to φ̃ = φ̄Equation of Motion· complexificatoin

SU(2)SU(2)ℂ
→
U(1)
U(1)ℂ

· coordinates (φ, φ̃)· analytic continuation of S

S[φ, φ̄]S[φ, φ̃]ℂP

1 corresponds to φ̃ = φ̄equations of motion

δS
δS
=
=0
δφ δφ̃

Equation of Motion· complexificatoin

SU(2)SU(2)ℂ
→
U(1)
U(1)ℂ

· coordinates (φ, φ̃)· analytic continuation of S

S[φ, φ̄]S[φ, φ̃]ℂP

1 corresponds to φ̃ = φ̄equations of motion

δS
δS
=
=0
δφ δφ̃

· x-independent ansatz

φ(τ, x) → φ(τ)φ̃(τ, x) → φ̃(τ)

Leading order contribution

Bion Solution· two conserved charges ( time shift, phase rotation )Bion Solution· two conserved charges ( time shift, phase rotation )solution (up to sym.)sn(ωτ, k)
φ = φ̃ = A
cn(ωτ, k)(Jacobi elliptic function)(A, !, k) : complex constants depending on (g, m, ϵ, β): integers labeling solutionsp≥00≤q≤p−1

Bion Solution· two conserved charges ( time shift, phase rotation )solution (up to sym.)sn(ωτ, k)
φ = φ̃ = A
cn(ωτ, k)(Jacobi elliptic function)(A, !, k) : complex constants depending on (g, m, ϵ, β): integers labeling solutionsp≥00≤q≤p−1· complex saddle points

for genericReal Bion Solution (p=1)• single bion solution

φ = φ̃ = A sinh ωτ (β → ∞): height
N

S

Real Bion Solution (p=1)• single bion solution

φ = φ̃ = A sinh ωτ (β → ∞): height
N

S

Real Bion Solution (p=1)• single bion solution

φ = φ̃ = A sinh ωτ (β → ∞): height
N

S(0)
Ereal ⇠ e

2m
g2

6= 0

even for ϵ = 0

there must be
another contributionComplex Bion Solution (p=1)
complex bion :not on the original contour

1 − φφ̃z=
1 + φφ̃complexified
height'=A'˜ =cosh !⌧

Complex Bion Solution (p=1)
complex bion :not on the original contour

1 − φφ̃z=
1 + φφ̃complexified
height'=A'˜ =cosh !⌧

Complex Bion Solution (p=1)
complex bion :not on the original contour

'=

A
'˜ =
cosh !⌧

1 − φφ̃z=
1 + φφ̃complexified
height(0)(0)
Ereal + Ecomplex = 0

for ✏ = 1

consistent
with SUSY

Kink profile of multi-bion· p : number of bions· q : label saddle points in p-bion sector
complex bion

real bion

complex bionNS

: multi bion solution

Quasi-Moduli Integral
det ΔF
Zp = dvolexp(−Seﬀ ) + 𝒪(g 2)∫
det ΔB
one-loop determinant“nearly flat” directions

bion effective action

Quasi-Moduli Integral
det ΔF
Zp = dvolexp(−Seﬀ ) + 𝒪(g 2)∫
det ΔB
one-loop determinant“nearly flat” directions

bion effective action(τi, ϕi) : positions and phases

Quasi-Moduli Integral
det ΔF
Zp = dvolexp(−Seﬀ ) + 𝒪(g 2)∫
det ΔB
one-loop determinant“nearly flat” directions

1

2

bion effective action(τi, ϕi) : positions and phases

3

4

5

6

kink gas with complexified quasi-moduli

Quasi-Moduli Integral
det ΔF
Zp = dvolexp(−Seﬀ ) + 𝒪(g 2)∫
det ΔB
one-loop determinant“nearly flat” directions

1

2

bion effective action(τi, ϕi) : positions and phases

3

4

5

6

kink gas with complexified quasi-moduli

Single Bion Contribution· quasi-moduli integral for p-bion contribution

det ΔF
Zp = dvolexp(−Seﬀ )∫det ΔB

Single Bion Contribution· quasi-moduli integral for p-bion contribution

det ΔF
Zp = dvolexp(−Seﬀ )∫det ΔB· complexified quasi-moduli integral
Which saddles? Which contours?

Single Bion Contribution· quasi-moduli integral for p-bion contribution

det ΔF
Zp = dvolexp(−Seﬀ )∫det ΔB· complexified quasi-moduli integral
Which saddles? Which contours?

Lefschetz thimble

Single Bion Contribution· quasi-moduli integral for p-bion contribution

det ΔF
Zp = dvolexp(−Seﬀ )∫det ΔB· complexified quasi-moduli integral
Which saddles? Which contours?

Lefschetz thimble

3d projection from ℂ2 ∋ (τ, ϕ)(p = 1)Single Bion Contribution· quasi-moduli integral for p-bion contribution

det ΔF
Zp = dvolexp(−Seﬀ )∫det ΔB· complexified quasi-moduli integral
Which saddles? Which contours?

