12 Two-Dimensional Supergravity

Transcript

1 PHY680 (Fall 207) Peter van Nieuwenhuizen 2 Two-Dimensional Supergravity We now discuss supergravity in two dimensions. In four dimensions there is a gauge action for local susy (which contains the Hilbert-Einstein action of general relativity and an action for the gravitino) but in two dimensions the Hilbert-Einstein action is a total derivative (a topological invariant) and the gravitino action vanishes (it contains a totally antisymmetric factor ρ [α ρ β ρ γ], which vanishes in two dimensions), so we will only discuss the matter coupling to the supergravity fields. Readers who have never studied supergravity will get here an introduction. As in the rigid case, there are two notations one can use: the covariant formalism with expressions such as δe α a = ɛγ a ψ α, and the light-cone formalism with + and indices. One can decompose both the curved and the flat vector indices into ± indices, but the simplest way is to use ± for the flat indices and α for the curved indices. So a vielbein field becomes then e α + and e α. Fermions have flat spinor indices (only in superspace fermions can also have curved spinor indices), so for them we also use + and indices. We shall first discuss supergravity in the covariant formulation; this is also the formulation one uses in 0 and dimensions. Afterwards we shall, in some cases, use light-cone indices. Readers may pick first one of them to study, but it is very useful to become familiar with both formalisms. We now first give the result for the covariant classical action, and afterwards comment on how it was obtained and what its symmetries are. In the covariant approach, the classical action describing the coupling of the matter fields X µ (σ, t), ψ µ (σ, t) and F µ (σ, t) to the supergravity fields e α a and ψ α A and is given by L = T [ e 2 hαβ α X µ β X ν e 4 ψ µ ρ α α ψ ν + e 2 F µ F ν + + e 2 ( χ αρ β ρ α ψ µ )( β X ν ) + e 6 ( ψ µ ψ ν )( χ α ρ β ρ α χ β ) ] η µν (2.) where e = det e a α. The gauge field of local supersymmetry is a real anticommuting spin 3 field, denoted by 2 χ A α(σ, t) with α = 0, and A = +,. (By the term spin 3 field we mean that 2 χa α has both a vector and a spinor index.) The index A is a spinor index and α a vector index in two dimensions. For notational simplicity we shall often suppress the spacetime index µ of the matter fields X µ, ψ µ and F µ. In all formulas below two µ and ν indices are always contracted with η µν. Furthermore, e = det e α a = det h αβ because h αβ = e α a e β b η ab, where a, b = 0, are flat bosonic indices. The matrix ρ α is equal to the constant matrix ρ a contracted with the inverse vielbein e a α. The field F is the real auxiliary field which we already encountered in the rigid susy action (in D = 4 the Wess-Zumino model has two such fields, usually denoted by F and G, but in D = 2 one needs only one auxiliary field). Because χ A α is the fermionic partner of e a α, it should be real in a real representation of the Dirac matrices ρ α, or in a general representation it satisfies the Majorana condition that the Dirac conjugate χ D χ iγ 0 (with constant γ 0 ) is equal to the Majorana conjugate ψ M ψ T C. However, if we count the number of bosonic and fermionic supergravity gauge fields minus the number

