1 Supersymmetry

Ref. Wess-Bagger, Supersymmetry and Supergravity

1.1 Spinor calculus

metric ηµν = diag(−1, 1, 1, 1)

The (proper)Lorentz group SO(1, 3) is generated by the rotations J and the boost

generators K. The combination J± iK satisfies the commutation relations of the SU(2)

algebra. Then SO(1, 3) is isomorphic to SL(2,C).

Weyl spinor two component spinor fields ψα (α = 1, 2) and ψα (α = 1, 2)

ψ′α =Mα

βψβ (1.1)

ψ′α = (M∗)α

βψβ (1.2)

Here M is a 2 × 2 matrix satisfying detM = 1. M∗ denotes the complex conjugate of

M , MT its transpose and M † its Hermite conjugate. The dotted spinor field transforms

under the (12, 0) representation of the Lorentz group. The undotted spinor field transforms

under the (0, 12) representation, which is the conjugate representation of (1

2, 0).

Since 12⊗ 1

2= 0⊕ 1 and the singlet representation 0 is anti-symmteric,

ϵαβχβηα, ϵαβχαηβ (1.3)

is Lorentz invariant inner product.

ϵαβ, ϵαβ: anti-symmetric symbol

ϵαβ = −ϵβα, ϵ12 = −ϵ12 = 1 (1.4)

ϵαβ = −ϵβα, ϵ12 = −ϵ12 = 1 (1.5)

Mαβ satisfies

ϵαβMαγMβ

δ = ϵγδ. (1.6)

ϵαβ, ϵαβ: inverse matrices of ϵαβ, ϵαβ

3

lowering and raising spinor indices

χα = ϵαβχβ, χα = ϵαβχβ (1.7)

ψα and ψα transform under SL(2,C) as

ψ′α = ψβ(M−1)βα (1.8)

ψ′α = ψβ(M∗−1)βα. (1.9)

Sigma matrices

σµαα = (−1, σ1, σ2, σ3) (1.10)

σµαα = (−1,−σ1,−σ2,−σ3) (1.11)

σµPµ =

(−P 0 + P 3 P 1 − iP 2

P 1 + iP 2 −P 0 − P 3

)(1.12)

det(σµPµ) = −ηµνP µP ν (1.13)

Exercise 1-1 Prove the follwoing formulas:

σµαα = ϵαβϵαβσµ

ββ(1.14)

σµαασµββ = −2δαβ δαβ

(1.15)

σµαασναα = −2ηµν (1.16)

σµσν + σν σµ = −2ηµν (1.17)

σµσν + σνσµ = −2ηµν (1.18)

Lorentz generators

σµν =1

4(σµσν − σν σµ) (1.19)

σµν =1

4(σµσν − σνσµ) (1.20)

Exercise 1.2 Check that 12σµν and 1

2σµν satsify the Lorentz algebra.

4

convention

ψχ ≡ ψαχα = χψ (1.21)

ψχ ≡ ψαχα = χψ (1.22)

Hermite conjugate

(ψχ)† = χψ (1.23)

Exercise 1-3

χσµψ = −ψσµχ (1.24)

(χσµψ)† = ψσµχ (1.25)

χσµσνψ = ψσµψµχ (1.26)

(χσµσνψ)† = ψσνσµχ (1.27)

Dirac spinor

ΨD =

(ψα

χα

)(1.28)

gamma matrices (Weyl representation)

γµ =

(0 σµ

σµ 0

)(1.29)

γµ, γν = −2ηµν14 (1.30)

5

1.2 SUSY algebra

Supersymmetry (SUSY): boson↔ fermion

N -extended SUSY generated by N -pair of supercharges

QIα, QαJ , I, J = 1, · · · , N (1.31)

fermionic (anticommuting + spinor)

N = 1 SUSY Qα, Qα, Pµ

Qα, Qα = 2σµααPµ (1.32)

Qα, Qβ = Qα, Qβ = 0 (1.33)

representation of SUSY

• massive representation

rest frame P µ = (M, 0, 0, 0)

aα =1√2M

Qα, a†α =1√2M

Qα (1.34)

aα, a†β = δαβ (1.35)

aα, aβ = a†α, a†β = 0 (1.36)

aα, a†β: anticommuting harmonic oscillator

|Ω⟩: ground state aα|Ω⟩ = 0

multiplets are made of four states

|Ω⟩a†α|Ω⟩ (α = 1, 2)

a†1a†2|Ω⟩

• massless representation

pµ = (E, 0, 0, E)

Qα, Qβ = 2(σ0αβ(−E) + σ3

αβE) = 2

(1 00 1

)(1.37)

6

a1 =1√2E

Q1, a†1 =1√2E

Q1 (1.38)

a1, a†1 = 1 (1.39)

a1, a1 = a†1, a† = 0 (1.40)

a†1 raise the helicity of a state by 12.

a1 lowers the helicity of a state by −12.

|Ωλ⟩: the ground state with helicty λ, a1|Ωλ⟩ = 0

two states

|Ωλ⟩ helicity λ

a†1|Ωλ⟩ helicity λ+ 12

In CPT invariant field theory, if a state with helicity λ, there exists also a state with

−λ.

scalar multiplet (two scalars + a Weyl fermion)

a†1|Ω0⟩ helicity 12

|Ω0⟩ a†1|Ω− 12⟩ helicity 0

|Ω− 12⟩ helicity −1

2

vector mulitplet (a gauge boson + a Weyl fermion)

a†1|Ω 12⟩ helicity 1

|Ω 12⟩ helicty 1

2

helicity 0

a†1|Ω−1⟩ helicity −12

|Ω−1⟩ helicity −1

helicity

7

1.3 Superspace

Extend spacetime to realize SUSY manifesty.

