Supersymmetry, Superspace, Superfields

Evgeny Ivanov (BLTP JINR, Dubna)

August 18, 2015

Outline

From symmetries to supersymmetry

Basic features of supersymmetry

On representations of supersymmetry

Superspace and superfields

Supersymmetric quantum mechanics

Summary

From symmetries to supersymmetry

Symmetries play the central role in physics: they underlie all theories ofinterest known to date:

I Gravity: based on the local diffeomorphism group of the space-time,Diff R4, xm ⇒ xm′(x);

I Maxwell theory and its generalization, Yang-Mills theory: based on thegauge groups U(1) and SU(n), with group parameters being arbitraryfunctions of the space-time point;

I Standard model, the unification of the electro-weak theory and quantumchromodynamics: [Gauge U(2)× SU(3)c ] × [Global Flavor SU(n)];

I String theory: the diffeomorphisms of the worldsheet (z , z);

I Supergravity, Superstrings, Superbranes: Supersymmetry, local, global,conformal , .... .

There are two vast classes of symmetries in the NatureI I. Internal symmetries: isotopic SU(2) ,flavour SU(n), etc. Their main

feature: they are realized as transformations of fields without affectingthe space-time coordinates. The generators are matrices acting onsome external indices of fields, no any x-derivatives are present.Example: realization of SU(2) on the doublet of fields ψi (x) (“neutron -proton”)

δψi (x) = iλa12

(σa)ki ψk (x) , [

12σa,

12σb] = iεabc

12σc ,

σaσb = δabI + iεabcσc ,

σa are Pauli matrices:

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

).

I II. Space-time symmetries: Lorentz, Poincaré and conformal groups.Generators in the realization on fields involve x-derivatives.Example: transformation of the scalar field ϕ(x) in the Poincaré group:

δϕ(x) := −i(cmPm + ω[mn]Lmn)ϕ(x) = −cm∂mϕ(x)− ω[mn] 12

(xm∂n − xn∂m)ϕ(x) ,

Pm =1i∂m , Lmn =

12i

(xm∂n − xn∂m) , m, n = 0, 1, 2, 3 .

I The primary fundamental symmetry principle is the invariance of theaction

S =

∫d4xL(φA, ∂φA, ψα, ...) , δS =

δSδφA

δφA = 0↔ δL = ∂mAm .

I Example: free Lagrangian of the scalar field

L(1)free =

12∂mφ(x)∂mφ(x)

transform under the Poincare group as

δωL(1)free = −1

2∂n(ωmnxm∂

sφ∂sφ) , δcL(1)free = −1

2cm∂m(∂nφ∂nφ),

whence the invariance of the relevant action follows.I In the systems with few scalar fields one can realize internal

symmetries. The free Lagrangian of one complex field

L(2)free = ∂mφ(x)∂mφ(x)

is invariant under U(1) symmetry

δφ = iλφ , δφ = −iλφ,

three real scalar fields can be joined into a triplet of the group SU(2)

L(3)free =

12∂mφa(x)∂mφa(x) , δφa = εabcλbφc ⇒ δL(3)

free = 0 .

I One more possibility to construct SU(2) invariant Lagrangian is to jointwo complex scalar fields into SU(2) doublet

L(4)free = ∂mφα(x)∂mφ

α(x) ,

δφα =i2λa(σa)βαφβ , δφα = − i

2λa(σa)αβ φ

β , ⇒ δL(4)free = 0 .

I Extending the sets of fields (and adding interaction terms), we canfurther enlarge internal symmetries.

I The characteristic feature of all these symmetries is that the groupparameters are ordinary commuting numbers, and so do not mixbosonic fields (Bose-Einstein statistics, integer spins 0, 1, . . . ) withfermionic fields (Fermi-Dirac statistics, half-integer spins 1/2, 3/2, . . . )

I The bosonic and fermionic parts of the Lagrangian are invariantseparately.

I Let us now consider a sum of the free Lagrangians of the masslesscomplex scalar field ϕ(x) and the Weyl fermionic field ψα(x)

Lφ+ψ = ∂mϕ∂mϕ−i4

[ψα(σm)αα∂mψ

α − ∂mψα(σm)ααψ

α],

where (σm)αα = (δαα, (σa)αα) are the so called sigma matrices, the

basic object of the spinor two-component formalism of the Lorentzgroup (they are invariant under simultaneous Lorentz transformation ofthe vector m,m = 0, 1, 2, 3 , and spinor α, α = 1, 2 indices).

I The evident symmetries of this Lagrangian are Poincaré and phaseU(1) symmetries which separately act on ϕ(x) and ψα(x).

I However, there is a new much less obvious symmetry. Namely, thisLagrangian transform by a total derivative under the followingtransformations mixing bosonic and fermionic fields

δϕ = −εαψα , δϕ = −ψαεα , δψα = 2i(σm)ααεα∂mϕ .