2 of local gauge symmetries, there occurs a mismatch e α a(4) ξ α (2) λ ab () = bosonic χ A α(4) ɛ A (2) = 2 fermionic (2.2) This seems to suggest that one should add one real scalar auxiliary field in the sector of supergravity gauge fields, and this is indeed one solution. e α a(4) ξ α (2) λ ab () = bosonic S = bosonic χ A α(4) ɛ A (2) = 2 fermionic (2.3) However, there is another solution to resolve the mismatch, namely adding new local gauge symmetries: Weyl gauge invariance with parameter λ W, and conformal local supersymmetry with parameter ɛ A cs. Then the counting becomes e α a(4) ξ α (2) λ ab () λ W () = 0 bosonic χ A α(4) ɛ A (2) ɛ A cs (2) = 0 fermionic (2.4) We encounter here two versions of supergravity theories: a non-conformal one (with S), and a conformal one (with Weyl and conformal supersymmetry). Also in 4 dimensions, we shall find these two versions of supergravity, but there the difference is even greater: there exist pure gauge actions, and the non-conformal version starts from the Einstein-Hilbert action of General Relativity, whereas the conformal version starts from the square of the Weyl tensor. We now first discuss all gauge symmetries of the action in (2.). The most difficult symmetry to discuss in detail is ordinary supersymmetry of the action. Then we discuss the boundary conditions for the supergravity case; as expected, they are curved space generalizations of the boundary conditions in flat space. Finally, we discuss closure of the gauge algebra, and here we shall see that there is non-closure on the gravitino, which can be remedied either by adding the auxiliary field S (giving the final formulation of non-conformal supergravity), or by recognizing the non-closure terms as the transformation law of the gravitino under local conformal supersymmetry with a complicated composite gauge parameter. The action in (2.) is invariant under the following 4 bosonic and 4 fermionic local symmetries: Einstein symmetry (general coordinate invariance). The gravitino field χ α is a covariant vector, δχ α = ( α ξ γ )χ γ + ξ γ γ χ α. The vielbein field e α a is also a covariant vector and transforms the same way: δe α a = ( α ξ γ )e γ a + ξ γ γ e α a. All vector indices are contracted and general relativity tells us to add a factor h = e to obtain an invariant worldsheet action. The coordinate X µ is a scalar field on the worldsheet, δx µ = ξ γ γ X µ, so the covariant derivative of X µ is its ordinary derivative, D α X µ = α X µ. Also spinors on the worldsheet transform like scalars under general coordinate 2

3 transformation, δψ = ξ γ γ ψ. local Lorentz symmetry. The index a of eα a indicates that eα a is a Lorentz vector and this vielbein field e a α transforms as δ L e a α = λ a be b α. The gravitino is a Lorentz spinor so δ L χ A α = 4 λab (ρ a ρ b ) A B χb α. The terms ψψ and χ α ρ β ρ α ψ and χ α ρ β ρ α χ β are each locally Lorentz invariant. (It is a good exercise to prove that ρ α ψ and χ α ρ β transform like a spinor, and a conjugate spinor, δ L (ρ α ψ) = ( 4 λab ρ a ρ b )ρ α ψ and δ L ( χ α ρ β ) = χ α ρ β ( 4 λab ρ a ρ b ).) In four dimensions one needs a covariant derivative D α ψ = α ψ + 4 ωab α ρ a ρ b ψ in the Dirac action, where the spin connection ω α ab depends on e a α and χ α and is the (composite) gauge field for local Lorentz invariance. In two dimensions the term with the spin connection would be proportional to ψρ α ρ [a ρ b] ψ ψρ α ρ 3 ψ ɛ αβ ψρβ ψ which vanishes because two arbitrary Majorana spinors ψ and ψ 2 satisfy the identity ψ ρ a ψ 2 = ψ 2 ρ a ψ. The term ψρ α α ψ is locally Lorentz invariant by itself (without spin connection), because under a local Lorentz transformation