12 Chapter 1. Overview• <strong>The</strong> number of bosons and fermions is equal.<strong>The</strong> proofs are easy and can be found in any of the introductions to supersymmetry listed inthe references.As always, the particles of the theory live in representations, now of the Poincaré superalgebra.To understand these representations, it’s convenient to start by looking at the representationsof the supersymmetry algebra. <strong>The</strong>y are called supersymmetry multiplets and typicallydescribe some particle and its superpartner. More about the representation theory of the supersymmetryalgebra can be found in Wess and Bagger [6, §II]; we go on to see how supersymmetrymultiplets are realized on superspace.Superfields. <strong>The</strong> next ingredient is the ‘super-’ version of fields. Superfields F(x µ , θ a ) aredefined on superspace. <strong>The</strong>y are defined via the Taylor expansion in θ a = 0, which looks likeF(x, θ) = F 0 (x) + ∑ θ a F a (x) + ∑ θ a θ b F ab (x)aa
1.2. <strong>Gauge</strong>: supersymmetry 13<strong>The</strong> definition only differs by one sign from the expression for Q a , so these new operators arealso fermionic derivations on superspace, and generate supertranslations. 6 <strong>The</strong> different signdirectly leads to{D a , D b } = −2 (γ µ ) ab P µ , {D a , Q b } = 0 . (1.25)<strong>The</strong> operators D a are called supercovariant derivatives. Since they anticommute with the Q a ,we can use them to impose further restrictions on the superfields in a way that’s compatible withsupersymmetry. Although we have only just introduced the D a , up to this point the theory iscompletely symmetric in Q a and D a . <strong>The</strong> distinction between them only arises once we chooseto use the D a to constrain the components of superfields.To get the simplest type of superfield, it’s convenient to use the Weyl basis in which thegamma matrices have the block decomposition (0.2):( )γ µ 0 σµ=¯σ µ ,0where each of the entries is a 2 × 2 matrix. Since the diagonal blocks only contain zeros, wecan read off from (1.25) that there are two pairs of anticommuting supercovariant derivatives:{D 1 , D 2 } = 0 and {D 3 , D 4 } = 0. This allows us to define two types of superfields: chiralsuperfields Φ, for which we impose D 3 Φ = D 4 Φ = 0, and antichiral superfields ¯Φ, satisfyingD 1 ¯Φ = D2 ¯Φ = 0. <strong>The</strong>se conditions relate some of the component fields with each other, cuttingdown the number of degrees of freedom of the superfields.In Chapter 3 we will see that (anti)chiral fields can be used to describe (anti)matter. Insuper qed (sqed), its component fields describe the electron and its superpartner, the selectron.Naturally, sqed should also include photons, i.e. gauge fields. In supersymmetric field theoriesthese are given by superfields which by definition satisfy a reality condition: vector superfields.In addition to a gauge field A µ they also contain a spinor (the photino).Extended supersymmetry. <strong>The</strong> ‘level’ of supersymmetry that we have discussed so farinvolves one four-component real (Majorana) spinor Q a worth of supercharges. We can extendour theory by adding more Majorana supercharges, provided we increase the number of odddirections in superspace by the same amount. A theory with N sets of supercharges Q N a thenhas N -extended supersymmetry, and the supersymmetry algebra (1.21) is modified to{Q M a , Q N b } = 2 δ MN (γ µ ) ab P µ with M, N = 1, · · ·, N .For us, only the cases N = 1 and N = 2 are relevant. In two dimensions, something specialhappens, and the N = 2 theories that we are interested in enjoy ‘(2, 2)’ supersymmetry. (<strong>The</strong>phrase ‘N = 2 supersymmetric with (2, 2) supersymmetry’ is often abbreviated by simply writing‘N = (2, 2)’.) We will get back to this in the next section and in more detail in Chapter 3.Supersymmetric vacua. To conclude our general discussion of supersymmetry we briefly saysomething about the exact methods that are available in supersymmetric theories. Supersymmetrictheories have a rich vacuum structure surviving quantum corrections, and supersymmetryimposes such a rigid structure that the behaviour of the theory in certain limits of the couplingconstants, such as the weak coupling limit, together with knowledge of the theory at somespecial values of these coupling constants, completely determines the theory for all values ofthe parameters [37]. <strong>The</strong>refore, supersymmetry allows one to use approximate methods in somespecial limits to find exact results. In Section 3.4.2 we will see an example of this principle.6 <strong>The</strong> reason that there are two kinds of differential operators generating translations in odd directions, Q aand D a, is a bit technical. <strong>The</strong> generators P µ of ordinary translations commute. This implies that the leftand right actions of P µ are both generated by P µ = −i∂ µ. Since supertranslations are fermionic, they obeyanticommutation relations. However, according to (1.21) the supercharges Q a do not anticommute with eachother. Thus, they give rise to different left and right actions, corresponding to Q a and D a respectively. (This isthe reason why we use sans symbols to distinguish the generators from their action on superfields.)
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