(xµ, θα, θα) (1.41)

where θα, θα are Grassmann odd coordinates (anticommuting numbers)

• θ2 := θθ = θαθα = −2θ1θ2

θ2 := θαθα = 2θ1θ2

•

θαθβ = −1

2ϵαβθ2 (1.42)

θαθβ =1

2ϵαβ θ2 (1.43)

Exercise 2-1 Show

(θσµθ)(θσν θ) = −1

2θ2θ2ηµν (1.44)

N = 1 SUSY transformation

θα → θα + ξα (1.45)

θα → θα + ξα (1.46)

xµ → xµ + iθσµξ − iξσµθ (1.47)

translation

xµ → xµ + ϵµ (1.48)

Pµ = −i∂µ (1.49)

iϵνPνxµ = ϵµ (1.50)

8

superderrivative

∂

∂θαθβ = δβα,

∂

∂θαθβ = δβα (1.51)

note that

ϵαβ∂

∂θβ= − ∂

∂θα(1.52)

supercharge

Qα =∂

∂θα− iσm

ααθα∂m (1.53)

¯Qα = − ∂

∂θα+ iθασm

αα∂m (1.54)

• ξQ+ ξ¯Q generates the translation in superspace.

(ξQ+ ξ¯Q)xν = −iξσν θ + iθσν ξ (1.55)

• anticommutation relations

Qα,¯Qα = 2iσµ

αα∂µ (1.56)

Qα, Qβ = ¯Qα,¯Qβ = 0 (1.57)

supercovariant derivative

Dα =∂

∂θα+ iσm

ααθα∂m (1.58)

Dα = − ∂

∂θα− iθασm

αα∂m (1.59)

Exercise 2-2 Check

Dα, Qβ = Dα,¯Qα = Dα, Qβ = Dα,

¯Qβ = 0 (1.60)

Dα, Dβ = Dα, Dβ = 0, Dα, Dα = −2iσµαα∂µ (1.61)

9

superfield a generic superfield:

F (x, θ, θ) = f(x) + θϕ(x) + θχ(x) + θθm(x) + θθn(x) + θσmθvm(x) + θθθλ(x) + θθθψ(x) + θθθθd(x)(1.62)

(ξQ+ ξQ)× F = ξϕ+ ξχ

+ θ(2ξm+ σmξ(vm + i∂mf)

)+ θ

(2ξn+ σmξ(−vm + i∂mf)

)+ θ2

(ξλ+

i

2ξσm∂mϕ

)+ θ2

(ξψ +

i

2ξσm∂mχ

)+ (θσnθ)

(−ψσnξ − ξσnλ+

i

2ξσmσn∂mχ− i

2∂mϕσ

nσmξ

)+ θ2θ

(2ξd− i

2σnσmξ∂mvn + iσmξ∂mn

)+ θ2θ

(2ξd+

i

2σnσmξ∂mvn + iσmξ∂mm

)+ θ2θ2

(i

2ξσm∂mλ+

i

2ξσm∂mψ

)(1.63)

δξf = ξϕ+ ξχ (1.64)

δξϕ = 2ξm+ σmξ(vm + i∂mf) (1.65)

δξχ = 2ξn+ σmξ(−vm + i∂mf) (1.66)

δξm = ξλ+i

2ξσm∂mϕ (1.67)

δξn = ξψ +i

2ξσm∂mχ (1.68)

δξvn = −ψσnξ − ξσnλ+i

2ξσmσn∂mχ− i

2∂mϕσ

nσmξ (1.69)

δξψ = 2ξd− i

2σnσmξ∂mvn + iσmξ∂mn (1.70)

δξλ = 2ξd+i

2σnσmξ∂mvn + iσmξ∂mm (1.71)

δξd =i

2ξσm∂mλ+

i

2ξσm∂mψ (1.72)

10

1.4 chiral superfield

chiral superfield DαΦ = 0

Φ(y, θ) = A(y) +√2θψ(y) + θ2F (y) (1.73)

ym = xm + iθσmθ (1.74)

Dαyn =

(− ∂

∂θα− iθασm

αα∂m

)(xn + iθσnθ)

= +iθασnαα − iθασn

αα = 0 (1.75)

A(y) = A(x+ iθσθ)

= A(x) + iθσmθ∂mA(x) +1

2(iθσmθ)(iθσnθ)∂m∂nA(x)

= A(x) + iθσmθ∂mA(x) +1

4θ2θ2∂m∂mA(x) (1.76)

where (??) is used.