ψρ α α ψ transforms into 4 ( αλ ab ) ψρ α ρ ab ψ which vanishes as we just explained. Note that in flat space the coordinates X µ do transform under rigid Lorentz transformations, but in curved space they are invariant under local Lorentz transformations. (The former are a combination of orbital and spin terms, but the latter contain only spin terms.) So, for example, β X µ does not transform under local Lorentz transformations. We shall later sometimes use a parameter λ L which is related to λ ab by λ ab = ɛ ab λ L. Weyl symmetry (local scale invariance). By requiring invariance of the action one deduces the correct transformation laws, δ W h αβ = λh αβ, δ W e α a = 2 λe α a, δ W X = 0, δ W ψ = 4 λψ, δ W F = 2 λf, δ W χ α = 4 λχ α. In a flat-space background with light-cone coordinates δ W h +,= = λ. Thus the Weyl transformations can be used to gauge away the helicity zero part of the worldsheet metric. (For invariance of the Dirac action use that γ α = e a α γ a with δ W e a α = 2 λe a α so δ W e = λe.) There is one small subtlety: varying the ψ in ψρ α α ψ into 4 λψ one might worry that a term with αλ is produced, but this term is multiplied by ψγ α ψ which vanishes for Majorana spinors as we discussed. super Weyl symmetry. Another name is conformal supersymmetry. This local symmetry is the fermionic counterpart of Weyl symmetry. (One sometimes calls it superconformal symmetry, but we reserve that term for the semilocal symmetry in flat space with fermionic parameters ɛ + (t + σ) and ɛ (t σ) which is the fermionic counterpart of the bosonic conformal symmetry whose parameters are ξ + (t + σ) and ξ (t σ).) There is a beautiful formalism behind super Weyl symmetry, but we only note that only the gravitino transforms: δ sw X = 0, δ sw e α a = 0, δ sw ψ = 0, δ sw F = 0, but δ sw χ α = ρ α ɛ sw. The invariance of the action follows from the identity ρ α ρ β ρ α = 0 in two dimensions. In fact, this symmetry explains the form of the last two terms in the action in (2.). In ± coordinates the gravitino transforms in flat space with h αβ = η αβ as follows: δ sw χ ++, = ɛ sw Another way of understanding why ψρ α ρ [a ρ b] ψ vanishes is to notice that one has four 2 2 matrices (including C = iγ 0 ), all of which are off-diagonal. The product of them is a diagonal hence one obtains only terms with ψ + ψ + and ψ ψ which vanish. 3

4 and δ sw χ,+ = ɛ + sw. Hence the super-weyl gauge symmetry can be used to gauge away the helicity parts of the gravitino, just as Weyl symmetry can be used to gauge away the helicity 0 part of ± 2 the graviton. local supersymmetry (supergravity) transformations. This is the least obvious local symmetry to prove. The local supersymmetry (susy) parameter is the local version of the rigid susy paramter we encountered before, so ɛ ± (t, σ). Since bosons transform into fermions and vice-versa, we expect δe α a ɛρ a χ α (another possibility δe α a ɛρ α χ a is nonlinear in fields and does not work. One can understand this from group theory). Since the gravitino is the gauge field of local supersymmetry it should transform as δχ α = α ɛ +..., and Einstein and local Lorentz symmetry promote this to δχ α = D α ɛ with a derivative D α which we must discuss further. The complete transformation rules are δ s X = 2 ɛ sψ ; δ s ψ = ρ α ( ˆD α X)ɛ s + F ɛ s ; δ s F = 2 ɛ sρ α ˆDα ψ δ s e α a = ɛ s ρ a χ α ; δ s χ α = D α (ˆω)ɛ s (2.5) These transformation rules for the matter fields X, ψ and F are local extensions and gravitational