Lefschetz thimble

3d projection from ℂ2 ∋ (τ, ϕ)(p = 1)Single Bion Contribution
1· single bion contribution to E = − lim log Z
β→∞ β
2

Γ(ε̃)
4πmR Γ(1 − mR)
4πmR
Ep=1 = − 2m
Γ(1 − ε̃) [ gR2 Γ(1 + mR) ] ( gR2 )

−2ε̃

with

ε̃ = ε + 1 ande ∓ε̃πi e

1
1
|
|
=
log
RΛ
gR2
π

− 4πmR
2
g+⋯

Single Bion Contribution
1· single bion contribution to E = − lim log Z
β→∞ β
2

Γ(ε̃)
4πmR Γ(1 − mR)
4πmR
Ep=1 = − 2m
Γ(1 − ε̃) [ gR2 Γ(1 + mR) ] ( gR2 )−2ε̃e ∓ε̃πi e

imaginary ambiguitywithε̃ = ε + 1 and

1
1
|
|
=
log
RΛ
gR2
π

− 4πmR
2g+⋯

Generalization to CPN-1 model
N-1 types of bions

Generalization to CPN-1 model
N-1 types of bions
single bion contributions

Generalization to CPN-1 model
N-1 types of bions
single bion contributions
jump of imaginary partN−1

22(1−ε̃)Γ(1 + mb R)
4πmb R1
ΔEp=1 = i
2πmb𝒜b
2
∑
Γ(1
−
ε̃)
Γ(1
−
m
R)
g
(
)
(
)
b
R
b=1

ma Γ(1 + (ma − mb)R) Γ(1 − mbR)𝒜b =∏ ma − mb Γ(1 − (ma − mb)R) Γ(1 + mbR)a≠b

4πmb R
exp −
gR2 )
(

Large-N limitama =··· ℤN symmetric twisted boundary conditionNRLarge-N limitama =··· ℤN symmetric twisted boundary conditionNRsingle bion contribution

1
(0)
(1)
(2) 2
− log Zp=1(ε̃) = Ep=1
+ Ep=1ε̃ + Ep=1ε̃ + ⋯β4ππi= 2NR | Λ |− ε̃ + log 2 + γ ±ε̃2 + ⋯
[
(
]
gR N
2)
2

Large-N limitama =··· ℤN symmetric twisted boundary conditionNRsingle bion contribution

1
(0)
(1)
(2) 2
− log Zp=1(ε̃) = Ep=1
+ Ep=1ε̃ + Ep=1ε̃ + ⋯β4ππi= 2NR | Λ |− ε̃ + log 2 + γ ±ε̃2 + ⋯
[
(
]
gR N
2)
2

1
4π
| Λ | = 2 exp − 2( gR N )R
2

fixed in the large-N limit

Large-N limitama =··· ℤN symmetric twisted boundary conditionNRsingle bion contribution

1
(0)
(1)
(2) 2
− log Zp=1(ε̃) = Ep=1
+ Ep=1ε̃ + Ep=1ε̃ + ⋯β4ππi= 2NR | Λ |− ε̃ + log 2 + γ ±ε̃2 + ⋯
[
(
]
gR N
2)
2

1
4π
| Λ | = 2 exp − 2( gR N )R
2

fixed in the large-N limit

survives in the large-N limit

1d limit ··· CPN-1 QM
ground state energy

E

(0)

= 0

one-point function

E

(1)

m
2
= − m coth 2 + gg

two-point function
E

(2)

=

2m
− 2p
(2)
2 ± πig
Epert −e4mp
+ γ + log 2 + 𝒪(g 2)
∑
( 2
)
g
[
]
p=1
∞

2m

complete agreement with exact results
obtained from Schrödinger equation

Route 2
Large-N Analysis

2d CPN-1 sigma model
2d 𝒩 = (2,2) SUSY CPN-1 sigma model1| 𝒟 ϕ | + | σϕA | + iD( | ϕA | − 2 ) + fermions + ℒsourceℒ=∑[ i A
]
g
N
A=1
N

2

2

2

2d CPN-1 sigma model
2d 𝒩 = (2,2) SUSY CPN-1 sigma model1| 𝒟 ϕ | + | σϕA | + iD( | ϕA | − 2 ) + fermions + ℒsourceℒ=∑[ i A
]
g
N
A=1
N

2

2

2

· Integrating out N chiral multiplets (ϕ A, ψlA, ψ rA)

2d CPN-1 sigma model
2d 𝒩 = (2,2) SUSY CPN-1 sigma model1| 𝒟 ϕ | + | σϕA | + iD( | ϕA | − 2 ) + fermions + ℒsourceℒ=∑[ i A
]
g
N
A=1
N

2

2

2

· Integrating out N chiral multiplets (ϕ A, ψlA, ψ rA)
effective action for vector multiplet
1 N
iD
Seﬀ = − Nlog sdet [Δ(Ai, σ, D, λ, ma, ϵ)] − 2
g
N
[N ∑
]
a=1