I One sees that the transformation parameters εα, εα have the dimensioncm1/2, so these transformations do not define an internal symmetry (therelevant group parameters are dimensionless). Moreover, for the actionto be invariant, these parameters should anticommute amongthemselves and with the fermionic fields, {ε, ε} = {ε, ε} = {ε(ε), ψ} = 0 ,and commute with the scalar field, [ε(ε), ϕ] = 0 .

I To see which kind of algebraic structure is behind this invariance oneneeds to consider the Lie bracket of two successive transformations onthe scalar ϕ(x)

(δ1δ2 − δ2δ1)ϕ = −(εα2 δ1ψα) + (εα1 δ2ψα)

= 2(ε1σ

m ε2 − ε2σm ε1)

(1i∂mϕ).

Thus the result is an infinitesimal 4-translation with the compositeparameter i (ε1σ

m ε2 − ε2σm ε1).

I Rewriting the ε variation in the form

δϕ = i(εαQα + εαQα

)ϕ ,

and taking into account that the spinor parameters anticommute withQα, Qα, we find that the above Lie bracket structure is equivalent to thefollowing anticommutation relations for the supergenerators

{Qα, Qβ} = 2 (σm)αβPm , Pm =1i∂

∂xm ,

{Qα,Qβ} = {Qα, Qβ} = 0 ,

[Pm,Qα] = [Pm, Qα] = 0 .

This is what is called N = 1 Poincaré superalgebra.

Basic features of supersymmetryThe full set of the (anti)commutation relations of the N = 1 Poincarésuperalgebra reads (Golfand, Lichtman, 1971; Volkov, Akulov, 1972; Wess,Zumino, 1974)

{Qα, Qβ} = 2 (σm)αβPm ,

{Qα,Qβ} = {Qα, Qβ} = 0 ,

[Pm,Qα] = [Pm, Qα] = 0 , (1)

[Jmn,Qα] = −12

(σmn) βα Qβ , [Jmn, Qα] =12

(σmn) βα Qβ ,

[Jmn,Ps] = i (ηnsPm − ηmsPn) ,

[Jmn, Jsq] = i (ηnsJmq − ηmsJnq + ηnqJsm − ηmqJsn) ,

[R,Qα] = Qα , [R, Qα] = −Qα [R,Pm] = [R, Jmn] = 0 .

Here Jmn are the full Lorentz group generators and R is a generator of anextra internal U(1) transformations (the so-called R symmetry). Also,

ηmn = diag(1,−1,−1,−1) , (σmn)βα =i2(σmσn − σnσm)β

α,

(σmn)βα =i2(σmσn − σnσm)β

α, σmαα = (δαα,−σaαα)

Some important common features and consequences of supersymmetry canbe figured out just from these (anti)commutation relations.

I The Poincaré superalgebra is an example of Z2-graded algebra: oneascribes parities ±1 to its all elements and the structure relationsshould respect these parities

[odd, odd] ∼ even , [even, odd] ∼ odd , [even, even] ∼ even .

The spinor generators Qα, Qα have the parity -1 and so are odd; allbosonic generators have the parity +1 and so are even.

I Lie superalgebras satisfy the same axioms as the Lie algebras, thedifference is that the relevant generators satisfy the graded Jacobiidentities. E.g.,

{[B1,F2],F3} − {[F3,B1],F2}+ [{F2,F3},B1] = 0 ,

[{F1,F2},F3] + [{F3,F1},F2] + [{F2,F3},F1] = 0 ,

where B1 is a bosonic generator and F1,F2 are fermionic ones.I Since the generators Qα, Qα are fermionic, irreducible SUSY multiplets

should unify bosons with fermions. Action of the spinor generators onthe bosonic state yields a fermionic state and vice versa.

I Since the translation operator Pm is non-vanishing on any field given onthe Minkowski space, the same should be true for the spinor generatorsas well. So any field should belong to a non-trivial supermultiplet.

I It follows from the relations [Pm,Qα] = [Pm, Qα] = 0 that[P2,Qα] = [P2, Qα] = 0. The operator P2 is a Casimir of the Poincarégroup, P2 = m2. So it is also a Casimir of the Poincaré supergroup.Hence all components of the irreducible supermultiplet should have thesame mass. No mass degeneracy between bosons and fermions isobserved, so supersymmetry should be broken in one or another way.

I In any representation of supersymmetry, such that the operator Pm isinvertible, there should be equal numbers of bosons and fermions.

I In any supersymmetric theory the energy P0 should be non-negative.Indeed, from the basic anticommutator it follows∑

α=1,2

(|Qα|2 + |Qα|2

)= 4P0 ≥ 0 .

