instanton calculus in r-r 3-form background and deformed n

instanton calculus

Instanton Calculus in R-R 3-form Backgroundand Deformed N=2 Super Yang-Mills Theory

Katsushi Ito

Tokyo Institute of Technology

July 13, 2009@Nagoya Univ,K.I., H. Nakajima and S. Sasaki, JHEP 0812 (2008) 113, arXiv:0811.3322

K.I., H. Nakajima, T. Saka and S. Sasaki, work in progress

Katsushi Ito instanton calculus

instanton calculus

Outline

1 Introduction

2 Multi-Instanton Calculus in N=2 SYM: a review

3 N=2 SYM in R-R 3-form background

4 Deformed Instanton Effective Action

5 Outlook


instanton calculus

Introduction

Introduction

In SUSY gauge theory, instanton plays an important role instudying strong coupling physics

gluino condensate, superpotential in N=1 SQCD

prepotential in N=2 SYM

Instanton Calculus in SUSY gauge theory

microscopic approach: ADHM construction of(super)instanton moduli space, Nekrasov’s formula

Low-energy effective theory: SUSY and Duality(Seiberg-Witten theory)

String theory appraoch: topological string, M-theory,D-instanton


instanton calculus

Introduction

Microscopic ApproachInstanton:

Euclidean spacetime

minimizing the action

S[A] = −1

2

∫

d4xtrNFmnFmn − iθg2

16π2

∫

d4xtrNFmnFmn

(anti)self-dual : Fmn = ±Fmn

topological number k = − g2

16π2

∫

d4xtrFmnFmn

Instanton solution

k = 1, BPST [Belavin-Polyakov-Schwartz-Tyupkin]

general k, ADHM [Atiyah-Drinfeld-Hitchin-Manin]

Path integral: zero modes M+ massive modes A′m [’t Hooft]

Z =

∫

DAme−S[A] =

∫

DMe−Seff [M ]DA′me

−S[A′]


instanton calculus

Introduction

Multi-instanton calculus in SUSY gauge theoryDorey-Hollowood-Khoze-Mattis, hep-th/0206063Nekrasov’s formulaInstanton partition function: intergral over (super) ADHM moduli

Z(a, ǫ) =

Z

DMe−Seff (M) ∼ exp“F inst(a,Λ) +O(ǫ)

ǫ1ǫ2

”

localization (sum over Young diagrams)

Noncommutative instanton (FI term) [xµ, xν ] = θµν

Ω-background ǫ1, ǫ2


instanton calculus

Introduction

Omega-background6-dim metric

ds26 = 2dzdz + (dxµ + Ωµdz + Ωµdz)2,

z = 1√2(x5 − ix6), z = 1√

2(x5 + ix6)

Ωµ ≡ Ωµνxν , Ωµ ≡ Ωµνxν , Ωµν = −Ωνµ, Ωµν = −Ωνµ

Ωµν =

0 ǫ1 0 0−ǫ1 0 0 00 0 0 ǫ20 0 −ǫ1 0

6-dim. N = 1 SYMDimensional rduction to 4 dim.→ N=2 SYM in the Ω-background


instanton calculus

Introduction

Superstring theory and Instanton Calculus

Low-energy effective theory: embedding SW curve in higherdimensional theory

(noncompact) Calabi-Yau manifold

M-theory

Microscopic instanton calculus in string theorySUSY gauge theory: Low-energy effective theory on D-branestype IIB superstring theory

N = 4 U(N) SYM: N D3-branesN = 2 U(N) SYM: N D3 at the orbifold

singularity of C2/Z2

Instanton: k D(-1) branes in N D3-branes[Douglas, Witten]


instanton calculus

Introduction

Microscopic Instanton Calculus vs D-instanton

ADHM moduli: massless open string modes connectingD-branes D3-D(-1), D(-1)-D(-1) NS, R-sector

