1 Aspects of Supersymmetry and its Breaking a b Thomas T. Dumitrescu ∗ and Zohar Komargodski † aDepartment of Physics, Princeton University, Princeton, New Jersey, 08544, USA bSchool of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, 08540, USA We describe some basic aspects of supersymmetric field theories, emphasizing the structure of various supersym- metry multiplets. In particular, we discuss supercurrents – multiplets which contain the supersymmetry current and the energy-momentum tensor – and explain how they can be used to constrain the dynamics of supersym- metric field theories, supersymmetry breaking, and supergravity. These notes are based on lectures delivered at the Carg´ese Summer School 2010 on “String Theory: Formal Developments and Applications,” and the CERN Winter School 2011 on “Supergravity, Strings, and Gauge Theory.” 1. Supersymmetric Theories In this review we will describe some basic aspects of SUSY field theories, emphasizing the structure 1.1. Supermultiplets and Superfields of various supermultiplets – especially those con- A four-dimensional theory possesses = 1 taining the supersymmetry current. In particu- supersymmetry (SUSY) if it contains Na con- 1 lar, we will show how these supercurrents can be served spin- 2 charge Qα which satisfies the anti- used to study the dynamics of supersymmetric commutation relation field theories, SUSY-breaking, and supergravity. ¯ µ Qα, Qα˙ = 2σαα˙ Pµ . (1) We begin by recalling basic facts about super- { } multiplets and superfields. A set of bosonic and Here Q¯ is the Hermitian conjugate of Q . (We α˙ α fermionic operators B(x) and F (x) fur- will use bars throughout to denote Hermitian con- i i nishes a supermultiplet{O if these} operators{O satisfy} jugation.) Unless otherwise stated, we follow the commutation relations of the schematic form conventions of [1]. In local quantum field theo- B F F ries, the basic objects of interest are well-defined Qα, i (x) j (x) + ∂ k (x) + , local operators. In SUSY field theories all such O ∼ O O · · · Q , F (x) B(x) + ∂ B(x) + , (3) operators must be embedded in multiplets of the " α Oi # ∼ Oj Ok · · · supersymmetry algebra, or supermultiplets. Con- and$ likewise%for the Q¯α˙ commutators, such that served currents furnish an important class of local the SUSY algebra (1) is satisfied. It is straightfor- operators. Of particular interest is the supersym- ward to show that a supermultiplet must contain metry current Sαµ, which satisfies equally many independent bosonic and fermionic operators (see, for instance, [2]). µ 3 0 ∂ Sαµ = 0 , Qα = d x Sα . (2) It is always possible (and very convenient) to ! B,F embed the component fields i (x) of a super- ∗T.D. is supported by a DOE Fellowship in High Energy O ¯ Theory, NSF grant PHY-0756966, and a Centennial Fel- multiplet in a superfield S(x, θ, θ). Here θα is an lowship from Princeton University. anti-commuting superspace coordinate. (For now †Z.K. is supported by DOE grant DE-FG02-90ER40542. we suppress any Lorentz indices carried by S.) Any opinions, findings, and conclusions or recommenda- The component fields are identified with the x- tions expressed in this material are those of the authors ¯ and do not necessarily reflect the views of the National dependent coefficients when S(x, θ, θ) is expanded Science Foundation. as a power series in θ, θ¯. The commutation rela- 2 tions (3) are succinctly encoded in the formula A general superfield does not furnish an irre- ducible representation of supersymmetry. To re- ξQ + ξ¯Q,¯ S = i ξ + ξ¯ ¯ S , (4) duce a supermultiplet, we impose supersymmet- Q Q ric constraints. This is most conveniently done in " # & ' which is the defining property of a superfield.3 terms of the superspace differential operators Here ξα is an arbitrary Grassmann parameter ∂ µ α˙ and α, ¯α˙ are the superspace differential oper- ¯ Dα = α + iσαα˙ θ ∂µ , atorsQ Q ∂θ ∂ D¯ = iσµ ∂ . (7) α˙ α˙ αα˙ µ ∂ µ −∂θ¯ − = iσ θ¯α˙ ∂ , α ∂θα αα˙ µ Q − These anti-commute with the supersymmetry ∂ ¯ = + iσµ ∂ . (5) generators , ¯ , and thus map superfields to α˙ ¯α˙ αα˙ µ α α˙ Q −∂θ superfields.Q Hence,Q any constraint written in ¯ Conversely, the defining property (4) implies that terms of Dα, Dα˙ is automatically supersymmet- the components of any superfield furnish a su- ric. We are now ready to begin exploring various permultiplet. Thus, supermultiplets and super- important supermultiplets. fields are in one-to-one correspondence, and we 1.2. Chiral Multiplets and Lagrangians will treat them synonymously. The most familiar representation of supersym- To see this in a little more