Φ(y, θ) = A(x) + iθσmθ∂mA(x) +1

4θ2θ2∂m∂mA(x)

+√2θψ(x) +

√2iθσmθ(θ∂mψ)(x) + θ2F (x)

= A(x) + iθσmθ∂mA(x) +1

4θ2θ2∂m∂mA(x)

+√2θψ(x) +

i√2θ2θσm∂mψ(x) + θ2F (x) (1.77)

In terms of chiral coordinates

Qα =∂

∂θα(1.78)

Qα = − ∂

∂θα+ 2iθασm

αα

∂

∂ym(1.79)

Dα =∂

∂θα+ 2iσm

ααθα ∂

∂ym(1.80)

Dα = − ∂

∂θα(1.81)

δξA =√2ξψ (1.82)

δξψ = i√2σmξ∂mA+

√2ξF (1.83)

δξF = i√2ξσm∂mψ (1.84)

11

antichiral superfield DαΦ = 0

Φ(y†, θ) = A(y†) +√2θψ(y†) + θ2F (y†)

= A(x)− iθσmθ∂mA(x) +1

4θ2θ2∂m∂mA(x)

+√2θψ(x) +

i√2θ2θσm∂mψ(x) + θ2F (x) (1.85)

y†m = xm − iθσmθ (1.86)

In terms of antichiral coordinates

Qα =∂

∂θα− 2iθασm

αα

∂

∂y†m(1.87)

Qα = − ∂

∂θα(1.88)

Dα =∂

∂θα(1.89)

Dα = − ∂

∂θα− 2iθασm

αα

∂

∂y†m(1.90)

δξA =√2ξψ (1.91)

δξψ = i√2σmξ∂mA+

√2ξF (1.92)

δξF = i√2ξσm∂mψ (1.93)

Exercise 2-3 Derive SUSY transformation for (A,ψ, F ) and (A, ψ, F )

• In a generic superfield F (x, θ, θ), the component field d(x) transforms as the total

derivative.

• For a chiral superfield F transforms as total derivative.

12

Grassmann integral ∫dθαθβ = δαβ,

∫dθα1 = 0 (1.94)

d2θ = −1

4dθαdθα, d2θ = −1

4dθαdθ

α (1.95)

∫d2θθ2 = 1,

∫d2θθ2 = 1 (1.96)

∫d2θd2θF (x, θ, θ) = d(x) (1.97)∫

d2θΦ(y, θ) = F (y) (1.98)

Lagrangian

L =

∫d2θd2θK(Φi, Φi) +

∫d2θW (Φi) +

∫d2θW (Φi) (1.99)

is SUSY invariant (up to total derivarive).

WZ model 1

L =

∫d2θd2θΦΦ +

∫d2θW (Φ) +

∫d2θW (Φ) (1.100)

with the superpotential

W (Φ) =m

2Φ2 +

g

3Φ3 (1.101)

ΦΦ∣∣∣θ2θ2

= FF +1

4∂m∂mAA+

1

4A∂m∂mA− 1

2∂mA∂mA+

i

2∂mψσ

mψ − i

2ψσm∂mψ

(1.102)

m

2Φ2

∣∣∣θ2

= m(AF − 1

2ψψ) (1.103)

g

3Φ3

∣∣∣θ2

= g(A2F − ψψA) (1.104)

L =i∂µψσµψ − ∂µA∂µA+ FF

+m(AF − 1

2ψψ) + (A2F − ψψA) +m(AF − 1

2ψψ) + (A2F − ψψA) (1.105)

1J. Wess and B. Zumino

13

R-symmetry

θ → e−iαθ, d2θ → e2iαd2θ (1.106)

θ → eiαθ, d2θ → e−2iαd2θ (1.107)

Φ(y, θ) → einαΦ(y, e−iαθ) (1.108)

Φ(y†, θ) → e−inαΦ(y†, e−iαθ) (1.109)

Φ has R-charge n∫d2θW (Φ) is R-inv. if W (Φ) has R-charge 2.

1.5 vector superfield

vector superfield: reality condition V = V †

V (x, θ, θ) = C(x) + iθχ(x)− iθχ(x)

+i

2θθ[M(x) + iN(x)]− i

2θθ[M(x)− iN(x)]

−θσmθvm(x) + iθθθ

[λ(x) +

i

2σm∂mχ(x)

]−iθθθ

[λ(x) +

i

2σm∂mχ(x)

]+

1

2θθθθ

[D(x) +

1

2∂m∂mC(x)

](1.110)

C, D, M , N : real scalars

vµ: real vector field

χ, χ, λ, λ: Weyl fermions

abelian case : V a real vector superfield

supergauge transformation

V → V ′ = V + Λ+ Λ† (1.111)

where Λ(y, θ) and Λ†(y, θ) are chiral and anti-chiral superfield, respectively.

Λ + Λ† = A+ A∗ +√2(θψ + θψ) + θθF + θθF ∗

+iθσmθ∂m(A− A∗) +i√2θθθσm∂mψ

+i

2θθθσm∂mψ +

1

4θθθθ∂m∂m(A+ A∗) (1.112)

14

The component fields transform as

C → C + A+ A∗ (1.113)

χ→ χ− i√2ψ (1.114)

χ→ χ+ i√2ψ (1.115)

M + iN →M + iN + F (1.116)

vn → vn − i∂n(A− A∗) (1.117)

λ→ λ (1.118)

λ→ λ (1.119)

D → D (1.120)

WZ gauge

VWZ = −θσmθvm(x) + iθθθλ(x)− iθθθλ(x) +1

2θθθθD(x) (1.121)

V 2WZ = −1

2θ2θ2vmvm (1.122)

V 3WZ = 0 (1.123)

superfield strength

Wα = −1

4D2DαV (1.124)

Wα = −1

4D2DαV (1.125)

In the WZ gauge

• Wα is a chiral superfield. Wα is anti-chiral.