NC instantons: NS-NS fields Bµν

Ω-background: closed string NS-NS background

Instanton Calculus in Closed String Backgroundgeneral R-R background

Ω-background vs R-R 3-form field strength background

R-R 5-form: Non(anti)commutative superspacevarious “NC” field theories (Deformed Field Theories)

deformation of instanton effective action

Flux compactification: non-perturbative correctionsstringy effects (Exotic, stringy instantons)


instanton calculus

Multi-Instanton Calculus in N=2 SYM: a review

Multi-Instanton Calculus in N = 2 SYM

Dorey-Hollowood-Khoze-Mattis, hep-th/0206063

1 Multi-instanton solution ADHM construction

2 Path-Integral around the instanton solution, zero modemeasure Hyper Kahler Quotient

3 Supersymmetric zero mode measureConstraint Instanton [Affleck]

4 linearlized formalism D-brane realization


instanton calculus


Notation

Euclidean spacetime xm = (x1, x2, x3, x4) (x4 = −ix0)Lorentz group SO(4) = SU(2) × SU(2): ψα, ψα

Dirac matrices: σnαα = (iτ1, iτ2, iτ3, 1), σααn = (−iτ1,−iτ2,−iτ3, 1)

τc: Pauli matricesYang-Mills TheoryGauge group G = SU(N)

Am: gauge field (N ×N 行列,A†m = −Am)

Fmn = ∂mAn − ∂nAm + g[Am, An]: field strengthaction:

S[A] = −1

2

∫

d4xtrNFmnFmn − iθg2

16π2

∫

d4xtrNFmnFmn

Fmn = 12ǫmnklFkl: dual field strength


instanton calculus


instanton (anti-instanton): self-dual (anti-self-dual)

Fmn = Fmn, (Fmn = −Fmn)

instanton number k = 0,±1,±2, · · ·

k = − g2

16π2

∫

d4xtrNFmnFmn

e.o.m. DmFnm = ±DmFnm = 0

minimizine the action S[A]

S[A] =8π2

g2|k| + ikθ =

−2πikτ k > 02πikτ k < 0

τ =4πi

g2+

θ

2π


instanton calculus


ADHM Construction

[Atiyah-Hitchin-Drinfeld-Manin]∆ = (∆λiα): (N + 2k) × 2k complex-valued matrix:

∆λiα = aλiα + bαλixαα ∆αλi = aαλ

i + xααbλiα

λ = 1, · · · , N + 2ki = 1, · · · , k: instanton indexα = 1, 2: spinor indexU = (Uαu): ((N + 2k) ×N matrix, u = 1, · · · , N gauge index)orthonormal basis of vectors of the null space ∆U = 0.

(An)uv = g−1Uλu ∂nUλv

Fmn: self-dual, instanton number k when ∆ satsifies the ADHMconstraints:

∆αλi ∆λjβ = δα

β(f−1)ij

f : x-dependent k × k Hermitian matrix.Katsushi Ito instanton calculus

instanton calculus


∆λiα are redundant variables: U(N + 2k) ×GL(k,C) symmetry

∆λiα =

(

ωujα

δijxαα + (a′αα)ij

)

=

I† Jz2 −B2 z1 −B1

−(z1 −B†1) z2 −B†

2

The ADHM constraints become

II† − J†J + [B2, B†2] + [B1, B

†1] = 0

IJ + [B1, B2] = 0

f−1 = II† + (z2 −B2)(z2 −B†2) + (z1 −B1)(z1 −B†

1)

B1, B2: k × k matrices, I†, J : N × k matricesz1 = x2 + ix1, z2 = x4 + ix3

residual symmetry U(k):Ba → gBag

†, I → gI, J → Jg†, (g ∈ U(k))


instanton calculus


instanton moduli space: Mk,N

Mk,N = (B1, B2, I, J),ADHM constraints /U(k)

(real) dimension 4k2 + 4Nk − 3k2 − k2 = 4Nk

noncompact, (small instanton) singurality

center Xn = 1k tra′n

Hyper-Kahler quotient M = ~µ−1(0)/G(~µ = 0: ADHM constraints, G = U(k))


instanton calculus


Path integral around instanton solution

An(x) = An(x;X) + δAn(x;X) An(x;X) : ADHM instanton

X: moduli parameters of k-instantons

∫

DA exp −S[A] =e−2πkiτ

g4kN

∫

Mk,N

ω1

det′∆(+)

