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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
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