• Wα and Wα are supergauge inv.

Wα(y, θ) = −iλα(y) + θαD − i

2(σmσnθ)α(∂mvn − ∂nvm)(y) + θ2(σm∂mλ)α(y) (1.126)

Wα(y†, θ) = iλα(y

†) + θαD(y† +i

2ϵαγ(σ

µσν θ)γ(∂mvn − ∂nvm)(y†)− ϵαβ θ

2(σµ∂µλ)(y†)

(1.127)

15

Execise 3.1 Show (1.126) and (1.127).

DαWα = DαWα (1.128)

Execise 3.2 Show (1.128)

16

2 N = 1 supersymmetric gauge theory

[WB chapter VII] In this section we review N = 1 supersymmetric gauge theory defined

in N = 1 superspace.

Abelian gauge field a vector superfield V (x, θ, θ)

supergauge transformation: V → V + iΛ− iΛ

chiral superfield Φℓ, anti-chiral superfield Φℓ transform as

Φℓ → Φ′ℓ = e−itℓΛΦℓ (2.1)

Φℓ → Φ′ℓ = eitℓΛΦℓ (2.2)

where tℓ is the U(1) charge of Φℓ. Then ΦℓeitℓVΦℓ is invariant under the supergauge

transformation.

In the WZ gauge

ΦetVΦ = ΦΦ + tΦV Φ +t2

2ΦV 2Φ (2.3)

Here ΦΦ has bee already calculated.

ΦV Φ = −i(θσnθ)vnAA

+ θ2θ(− 1√

2σnψvnA+ i

√2λAA

)+ θ2θ

( 1√2σnψvnA− i

√2λAA

)+ θ2θ2

(− i

2∂mAAv

m +i

2∂mAAv

m +1

2(ψσnψ)vn −

i√2ψλA+

i√2Aλψ +

1

2DAA

)(2.4)

ΦV 2Φ = −1

2θ2θ2vnvnAA (2.5)

Exercise 4.1 Confirm (2.4).

17

The Lagrangian is

L =

∫d2θd2θΦℓe

tℓVΦℓ +1

4

∫d2θWαWα +

1

4

∫d2θWαW

α

+

∫d2θW (Φℓ) +

∫d2θW (Φℓ) (2.6)

= FℓFℓ − (∂m − itℓvm)Aℓ(∂m + itℓvm)Aℓ

+i

2(∂m − itℓv

m)ψℓσmψℓ −

i

2ψℓσ

m(∂m − itℓvm)ψℓ

+1

2D2 − 1

4FmnFmn −

i

2λσm∂mλ− i

2λσm∂mλ

+ tℓ

(− i√

2ψℓλAℓ +

i√2Aℓλψℓ +

1

2DAℓAℓ

)+ interaction terms (2.7)

where W (Φℓ) must be gauge invariant.

SQED SUSY generalization of QED is constructed in terms of two chiral superfields

Φ+, Φ− with charge +e and −e. The Lagrangian is

L =

∫d2θd2θΦ+e

eVΦ+ + Φ−e−eVΦ− +

1

4

∫d2θWαWα +

1

4

∫d2θWαW

α

+

∫d2θmΦ+Φ− +

∫d2θmΦ+Φ−

= F+F+ + F−F− − (∂m − ievm)A+(∂m + ievm)A+ − (∂m + ievm)A−(∂m − ievm)A−

+i

2(∂m − ievm)ψ+σ

mψ+ − i

2ψ+σ

m(∂m − ievm)ψ+ +i

2(∂m + ievm)ψ−σ

mψ− − i

2ψ−σ

m(∂m + ievm)ψ−

+1

2D2 − 1

4FmnFmn −

i

2λσm∂mλ− i

2λσm∂mλ

− ie√2(A+ψ+λ− A+ψ+λ− A−ψ−λ+ A−ψ−λ) +

e

2D(A+A+ + barA−A−)

+m(A+F− + A−F+ + A+F− + A−F+ − ψ+ψ− − ψ+ψ−) (2.8)

Lie algebra Lie algebra g with generators ta:

[ta, tb] = itabctc

trtatb = kδab (2.9)

ta are hermitian matrices. The structure constants tabc are totally antisymmetric.

18

vector superfields V are expanded in ta as

V = V ata. (2.10)

supergauge transformation Let Φ, Φ be the matter fields which belong to a repre-

sentation of g with matrix ta. They transforms as

Φ → e−iΛΦ Φ† → Φ†eiΛ†

(2.11)

where Λ(y, θ), Λ†(y†, θ) are chiral (anti-chiral) and expanded as Λ = Λata and Λ† = Λata.