∆(+) = −D/D/, ω: volume form on Mk,N

∫

Mk,N

ω =Ck

VolU(k)

∫

d4k(N+k)a |detk2L|k2∏

r=1

3∏

c=1

δ

(

1

2trk(T rτcβ

αaαaβ)

)

T r: U(k) generatorsLΩ = 1

2

wαwα,Ω

+ [a′n, [a′a,Ω]] (Ω: k × k Hermitian matrix)


instanton calculus


Instanton Calculus in N = 2 Supersymmetric Yang-Mills TheoryΦ = ( φ√

2, ψ): chiral multiplet, φ = φ1 − iφ2

V = (Am, λ): vector multiplet

S =

Z

d4xtrNn

−1

2F 2

mn − iθg2

16π2Fmn ∗ Fmn − 2Dnλσnλ− 2Dnψσnψ

+Dnφ†Dnφ+ 2igψ[φ, λ] + 2ig[φ†, λ]ψ +

1

4g2[φ, φ†]2

o

Solve eq. of motion perturbatively in g, in the instantonbackground

The solution is not exact but it is good approximation wheneffective coupling constant is small (Constrained Instanton)

fermion: λA = (λ1, λ2) = (ψ, λ) has zero modes in theinstanton background

scalar: b.c. φ→ a (x→ ∞) EOM D2φ = −gǫABλAλB


instanton calculus


fermionic zero modes: D/λA = 0

λα = g−12(

UMf bαU − UbαfMU)

M = (Mλi) =

(

µuj

(M′α)ij

)

: (N + 2k) × 2k matrix

fermionic ADHM constraints

Mλi aλαj + aiαλMλj = µu

i wuαj + wiαuµuj + [M′α, a′αα]ij = 0

M′α = M′

α

2k(2k +N) − 2k2 = 2kN zero modes for each fermion λA

ξA = i4 trkM

′A: supertranslation

tangent vector of TMk,N


instanton calculus


Path-integral around the instanton background

∫

DADλADλADφae−S =

Λ2kN

g8kN

∫

Mk,N

ωe−S(0)eff

Λ = µ2Ne2πiτ : scale parametersupersymmetric volume form

Z

Mk

ω =ck

volU(k)

Z

d4k(N+k)a2

Y

A=1

d2k(N+k)MA|detL|−1

×k2

Y

r=1

n

3Y

c=1

δ

„

1

2trkT

rτ cα

βaβaα

« 2Y

A=1

2Y

α=1

δ“

trkTr(MAaα + aαMA)

”o

S(0)eff =

1

κ

Z

d4x Tr

»

∇µϕ(0)∇µϕ(0) − i√

2Λ

(0)I [ϕ(0),Λ

(0)I]

–

,


instanton calculus


Linealized Formalism: Introduce auxiliary fieldsχa: (a = 1, 2 ) Hermitian k × k matrices (detL)Dc: c = 1, 2, 3 Hermitian k × k matrices (δ(ADHM))ψα

A: (A = 1, 2) Grassmann k × k matrices (δ(fermionic ADHM))

Zk =π−4c

(N)k

volU(k)

∫

dadDdχ

2∏

A=1

dMAdψAe−S−SL.M.

S = 4π2trk

|wαχa + φ0awα|2 − [χa, a

′n]2

+i

2µA(µAχ

† + φ0†µA) +i

2M′AM′

Aχ†

SL.M. = −4π2trk

ψαA(MAaα + aαMA) + ~D~τ α

βaβaα

Seff = S + SL.M.: instanton effective action


instanton calculus


String theory interpretation of ADHM modulik D(-1)+N D3-branes (at orbifold singularity )

D(-1)-D(-1) k × k

NS-sector a′m, χ,~D

R-sector M′

αA, ψα

A

D(-1)-D3 k ×N , D3-D(-1) N × k

twisted NS-sector wα, wα

twisted R-sector µA, µA

Disk amplitudes of the moduli vertex operators→ Low-energy effective action on D(-1) Sstr

Seff = Sstr


instanton calculus


Field Theory:

