4 Superspace and Superfields

4 Superspace and superﬁelds

The usual space-time Lagrangian formulation is not the most convenient one for describing supersymmetric field theories. This is because in ordinary space-time supersymmetry is not manifest. In fact, an extension of ordinary space-time, known as superspace,happenstobethebestandmostnaturalframeworkinwhichto formulate supersymmetric theories. Basically, the idea of (N =1)superspaceisto µ enlarge the space-time labelled with coordinates x ,associatedtothegeneratorsPµ, by adding 2 + 2 anti-commuting Grassman coordinates ✓↵, ✓¯↵˙ ,associatedtothesu- persymmetry generators Q↵, Q¯↵˙ ,andobtainaeightcoordinatesuperspacelabelled µ by (x ,✓↵, ✓¯↵˙ ). In such apparently exotic space many mysterious (or hidden) properties of supersymmetric field theories become manifest. As we will see, at the price of learning a few mathematical new ingredients, the goal of constructing supersymmetric field theories will be gained much easily, and within a framework where many classical and quantum properties of supersymmetry will be more transparent. In this lecture we will introduce superspace and superfields. In subsequent lec- tures we will use this formalism to construct supersymmetric field theories and study their dynamics.

4.1 Superspace as a coset

Let us start recalling the relation between ordinary Minkowski space and the Poincar´e group. Minkowski space is a four-dimensional coset space deﬁned as

ISO(1, 3) = , (4.1) M1,3 SO(1, 3) where ISO(1, 3) is the Poincarégroup and SO(1, 3) the Lorentz group. The Poincaré group ISO(1, 3) is nothing but the isometry group of this coset space, which means that any point of can be reached from the origin with a Poincarétrans- M1,3 O formation. This transformation, however, is defined up to Lorentz transformations. Therefore, each coset class ( apointinspace-time)hasaunique representative ⌘ which is a translation and can be parametrized by a coordinate xµ

µ xµ e(x Pµ) . (4.2) ! Superspace can be defined along similar lines. The first thing we need to do is to extend the Poincarégroup to the so-called superPoincarégroup. In order to do this,

1 given that a group is the exponent of the algebra, we have to re-write the whole supersymmetry algebra in terms of commutators, namely as a Lie algebra. This is easily achieved by introducing a set of constant Grassmann numbers ✓↵, ✓¯↵˙ which anti-commute with everything fermionic and commute with everything bosonic

↵ ↵ ✓ ,✓ =0 , ✓¯ , ✓¯˙ =0 , ✓ , ✓¯˙ =0. (4.3) { } { ↵˙ } { } This allows to transform anti-commutators of the supersymmetry algebra into commutators, and get

µ ✓Q, ✓¯Q¯ =2✓ ✓¯Pµ , [✓Q, ✓Q]= ✓¯Q,¯ ✓¯Q¯ =0, (4.4) ⇥ ⇤ ⇥ ⇤ where as usual ✓Q ✓↵Q , ✓¯Q¯ ✓¯ Q¯↵˙ .Thisway,onecanwritethesupersymmetry ⌘ ↵ ⌘ ↵˙ algebra solely in terms of commutators. Exponentiating this Lie algebra one gets the superPoincarégroup. A generic group element can then be written as 1 G(x, ✓, ✓¯,!)=exp(ixP + i✓Q + i✓¯Q¯ + i!M) , (4.5) 2 µ where xP is a shorthand notation for x Pµ and !M ashorthandnotationfor µ⌫ ! Mµ⌫. The superPoincarégroup, mathematically, is Osp(4 1). Let us open a brief paren- | thesis and explain such notation. Let us define the graded Lie algebra Osp(2p N) | as the grade one Lie algebra L = L0 L1 whose generic element can be written as amatrixofcomplexdimension(2p + N) (2p + N) ⇥ AB (4.6) CD! where A is a (2p 2p)matrix,B a(2p N)matrix,C a(N 2p)matrixandD a ⇥ ⇥ ⇥ (N N) matrix. An element of L0 respectively L1 has entries ⇥ A 0 0 B respectively (4.7) 0 D! C 0 ! where

T A ⌦(2p) +⌦(2p)A =0

T D ⌦(N) +⌦(N)D =0

T C =⌦(N)B ⌦(2p)

2 and 2 T T ⌦(2p) = I , ⌦(N) =⌦(N) , ⌦(2p) = ⌦(2p) . (4.8) This implies that the matrices A span a Sp(2p, C)algebraandthematricesD a O(N,C)algebra.Thereforewehavethat

L0 = Sp(2p) O(N) , (4.9) ⌦ hence the name Osp(2p N)forthewholesuperalgebra.Agenericelementofthe | superalgebra has the form a l Q = q ta + q tl , (4.10) where ta L0 and tl L1 are a basis of the corresponding vector spaces, and we 2 2 a l have introduced complex numbers q for L0 and Grassman numbers q for L1 (recall why and how we introduced the fermionic parameters ✓↵, ✓¯↵˙ before). Taking now p =2wehavethealgebraOsp(4 N). This is not yet what we are | after, though. The last step, which we do not describe in detail here, amounts to take the so-called Inonu-Wigner contraction. Essentially, one has to rescale (almost) all generators by a constant 1/e¯, rewrite the algebra in terms of the rescaled generators and take the limite ¯ 0. What one ends up with is the N-extended supersymmetry ! algebra in Minkowski space we all know, dubbed Osp(4 N), where in the limit one | gets the identiﬁcation