I Rigid supersymmetry, with constant parameters, implies the translationinvariance. Gauge supersymmetry, with the parameters being arbitraryfunctions of the space-time point, implies the invariance under arbitrarydiffeomorphisms of the Minkowski space. Hence the theory of gaugedsupersymmetry necessarily contains gravity. This is supergravity. Itsbasic gauge fields are graviton (spin 2) and gravitino (spin 3/2).

Supersymmetry allows one to evade the famous Coleman-Mandula theoremabout impossibility of non-trivial unification of the space-time symmetries withthe internal ones. It states that any symmetry of such type (in dimensions≥ 3), under the standard assumptions about the spin-statistic relation, isinevitably reduced to the direct product of the Poincaré group and the internalsymmetry group.The arguments of this theorem do not apply to superalgebras, when onedeals with both commutation and anticommutation relations. Haag,Lopushanski, Sohnius (1975) showed that the most general superextensionof the Poincaré group algebra is given by the following relations

{Q iα, Qβk} = 2δi

k (σm)αβPm ,

{Q iα,Q

jβ} = εαβZ ij , {Qαi , Qβj} = εαβZij ,

[T ij ,Q

kα] = −i

(δk

j Q iα −

1N δi

j Qkα

), [T i

j , Qαk ] = i(δi

k Qαj −1N δi

j Qαk

),

[T ij ,T

kl ] = i

(δi

l Tkj − δk

j T il

),

where T ij ((T i

j )† = −T ji , T i

i = 0) are generators of the group SU(N ) Thegenerators Z ij = −Z ji , Zij = −Zji are central charges, they commute with allgenerators except the SU(N ) ones

[Z ,Z ] = [Z , Z ] = [Z ,P] = [Z , J] = [Z ,Q] = [Z , Q] = 0 .

The relevant supergroup is called N -extended Poincaré supergroup.

Due to the property that the spinor generators Q iα, Qβk carry the internal

symmetry indices, the supermultiplets of extended supersymmetries joinfields having not only different statistics and spins, but also belonging todifferent representations of the internal symmetry group U(N ). In otherwords, in the framework of extended supersymmetry the actual unification ofthe space-time and internal symmetries comes about. The relevantsupergravities involve, as a subsector, gauge theories of internal symmetries,i.e. they yield non-trivial unification of the Einstein gravity with Yang-Millstheories.

An important ingredient of supersymmetric theories is the auxiliary fields.They ensure the closedness of the supersymmetry transformations off massshell.

Let us come back to the realization of N = 1 supersymmetry on the fieldsϕ(x), ψα(x) and calculate Lie bracket of the odd transformations on ψα(x):

(δ1δ2 − δ2δ1)ψα = −2i(ε1σ

m ε2 − ε2σm ε1)∂mψα + 2i

[ε1αε2α(σm)αβ∂mψβ − (1↔ 2)

].

The first term in the r.h.s. is the translation one, as for ϕ(x). However, thereis one extra term. It is clear that the Lie bracket on all members of thesupermultiplet should have the same form, i.e. reduce to translations. Thecondition of vanishing of the 2-nd term is

σm∂mψ = σm∂mψ = 0 .

But this is just the free equation of motion for ψα(x). Thus N = 1 SUSY isclosed only on-shell, i.e. modulo equations of motion.

How to achieve the off-shell closure? The solution is to introduce a new fieldF (x) of non-canonical dimension cm−2 and to extend the free action of ϕ,ψα:

Lφ+ψ+F = ∂mϕ∂mϕ−i4

[ψα(σm)αα∂mψ

α − ∂mψα(σm)ααψ

α]

+ FF .

It is invariant, up to a total derivative, under the modified transformationshaving the correct closure for all fields:

δφ = −εαψα , δψα = −2i(σm)ααεα∂mφ− 2εαF , δF = −i εα(σm)αα∂mψ

α .

The auxiliary fields satisfy the algebraic equations of motion

F = F = 0 .

After substitution of this solution back in the Lagrangian and SUSYtransformations, we reproduce the previous on-shell realization. They do notpropagate also in the quantum case, possessing delta-function propagators.

The only (but very important!) role of the auxiliary fields is just to ensure thecorrect off-shell realization of supersymmetry, such that it does not dependon the precise choice of the invariant Lagrangian, like in the cases of ordinarysymmetries.

The simplest non-trivial choice is

Lwz = ∂mφ∂mφ−i4

[ψα(σm)αα∂mψ

α − ∂mψα(σm)ααψ

α]

+ FF

+

[m(φF − 1

4ψψ

)+ g

(φ2F − 1

2φψψ

)+ c.c.

].

This model was the first example of renormalizable supersymmetricquantum field theory and it is called the Wess-Zumino model, after names ofits discoverers. The Lagrangian Lwz is invariant under the sametransformations as the free Lagrangian we have considered before.