N = 2 SYM:(N D3 at oribfold singularity C

2/Z2)

ADHM construction=⇒instanton effective action Seff

String Theory:

N D3+k D(-1) at C2/Z2

low-energy effective theory on D(-1)

=⇒instanton effective action Sstr

Seff = Sstr


instanton calculus

N=2 SYM in R-R 3-form background

deformed action in the R-R bacground

N (fractional) D3-branesN = 4: R

10

N = 2: R6 × C

2/Z2

N = 1: R4 × C

3/Z2 × Z2

NSR formalsimvertex operators (massless fields(open), R-R field(closed))disk amplitudes including a R-R vertex operatorzero-slope limit α′ → 0

Interaction terms =⇒ Low-energy effective Lagrangian

A A

A A

A

A

A

A A

+ + +


instanton calculus


N = 4 super Yang-Mills theory

N D3-branes (x0, x1, x2, x3)

Lorentz group SO(10) → SO(4) × SO(6)

spin fieldsSλ → (SαSA, SαS

A)

Sα, Sα (α, α = 1, 2): four-dimensional spinorsSA, SA (A = 1, · · · , 4): six-dimensional spinors

Gamma matrices Γλ → σµαα, σ

µαα,ΣaAB , ΣaAB

ΣaAB =(

η3,−iη3, η2,−iη2, η1, iη1)

,

massless fields

gauge fields Aµ, scalar fields ϕa gaugino ΛAα , ΛA

α

vertex operators: VA, Vϕ, VΛ, VΛ


instanton calculus


Orbifold

N=2 SYM Aµ, ϕ1, ϕ2, Λ1, Λ2

Z2 : (x4, · · · , x9) → (x4, x5,−x6, · · · ,−x9)

N=1 SYM Aµ, Λ

Z2 : (x4, · · · , x9) → (x4, x5,−x6, · · · ,−x9)

Z2 : (x4, · · · , x9) → (−x4,−x5, x6, x7,−x8,−x9)


instanton calculus


R-R background

R-R sector : massless fields Fλλ′Sλ(z)Sλ′

(z)λ = (αA), Fλλ′ → FαβAB 6= 0, Fα

αA

B = Fαα

AB = FαβAB = 0

No backreaction to the EM tensor: flat spacetime

V(−1/2,−1/2)F (z, z) = (2πα′)

32FαβABSαSAe

− 12φ(z)Sβ SBe

− 12φ(z)

Decomposition of R-R field strength

FαβAB = F [αβ][AB] + F [αβ](AB) + F (αβ)[AB] + F (αβ)(AB)

= Faǫαβ(Σa)AB + Fabcǫ

αβ(Σabc)AB

+Fµνa(σµν)αβ(Σa)AB + Fµνabc(σ

µν)αβ(Σabc)AB

(Σabc)AB ≡ (Σ[aΣbΣc])AB

(σµν)αβ = 14 (σµσν − σν σµ)αγε

γβ


instanton calculus


Classification of the R-R field strength FαβAB

F (αβ)(AB) ∼ Fµνabc (S,S)-type (5-form field strength)Graviphoton background

F (αβ)[AB] ∼ Fµνa (S,A)-type (R-R 3-form)

F [αβ](AB) ∼ Fabc (A,S)-type (R-R 3-form)

F [αβ][AB] ∼ Fa (A,A)-type (R-R 1-form)

Scaling Condition: α′ → 0

(2πα′)n/2FαβAB ≡ CαβAB = Mass−n+2

n = 3 case: C has dimension −1 (non(anti)commutativesuperspace) [θ] = M−1/2

n = 1 case: C has dimension +1 (Ω-background)

(A,∗)-type deformation preserves the Lorentz symmetryKatsushi Ito instanton calculus

instanton calculus


Deformed Action

(2πα′)3/2FαβAB = CαβAB = fixed

N = 1 (S,S) [Billo-Frau-Pesando-Lerda 0402160]N = 2 (S,S) [Ito-Sasaki 0608143]N = 4 (S,S) [Ito-Kobayashi-Sasaki 0612267]