A P ,M D ZIJ B,C Q , Q¯ . (4.11) ! µ µ⌫ ! ! I I while all other generators vanish. Taking N = 1 one finally gets the unextended supersymmetry algebra Osp(4 1). | Given the generic group element of the superPoincarégroup (4.5), the N =1 superspace is defined as the (4+4 dimensional) group coset

Osp(4 1) 4 1 = | (4.12) M | SO(1, 3) where, as in eq. (4.1), by some abuse of notation, both factors above refer to the groups and not the algebras. A point in superspace (point in a loose sense, of course, given the non-commutative nature of the Grassman parameters ✓↵, ✓¯↵˙ ) gets identiﬁed with the coset representative corresponding to a so-called super-translation through the one-to-one map

µ ¯ ¯ xµ,✓ , ✓¯ e(x Pµ) e(✓Q+✓Q) . (4.13) ↵ ↵˙ ! 3 The 2+2 anti-commuting Grassman numbers ✓↵, ✓¯↵˙ can then be thought of as coordinates in superspace (in four-component notation they correspond to a Marojana spinor ✓). For these Grassman numbers all usual spinor identities hold. Thus far we have introduced what is known as N =1superspace.Ifdiscussing extended supersymmetry one should introduce, in principle, a larger superspace. There exist (two, at least) formulations of N = 2 superspace. However, these formulations present some subtleties and problems whose discussion is beyond the scope of this course. And no formulation is known of N =4superspace.Inthis course we will use N = 1 superspace even when discussing extended supersymmetry, as it is typically done in most of the literature.

4.2 Superﬁelds as ﬁelds in superspace

Superfields are nothing but fields in superspace: functions of the superspace co- µ ordinates (x ,✓↵, ✓¯↵˙ ). Since ✓↵, ✓¯↵˙ anticommute, any product involving more than two ✓’s or two ✓¯’s vanishes: given that ✓ ✓ = ✓ ✓ ,wehavethat✓ ✓ =0for ↵ ↵ ↵ ↵ = and therefore ✓↵✓✓ = 0, since at least two indices in this product are the same. Hence, the most general superfield Y = Y (x, ✓, ✓¯)hasthefollowingsimple Taylor-like expansion

Y (x, ✓, ✓¯)=f(x)+✓ (x)+✓¯¯(x)+✓✓ m(x)+✓¯✓¯ n(x)+

µ + ✓ ✓¯ vµ(x)+✓✓ ✓¯¯(x)+✓¯✓✓⇢¯ (x)+✓✓ ✓¯✓¯ d(x) . (4.14)

Each entry above is a field (possibly with some non-trivial tensor structure). In this sense, a superfield it is nothing but a finite collection (a multiplet) of ordinary fields. We aim at constructing supersymmetric Lagrangians out of superfields. In such Lagrangians superfields get multiplied by each other, sometime we should act on them with derivatives, etc... Moreover, integration in superspace will be needed, eventually. Therefore, it is necessary to pause a bit and recall how operations of this kind work for Grassman variables. Derivation in superspace is defined as follows

@ ↵ ↵ @ ↵˙ ↵˙ ˙ @ and @ = ✏ @ , @¯ and @¯ = ✏ @¯˙ , (4.15) ↵ ⌘ @✓↵ ↵˙ ⌘ @✓¯↵˙ where ¯ ¯˙ ˙ ¯ ¯↵˙ @↵✓ = ↵ , @↵˙ ✓ = ↵˙ ,@↵✓˙ =0 , @ ✓ =0. (4.16)

4 ¯ ¯ For a Grassman variable ✓ (either ✓1,✓2, ✓1˙ or ✓2˙ in our case), integration is deﬁned as follows d✓ =0 d✓ ✓ =1. (4.17) Z Z This implies that for a generic function f(✓)=f0 + ✓f1,thefollowingresultshold

d✓f(✓)=f , d✓ (✓)f(✓)=f = @,✓= (✓) . (4.18) 1 0 ! Z Z Z These relations can be easily generalized to N = 1 superspace, provided 1 1 d2✓ d✓1d✓2 ,d2✓¯ d✓¯2˙ d✓¯1˙ . (4.19) ⌘ 2 ⌘ 2 With these deﬁnitions one can prove the following useful identities

d2✓✓✓= d2✓¯ ✓¯✓¯ =1, d2✓d2✓✓✓¯ ✓¯✓¯ =1 Z Z Z 2 1 ↵ 2 1 ↵˙ ˙ d ✓ = ✏ @ @ , d ✓¯ = ✏ @¯ @¯˙ . (4.20) 4 ↵ 4 ↵˙ Z Z Another crucial question we need to answer is: how does a superfield transform under supersymmetry transformations? As it is the case for all operators of the Poincaré algebra (translations, rotations and boosts), we want to realize the supersymmetry generators Q↵, Q¯↵˙ as di↵erential operators. In order to make this point clear, we will use momentarily calligraphic letters for the abstract operator and latin ones for the representation of the same operator as a di↵erential operator in field space. Let us first recall how the story goes in ordinary space-time and consider a translation, generated by with infinitesimal parameter aµ,onafield(x). This Pµ is defined as

ia ia µ (x + a)=e P (x)e P = (x) ia [ ,(x)] + ... . (4.21) Pµ On the other hand, Taylor expanding the left hand side we get

µ (x + a)=(x)+a @µ(x)+... (4.22)

Equating the two right hand sides we then get

[(x), ]= i@ (x) P (x) , (4.23) Pµ µ ⌘ µ where is the generator of translations and P is its representation as a di↵erential Pµ µ operator in ﬁeld space (recall that @µ is an operator and from (@µ)⇤ = @µ one gets