After eliminating the auxiliary fields by their algebraic equations of motion,

F = −(mφ+ gφ2), F = −(mφ+ gφ2),

we obtain the on-shell Lagrangian of the W.-Z. model as

Lwz = ∂mφ∂mφ−i4

[ψα(σm)αα∂mψ

α − ∂mψα(σm)ααψ

α]

−(

mφ+ gφ2)(

mφ+ gφ2)− m

4(ψψ + ψψ

)− g

2(φψψ + φψψ

).

The first term in the second line contains the mass term of φ and therenormalizable ∼ φ3 and ∼ φ4 self-interaction

−m2φφ−mg(φφ2 + φφ2

)− g2 φ2φ2 ,

the second term is the mass term of fermions and the third one is a kind ofYukawa coupling of fermions to the scalar field. It is supersymmetry whichrequires the masses of bosons and fermions to be equal and the couplingconstant of the φ4 interaction to be the square of that of the Yukawa coupling.

The Wess-Zumino model Lagrangian was originally found by the “trying anderror” method. The systematic way of constructing invariant off-shellLagrangians is the superfield method we will discuss in the second Lecture.

Using this systematic method, one can equally construct more generalLgarangians of the fields (φ, ψα,F ), invariant under the same linear off-shellN = 1 supersymmetry transformations. After eliminating the auxiliary fieldsfrom these Lagrangians by their equations of motion, we will obtain theLagrangians in terms of the physical fields (φ, ψα) only. These physicalLagrangians are invariant under the nonlinear on-shell N = 1 supersymmetrytransformations the precise form of which depends on the form of the on-shellLagrangian, though it is uniquely specified by the off-shell Lagrangian.

To summarize, the fields (φ, ψα,F ) form the set closed under the off-shellN = 1 supersymmetry transformations, and it is impossible to select anylesser closed set of fields in it. Thus these fields constitute the simplestirreducible multiplet of N = 1 supersymmetry. It is called scalar N = 1supermultiplet.

On representations of supersymmetrySupersymmetry is an extension of the Poincaré symmetry. Hence itsrepresentations should be the appropriate extensions of the representationsof the Poincaré group.

I The irreducible representations of the Poincaré group in the case ofnon-zero mass are characterized by the two quantum numbers, themass m 6= 0 and spin s. They are the eigenvalues of the two Casimiroperators, the square momentum operator, C1 = P2 = PmPm,

P2|m, s >= m2|m, s > ,

I and the square spin operator C2 = −W mWm, where W m = 12ε

npqmPnJpq

is the Pauli-Lubanski vector,

C2|m, s >= m2s(s + 1)|m, s > .

I In the massless case the irreducible representations are characterizedby the helicity ±|λ| which is, roughly speaking, a projection of the spinof the particle on the direction in which the massless particle ispropagating. The helicity can take integer or semi-integer values,λ = 0,±1,±2, . . . for bosons and λ = ±1/2,±3/2, . . . for fermions.

I In the case of supersymmetry the operator P2 = −� = −∂m∂m

commutes with all generators and so is still Casimir operator. Hence allmembers of the irreducible massive supermultiplet should have equalmasses.

I The second Casimir of the Poincaré group also admits asuperextension,

C2 =12

W [mn]W[mn] , W [mn] = W mPn − W nPm ,

W m = W m +18

(σm)αβ [Qα, Qβ ]. (2)

It commutes with all generators of the superalgebra Poincarè and canbe shown to take the following value on each member of the irreduciblerepresentation

C2 = m4Y (Y + 1)

I The new quantum number Y takes integer and semi-integervalues,Y = 0, 1

2 , 1,32 , . . ., and it is called superspin. The spin content of

any irreducible massive supermultiplet of the N = 1 superalgebraPoincaré is related to Y as

(Y , Y + 1

2 , Y − 12 , Y

).

I For the scalar multiplet Y = 0, and it consists of the states with(s = 0, s = 1/2, s = 0) which correspond to the set of fields(φ(x), ψα(x),F (x)) we met before.

I In N = 1 supersymmetry the massless supermultiplets are formed bypairs of states with the adjacent helicities, |λ〉, |λ− 1/2〉. In N = 2supersymmetry the massless supermultiplets consist of the states withthe helicities |λ〉, |λ− 1/2〉2, |λ− 1〉, etc.

I The multiplets with opposite helicities can be obtained via CPTconjugation. Of special interest are the so called “self-conjugated”multiplets which, from the very beginning, involve the full spectrum ofhelicities from λ to −λ. Equating

λ−N/2 = −λ ⇒ λ = N/4 , (3)

find that, until N = 8, there exist the following self-conjugatedmassless supermultiplets

N = 2 matter multiplet: 1/2, (0)2,−1/2;

N = 4 gauge multiplet: 1, (1/2)4, (0)6, (−1/2)4,−1;

N = 8 supergravity multiplet: 2, (3/2)8, (1)28, (1/2)56, (0)70,

(−1/2)56, (−1)28, (−3/2)8,−2.

