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4 Superspace and Superfields

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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) prop- erties of supersymmetric field theories become manifest. As we will see, at the price of learning a few mathematical new ingredients, the goal of constructing supersym- metric 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 Poincar´egroup and SO(1, 3) the Lorentz group. The Poincar´e 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 Poincar´etrans- 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 Poincar´egroup to the so-called superPoincar´egroup. 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 ↵ β ↵ ✓ ,✓ =0 , ✓¯ , ✓¯˙ =0 , ✓ , ✓¯˙ =0. (4.3) { } { ↵˙ β} { β} This allows to transform anti-commutators of the supersymmetry algebra into com- mutators, 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 superPoincar´egroup. 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 superPoincar´egroup, 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 superPoincar´egroup (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 represen- tative 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 coor- dinates 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 ↵ β − β ↵ ↵ β ↵ = β and therefore ✓↵✓β✓γ = 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)+✓¯χ¯(x)+✓✓ m(x)+✓¯✓¯ n(x)+ µ + ✓ ✓¯ vµ(x)+✓✓ ✓¯λ¯(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 @ ↵ ↵ @ ↵˙ ↵˙ β˙ @ and @ = ✏ @ , @¯ and @¯ = ✏ @¯˙ , (4.15) ↵ ⌘ @✓↵ − β ↵˙ ⌘ @✓¯↵˙ − β where β β ¯ ¯β˙ β˙ ¯ ¯↵˙ β @↵✓ = δ↵ , @↵˙ ✓ = δ↵˙ ,@↵✓β˙ =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✓ δ(✓)f(✓)=f = @,✓= δ(✓) . (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 ↵˙ β˙ d ✓ = ✏ @ @ , d ✓¯ = ✏ @¯ @¯˙ . (4.20) 4 ↵ β −4 ↵˙ β 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 Poincar´e 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,onafieldφ(x). This Pµ is defined as ia ia µ φ(x + a)=e− P φ(x)e P = φ(x) ia [ ,φ(x)] + ... (4.21) − Pµ On the other hand, Taylor expanding the left hand side we get µ φ(x + a)=φ(x)+a @µφ(x)+... (4.22) Equating the two right hand sides we then get [φ(x), ]= i@ φ(x) P φ(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 δ φ = φ(x + a) φ(x)=iaµP φ. (4.24) a − µ Notice that here and below we are using right multiplication, when acting on fields.
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