Being Φ†eVΦ to be invariant under the gauge transformation, eV should transform as

eV → eV′= e−iΛ†

eV eiΛ. (2.12)

Using the Hausdorff’s formula [Takeuchi]

exety = ex+tu+O(t2) (2.13)

u =

(adx

eadx − 1+ adx

)y (2.14)

Using the identity

z

ez − 1= −z

2+z

2coth

(z2

)(2.15)

we find

eAeB = eA+LA/2[B+coth(LA/2)·B]+··· (2.16)

with LA/2 ·B := [A2, B], and assuming Λ and Λ† are infinitesimal parameters, we obtain

δV = V ′ − V = iLV2

[(Λ + Λ) + cothLV

2(Λ− Λ)

](2.17)

Since

cothx = x−1 +1

3x− 1

45x3 +

2

945x5 +O(x7) (2.18)

we have

δgaugeVWZ =i

2[VWZ ,Λ + Λ] + i(Λ− Λ) +

i

12

[VWZ ,

[VWZ ,Λ− Λ

]](2.19)

19

WZ gauge The vector superfield in the WZ gauge is

V = VWZ = −θσmθvm(x) + iθθθλ(x)− iθθθλ(x) +1

2θθθθD(x) (2.20)

V 2WZ = −1

2θθθθvmvm

V 3WZ = 0 (2.21)

For the gauge transformation preserving the WZ gauge

Λ = a(y), Λ = a(y†) (2.22)

with real a. Then (2.19) becomes

δgaugeVWZ =1

2[VWZ , a(y) + a(y†)] + i(a(y)− a(y†)) +

i

12(θσnθ)(θσmθ)[vn, [vm, a(y)− a(y†)]]

= (θσmθ)(−2∂ma− [vm, a]

)(x) + θθθ

(iλ(x) + i[λ, a]

)+ θθθ

(−iλ(x)− i[λ, a]

)+ θθθθ

(12D(x) +

1

2[D, a]

)(2.23)

The gauge transformation for component fields are

δvµ = 2∂µa+ [vµ, a] (2.24)

δλ = [λ, a] (2.25)

δλ = [λ, a] (2.26)

δD = [A, a] (2.27)

The superfield strength

Wα = −1

4DDe−VDαe

V ,

Wα = −1

4DDe−V Dαe

V (2.28)

In the WZ gauge,

Wα = −iλα(y) +[δβαD(y)− iσµνβ

α Fµν(y)]θβ + θθσµ

ααDµλα(y)

Wα = iλα(y) +[ϵαβD(y) + iϵαγ(σ

µν)γβFµν(y)

]θβ − ϵαβθθσ

µβαDµλα(y) (2.29)

20

where

Fmn = ∂mvn − ∂nvm +i

2[vm, vn] (2.30)

Dmλ = ∂mλ+i

2[vm, λ] (2.31)

is the field strength of the gauge field vn.

The gauge transformation of the superfield strength

Wα → e−iΛWαeiΛ, Wα → e−iΛ†

WαeiΛ†

(2.32)

Wα = −1

4D2(e−VDαe

V )

→ −1

4D2(e−iΛe−V eiΛ

†Dα(e

−iΛ†eV eiΛ))

= −1

4D2(e−iΛe−VDα(e

V eiΛ))

= −1

4e−iΛD2(e−VDαe

V )eiΛ − 1

4e−iΛD2Dαe

iΛ = e−iΛWαeiΛ (2.33)

Lagrangian

L =

∫d2θd2θΦ†eVΦ +

1

16kg2tr

(∫d2θWαWα +

∫d2θWαW

α

)(2.34)

In terms of component fields, the Lagrangian becomes

L =1

16kg2tr(−4iλσµDµλ− F µνFµν + 2D2

)+ FF − iψσµDµψ −DµADµA+

1

2ADA+

i√2(Aλψ − ψλA) (2.35)

where

Fmn = ∂mvn − ∂nvm +i

2[vm, vn]

Dnλ = ∂nλ+i

2[vn, λ]

Dµψ = ∂nψ +i

2vnψ (2.36)

21

After rescaling V → 2gV and writing in the basis A = Aata etc. the Lagrangian

becomes

L = −1

4vamnv

amn − iλaσmDmλ

a +1

2DaDa −DmA

†DmA

−iψσmDmψ + F †F + i√2g

(A†T aψλa − λaψT aA

)+gDaA†T aA (2.37)

where

DmA = ∂mA+ igvamTaA

Dmψ = ∂mψ + igvamTaψ

Dmλa = ∂mλ

a − gtabcvbmλc

vamn = ∂mvan − ∂nv

am − gtabcvbmv

cn (2.38)

complex coupling Introduce complex coupling

τ =Θ

2π+

4π

g2i (2.39)

Θ denotes the vacuum θ-angle. The Lagrangian can be written as

L =

∫d2θd2θΦ†eVΦ +

1

16kπ4itr

(∫d2θτW αWα −

∫d2θτ WαW

α

)=

1

16kg2tr

(−4iλσµDµλ− F µνFµν +

Θg2

16π2ϵmnpqFmnFpq + 2D2

)+ FF − iψσµDµψ −DµAD

µA+1

2ADA+

i√2(Aλψ − ψλA) (2.40)

The rescaling V → 2gV leads to

L = −1

4F amnF

amn +

Θg2

32π2F amnF

amn − iλaσmDmλa +

1

2DaDa −DmA

†DmA

−iψσmDmψ + F †F + i√2g

(A†T aψλa − λaψT aA

)+gDaA†T aA (2.41)

The Θ term is a toplogical term and plays an important role in non-perturbative physics.