I For N > 8 the massless supermultiplets would include helicities > 2.The relevant theories are called “higher-spin theories” and, forself-consistency at the full interaction level, should include the wholeinfinite set of such spins (helicities). Such complicated theories are nowunder intensive study, but they are out of the scope of my lectures.

Superspace and superfieldsI When considering one or another symmetry and constructing physical

models invariant w.r.t. it, it is very important to find out the proper spaceand/or the fundamental multiplet on which this symmetry is realized inthe most natural and simplest way.

I The Poincaré group has a natural realization in the Minkowski spacexm,m = 0, 1, 2, 3 as the group of linear rotations and shifts of xm

preserving the flat invariant interval ds2 = ηmndxmdxn. Analogously,supersymmetry has a natural realization in the Minkowski superspace.

I The translation generators Pm can be realized as shifts ofxm, xm′ = xm + cm. In the case of N = 1 supersymmetry we haveadditional spinor generators Qα, Qα and anticommuting parametersεα, εα. Then it is natural to introduce new spinor coordinates θα, θα

having the same dimension cm1/2 as the spinor parameters and torealize the spinorial generators as shifts of these new coordinates

θα′ = θα + εα , θα′ = θα + εα .

I The extended manifold

M(4|4) =(

xm , θα , θα),

is called N = 1 Minkowski superspace (Salam, Strathdee, 1974).

I Its natural generalization is

M(4|4N ) =(

xm , θαi , θα i),

and it is called N extended Minkowski superspace.I The spinor coordinates are called odd or Grassmann coordinates and

have the Grassmann parity −1, while xm are even coordinates havingthe Grassmann parity +1

[θαi , xm] = [θα i , xm] = 0 , {θαi , θβk } = {θαi , θβ k} = 0 .

The spinor coordinates also anticommute with the parameters εα, εα.I Since two superstranslations yield a shift of xm, they should be

non-trivially realized on xm. In the N = 1 case:

xm′ = xm − i(εσmθ − θσm ε) , (δ1δ2 − δ2δ1)xm = 2i(ε1σm ε2 − ε2σ

m ε1).

(an analogous transformation takes place in the general case of Nextended supersymmetry).

I Superfields are functions on superspace, such that they have definitetransformation properties under supersymmetry. The general scalarN = 1 superfield is Φ(x , θ, θ) with the following transformation law

Φ′(x ′, θ′, θ′) = Φ(x , θ, θ) .

I The most important property of superfield is that its series expansion inGrassmann coordinates terminates at the finite step. The reason is thatthese coordinates are nilpotent, because they anticommute. E.g.,{θα, θβ} = 0⇒ θ1θ1 = θ2θ2 = 0 . Then

Φ(x , θ, θ) = φ(x) + θα ψα(x) + θα χα(x) + θ2 M(x) + θ2 N(x)

+ θσmθAm(x) + θ2 θα ρα(x) + θ2 θα λα(x) + θ2 θ2 D(x) ,

where θ2 := θαθα = εαβθαθβ , θ2 = θαθ

α = εαβ θβ θα , ε12 = ε12 = 1.

I Here one deals with the set of 8 bosonic and 8 fermionic independentcomplex component fields. The reality condition

(Φ) = Φ

implies the following reality conditions for the component fields

φ(x) = φ(x) , χα(x) = ψα(x) , M(x) = N(x) , Am(x) = Am(x) ,

λα(x) = ρα(x) , D(x) = D(x) .

They leave in Φ just (8 + 8) independent real components.

I The transformation law Φ′(x , θ, θ) = Φ(x − δx , θ − ε, θ − ε) implies

δΦ = −εα ∂Φ

∂θα− εα

∂Φ

∂θα− δxm ∂Φ

∂xm ≡ i(εαQα + εαQα

)Φ ,

Qα = i∂

∂θα+ θα(σm)αα

∂

∂xm , Qα = −i∂

∂θα− θα(σm)αα

∂

∂xm ,

{Qα, Qα} = 2Pm , {Qα,Qβ} = {Qα, Qβ} = 0 , Pm =1i∂

∂xm .

I The relevant component transformations are read off from the formulaδΦ = δφ+ θαδψα + . . .+ θ2θ2δD. They are

δφ = −εψ − εχ , δψα = −i(σm ε)α∂mφ− 2εαM − (σm ε)αAm , . . . ,

δD =i2∂mρσ

m ε− i2εσm∂mλ .

I These transformations uniformly close on xm translations without use ofany dynamical equations. However, the supermultiplet of fieldsencompassed by Φ(x , θ, θ) is reducible: it contains in fact both thescalar and gauge N = 1 supermultiplets (superspins Y = 0 andY = 1/2). How to describe irreducible supermultiplets in the superfieldlanguage?