5 that (@ )† = @ ;henceP is indeed hermitian). Therefore, a translation of a ﬁeld µ µ µ by parameter aµ induces a change on the ﬁeld itself as

= (x + a) (x)=iaµP . (4.24) a µ Notice that here and below we are using right multiplication, when acting on fields. We now want to apply the same procedure to a superfield. A translation in superspace (i.e. a supersymmetry transformation) on a superfield Y (x, ✓, ✓¯)bya quantity (✏↵, ✏¯↵˙ ), where ✏↵, ✏¯↵˙ are spinorial parameters, is defined as

i ✏ +¯✏ ¯ i ✏ +¯✏ ¯ Y (x + x,✓ + ✓,✓¯ + ✓¯)=e ( Q Q)Y (x, ✓, ✓¯)e ( Q Q) , (4.25) with Y (x, ✓, ✓¯) Y (x + x, ✓ + ✓,✓¯ + ✓¯) Y (x, ✓, ✓¯)(4.26) ✏,✏¯ ⌘ the variation of the superﬁeld under the supersymmetry transformation. What is the explicit expression for x, ✓,✓¯? Why are we supposing here x = 6 0, given we are not acting with the generator of space-time translations ,but Pµ just with supersymmetry generators? What is the representation of and ¯ as Q Q di↵erential operators? In order to answer these questions we should ﬁrst recall the Baker-Campbell- Hausdor↵formula for non-commuting objects which says that

1 1 eAeB = eC where C = C (A, B)(4.27) n! n n=1 X with 1 1 C = A + B,C=[A, B] ,C= [A, [A, B]] [B,[B,A]] ... . (4.28) 1 2 3 2 2 Eq. (4.25) can be written as

i ✏ +¯✏ ¯ i x +✓ +✓¯ ¯ i x +✓ +✓¯ ¯ i ✏ +¯✏ ¯ Y (x + x,✓ + ✓,✓¯ + ✓¯)=e ( Q Q)e ( P Q Q)Y (0; 0, 0)e ( P Q Q)e ( Q Q) (4.29) Let us now evaluate the last two exponentials. We have

exp i x + ✓ + ✓¯ ¯ exp i ✏ +¯✏ ¯ = { P Q Q } { Q Q } 1 1 =exp ixµ + i(✏ + ✓) + i(¯✏ + ✓¯) ¯ ✓¯ ¯,✏ ✓ , ✏¯ ¯ { Pµ Q Q2 Q Q 2 Q Q } =exp ixµ + i(✏ + ✓) + i(¯✏ + ✓¯) ¯ + ✏⇥µ✓¯ ⇤✓µ✏¯ ⇥ ⇤ { Pµ Q Q Pµ Pµ} =exp i(xµ + i✓µ✏¯ i✏µ✓¯) + i(✏ + ✓) + i(¯✏ + ✓¯) ¯ (4.30) { Pµ Q Q} 6 which means that xµ = i✓µ✏¯ i✏µ✓¯ ↵ ↵ 8✓ = ✏ (4.31) <>✓¯↵˙ =¯✏↵˙

This answers the ﬁrst question.:> Notice the ✏, ✏¯-depending piece in xµ.Thisisneededtobeconsistentwiththe supersymmetry algebra, Q , Q¯ P :twosubsequentsupersymmetrytransfor- ↵ ↵˙ ⇠ µ mations generate a space-time translation. This answers the second question. We can now address the third question and look for the representation of the supersymmetry generators and ¯ as di↵erential operators. To see this, let us Q↵ Q↵˙ consider eq. (4.26) and, recalling eqs. (4.31), let us ﬁrst Taylor expand the right hand side which becomes

Y (x, ✓, ✓¯)=Y (x, ✓, ✓¯)+i ✓µ✏¯ ✏µ✓¯ @ Y (x, ✓, ✓¯)+ ✏,✏¯ µ + ✏↵@ Y (x, ✓, ✓¯)+¯ ✏↵˙ @¯ Y (x, ✓, ✓¯)+ Y (x, ✓, ✓¯) ↵ ↵˙ ··· = ✏↵@ +¯✏↵˙ @¯ + i ✓µ✏¯ ✏µ✓¯ @ + ... Y (x, ✓, ✓¯) (4.32) ↵ ↵˙ µ ⇥ ⇤ On the other hand, from eq. (4.25) we get

Y (x, ✓, ✓¯)= 1 i✏ i✏¯ ¯ + ... Y (x, ✓, ✓¯) 1+i✏ + i✏¯ ¯ + ... Y (x, ✓, ✓¯) ✏,✏¯ Q Q Q Q = i✏↵ ,Y(x, ✓, ✓¯) + i✏¯↵˙ ¯ ,Y(x, ✓, ✓¯) + ... , (4.33) Q↵ Q↵˙ ⇥ ⇤ ⇥ ⇤ (recall that i✏¯Q¯ i✏¯ Q¯↵˙ = i✏¯↵˙ Q¯ ). Deﬁning ⌘ ↵˙ ↵˙ [Y, ] Q Y, Y, ¯ Q¯ Y, (4.34) Q↵ ⌘ ↵ Q↵˙ ⌘ ↵˙ ⇥ ⇤ the previous result implies that the supersymmetry variation of a superﬁeld by parameters ✏, ✏¯ is represented as

✏,✏¯Y = i✏Q + i✏¯Q¯ Y. (4.35) Comparing with eq. (4.32) we get the following expression for the di↵erential operators Q↵, Q¯↵˙ µ ˙ Q↵ = i@↵ ✓¯ @µ ↵˙ (4.36) ¯ ¯ µ ( Q↵˙ =+i@↵˙ + ✓ ↵˙ @µ Notice that, consistently, Q = Q¯ (recall that (µ ) = µ ). ↵† ↵˙ ↵˙ † ↵˙

7 One can check the validity of the expressions (4.36) by showing that the two di↵erential operators close the supersymmetry algebra, namely that

µ Q ,Q = Q¯ , Q¯ ˙ =0, Q , Q¯ ˙ =2 P . (4.37) { ↵ } { ↵˙ } { ↵ } ↵˙ µ To close this section we can now give a more precise definition of a superfield: a superfield is a field in superspace which transforms under a super-translation accord- ing to eq. (4.25). This implies, in particular, that a product of superfields is still a superfield.