Fmn =1

2ϵmnpqF

pq (2.42)

22

SUSY transformation The Lagrangian (2.40) is invariant under the SUSY transfor-

mation:

δξvm = iξσmλ+ iξσmλ

δλ = iξD + σmnξFmn

δξλ = −iξD + σmnξFmn

δξD = −ξσmDmλ+ ξσnDnλ (2.43)

where

Fmn = ∂mvn − ∂nvm +i

2[vm, vn]

Dnλ = ∂nλ+i

2[vn, λ]

Dnλ = ∂nλ+i

2[vn, λ] (2.44)

δξA =√2ξψ

δξψ =√2iσmξDmA+

√2ξF

δξF =√2iξσmDmψ + iξλA (2.45)

where DmA = ∂mA+ i2vmA and Dmψ = ∂mA+ i

2vmψ

δξA =√2ξψ

δξψ = i√2σmξDmA+

√2ξF

δξF =√2iξσmDmψ − iAξλ (2.46)

After rescaling V → 2gV , the SUSY transformation takes the form:

δξvm = iξσmλ+ iξσmλ

δλ = iξD + σmnξFmn

δξλ = −iξD + σmnξFmn

δξD = −ξσmDmλ+ ξσnDnλ (2.47)

23

δξA =√2ξψ

δξψ =√2iσmξDmA+

√2ξF

δξF =√2iξσmDmψ + 2giξλA (2.48)

δξA =√2ξψ

δξψ = i√2σmξDmA+

√2ξF

δξF =√2iξσmDmψ − 2giAξλ (2.49)

24

3 N = 2 Super Yang-Mils Theory

3.1 N = 2 SUSY

QAα , QβB (A,B = 1, 2)

index A: SU(2) doublet

QAα , QβB = 2σµ

αβPµδ

AB (3.1)

QAα , Q

Bβ = 2

√2ϵαβϵ

ABZ (3.2)

QαA, QβB = 2√2ϵαβϵABZ (3.3)

Z, Z: central charge

Z, Z commute with Pµ and Q, Q

massless representation representation without central charge P µ = (E, 0, 0, E)

aA =1√2E

QA1 , a†A =

1√2E

QA1

(3.4)

• N = 2 vector multiplet (a vector field +two Weyl fermions+two real scalars)

a†1a†2|Ω0⟩ helicity 1

a†A|Ω0⟩ helicity 12

|Ω0⟩ a†1a†2|Ω−1⟩ helicity 0

a†A|Ω−1⟩ helicity −12

|Ω−1⟩ helicity −1

• N = 2 hypermutiplet (4 real scalars +two Weyl fermions)

a†1a†2|Ω− 1

2⟩ a†1a

†2|Ω− 1

2⟩ helicity 1

2

a†A|Ω− 12⟩ a†A|Ω− 1

2⟩ helicity 0

|Ω− 12⟩ |Ω− 1

2⟩ helicity −1

2

massive representation P µ = (M, 0, 0, 0)

25

representation with central charge

aα :=1

2(Q1

α + ϵαβ(Q2β)

†) (3.5)

a†α =1

2((Q1

α)† + ϵαβQ

2β) (3.6)

bα =1

2(Q1

α − ϵαβ(Q2β)

†) (3.7)

b†α =1

2((Q1

α)† − ϵαβQ

2β) (3.8)

assume that Z is real

aα, a†β = (M +√2Z)δαβ (3.9)

bα, b†β = (M −√2Z)δαβ (3.10)

other anticommutators = 0 (3.11)

Exercise 5.1 Prove (3.9) and (3.10).

positive definiteness of the Hilbert space: Bogomolnyi bound

M ≥√2|Z| (3.12)

3.2 N = 2 superspace

N = 2 vector multiplet (vµ, λ, ψ,A)

N = 1 vector multiplet (vµ, λ)+N = 1 chiral multiplet Φ = (A,ψ) in the adjoint

representation

adjoint representation dimg × dimg matrices

(T a)ac := itabc (3.13)

satisfy the commutation relations

[T a, T b] = itabcT c (3.14)

Let Φ = (Φa) be a dimg chiral superfield. Φ := Φata

Λ = ΛaT a, Λ = Λata (3.15)

26

The infinitesimal supergauge transformation

ΛΦ = ΛaitbacΦc (3.16)

is equivalent to

[Λ,Φ] = iΛaΦctacbtb (3.17)

inner product

1

ktrΛΛ′ = ΛaΛ

′a (3.18)

Φ†eV Φ =1

ktrΦ†eVΦe−V (3.19)

Lagrangian

L =

∫d2θd2θΦ†eV Φ +

1

32kπImtr

(∫d2θτW αWα

)(3.20)

=1

16kg2tr

(−4iλσµDµλ− F µνFµν +

Θg2

16π2ϵmnpqFmnFpq + 2D2

)+ ¯FF − i ¯ψσµDµψ −Dµ

¯ADµA+1

2¯ADA+

i√2( ¯Aλψ − ¯ψλA) (3.21)