I An important element of the superspace formalism are spinor covariantderivatives

Dα =∂

∂θα+ i θα(σm)αα

∂

∂xm , Dα = − ∂

∂θα− iθα(σm)αα

∂

∂xm ,

{Dα, Dα} = −2i(σm)αα∂m , {Dα,Dβ} = {Dα, Dβ} = 0 .

DαDβDγ = 0 because of the nilpotency : D1D1 = D2D2 = 0 .

I The covariant derivatives anticommute with supercharges,{D,Q} = {D, Q} = 0, so DαΦ and DαΦ are again superfields, e.g.,δDαΦ = DαδΦ = Dαi

(εαQα + εαQα

)Φ = i

(εαQα + εαQα

)DαΦ .

I Now, it becomes possible to define the “irreducible” superfields.Analogy: in Minkowski space the vector field Am is known to carry twoPoincaré spins 1 and 0. The irreducible components are distinguishedby imposing on Am the supplementary differential conditions

∂mAm = 0↔ spin 1 , ∂mAn − ∂nAm = 0↔ spin 0 .

I Analogous conditions can be imposed on the superfield Φ in order tosingle out the irreducible multiplets with the superspins 0 and 1/2.These conditions are defined with the help of the covariant spinorderivatives.

I The simplest conditions are the chirality or anti-chirality ones

(a) DαΦL(x , θ, θ) = 0 , or (b) DαΦR(x , θ, θ) = 0 .

I Eq. (a), e.g., implies

ΦL(x , θ, θ) = ϕL(xL, θ) = φ(xL) + θαψα(xL) + θθF (xL) ,

xmL = xm + iθσmθ ,

i.e. we are left with the independent fields φ, ψα,F .I From the general transformation laws of the component fields it follows

that this set is closed under N = 1 supersymmetry:

δφ = −εψ , δψα = −2i(σm ε)α∂mφ− 2εαF , δF = −i εσm∂mψ .

This is just the transformation law of the scalar N = 1 supermultiplet.I The geometric interpretation: the coordinate set (xm

L , θα) is closed

under N = 1 supersymmetry:

δxmL = 2iθσm ε , δθα = εα .

It is called left-chiral N = 1 superspace.I In the basis (xm

L , θα, θα) the covariant derivative Dα is “short”,

Dα = − ∂∂θα , and the chirality condition (a) is reduced to the Grassmann

Cauchy-Riemann conditions:

DαΦL(xL, θ, θ) = 0 ⇒ ∂

∂θαΦL = 0 ⇒ ΦL = ϕL(xL, θ) .

Superfield actionsI Having superfields, one can construct out of them, as well as out of

their vector and covariant spinor derivatives, scalar superfieldLagrangians. Any local product of superfields is again a superfield:

L = L(Φ,DαΦ, DαΦ, ∂mΦ, . . .) , δL = i(εαQα + εαQα

)L .

I It is easy to realize that the variation of the higher component of anyexpansion of any superfield is a total derivative. Then one takes thehigher component field in the θ expansion of the superfield Lagrangianand integrates it over Minkowski space. It will be just an action invariantunder N = 1 supersymmetry!

I A manifestly covariant way to write supersymmmetric actions is to usethe Berezin integral. It is equivalent to differentiation in Grassmanncoordinates. In the considered case of N = 1 superspace it is definedby the rules∫

d2θ (θ)2 = 1 ,∫

d2θ (θ)2 = 1 ,∫

d2θd2θ (θ)4 = 1 , (θ)4 ≡ (θ)2(θ)2

Hence the Berezin integration yields an efficient and manifestlysupersymmetric way of singling out the coefficients of the highest-orderθ monomials in the superfield Lagrangians.

I The simplest invariant action of chiral superfields producing the kineticterms of the scalar multiplet is as follows

Skin =

∫d4xd4θ ϕ(xL, θ)ϕ(xR , θ) , xm

R = (xmL ) = xm − iθσmθ .

After performing integration over Grassmann coordinates, one obtains

S ∼∫

d4x(∂mφ∂mφ−

i2ψσm∂mψ + FF

).

I The total Wess-Zumino model action is reproduced by adding, to thiskinetic term, also potential superfield term

Spot =

∫d4xLd2θ

(g3ϕ3 +

m2ϕ2)

+ c.c. .

I This action is the only renormalizable action of the scalar N = 1multiplet. In principle, one can construct more general actions, e.g., theaction of Kähler sigma model and generalized potential term

Skin =

∫d4xd4θK

[ϕ(xL, θ), ϕ(xR , θ)

], Spot =

∫d4xLd2θP(ϕ) + c.c. .

I The multiplet with the superspin Y = 1/2 is described by the gaugesuperfield V (x , θ, θ) possessing the gauge freedom

δV (x , θ, θ) = i[λ(xm − iθσmθ, θ)− λ(xm + iθσmθ, θ)],

where λ(xL, θ) is an arbitatry chiral superfield parameter.I Using this freedom, one can choose the so called Wess-Zumino gauge

VWZ (x , θ, θ) = 2 θσmθAm(x) + 2i θ2θα ψα(x)− 2iθ2θα ψα(x) + θ2θ2 D(x) .