4.3 Supersymmetric invariant actions - general philosophy

Having seen that a supersymmetry transformation is simply a translation in superspace, it is now easy to construct supersymmetric invariant actions. In order for an action to be invariant under superPoincarétransformations it is enough that the Lagrangian is Poincaréinvariant (actually, it should transform as a scalar density) and that its supersymmetry variation is a total space-time derivative. Here is where the formalism we have introduced starts to manifest its powerful- ness. The basic point is that the integral in superspace of any arbitrary superfield is a supersymmetric invariant quantity. In other words, the following integral

d4xd2✓d2✓¯ Y(x, ✓, ✓¯)(4.38) Z is manifestly supersymmetric invariant, if Y is a superﬁeld. This can be proven as follows. The integration measure is translationally invariant by construction since

d✓✓ = d(✓ + ⇠)(✓ + ⇠)=1 (4.39) Z Z This implies that

4 2 2 4 2 2 ✏,✏¯ d xd ✓d ✓¯ Y(x, ✓, ✓¯)= d xd✓d✓¯ ✏,✏¯Y (x, ✓, ✓¯) . (4.40) Z Z Now, using eqs. (4.35) and (4.36) we get

Y = ✏↵@ Y +¯✏ @¯↵˙ Y + @ i ✏✓¯ ✓✏¯ Y . (4.41) ✏,✏¯ ↵ ↵˙ µ Integration in d2✓d2✓¯ kills the ﬁrst two terms⇥ since they do not ⇤ have enough ✓’s or ✓¯’s to compensate for the measure, and leaves only the last term, which is a total derivative. In other words, under supersymmetry transformations the integrand in

8 eq. (4.40) transforms as a total space-time derivative plus terms which get killed by integration in superspace. Hence the full integral is supersymmetric invariant

4 2 2 ✏,✏¯ d xd ✓d ✓¯ Y(x, ✓, ✓¯)=0. (4.42) Z Supersymmetric invariant actions are constructed this way, i.e. by integrating in superspace a suitably defined superfield. Such superfield, call it ,shouldnotbe A generic, of course. It should have the right structure to give rise, upon integration on Grassman coordinates, to a Lagrangian density, which is a real, dimension-four operator, transforming as a scalar density under Poincarétransformations. The end result will be a supersymmetric invariant action S = d4xd2✓d2✓¯ (x; ✓, ✓¯)= d4x ((x), (x),A (x),...) . (4.43) S A L µ Z Z Let us emphasize again: one does not need to prove to be invariant under su- S persymmetry transformations. If it comes from an integral of a superfield in superspace, this is just automatic: by construction, the Lagrangian on the r.h.s. of L eq. (4.43), an apparently innocent-looking function of ordinary fields, is guaranteed to be Poincaréand supersymmetric invariant, up to total space-time derivatives. The superfield will be in general a product of superfields (recall that a product A of superfields is still a superfield). However, the general superfield (4.14) cannot be the basic object of this construction: it contains too many field components to correspond to an irreducible representation of the supersymmetry algebra. We have to put (supersymmetric invariant) constraints on Y and restrict its form to contain only a subset of fields. Being the constraint supersymmetric invariant this reduced set of fields will still be a superfield, and hence will carry a representation of the supersymmetry algebra. In what follows, we will start discussing two such constraints, the so-called chiral and real constraints. These will be the relevant ones for our purposes, as they will lead to chiral and vector superfields, the right superfields to accommodate matter and radiation, respectively.

4.4 Chiral superﬁelds

One can construct covariant derivatives D↵, D¯ ↵˙ deﬁned as

µ ˙ D↵ = @↵ + i ✓¯ @µ ↵˙ (4.44) ¯ ¯ µ ( D↵˙ = @↵˙ + i✓ ↵˙ @µ

9 and which anticommute with the supersymmetry generators Q↵, Q¯↵˙ .Moreprecisely we have

µ µ D , D¯ ˙ =2i @ = 2 P , (4.45) { ↵ } ↵˙ µ ↵˙ µ

D ,D or Q or Q¯ ˙ =0 (similarlyforD¯ ) . (4.46) { ↵ } ↵˙ This implies that

✏,✏¯ (D↵Y )=D↵ (✏,✏¯Y ) . (4.47) Therefore, if Y is a superﬁeld, that is a ﬁeld in superspace transforming as dictated by eq. (4.25) under a supersymmetry transformation, so is D↵Y .Thismeansthat

D↵Y = 0 is a supersymmetric invariant constraint we can impose on a superﬁeld Y to reduce the number of its components, while still having the ﬁeld carrying a representation of the supersymmetry algebra (the same holds for the constraint

D¯ ↵˙ Y =0).