L =1

ktr

(∫d2θd2θΦ†eVΦe−V +

1

32πIm

∫d2θτW αWα

)=

1

16kg2tr

(−4iλσµDµλ− F µνFµν +

Θg2

16π2ϵmnpqFmnFpq + 2D2

)+

1

ktr

(FF − iψσµDµψ −DµAD

µA+1

2A[D,A] +

i√2A[λ, ψ]− ψ[λ, A]

)(3.22)

Rescale V → 2V , Φ → Φg

L =1

ktr 1

g2

∫d2θd2θΦ†e2VΦe−2V +

1

8πIm

∫d2θτW αWα

=

1

kg2tr(−iλσµDµλ− 1

4F µνFµν +

Θg2

32π2FµνFµν +

1

2D2

+ FF − iψσµDµψ −DµADµA+D[A, A] + i

√2A[λ, ψ]− [ψ, λ], A]

)(3.23)

27

potential term

1

g2tr(

1

2D2 +D[A, A] + FF ) → − 1

g2tr[A, A]2 (3.24)

spontaneous symmetry breaking, Higgs mechanism → massive gauge boson

flat direction

N = 1 SUSY manifest

N = 2 superspace N = 2 superspace2

(xµ, θ, θ, θ, ¯θ) (3.25)

N = 2 superfield F (xµ, θ, θ, θ, ¯θ)

supercharge

Qα =∂

∂θα− iσm

ααθα∂m (3.26)

Qα = − ∂

∂θα+ iθασm

αα∂m (3.27)

Qα =∂

∂θα− iσm

αα¯θα∂m (3.28)

¯Qα = − ∂

∂ ¯θα+ iθασm

αα∂m (3.29)

supercovariant derivative

D1α := Dα =

∂

∂θα+ iσm

ααθα∂m (3.30)

D1α := Dα = − ∂

∂θα− iθασm

αα∂m (3.31)

D2α := Dα =

∂

∂θα+ iσm

αα¯θα∂m (3.32)

D2α := ¯Dα = − ∂

∂ ¯θα− iθασm

αα∂m (3.33)

2R. Grimm, M. Sohnius, J. Wess, Nucl. Phys. B133 (1978) 275-284

28

chiral superfield: DαΨ = 0, ¯DαΨ = 0

Ψ = Ψ(1)(y, θ) +√2θΨ(2)(y, θ) + θ2Ψ(3)(y, θ) (3.34)

yµ = xµ + iθσµθ + iθσµ ¯θ

[Bilal, 9601007]

Ψ(2)α (y, θ) = iWα(y, θ), (3.35)

Ψ(3)(y, θ) = e−2V (y−iθσθ,θ,θ)Φ(y − iθσθ, θ, θ)e2V (y−iθσθ,θ,θ)∣∣∣θθ

(3.36)

L =1

8πkImtr

∫d2θd2θτΨ2

=1

8πkImtr

∫d2θd2θθ2(2τΨ(1)Ψ(3) − τΨ(2)Ψ(2))

=1

8πkImtr

(∫d2θ2τΦe−2V Φe2V

∣∣∣θθ+ τW αWα

)(3.37)

F-term in the sense of N = 2 superfield

N = 2 SUSY The Lagrangian (3.23) is invariant under theN = 1 SUSY transformation

δξvm = iξσmλ+ iξσmλ

δλ = iξD + σmnξFmn

δξλ = −iξD + σmnξFmn

δξD = −ξσmDmλ+ ξσnDnλ (3.38)

δξA =√2ξψ

δξψ =√2iσmξDmA+

√2ξF

δξF =√2iξσmDmψ + i[ξλ, A] (3.39)

δξA =√2ξψ

δξψ = i√2σmξDmA+

√2ξF

δξF =√2iξσmDmψ − i[A, ξλ] (3.40)

29

By rescaling V → 2V and Φ → Φ/g

δξvm = iξσmλ+ iξσmλ

δλ = iξD + σmnξFmn

δξλ = −iξD + σmnξFmn

δξD = −ξσmDmλ+ ξσnDnλ (3.41)

δξA =√2ξψ

δξψ =√2iσmξDmA+

√2ξF

δξF =√2iξσmDmψ + 2gi[ξλ, A] (3.42)

δξA =√2ξψ

δξψ = i√2σmξDmA+

√2ξF

δξF =√2iξσmDmψ − i2g[A, ξλ] (3.43)

The other SUSY transformation is obtained by replacing

(λ, ψ) → (ψ,−λ) (3.44)

δηvm = iησmψ + iησmψ

δηψ = iηD + σmnηFmn

δηψ = −iηD + σmnηFmn

δηD = −ησmDmψ + ησnDnψ (3.45)

δηA = −√2ηλ

δηλ = −√2iσmηDmA−

√2ηF

δηF = −√2iησmDmλ+ 2gi[ξψ, A] (3.46)

30

δηA = −√2ξλ

δηλ = −i√2σmηDmA−

√2ηF

δηF = −√2iησmDmλ− i2g[A, ηψ] (3.47)

3.3 Supercurrents and central charge

Noether’s theorem If the theory has continuous symmetry there exists a conserved

current. The action

S =

∫d4xL(ϕ, ∂µϕ) (3.48)

is invariant under the infinitesimal transformation

ϕ(x) → ϕ′(x) = ϕ(x) + αδϕ(x). (3.49)