I Thus in the WZ gauge we are left with the irreducible set of fieldsforming the gauge (or vector) off-shell supermultiplet: the gauge fieldAm(x), A′m(x) = Am + ∂mλ(x), the fermionic field of gaugino ψα(x),ψα(x) and the auxiliary field D(x).

I The invariant action is written as an integral over the chiral superspace

SN=1gauge =

116

∫dζL (WαWα) + c.c. , Wα = −1

2D2DαV , DαWα = 0 .

I Everything is easily generalized to the non-abelian case. Thecorresponding component off-shell action reads

S =

∫d4x Tr

[−1

4F mnFmn − iψσmDmψ +

12

D2].

I What about superfield approach to higher N supersymmetries? Thedifficulties arise because the relevant superspaces contain too many θcoordinates and it is a very complicated problem to define thesuperfields which would correctly describe the relevant irreps.

I For N = 2, the off-shell gauge multiplet contains the vector gauge fieldAm(x), the complex scalar physical field ϕ(x), the SU(2) doublet ofWeyl fermions ψi

α(x), ψα i (x) and the auxiliary real SU(2) triplet D(ik)(x).I There is no simple way to define N = 2 analog of the N = 1 gauge

prepotential V . However, one can define the appropriate covariantsuperfield strength W . In the abelian case,

(a) DiαW = 0 , (b) DαiDk

αW = DiαDαk W .

The invariant action is an integral over chiral N = 2 superspace

S ∼∫

d4xLd4θW 2 + c.c. .

I What about maximaly extended N = 4 super Yang-Mills? It has nosuperfield formulation with all N = 4 supersymmetries being manifestand off-shell. There is N = 1 superfield formulation with one gaugesuperfield and three chiral superfields; N = 2 formulation in terms ofN = 2 gauge superfield and one hypermultiplet. The latter possessesan off-shell formulation only in the N = 2 harmonic superspace(Galperin, Ivanov, Kalitzin, Ogievetsky, Sokatchev, 1984).

Supersymmetric quantum mechanicsI Quantum mechanics = one-dimensional field theory. The relevant

supersymmetry can be understood as the d = 1 reduction ofhigher-dimensional Poincaré supersymmetry. More generally, theN -extended d = 1 “Poincaré” supersymmetry can be defined by the(anti)commutation relations

{Qm,Qn} = 2δmn H , [H,Qm] = 0 ,Qm = Qm , m = 1, . . .N .

I The associated systems are models of supersymmetric quantummechanics (SQM) (Witten, 1981, 1982), with H as the relevantHamiltonian. The SQM models have a lot of applications in variousphysical and mathematical domains.

I We will deal with the simplest non-trivial N = 2, d = 1 supersymmetry

Q =1√2

(Q1 + iQ2) , Q =1√2

(Q1 − iQ2) ,

{Q, Q} = 2H , Q2 = Q2 = 0, [H,Q] = [H, Q] = 0 .It is also instructive to add the commutators with the generator J of thegroup O(2) ∼ U(1) which is the automorphism group of the N = 2superalgebra:

[J,Q] = Q , [J, Q] = −Q , [H, J] = 0 .

d = 1 SuperspaceI N = 2, d = 1 superspace:

M(1|2) = (t , θ, θ) , δθ = ε , δθ = ε , δt = i(εθ + εθ) .

I N= 2 covariant spinor derivatives:

D = ∂θ − i θ∂t , D = −∂θ + iθ∂t , {D, D} = 2i∂t .

I The simplest superfield is the real one, Φ(t , θ, θ),

Φ′(t ′, θ′, θ′) = Φ(t , θ, θ) ⇒ δΦ = −δt∂t Φ− ε∂θΦ− ε∂θΦ .

I On the component fields appearing in the θ expansion of Φ,

Φ(t , θ, θ) = x(t) + θψ(t)− θψ(t) + θθy(t) ,

N= 2 supersymmetry acts as

δx = εψ − εψ , δψ = ε(i x − y) , δψ = −ε(i x + y) , δy = i(εψ + ε ˙ψ) .

I The superfield Φ(t , θ, θ) encompasses the irreducible N = 2, d = 1multiplet (1, 2, 1) . Other N = 2, d = 1 multiplets exist as well, e.g.,(2, 2, 0) .

I The simplest invariant superfield action containing interaction reads

S(N=2) =

∫dtd2θ

[DΦ DΦ + W (Φ)

].

Here W (Φ) is the superpotential. After integrating over Grassmanncoordinates, we obtain

S(N=2) =

∫dt[

x2 − i(

˙ψψ − ψψ)

+ y2 + y∂x W (x) + (ψψ)∂2x W (x)

].