Recall the generic expression (4.14) for Y and consider @¯↵˙ Y :thishasfewer components with respect to Y itself, since, for instance, there is no ✓✓✓¯✓¯ term. However ¯ µ @↵˙ ,✏Q = ✏ ↵˙ @µ . (4.48)

This implies that a supersymmetry⇥ transformation⇤ on @¯↵˙ Y would generate a ✓✓ ✓¯✓¯ term. Hence @¯↵˙ Y is not a true superfield in the sense of providing a basis for arepresentationofsupersymmetry.Ontheotherhand,thecovariantderivatives defined in (4.44) anticommute with Q and Q¯. Hence, if Y is a superfield, D↵Y,D¯ ↵˙ Y are also superfields (and so is @µY ,sincePµ commutes with Q and Q¯). A chiral superfield is a superfield such that

D¯ ↵˙ =0. (4.49)

Seemingly, an anti-chiral superﬁeld is a superﬁeld such that

D↵ =0. (4.50)

Notice that if is chiral, ¯ isanti-chiral.Thisimpliesthatachiralsuperfield cannot be real (i.e. ¯ =).Indeed,inthiscaseitiseasytoshowthatitshouldbe aconstant.Takingthehermitianconjugateofeq.(4.49)onewouldconcludethat the field would also be anti-chiral. Acting now on it with the anticommutator in eq.(4.45) one would get @µ= 0. This is the superfield analogue of what we have seen in the previous lecture, when we constructed the chiral multiplet.

10 We would like to find the most general expression for a chiral superfield in terms of ordinary fields, as we did for the general superfield (4.14). To this aim, it is useful to define new coordinates

yµ = xµ + i✓µ✓¯ , y¯µ = xµ i✓µ✓¯ . (4.51) It easily follows that

¯ ¯ µ ¯ µ D↵˙ ✓ = D↵˙ y =0 ,D↵✓˙ = D↵y¯ =0. (4.52)

µ Recalling the deﬁnition (4.49) this implies that depends only on (y ,✓↵)explicitly, µ but not on ✓¯↵˙ (the ✓¯-dependence is hidden inside y ). In this (super)coordinate system the chiral constraint is easily solved by

(y, ✓)=(y)+p2✓ (y) ✓✓F(y) . (4.53) Taylor-expanding the above expression around x we get for the actual (x, ✓, ✓¯)

¯ µ ¯ i µ ¯ 1 ¯¯ (x, ✓, ✓)=(x)+p2✓ (x)+i✓ ✓@µ(x) ✓✓F(x) ✓✓@µ (x) ✓ ✓✓✓✓⇤(x) , p2 4 (4.54) µ ¯ which can also be conveniently recast as (x, ✓, ✓¯)=ei✓ @µ✓(x, ✓). We see that, as expected, this superfield has less components than the general superfield Y ,and some of them are related to each other. The chiral superfield (4.54) is worth its name, since it is a superfield which encodes precisely the degrees of freedom of the chiral multiplet of fields we have previously constructed. On-shell, it corresponds to a N =1multipletofstates, hence carrying an irreducible representation of the N =1supersymmetryalgebra. Asimilarstoryholdsforananti-chiralsuperfield¯ forwhichwewouldget

¯(x, ✓, ✓¯)=¯(¯y)+p2✓¯ ¯(¯y) ✓¯✓¯F¯(¯y) (4.55) ¯ ¯¯ µ ¯ ¯ ¯¯ ¯ i ¯¯ µ ¯ 1 ¯¯ ¯ = (x)+p2✓ (x) i✓ ✓@µ(x) ✓✓F (x)+ ✓✓✓ @µ (x) ✓✓✓✓⇤(x) . p2 4 Let us now try and see how does a chiral (or anti-chiral) superﬁeld transform under supersymmetry transformations. This amounts to compute

✏,✏¯(y; ✓)= i✏Q + i✏¯Q¯ (y; ✓)(4.56)

(and similarly for ¯). To compute eq. (4.56) it is convenient to write the di↵erential µ operators Q↵, Q¯↵˙ in the (y ,✓↵, ✓¯↵˙ )coordinatesystem.Thisamountstotradethe

11 µ partial derivatives taken with respect to (x ,✓↵, ✓¯↵˙ )forthosetakenwithrespectto µ the new system (y ,✓↵, ✓¯↵˙ )andplugthisintoeqs.(4.36).Theﬁnalresultreads

Qnew = i@ ↵ ↵ (4.57) ¯new ¯ ↵ µ @ (Q↵˙ = i@↵˙ +2✓ ↵↵˙ @yµ

Plugging these expressions into eq. (4.56) one gets

˙ @ (y; ✓)= ✏↵@ +2i✓↵µ ✏¯ (y; ✓) ✏,✏¯ ↵ ↵˙ @yµ ✓ ◆ @ @ = p2✏ 2✏✓F +2i✓µ✏¯ + p2✓ (4.58) @yµ @yµ ✓ ◆ @ @ = p2✏ + p2✓ p2✏F + p2iµ✏¯ ✓✓ ip2¯✏¯µ . @yµ @yµ ✓ ◆ ✓ ◆ Therefore, the final expression for the supersymmetry variation of the di↵erent field components of the chiral superfield reads

= p2✏ µ 8 ↵ = p2i( ✏¯)↵@µ p2✏↵F (4.59) µ <> F = ip2@µ ✏¯

It is left to the reader to:> derive the corresponding expressions for an anti-chiral superﬁeld. In this case, one should write the generators Q↵, Q¯↵˙ as functions of µ (¯y ,✓↵, ✓¯↵˙ ).