The Lagrangian changes by the total derivative

δL = α∂µXµ (3.50)

δL =∂L∂ϕ

αδϕ+∂L

∂(∂µϕ)αδ∂µϕ

= ∂µ

(∂L

∂(∂µϕ)αδϕ

)+−∂µ(

∂L∂(∂µϕ)

) +∂L∂ϕ

αδϕ

∼ ∂µ

(∂L

∂(∂µϕ)δϕ

)(3.51)

∂µ

(∂L

∂(∂µϕ)αδϕ− αXµ

)= 0 (3.52)

αJµ :=∂L

∂(∂µϕ)αδϕ− αXµ (3.53)

is a conserved current.

31

energy-momentum tensor For the spacetime translation xµ → x′µ = xµ − aµ. La-

grangian is scalar: L′(x− a) = L(x)

δL = L′(x)− L(x) = aµ∂µL (3.54)

The conserved current

aνTµν =∂L

∂(∂µϕ)aν∂νϕ− aνδµνL (3.55)

energy-momentum tensor

Tµν =∂L

∂(∂µϕ)∂νϕ− ηµνL (3.56)

SUSY currents For the SUSY transformation:

θα → θα + ξα (3.57)

θα → θα + ξα (3.58)

xµ → xµ + iθσµξ − iξσµθ (3.59)

In the superfield Lagrangian , the D-term∫d2θd2θF (x, θ, θ) = d with F = · · ·+θ2θΛ+

θ2θΛ + θ2θ2d changes as

i

2ξσµ∂µΛ +

i

2ξσµ∂µΛ (3.60)

The F-term∫d2θΦ = F ′ with Φ = ...+ θψ′ + θ2F ′ changes as

iξσµ∂µψ′ (3.61)

Then the supercurrents Sµ, Sµ are defined as

ξSµ + ξSµ =∂L

∂(∂µϕ)δξϕ− i

2ξσµΛ− i

2ξσµΛ− iξσµψ′ − iξσµψ′ (3.62)

WZ model

Sµ =√2σν σµψ∂

νA− i√2σµψ

∂W

∂A(3.63)

Sµ =√2σνσµψ∂

νA− i√2σµψ

∂W

∂A(3.64)

32

N = 1 U(1)

Sµ = −iσρσσµλFρσ (3.65)

Sµ = −iσρσσµλFρσ (3.66)

N = 1 gauge theory

ξSµ + ξSµ =1

4g2ktr−iξσρσσµλFρσ − iξσµν σµλFµν

+√2ξσν σµψDνA+

1√2ξσµAλA+

√2ξσν σµψDνA− 1√

2ξσµAλA (3.67)

Rescaling V → 2gV

Sµ = − i

ktrσρσσµλFρσ +

√2σν σµψD

νA+√2gσµ(AλA) (3.68)

Sµ = − i

ktrσρσσµλFρσ +

√2σνσµψD

νA−√2gσµ(AλA) (3.69)

Using the idenities

σaσbσc = ηacσb − ηbcσa − ηabσc + iεabcdσd

σaσbσc = ηacσb − ηbcσa − ηabσc − iεabcdσd (3.70)

Sµ = − i

ktrσνλ(−Fµν − iFµν) +

√2σν σµψD

νA+√2gσµ(AλA) (3.71)

Sµ = − i

ktrσνλ(−Fµν + iFµν) +

√2σνσµψD

νA−√2gσµ(AλA) (3.72)

central charge 3 For N = 2 super Yang-Mills theory

Sµ =1

ktrσνλ(iFµν − Fµν) +

√2σν σµψD

νA+√2gσµλ[A, A] (3.73)

Sµ =1

ktrσνλ(iFµν + Fµν) +

√2σνσµψD

νA−√2gσµλ[A, A] (3.74)

Another set of supercurrents:

Sµ =1

ktrσνψ(iFµν − Fµν)−

√2σν σµλD

νA+√2gσµψ[A, A] (3.75)

¯Sµ =1

ktrσνψ(iFµν + Fµν)−

√2σνσµλD

νA−√2gσµψ[A, A] (3.76)

3E.Witten and D.I.Olive

33

Qα :=

∫d3xS0

α, Qα :=

∫d3xS0

α (3.77)

Qα, Qβ = −2√2ϵαβ

1

ktr

∫d3x(iF 0i − F 0i)DiA (3.78)

electric charge

Qe =1

⟨A⟩

∫d3x∂i(F

a0iAa) (3.79)

Qm =1

⟨A⟩

∫d3x∂i(F

a0iAa) (3.80)

Z = i⟨A⟩(Qe + iQm) = i2√2⟨A⟩g(ne + τnm) (3.81)

Qe = neg, Qm = nmgm (3.82)

and the Dirac quantization condition:

ggm = 4π (3.83)

In the presence of the θ-term, electric charge shifts to

Qe → Qe +θg2

8π2Qm (3.84)

(Witten effetcs4)

After rescaling A→ A/g, we have

Z = a(ne + τnm) = ane + aDnm (3.85)

where

aD = τa (3.86)

4E. Witten, Phys. Lett. B86 (1979) 283

34