(2πα′)1/2FαβAB = CαβAB = fixed

N = 1 (A,S) [Ito-Nakajima-Sasaki,0705.3532]N = 2 (S,A) [Billo-Frau-Fucito-Lerda, 0606013](A,S) [Ito-Nakajima-Sasaki]N = 4 (S,A), (A,S) [Ito-Nakajima-Sasaki, 0705.3532](S,S)-type: no deformation


instanton calculus


(S,A)-deformed N=2 SYM

L = L0 + LC

L0 =1

κTr

»

−1

4FµνF

µν +iθg2

32π2Fµν F

µν −DµϕDµϕ− 1

2g2[ϕ, ϕ]2

−iΛIα(σµ)αβDµΛβI +

i√2g ΛI [ϕ,ΛI ] − i√

2g ΛI [ϕ, Λ

I ]

–

.

LC =1

κTr

»

ig(Cµνϕ+ Cµνϕ)Fµν +i√2gΛα

IΛβIC(αβ) +

1

2g2(Cµν ϕ+ Cµνϕ)2

–

.

Cαβ = 4√

2π(2πα′)12F (αβ)12, Cαβ = 4

√2π(2πα′)

12F (αβ)34.


instanton calculus


N = 2 SYM in Ω-background

L(Ω, Ω) = L0 + δL(Ω, Ω),

δL(Ω, Ω)

=1

kTr

h

gFµνDµϕΩν + gFµνD

µϕΩν

+ ig2Dµϕ[ϕ, ϕ]Ωµ + ig2Dµϕ[ϕ, ϕ]Ωµ

−g2FµρFνρΩµΩ

ν+g2

2DµϕDν ϕΩµΩν − g2DµϕDνϕΩµΩ

ν+g2

2DµϕDνϕΩ

µΩ

ν

+ig3[ϕ, ϕ]FµνΩµΩν

− g√2ΛαIDµΛαIΩ

µ − g√2ΛαIDµΛ

αIΩµ − g√

2Λα

IΛβIΩ(αβ) − g√

2ΛαIΛβ

IΩ(αβ)

–

+O(Ω3,Ω3).

Different from the action in the (S,A) type R-R 3-form background


instanton calculus

Deformed Instanton Effective Action

Deformed EOM

anti-self-dual: F(+)µν = 0 not deformed

self-dual: F(−)µν = 0 deformed

Deformed EOM

D2ϕ− i√

2gΛIΛI − g2

h

ϕ, [ϕ, ϕ]i

+ igFµνCµν + g2ϕCµνC

µν + g2ϕCµνCµν = 0,

D2ϕ+ i√

2gΛIΛI − g2h

ϕ, [ϕ, ϕ]i

+ igFµνCµν + g2ϕCµνC

µν + g2ϕCµνCµν = 0,

(σµ)αβDµΛIβ +

√2g[ϕ,ΛIα] +

√2g CαβΛβ

I = 0,

(σµ)αβDµΛIβ −

√2g[ϕ,Λ

Iα] = 0,

Dµ

`

Fµν − 2igϕCµν − 2igϕCµν´

−ig[ϕ,Dν ϕ] − ig[ϕ,Dνϕ] − g(σν)αβΛIα,ΛI

β = 0.


instanton calculus


Deformed ADHM construction

Aµ = g−1A(0)µ + gA(1)

µ + · · · ,

ΛI = g−12 Λ(0)I + g

32 Λ(1)I + · · · , ΛI = g

12 Λ

(0)I + g

52 Λ

(1)I + · · · ,

ϕ = g0ϕ(0) + g2ϕ(1) + · · · , ϕ = g0ϕ(0) + g2ϕ(1) + · · · .