4.5 Real (aka vector) superﬁelds

In order to have gauge interactions we clearly need to find some new supersymmetric invariant projection which saves the vector field vµ in the general expression (4.14) and makes it real (this was not the case for the chiral projection, for which the vector component is @ ). The right thing to do is to impose a reality condition on the ⇠ µ general superfield Y . Indeed, under hermitian conjugation, Y Y¯ ,onehasthat ! vµ v¯µ;soimposingarealitycondition,notonlythevectorcomponentsurvives ! as a degrees of freedom, but becomes real. Areal(akavector)superfieldV is a superfield such that

V = V.¯ (4.60)

12 Looking at the general expression (4.14) this leads to the following expansion for V i V (x, ✓, ✓¯)=C(x)+i✓(x) i ✓¯¯(x)+✓µ✓¯v + ✓✓ (M(x)+iN(x)) µ 2 i i ✓¯✓¯(M(x) iN(x)) + i✓✓✓¯ ¯(x)+ ¯µ@ (x) (4.61) 2 2 µ ✓ ◆ i 1 1 i ✓¯✓✓¯ (x)+ µ@ ¯(x) + ✓✓✓¯✓¯ D(x) @2C(x) . 2 µ 2 2 ✓ ◆ ✓ ◆ Notice that, as such, this superﬁeld has 8B +8F degrees of freedom. The next step is to introduce the supersymmetric version of gauge transformations. As we shall see, after gauge ﬁxing, this will reduce the number of o↵-shell degrees of freedom to

4B +4F ,whichbecome2B +2F on-shell (for a massless representation), as it should be the case for a massless vector multiplet of states. First notice that + ¯ isavectorsuperfield,ifisachiralsuperfield.Second, notice that under V V ++¯ (4.62) ! the vector v in V transforms as v v @ (2 Im). This is precisely how an µ µ ! µ µ ordinary (abelian) gauge transformation acts on a vector field. Therefore, eq. (4.62) is a natural definition for the supersymmetric version of a gauge transformation. Under eq. (4.62) the component fields of V transform as

C C +2Re ! ip2 8 ! >M M 2ImF > > ! >N N +2ReF (4.63) > ! <>D D ! > ! > µ µ >v v 2 @µIm > ! > where the components of have:> been dubbed (, ,F ). From the transformations above one sees that properly choosing , namely choosing C i N M Re = , = , ReF = , ImF = . (4.64) 2 p2 2 2 one can gauge away (namely put to zero) C,M,N,.Thechoiceaboveiscalled Wess-Zumino gauge. In this gauge a vector superfield can be written as 1 V = ✓µ✓¯v (x)+i✓✓ ✓¯¯(x) i✓¯✓✓¯ (x)+ ✓✓ ✓¯✓¯D(x) . (4.65) WZ µ 2 13 Therefore, taking into account gauge invariance (that is the redundancy of one of the vector degrees of freedom, the one associated to the transformation v µ ! v @ (2 Im)), we end-up with 4 +4 degrees of freedom o↵-shell. As we shall see µ µ B F later, D will turn out to be an auxiliary field; therefore, by imposing the equations of µ motion for D,thespinor and the vector v ,onewillendupwith2B +2F degrees of freedom on-shell. Since we like to formulate gauge theories keeping gauge invariance manifest o↵-shell, the WZ gauge is defined as a gauge where C = M = N = =0, but no restrictions on vµ. This way, while remaining in the WZ gauge, we still have the freedom to do ordinary gauge transformations. In other words, once in the WZ gauge, we can still perform a supersymmetric gauge transformation (4.62) with parameters = ¯, =0,F =0. Let us end this section with two important comments. First notice that in the

WZ gauge each term in the expansion of VWZ contains at least one ✓.Therefore 1 V 2 = ✓✓✓¯✓¯v vµ ,Vn =0 n 3 . (4.66) WZ 2 µ WZ These identities will simplify things a lot when it comes to construct supersymmetric gauge actions. Second, notice that the WZ gauge is not supersymmetric. In other words, it does not commute with supersymmetry. Acting with a supersymmetry transformation on a vector superﬁeld in the WZ gauge, one obtains a new superﬁeld which is not in the WZ gauge. Hence, when working in this gauge, after a supersymmetry transformation, one has to do a compensating supersymmetric gauge transformation (4.62), with a properly chosen , to come back to the WZ gauge. We leave to the reader to check this.

4.6 (Super)Current superﬁelds

The two superfields described above are what we need to describe matter and radiation in a supersymmetric theory, if we are not interested in gravitational interactions. However, in a supersymmetric theory, also composite operators should sit in superfields. These can be, e.g. chiral superfields, but there are at least two other classes of superfields which accommodate important composite operators. They are those describing conserved currents and the supersymmetry current (supercurrent for short), respectively, the latter being ubiquitous in a supersymmetric QFT, as this is the current associated to the supersymmetry charge itself. Both these superfields turn out to be real superfields, as the superfield described in the previous

14 section, but current conservation implies extra supersymmetric invariant conditions they should satisfy which make them a particular class of real superﬁelds. In what follows, we will brieﬂy describe both of them.