covariantizations of the rules of rigid susy in flat space we studied before. The derivatives D α (ˆω) act on spinors as D α (ˆω)ɛ s = α ɛ s + 4 ˆω α ab ρ a ρ b ɛ s but ˆω α ab is the supercovariant spin connection : the usual spin connection ω α ab (e) of general relativity plus three torsion terms of the form χρχ such that the variation of ˆω under a susy transformation does not contain any α ɛ s terms (only ɛ s terms 2 ). In the Dirac action in (2.) this spin connection cancels, but we need it if we want to evaluate the local gauge algebra. The supercovariant derivatives ˆD α X and ˆD α ψ are defined as follows ˆD α X = α X 2 χ αψ ˆD α ψ = D α (ˆω)ψ ρ β ( ˆD β X)χ α F χ α (2.6) The derivative D α (ˆω)ψ is the same as in D α (ˆω)ɛ s. With ± indices it reads D α (ˆω)ψ ± = α ψ ± + 2 ˆω α0(ρ ρ 0 ψ) ± = α ψ ± ± 2 ˆω αψ ± with ˆω α0 = ˆω α. (2.7) It is fairly easy to check that δ s ˆDα X contains no α ɛ s terms. The expression for ˆD α ψ is less obvious, but 2 The explicit form (for vielbeins and gravitinos transforming as in (2.5)) is ˆω α bc e bβ e cγ ˆω αβγ = { 2 e aγ( α e a β βe a α χ α ρ a χ β ) β γ} 2 e aα( β e a γ γ e a β χ βρ a χ γ ). Hence ˆω αβγ = ω αβγ (e) + 2 ( χ αρ β χ γ χ α ρ γ χ β + χ β ρ α χ γ ). Note that the combination ( α e a β βe a α 2 χ α ρ a χ β ) is supercovariant. Thus ˆω αβγ is supercovariant and a good Einstein tensor, but a connection for local Lorentz invariance. Under Weyl rescalings one has δ W (λ)ˆω αab = 2 e αa b λ 2 e αb a λ. 4

5 contemplating it for a while the reader may note that the last two terms are obtained from the rigid susy rule δ s ψ by replacing everywhere ɛ by χ α and by replacing β X by ˆD β X. The final result for ˆD α ψ is also supercovariant: δ s ˆDα ψ does not contain α ɛ terms. As an exercise in the technique of Fierz rearrangements (see (2.5)), one may show that the last two terms in the action can be rewritten as follows e ( χ 2 α ρ β ρ α ψ ) ( β X χ 4 βψ ) (2.8) (It is easier to work backwards, and to show that the second term in (2.8) yields the last term in (2.).) Because of the factor the sum of these two terms in the action is not supercovariant, but the ψ- and χ- 4 field equations are supercovariant. (Variation of (2.8) with respect to ψ or χ gives an extra factor 2 in the second term because both ψ s and χ s appear symmetrically, as is manifest from (2.).) This explains the relative normalization of the last two terms in the action. We shall not explicitly check in all detail that this action is locally supersymmetric. For the interested reader we provide this calculation in an appendix. However, we shall show why the last two terms in (2.) are necessary. For a similar approach to supergravity in four dimensions which starts at the matter-coupled end, see []. The approach which starts at the gauge action end was given in [2]. The two dimensional action in (2.) was presented in [3]. If one varies the rigid susy action in flat space T L = 2 α X α X 4 ψρ α α ψ + 2 F 2 (2.9) under the transformations with a local susy parameter ɛ(σ, t), and disregards boundary terms for the time being, one obtains the Noether current times α ɛ [( T δs = X) 2 ɛψ ψρ ( α 2 α ρ β ɛ β X )] d 2 σ = ( 2 ψρ α ρ β α ɛ ) β Xd 2 σ (2.0) Introducing the gauge field χ A α for local susy, it should transform, as any gauge field, into the derivative of the parameter plus more δχ α = α ɛ +... (2.) We can then cancel the variation in (2.0) by adding the following term to the action L(Noether) = 2 ( ψρ α ρ β χ α ) β X (2.2) because the variation δχ α = α ɛ in this term precisely cancels the variation in (2.0). Since ψρ α ρ β χ α = χ α ρ β ρ α ψ, we have now understood the need for the one but last term in (2.). However, in the variation of L(Noether) there is a new term proportional to ɛ, namely, when we vary 5