2d CPN-1 sigma model
2d 𝒩 = (2,2) SUSY CPN-1 sigma model1| 𝒟 ϕ | + | σϕA | + iD( | ϕA | − 2 ) + fermions + ℒsourceℒ=∑[ i A
]
g
N
A=1
N

2

2

2

· Integrating out N chiral multiplets (ϕ A, ψlA, ψ rA)
effective action for vector multiplet
1 N
iD
Seﬀ = − Nlog sdet [Δ(Ai, σ, D, λ, ma, ϵ)] − 2
g
N
[N ∑
]
a=1
2

large-N limit : N → ∞ with g N ··· fixedLarge-N Saddle Point
constant ansatz

Large-N Saddle Point
constant ansatz
∞
N
δSeﬀ
1
∝
∑ ∑R
δD const
n=−∞ A=1

1(n/R + mA + Ax)2 + | σ |2 − g 2 NϵmA + iD

⋮

1
− 2
g

Large-N Saddle Point
constant ansatz
∞
N
δSeﬀ
1
∝
∑ ∑R
δD const
n=−∞ A=1

1(n/R + mA + Ax)2 + | σ |2 − g 2 NϵmA + iD

1
− 2
g

⋮

A
··· ℤN symmetric twisted boundary conditionmA =NR

Large-N Saddle Point
constant ansatz
∞
N
δSeﬀ
1
∝
∑ ∑R
δD const
n=−∞ A=1

1(n/R + mA + Ax)2 + | σ |2 − g 2 NϵmA + iD

1
− 2
g

⋮

A
··· ℤN symmetric twisted boundary condition
mA =
NR
A
f (mA) =f∑∑ ( NR )
A=1
A=1
N

N

large-N

NR

1
R

∫0

dp f(p)Generating Function
δSeﬀ
eom
=0 ⋯
δD

large-N saddle point solution

Generating Function
δSeﬀ
eom
=0 ⋯
δD

large-N saddle point solution

Leading order part of generating function

1− log Z(ε) = E (0) + E (1) ε + E (2) ε 2 + ⋯β= Ssol(ϵ) + 𝒪(N 0)Generating Function
δSeﬀ
eom
=0 ⋯
δD

large-N saddle point solution

Leading order part of generating function

1− log Z(ε) = E (0) + E (1) ε + E (2) ε 2 + ⋯β= Ssol(ϵ) + 𝒪(N 0)
E (0) = Ssol(ϵ = 0) = 0
E(1)··· SUSY vacuum2= ∂ϵSsol |ϵ=0 = − 2NR | Λ | ··· agree with bion

1d Limit ··· CPN-1 QM
12πR
A
R → 0 limit with mA = M and g 2 = g 2 ··· fixedN
1d2d1d Limit ··· CPN-1 QM
12πR
A
R → 0 limit with mA = M and g 2 = g 2 ··· fixedN
1d
2d
Leading order part of generating function
1− log Z(ε) = Ssol(ϵ) + 𝒪(N 0)βNM2Ssol(ϵ) =ϵ(ϵ + 2) g N − M coth 2
(
4
g N)

1d Limit ··· CPN-1 QM
12πR
A
R → 0 limit with mA = M and g 2 = g 2 ··· fixedN
1d
2d
Leading order part of generating function
1− log Z(ε) = Ssol(ϵ) + 𝒪(N 0)βNM2Ssol(ϵ) =ϵ(ϵ + 2) g N − M coth 2(
4g N)
E (0) = Ssol(ϵ = 0) = 0
E(1)··· SUSY vacuumNM
agree with
2
= ∂ϵSsol |ϵ=0 =g N − M coth 2···(exact) bion result
2 (
g N)

Two Point Funciton in CPN-1 SQM
large-N analysis
∂2
N
M
2
0
E(ϵ
=
0)
=
g
N
−
M
coth
+
𝒪(N
)
∂ϵ 2
2 (
g2N )

· no imaginary ambiguity

(2)
Epert

N 2
=
g N
2

not consistent with bion and Schrdinger eq.

Two Point Funciton in CPN-1 SQM
large-N analysis
∂2
N
M
2
0
E(ϵ
=
0)
=
g
N
−
M
coth
+
𝒪(N
)
∂ϵ 2
2 (
g2N )

· no imaginary ambiguity

(2)
Epert

N 2
=
g N
2

not consistent with bion and Schrdinger eq.
bion and Schrödinger eq.
∞t Γ (1 − t N/2M)
∂2
−
2
g2
E
(ϵ
=
0)
=
N
g
N
−
Γ(N
+
1)
dt
e
pert
2
∫0
∂ϵ
Γ (N − t N/2M) ][Summary· bion contributions in CPN-1 sigma model· large-N limit with ZN symmetric twisted boundary condition· some of bion contributions are correctly reproduced· non-trivial resurgence structure has not yet been reproducedSummary· bion contributions in CPN-1 sigma model· large-N limit with ZN symmetric twisted boundary condition· some of bion contributions are correctly reproduced· non-trivial resurgence structure has not yet been reproduced

future problem