detail, consider the metry is the chiral multiplet. It is the basic build- component expansion of a general scalar super- ing block which enables us to write SUSY La- field: grangians describing only scalars and fermions. The bottom component of a chiral multiplet is an- S(x, θ, θ¯) = C(x)+iθαψ (x)+ +θ2θ¯2D(x) . (6) ¯ α · · · nihilated by Qα˙ . For example, the bottom com- ponent φ(x) of a scalar chiral multiplet satisfies As explained above, the defining property (4) de- termines the commutation relations of the super- Q¯α˙ , φ(x) = 0 . (8) charges Qα, Q¯α˙ with the component fields. These commutators show that the supercharges act as The" multiplet# obtained from φ(x) by acting with raising operators for the component fields.4 This the supercharges is organized in a superfield ¯ has two important consequences: which satisfies the constraint Dα˙ Φ = 0. This con- straint can be solved in components: The superfield S(x, θ, θ¯) can be constructed 2 • from its bottom component C(x) by apply- Φ = φ(y) + √2θψ(y) + θ F (y) , (9) ing the supercharges. Thus, any local oper- where yµ = xµ + θσµθ¯. We immediately note two ator can be embedded in the bottom com- key properties of chiral superfields: ponent of a superfield. However, it is not always possible to embed an operator in a Any function which depends on the chiral higher component. This will play a crucial • superfields Φi, but not their complex con- role in our analysis of various supercurrents. jugates, is again a chiral superfield. Such a function is said to be holomorphic in the Φi. The SUSY variation of the top compo- • The SUSY variation of F (x) is a total nent D(x) of any superfield is always a total • derivative. This fact will enable us to write derivative. supersymmetric Lagrangians. From (9) we see that Φ contains a complex scalar φ, a Weyl fermion ψ , and another com- 3 The factor of i in (4) is necessary for Hermiticity in α Minkowski space. plex scalar F which will turn out to be a non- 4 For instance, [Qα, C] = ψα. propagating auxilliary field. Among other things, − 3 F ensures that the chiral multiplet has four (real) taken to vanish.) After solving for the auxiliary bosonic degrees of freedom to match the four fields F i, which are now non-zero, the component fermionic degrees of freedom coming from ψα. We Lagrangian takes the form will abbreviate this by saying that Φ is a 4 + 4 ¯j µ i i µ ¯j L = g ¯ φ, φ¯ ∂ φ¯ ∂ φ + ig ¯ ∂ ψ σ ψ¯ multiplet. As advertised, Φ has exactly the right − ij µ ij µ field content to describe a theory of scalars and i k l µ ¯j 1 i k ¯j ¯l + ig ¯Γ &∂ φ ' ψ σ ψ¯ + R ¯ ¯ψ ψ ψ¯ ψ¯ (14). fermions, and we would like to write a Lagrangian ij kl µ 4 ijkl for such a theory. & ' This theory is known as the supersymmetric non- Up to total derivatives, a SUSY Lagrangian L linear σ-model. The bosonic part is an ordi- must be a real scalar whose variation under su- nary σ-model, whose target space is an N com- persymmetry is a total derivative. We can thus plex dimensional manifold with metric g ¯. Since take ij the Lagrangian (14) does not have a potential L = D + F + F¯ , (10) term, any point φi on the target manifold is a supersymmetric vacuum. (Recall that it follows where D is the top component of a real scalar from the SUSY algebra (1) that supersymmetric 2 superfield K = K¯ and F is the θ -component of vacua have zero energy; vacua with positive en- a chiral superfield W . This ensures that L is ergy spontaneously break SUSY.) In supersym- real and supersymmetric. For reasons that will metric theories, it is customary to refer to such be explained below, K is known as the K¨ahler manifolds of vacua as moduli spaces. As usual, potential, and W is called the superpotential. A the scalars φi should be thought of as coordinates particularly simple choice is to take K = ΦΦ¯ on the target space. This is consistent with the and W = 0. It is standard to pick out different fact that the theory is invariant under holomor- components of a superfield using Grassmann in- phic field redefinitions of the form tegration. For example, we can use d4θ to pick 2 i i j out the top component of a superfield, or d θ Φ" = f (Φ ) . (15) to pick out the θ2 component. We th(us write ( Such field redefinitions can be thought of as co- L = d4θ ΦΦ¯ , (11) ordinate changes on the target manifold under which the metric transforms in the usual tenso- ! up to total derivatives. In components: rial way. In the supersymmetric σ-model, the metric is L = ∂µφ∂¯ φ + i∂ ψσµψ¯ + F¯F . (12) determined by the K¨ahler potential: − µ µ This describes a free complex scalar φ, a free gi¯j = ∂i∂¯jK .