6 X according to δx = 2 ɛψ we find ( ) δl(noether) = 4 ψρ α ρ β χ α ( β ɛ)ψ + (2.3) The terms denoted by come from varying all other fields in L N and do not contain ɛ. We claim that this term can be canceled by adding another term to the action 3 : ( ) L(last term) = 8 ψρ α ρ β χ α ( χβ ψ) (2.4) We again replaced β ɛ by χ β, but this time we added an extra factor because there are two χ fields in 2 this term and they yield the same variation as we now show. The spinors in L(last term) can be recoupled by a Fierz rearrangement 4 which couples the two gravitinos to each other ( ψρ α ρ β χ α )( χ β ψ) = I 2 ( ψo I ψ)( χ β O I ρ α ρ β χ α ) (2.5) The basic idea is to use a complete set of matrices, which is given for 2 2 matrices by the set I, ρ a with a = 0,, and ρ 3. The set of matrices O I O I is, ρ a ρ a and ρ 3 ρ 3 where ρ a = η ab ρ b and ρ 3 = ρ 3. However, ψρ a ψ = ψρ 3 ψ = 0 (2.6) So only O I = I contributes, and one finds L(last term) = 6 ψψ( χ β ρ α ρ β χ α ) (2.7) The two gravitino fields indeed appear symmetrically and L(last term) is indeed the last term in (2.). The structure of the two last terms in (2.) is the same as that of the coupling of a complex scalar in QED to the electromagnetic field: there is one term of the form (gauge field) (matter current) and a second term of the form (gauge field) 2 (matter field) 2. There are many more variations to consider, but all variations cancel: the action is locally supersymmetric. The beginner may try his or her luck with the variations of e and h αβ in the first term in (2.) (which yield a result proportional to the stress tensor of X), which should cancel the variation of ψ in the Noether term. An easier exercise is to show that all terms with two F fields cancel. A more demanding 3 It might seem that there is another way to get rid of the ɛ term in (2.3): by partial integration. That would not lead to the 4-fermion term in (2.4), but we need this term in order to obtain a supercovariant field equation for ψ and for χ α. Thus we do not partially integrate. 4 Given four spinors χ, ψ, λ, φ and a coupling χmψ λnφ, one can write this expression as χ A T A Bφ B. The matrix T A B can be expanded in a complete set of Dirac matrices as T A B = (O I ) A B 2 tr(o IT ) where O I is the set {I, ρ a, ρ 3 }. To check this formula one may contract with (O J ) B A and take the trace. Substitution of this expression for T A B yields the Fierz rearrangement formula. Fierz used it in his studies of the coupling of fermions in the 4-fermi theory of weak interactions, see [4]. 6

7 exercise is to show that the variations of e and e a α and the variation δψ = ρ α ɛ( χ 2 αψ) in the second term in (2.) are canceled by the variation of X in the Noether term and the variation of the gravitinos in the last term. (Hint: couple ψ to α ψ by a Fierz rearrangement and use ρ γ ρ α ρ β ρ β ρ γ ρ α = 2η αβ ρ γ 2η βγ ρ α.) In supergravity one usually considers spaces without boundaries, but for open string theory one must make sure that the contributions to the variation of the action from boundary terms cancel. Requiring this cancellation determines the boundary conditions for the various fields. We now