A(0)µ = −iU∂µU,

Λ(0)Iα = U(MIf bα − bαfMI

)U,

ϕ(0) = −i√

2

4ǫIJ UMIfMJU + U

(

φ 00 χ12 + 1kC

)

U,

ϕ(0) = U

(

φ 00 χ12 + 1kC

)

U.


instanton calculus


Deformed Instanton Effective action

S =8π2k

g2+ ikθ + g0S

(0)eff + O(g2)

S(0)eff =

2π2

κtrk

[

−2(

[χ, a′µ] − Cµνa′ν

)(

[χ, a′µ] − Cµρa′ρ

)

+ (χwα − wαφ)(wαχ− φwα) + (χwα − wαφ)(wαχ− φwα)

− i

√2

2µIǫIJ(µJ χ− φµJ ) − i

√2

4M′αIǫIJ

(

[χ,M′Jα ] − C(αβ)M′βJ

)

+1

2CµνCµρ(δν

ρwαwα + 4a′νa′ρ)

]

+ SADHM,


instanton calculus


Instanton effective action from D3/D(-1) system

[Billo-Frau-Fucito-Lerda, 0606013]

D(-1)-D(-1): a′, χ, D (NS) M′, ψ (R)

D3-D(-1): w, w (twisted NS), µ, µ (twisted R)

S(0)str =

2π2

κtrk

[

−2(

[χ, a′µ] − Cµνa′ν

)(

[χ, a′µ] − Cµρa′ρ

)

+ (χwα − wαφ)(wαχ− φwα) + (χwα − wαφ)(wαχ− φwα)

− i

√2

2µIǫIJ(µJ χ− φµJ ) − i

√2

4M′αIǫIJ

(

[χ,M′Jα ] − C(αβ)M′βJ

)

]

+ SADHM.

This is the same action derived from the N = 2 SYM in theΩ-background.


instanton calculus


Field Theory:

N = 2 SYM in R-R 3-form background:SN=2

deformed

ADHM =⇒ S(0)eff

String Theory:

N D3+k D(-1) in R-R3-form backgroundlow-energy effective theory

on D(-1) =⇒deformed action S

(0)str

Field Theory in Ω-

background

SΩspacetime

ADHM =⇒ S(0)Ωeff

spacetime action: SN=2deformed 6= SΩ

spacetime:

instanton effective action: S(0)eff 6= S

(0)str = S

(0)Ωeff


instanton calculus


A puzzle

S(0)eff 6= S

(0)str

instanton effective action

In D3/D(-1), tra′m couples to the R-R field.EOM → tra′m = 0 (stabilized due to R-R fields)In D3-system without D(-1), instanton translational zero modedoes not couple to R-R fields

spacetime action

Sdeformed 6= SΩ

Omega-background breaks the translational symmetryR-R 3-form background keeps the translational symmetry


instanton calculus


Possible solution

In order to recover the string instanton effective action we need toadd a term

δL = − g2

16κCρσCρσ|x|2Tr [FµνFµν ]

to the deformed N = 2 Lagrangian L0 + LC .

ADHM → S(0)eff + δS

(0)eff = S

(0)str

L0 + LC + δL ∼ LΩ in the instanton backgroundEOMs of Aµ, ϕ, ϕ and Λ are the same but EOM of Λ isdifferent. But its contribution to the instanton effective actionis higher order in g.

string theory derivation? (backreaction to geometry frominstanton)


instanton calculus


Conclusion:

Field Theory:

N = 2 SYM in R-R 3-form background:SN=2

deformed + δSADHM=⇒ S

(0)eff + δS

(0)eff

String Theory:

N D3+k D(-1) in R-R3-form backgroundlow-energy effective theory

on D(-1) =⇒deformed action S

(0)str

Field Theory in Ω-

background

SΩspacetime

ADHM =⇒ SΩeff

spacetime action: SN=2deformed ∼ SΩ

spacetime:

instanton effective action: Seff + δS(0)eff = S

(0)str = S

(0)Ωeff


instanton calculus

Outlook

Outlook

N=4 deformed instanton effective action in R-R 3-formbackground Cµνa [KI-Nakajima-Saka-Sasaki]10-dim Omega background

(A,S) deformed action : mass deformation

D7-D3-D(-1) [Gava-Narain-Sarmadi, Gutperle]type I/Heterotic dualityD7/D(-1) in type I’: exotic instantons[Billo-Frau-Gallot-Lerda-Pesando]

fermionic T-duality

gravity dual

quiver, flux compactification


instanton calculus in r-r 3-form background and deformed n

Documents