4.6.1 Internal symmetry current superﬁelds

Because of Noether theorem, in a local QFT any continuous symmetry is associated µ to a conserved current jµ satisfying @ jµ =0,andtothecorrespondingconserved charge Q defined as Q = d3xj0. Here we are referring to non-R symmetries; R-symmetry will be discussedR later. As any other operator, in a supersymmetric theory a conserved current should be embedded in a superfield. It turns out that this is a real scalar superfield J satisfying the following extra constraint

D2 = D¯ 2 =0. (4.67) J J Arealsuperfieldsatisfyingtheconstraintaboveiscalledlinear superfield. Working alittlebitonecanshowthatarealsuperfieldsubjecttotheconditions(4.67)has the following component expression 1 1 1 = J(x)+i✓j(x) i✓¯¯j(x)+✓µ✓¯j (x)+ ✓2✓¯¯µ@ j(x) ✓¯2✓µ@ ¯j(x)+ ✓2✓¯2 J(x) , J µ 2 µ 2 µ 4 ⇤ (4.68) where J is a real scalar and j↵ aspinor.Byimposingeq.(4.67)ontheabove µ expression one easily sees that the current jµ satisfies @ jµ =0,i.e.isaconserved current. So the constraint (4.67) is indeed the correct supersymmetric generaliza- tion of current conservation. Note that while the condition (4.67) is compatible with supersymmetry, as it should (both D2 and D¯ 2 commute with supersymmetry transformations), it stands on a slightly di↵erent footing with respect to the conditions (4.49) and (4.60). The latter constrain the dependence of a superfield as a function of the fermionic coordinates (✓↵, ✓¯↵˙ ), but they do not say anything about space- time dependence. On the contrary, eq. (4.67) constrains the space-time dependence of some of the fields imposing di↵erential equations in x-space, one obvious exam- µ ple being the conservation equation @ jµ =0.Inthissense,(4.67)isanon-shell constraint. A few comments are in order. First notice that, as compared to a general real superfield (4.61), a linear superfield has less independent components. This is due to the extra condition (4.67) a linear superfield has to satisfy. Another comment

15 regards the spin content of .Oneconditionthat should (and does) satisfy is J J that it should not contain fields with spin higher than one. If this were the case, one could not gauge the current jµ without introducing higher-spin gauge fields, something which is expected not to be consistent in a local interacting QFT with rigid supersymmetry (recall our discussion in the previous lecture). This implies that should be a real scalar superfield, namely its lowest component J should be J a scalar. Finally, it may worth notice that the detailed structure of is not uniquely J fixed, but in fact defined up to Schwinger terms entering the current algebra. This can be understood as follows. Because the conserved charge Q is a non-R symmetry charge, it commutes with supersymmetry generators, [Q↵,Q]=0.Thisinturn implies that in the current algebra

[Q ,j ]= , (4.69) ↵ µ O↵µ the operator should be an operator which vanishes when acting with @ ,because O↵µ µ so is jµ,anditshouldalsobeatotalspace-timederivativeforµ =0,say ↵0 = ⌫ O @ A↵⌫,sothatitintegratestozero,becausesohappenstothelefthandsidesince

4 d x [Q↵,j0]= dt [Q↵,Q]=0. (4.70) Z Z An operator of this kind is known as Schwinger term. Di↵erent Schwinger terms provide di↵erent completions of the linear superfield , which is hence not univocally J defined. The superfield defined in eq. (4.68) is one possible such completions, for which = 2i( ) @⌫j .Thiscanbeeasilycheckedusingeqs.(4.34)-(4.35). O↵µ µ⌫ ↵

4.6.2 Supercurrent superﬁelds

While currents associated to internal symmetries might or might not be there, in any supersymmetric theory there always exists, by deﬁnition, a conserved current, the supersymmetry current S↵µ,associatedtotheconservationofthefermioniccharge µ Q↵,forwhich@ S↵µ =0.Intermsofthesupercurrent,thesupersymmetrycharge 3 0 is Q↵ = d xS↵ .Suchsupercurrentshouldbeembeddedinasuperﬁeld. An equationR analogous to eq. (4.69) is imposed by the supersymmetry algebra, which reads Q¯ ,S =2µ T + , (4.71) ↵˙ ↵⌫ ↵↵˙ µ⌫ O↵↵⌫˙ where T is the (conserved) energy-momentum tensor and is again a Schwinger µ⌫ O↵↵⌫˙ term. Note that now the ⌫ = 0 component of the left hand side does not integrate to

16 zero but in fact it is proportional to dt Pµ by the supersymmetry algebra, namely 4 0 to d xTµ .Thisiswhy,ontopofaSchwingerterm,theenergy-momentumtensorR appearsR on the right hand side of eq. (4.71). This also shows that the supercurrent and the energy-momentum tensor sit in the same superfield, Tµ⌫ being the highest spin field of the representation (otherwise, it would be problematic coupling supersymmetry with gravity). This is the current operators counterpart of the fact that the graviton and the gravitino sit in the same multiplet. The arbitrariness of the Schwinger term gives rise, as before, to di↵erent possible completions of the superfield. The most known such completions is due to Ferrara and Zumino. The FZ supermultiplet can be described by a pair of superfields ( ,X) Jµ satisfying the relation 2 D¯ ↵˙ µ = D X, (4.72) ↵↵˙ Jµ ↵ with being a real vector superfield, and X achiralsuperfield,D¯ X =0.The Jµ ↵˙ same comment we made on the on-shell nature of the condition (4.67) holds also in this case. From the defining equation above one can work out the component expression of these two superfields. They read