study such boundary terms. The variation of the classical supergravity action under local susy transformations contains the following boundary contributions (this is shown in the appendix) δs = α [ e 2 ɛψhαβ β X + e 4 ɛ(/ X)ρα ψ e 4 ψρ α ɛf ] 8 e( ψρ α ρ β ɛ)( ψχ β ) (2.8) Although this expression results from a direct calculation, it is simply the covariantization of the flat-space result except for the last term. For the open string, the first term vanishes provided h β β X = 0 at σ = 0, π (2.9) This is the curved space generalization of σ X = 0 at σ = 0, π in flat space. The next term can be written as a sum of two terms, using / Xρ α = h αβ β X + 2 [ρβ, ρ α ] β X where α =. The first term vanishes again due to (2.9), while the second term yields 4 e ɛ ( 2 [ρβ, ρ ] β X ) ψ = 8 e ɛ(eβ b e a e β ae b)ρ b ρ a β Xψ = 8 e ɛ( 2e ρ 3 t X)ψ = 4 ( ɛρ3 ψ) t X (2.20) = i 4 (ɛt ρ ψ) t X = i 4 (ɛ+ ψ + ɛ ψ + ) t X We see that the flat-space boundary conditions ɛ + ψ + = ɛ ψ at σ = 0 and σ = π are also the boundary conditions in curved space. The last term in (2.8) becomes proportional to ψψ after a Fierz rearrangement α [( ψγ α γ β ɛ)( ψχ β )] = α [( 2 ψψ)( ɛγ β γ α χ β )] (2.2) and also ψψ vanishes at σ = 0 and σ = π due to ψ + = ±ψ at the boundary. Finally, the term with F is proportional to (ɛ + ψ + + ɛ ψ )F, and since ɛ + ψ + ɛ ψ vanishes at the endpoints, the combination ɛ + ψ + +ɛ ψ cannot vanish, hence the boundary condition on F reads F = 0 and at σ = 0 and σ = π, which is also the same boundary condition as in flat space. The final conclusion is that in local supersymmetry (supergravity) the boundary terms in the variation of the action cancel provided h β β X = 0, ψ + = ±ψ, ɛ + ψ + = ɛ ψ and F = 0 at σ = 0 and σ = π. 7

8 Let us now consider the local susy algebra. On X one finds [δ(ɛ 2 ), δ(ɛ )]X = 2 ɛ (ρ α Dα Xɛ 2 + F ɛ 2 ) ( 2) = 2 (ɛ ρ α ɛ 2 )( α X 2 χ αψ) + 2 F ɛ 2ɛ ( 2) (2.22) = (ɛ ρ α ɛ 2 ) α X + 2 ( ɛ ρ α ɛ 2 ψ α )ψ (2.23) which is an Einstein transformation with composite parameter ξ α = ɛ ρ α ɛ 2 and a local susy transformation with composite parameter ɛ = ɛ ρ α ɛ 2 χ α = ξ α χ α. However, on the vielbein also a local Lorentz transformation is produced, as we now show. We begin with [δ(ɛ 2 ), δ(ɛ )]e α a = ɛ ρ a (D α ( ω)ɛ 2 ) ( 2) = ɛ ρ a α ɛ (ɛ ρ a ρ b ρ c ɛ 2 ) ω α bc ( 2) = α (ɛ ρ a ɛ 2 ) + (ɛ ρ c ɛ 2 ) ω α ac (2.24) We used that in two dimensions ɛ ρ a ρ b ρ c ɛ 2 is equal to δ a bɛ ρ c ɛ 2 δ a cɛ ρ b ɛ 2 + η bc ρ a. To obtain the expected Einstein transformation we replace ρ a in the first term by ρ β e a β and thus ɛ ρ a ɛ 2 = ξ β e a β, and find α ( ξ β e β a ) + ξ c ω α ac = ( α ξβ )e β a + ξ β α e β a + ξ c ω α ac + ξ β β e α a ξ β β e α a ξ β ω β a b e α b + ξ β ω β a b e α b (2.25) We added in the second line twice a pair of terms whose sum vanishes in order to obtain an Einstein transformation and covariant derivatives with D α ( ω) acting on a vielbein field. The two terms in the first column give the Einstein transformation, while the four terms in the next two columns give the covariant curl of the vielbein, and finally the last term is a local Lorentz transformation of the vielbein ( ξ β ω a β b ) [δ(ɛ 2 ), δ(ɛ )]e a α = δ E ( ξ α )e α a + ξ β (D α ( ω)e β a D β ( ω)e α a ) + δ ll ( ξ β ω β a b )e α a (2.26) Using the torsion equation D α ( ω)e β a D β ( ω)e α a = χ α ρ a χ β, the term with the curl yields ξ β χ α ρ a χ β, which is the expected local susy transformation (the same local susy transformation as on X) ξ β χ α ρ a χ β = ξ β χ β ρ a χ α = δ S ( ξ β χ β )e α a (2.27) However, if we evaluate the local susy commutator on the gravitino, we do not find closure: there are extra terms proportional to the gravitino curl χ αβ D α ( ω)χ β D β ( ω)χ α (2.28) We should have expected this non-closure because, as we showed in (2.2), the number of bosonic field 8