I The next step is to eliminate the auxiliary field y by its algebraicequation of motion

y = −12∂x W .

The on-shell action is

S(N=2) =

∫dt[

x2 − i(

˙ψψ − ψψ)− 1

4(∂x W )2 + (ψψ)∂2

x W (x)

].

I The action is invariant under the transformations

δx = εψ − εψ , δψ = ε(i x +12∂x W ) , δψ = −ε(i x − 1

2∂x W ) .

Hamiltonian formalism and quantizationI The quantum Hamiltonian obtained in a standard way from the

canonical one reads

H =14

[p2 +

(dWdx

)2]− 1

2d2Wdx2

(ψ ˆψ − ˆψψ

),

where we have Weyl-ordered the fermionic term. The superchargescalculated by the Noether procedure and then brought into the quantumform through passing to the operators are

Q = ψ

(p + i

dWdx

), Q = ˆψ

(p − i

dWdx

).

I The algebra of the basic quantum operators is

[x , p] = i , {ψ, ˆψ} = 12 .

Using it, we can calculate the anticommutators of the quantumsupercharges and check that they form N = 2, d = 1 superalgebra

{Q, Q} = 2H , {Q,Q} = {Q, Q} = 0 .

I By the graded Jacobi identities:

[Q,H] = [Q,H] = 0 .

Spectral problem: double degeneracyI We use the standard realization for p, p = 1

i∂∂x , and the Pauli-matrix

realization for the fermionic operators

ψ =1

2√

2(σ1 + iσ2) , ˆψ =

12√

2(σ1 − iσ2) , ψ ˆψ − ˆψψ = 1

2 σ3 .

Then the Hamiltonian and supercharges become 2× 2 matrices

H =14

[−∂x

2 + (Wx )2]( 1 0

0 1

)− 1

4Wxx

(1 00 −1

)Q = − i√

2

(0 10 0

)(∂x −Wx ) , Q = − i√

2

(0 01 0

)(∂x + Wx ) .

I Thus the wave functions form a doublet and, taking into account theconditions [Q,H] = [Q,H] = 0, the relevant matrix spectral problem is

H(

ψ+

ψ−

)= λ

(ψ+

ψ−

).

I It is equivalent to the two ordinary problems

H± = λψ± , H± = −14

(∂x ∓Wx )(∂x ±Wx ).

I Using the intertwining property

H−(∂x + Wx ) = (∂x + Wx )H+ , H+(∂x −Wx ) = (∂x −Wx )H− ,

now it easy to show that the states

Q(

ψ+

ψ−

)=

(−i(∂x −Wx )ψ−

0

), Q

(ψ+

ψ−

)=

(0

−i(∂x + Wx )ψ+

)are the eigenfunctions of H+ and H− with the same eigenvalue λ. Thuswe observe the double degeneracy of the spectrum. This doubledegeneracy is the most chracteristic feature of the N = 2supersymmetry in d = 1 (and of any higher N supersymmetry).

I In general, the Hilbert space of quantum states of N = 2 SQM isdivided into the following three sectors

(a) Ground state : QΨ0 = QΨ0 = HΨ0 = 0 ,

(b) HΨ1 = EΨ1 , QΨ1 6= 0 , QΨ1 = 0 ,

(c) HΨ2 = EΨ2 , QΨ2 6= 0 , QΨ2 = 0 .

I One can argue that many QM models with the double degeneracy ofthe energy spectrum can be identified with some N = 2 SQM model.

Summary

I Supersymmetry between fermions and bosons is a new uinusualconcept in the mathematical physics which allowed to construct a lot ofnew theories with remarkable and surprising features: supergravities,superstrings, superbranes, N = 4 super Yang-Mills theory (the firstexample of the ultraviolet-finite quantum field theory), etc. It alsoallowed to establish unexpected relations between these theories, e.g.,the AdS/CFT (or “gravity/gauge”) correspondence, AGTcorrespondence, etc.

I It predicts new particles (superpartners) which still await theirexperimental discovery.

I The natural approach to supersymmetric theories is the superfieldmethods.

L. Mezincescu, V.I. Ogievetsky, Symmetry between bosonsand fermions and superfields, Usp. Fiz. Nauk, 117 (1975)937.

J. Wess, J. Bagger, Supersymmetry - Supergravity,

P. West, Introduction to Supersymmetry and Supergravity,World Scientific, 1990.

J. Gates et al, Superspace, or 1001 Lessons inSupersymmetry, Benjamin/Cummings, Reading, MA, 1983.

I. Buchbinder, S. Kuzenko, A walk through superspace,Bristol, 1998, 656 p.

A. Galperin, E.I., V. Ogievetsky, E. Sokatchev, Harmonicsuperspace, CUP, 2001.

THANK YOU FOR YOUR ATTENTION!