1 1 i 2 i 2 =j + ✓ S S¯ + ✓¯ S¯ + ¯ S + ✓ @ x⇤ ✓¯ @ x Jµ µ µ 3 µ µ 3 µ 2 µ 2 µ ✓ ◆ ✓ ◆ (4.73) 2 1 + ✓⌫✓¯ 2T ⌘ T + " @⇢j + ... µ⌫ 3 µ⌫ 2 µ⌫⇢ ✓ ◆ and 2 2 X = x + ✓S + ✓2 T + i@µj + ... , (4.74) 3 3 µ ✓ ◆ where ... stand for the supersymmetric completion and we have deﬁned the trace operators T T µ and S µ S¯↵˙ . All in all, the FZ superﬁeld contains a (in gen- ⌘ µ ↵ ⌘ ↵↵˙ µ eral non-conserved) R-current jµ,asymmetricandconservedTµ⌫,aconservedS↵µ, and a complex scalar x.Fromtheaboveexpressiononecanalsoseethatwhenever

X vanishes the current jµ becomes conserved and all trace operators vanish. In this case the theory is conformal and jµ becomes the always present (and conserved) superconformal R-current. We will have more to say on this issue later. For theories with an R-symmetry (be it preserved or spontaneously broken), there exists an alternative supermultiplet accommodating the energy-momentum tensor and the supercurrent, the so-called multiplet. It turns out this is again R deﬁned in terms of a pair of superﬁelds ( , )whichnowsatisfyadi↵erenton- Rµ ↵

17 shell condition

2 D¯ ↵˙ µ = , (4.75) ↵↵˙ Rµ ↵ where is a real vector superfield and achiralsuperfieldwhich,besidesD¯ = Rµ ↵ ↵˙ ↵ 0, also satisfies the identity D¯ ¯↵˙ D↵ =0.Thisimplies,inturn,that@µ =0, ↵˙ ↵ Rµ from which it follows that the lowest component of is now a conserved current, Rµ the R-current jR. The component expression of the superfields making-up the µ R multiplet reads

1 = jR + ✓S + ✓¯S¯ + ✓⌫✓¯ 2T + " (@⇢j + C⇢) + ... (4.76) Rµ µ µ µ µ⌫ 2 µ⌫⇢ ✓ ◆ and

= 2S 4T +2i (⇢¯⌧ ) C ✓ +2i✓2⌫ @ S¯↵˙ + ... (4.77) ↵ ↵ ↵ ↵ ⇢⌧ ↵↵˙ ⌫ where again ... stand for the supersymmetric completion, and Cµ⌫ is a closed two- R form. That jµ is an R-current can be easily seen noticing that the current algebra R now reads Q↵,jµ = S↵µ. Taking the time-component and integrating, this implies R that dt ⇥Q↵,Q ⇤ = dt Q↵,whichiswhatisexpectedforaR-symmetry,recall eq. (2.72).R ⇥ Notice,⇤ finally,R that when X = 0, the FZ multiplet (4.73) becomes a (special instance of an) R-multiplet. Indeed its lowest component jµ becomes now the conserved superconformal R-current. The FZ and multiplets are the more common supercurrent multiplets. How- R ever, there are instances in which a theory does not admit a R-symmetry (and hence the multiplet cannot be defined) and the FZ multiplet is not a well-defined oper- R ator, e.g. it is not gauge invariant. In these cases, one should consider yet another multiplet where the supercurrent can sit, the so-called multiplet, which is bigger S than the two above. We will not discuss the multiplet here, and refer to the S references given at the end of this lecture. On the contrary, there exist theories in which both the FZ and the multiplets can be defined. In such cases it turns out R the two are related by a so-called shift transformation defined as 1 1 3 = + ¯↵↵˙ D , D¯ U,X= D¯ 2U,= D¯ 2D U, (4.78) Rµ Jµ 4 µ ↵ ↵˙ 2 ↵ 2 ↵ ⇥ ⇤ where U is a real superfield associated to a non-conserved (and non-R) current. We will encounter examples of current and supercurrent multiplets in later lec- tures.

18 4.7 Exercises

1. Prove identities (4.20).

2. Check that the di↵erential operators Q↵ and Q¯↵˙ (4.36) close the supersymmetry algebra (4.37). Hint:recallthatall✓’s and ✓¯’s anti-commute between themselves, and that

@ @aj @ ai,aj =0 aj = aj , (4.79) { } ! @ai @ai @ai which implies that, e.g.

˙ ˙ @ , ✓¯˙ =0, @ ,✓ = , @¯ , ✓¯ = . (4.80) { ↵ } { ↵ } ↵ { ↵˙ } ↵˙

3. Check that the covariant derivatives D↵ and D¯ ↵˙ (4.44) anticommute between themselves and with the supercharge operators (4.36).

4. Compute how the ﬁeld components of an anti-chiral superﬁeld transform under supersymmetry transformations. Show that if = ¯ onegetsthe hermitian conjugate of the transformations (4.59).

5. Compute the supersymmetric variation of a vector superfield in the WZ gauge, and find the explicit form of the chiral superfield which, via a compensating gauge transformation (4.62), brings the vector superfield back to WZ gauge.

References

[1] A. Bilal, Introduction to supersymmetry, Chapter 4, arXiv:hep-th/0101055.

[2] L. Castellani, R. D’Auria and P. Fr`e, Supergravity And Superstrings: A Geomet- ric Perspective. Vol. 1: Mathematical Foundations,ChapterII.2,Singapore: World Scientiﬁc (1991).

[3] J. D. Lykken, Introduction to Supersymmetry, Chapters 2 and 3, arXiv:hep- th/9612114.

[4] Z. Komargodski and N. Seiberg, Comments on Supercurrent Multiplets, Super- symmetric Field Theories and Supergravity, section 1, arXiv:1002.2228 [hep-th].

[5] T. T. Dumitrescu and Z. Komargodski, Aspects of supersymmetry and its break- ing, sections 2 and 4, Nucl. Phys. Proc. Suppl. 216 (2011) 44.