9 components minus the number of bosonic gauge symmetries is not equal to the number of fermionic field components minus the number of fermionic symmetries e α A(4) Einstein(2) local Lorentz() = bosonic ψ α A (4) local susy(2) = 2 fermionic (2.2) We mentioned that one can obtain closure either by adding a real scalar auxiliary field S, or by adding further bosonic and fermionic conformal gauge symmetries. We shall first discuss the nonconformal model, and then the conformal model. In string theory, the latter is used. In the nonconformal model, the auxiliary field can only appear in the transformation law of the gravitino, and only as follows δ(ɛ)χ α = D α ( ω)ɛ + pγ α Sɛ (2.29) where p is a constant we shall fix. The new auxiliary field S transforms as follows under local susy δ(ɛ)s = ρ α ρ β (D α ( ω)χ β D β ( ω)χ α ) ρ α ρ β χ αβ (2.30) We can always scale S to obtain this normalization. The local susy commutator acting on the gravitino yields [δ(ɛ 2 ), δ(ɛ )]χ α = δ(ɛ 2 ) [D α ( ω)ɛ + psγ α ɛ ] ( 2) = ( 4 ρab ɛ )(δ(ɛ 2 ) ω αab ) + (pγ α ɛ )(ɛ 2 ρ ab χ ab ) ( 2) (2.3) Using δ(ɛ) ω αab = 2 (ɛρ bχ αa ɛρ a χ αb ɛρ α χ ab ) (2.32) where χ αa = χ αβ e a β and χ ab = χ αβ e a α e b β, we get 8 ρab ɛ (2ɛ 2 ρ b χ αa ɛ 2 ρ α χ ab ) + (pρ α ɛ )(ɛ 2 ρ ab χ ab ) ( 2) (2.33) where ρ ab denotes the antisymmetric part of ρ a ρ b, so ρ ab = 2 (ρa ρ b ρ b ρ a ). In 2 dimensions, a totally antisymmetric tensor with 3 indices vanishes, hence ρ b χ αa ρ a χ αb + ρ α χ ab = 0, and thus the expression inside the first parenthesis simplifies to 2ɛ 2 ρ α χ ab. Fierzing yields (ɛ 8 2O I ɛ )(ρ ab O I ρ α χ ab 4pρ α O I ρ ab χ ab ) (2.34) Only O I = ρ c and O I = ρ 3 can contribute, but the contribution from O I = ρ 3 vanishes if p is fixed as 9

10 p = 4. Then the contribution from OI = γ c reads (ɛ 4 2γ c ɛ )(ρ ab ρ c ρ α ρ α ρ c ρ ab )χ ab (2.35) We substitute ρ ab ρ c ρ α ρ α ρ c ρ ab = 2δ b c (ρ a ρ α + ρ α ρ a ) = 4δ b ce α a, and arrive at For closure of the gauge algebra on χ α, this should be equal to (ɛ 2 γ b ɛ )χ αb = ξ β (D β ( ω)χ α D α ( ω)χ β ) (2.36) δ E ( ξ β )χ α + δ S ( ξ β χ β )χ α + δ ll ( ξ α ω ab α )χ α = ξ ( ) β β χ α + ( α ξβ )χ β α ( ξ β χ β ) + ω 4 α ab ρ ab ( ξ β χ β ) = ξ β ( β χ α α χ β 4 ω α ab ρ ab χ β + 4 ω β ab ρ ab χ α ) + ξ β ω β ab 4 ρ abχ α = ξ β (D β ( ω)χ α D α ( ω)χ β ) (2.37) which agrees with (2.36). Hence, the local gauge algebra also closes on the gravitino if one adds the auxiliary field S. We leave the closure of the matter fermion ψ as an exercise. We now discuss the case of conformal supergravity. In that case, the local susy commutator on the gravitino is given by only the first term in (2.34) [δ(ɛ 2 ), δ(ɛ )]χ α = 4 (ɛ 2O I ɛ )(ρ ab O I ρ α χ ab ) (2.38) We shall rewrite the terms on the right-hand side which violate closure as ρ α X where X is a complicated expression, and then we shall interpret ρ α X as a conformal local susy transformation of the gravitino with composite parameter X ρ α X = δ cs (X)χ α (2.39) To achieve this we use ρ ab ɛ ab ρ 3, with ɛ 0 =. The term with O I = ρ 3 is then indeed proportional to ρ α, whereas the term with O I = ρ c yields ρ ab ρ c ρ α χ ab = 2ρ a ρ α χ ac = 2ρ α ρ a χ ac + 4χ αc (2.40) The first term gives another local conformal susy transformation while the last term agrees with (2.37). 0

11 A The complete proof of local susy invariance In this appendix we give the full analysis of the cancellation of all variations of the action of d = 2 supergravity. The cancellation of the leading terms in the variations are easy to spot, but for the remaining terms one repeatedly needs to use the Fierz rearrangement formula. It requires time to build up the experience to master this technology, so beginners might only have a look at the leading terms in the variations (the terms with X, all terms with F, and the terms with the stress tensor of X). cancellation of the terms with ɛ is also worthwhile studying, as it is of course the foundation for the Noether procedure. But it is advisable to stay initially away from all 4-fermion terms in the variations. (In the first article on supergravity in 976, a computer was used to handle them! In d = 4 supergravity one can now give a rather simple but complete proof using the so-called.5-order formalism, but that requires the presence of a gauge action, and in d = 2 there is of course no gauge action. Hence the analysis for the case of d = 2 which follows is even today more complicated than for the case of d = 4.) The Lagrangian density and the transformation laws of local susy are as follows T L = hh 2 αβ α X β X e 4 ψρα α ψ + 2 F 2 + e 2 δx = ( 2 ɛψ; δψ = ρα α X ) 2 χ αψ ɛ + F ɛ δf = 2 ɛρα [ D α (ω 0 )ψ ρ β ( β X 2 (χ βψ)χ α ) F χ α ] ( χα ρ β ρ α ψ ) ( β ) 4 χ βψ The (A.) (A.2) (A.3) δe α a = ɛρ a χ α ; δχ α = D α ( ω)ɛ (A.4) Note that the combination β X 4 χ βψ is not supercovariant, but it will turn into the supercovariant derivative D β X = β X 2 χ βψ when we vary χ α or ψ. Note also the torsionless connection ω (0) in δf ; we need it to cancel another term with D α (ω 0 )ψ obtained by replacing α ψ in the action by D α (ω 0 )ψ. (Recall that the term with ω (0) cancels in the action, but it simplifies the analysis of the variations if we use D α (ω (0) ) instead of α when we partially integrate.) We now record the variations of the four terms in (A.) and discuss how they cancel each other. Variations of Term I: ( hh ) [ (ɛψ) 2 α αβ β X + α ] hh αβ δx β X + e ( ɛρ α χ β) α X β X e(ɛρ χ) [ 2 α h αβ α X β X ] (A.5) The leading variation is e ɛψ X, and the last two terms contain the stress tensor of X. 2

12 Variations of Term II: e 2 ψρα D α (ω 0 )[ρ β ( β X 2 (χ βψ)ɛ) + F ɛ] α [ e 4 δψρα ψ] + e 4 (ɛρα χ a )(ψρ a D α (ω 0 )ψ) e 4 (ɛρ χ)(ψρα D α (ω 0 )ψ) (A.6) We need not vary ω 0 in the Dirac action because ω (0), and hence also δω 0, is multiplied by ψρ c ψ which vanishes. The leading term (from varying ψ into β X) is e ψ( X)ɛ. It cancels the leading term in (A.5). 2 The last two terms contain the stress tensor of ψ. Omitting the two total derivatives we are left with Sum of variations of Term I + Term II: e(ɛρ α χ β )( α X β X) e 2 (ɛρ χ)(hαβ α X β X) e 2 (ψρα ρ β D α (ω 0 )ɛ) β X + e 4 ψρα D α (ω 0 )[(χ β ψ)ɛ] + e 4 (ɛρα χ β )(ψρ β D α (ω 0 )ψ) e 4 (ɛρ χ)(ψρα D α (ω 0 )ψ) (A.7) e 2 F ɛρa D α (ω 0 )ψ α [ e 2 ψρα F ɛ] We next record the variation of the last term. Variation of Term IV: e (D 2 α( ω)ɛ)(ρ β ρ α ψ)( β X χ 2 βψ) + eχ 2 αρ β ρ α [ρ γ ( γ X χ 2 γψ)ɛ + F ɛ]( β X χ 2 βx) + e(χ 4 αρ β ρ α ψ)(ψd β ( ω)ɛ + ɛd β ( ω)ψ) (A.8) + O(χ 2 ) and O(χ 3 ) terms from the vielbein variations in term IV. We observe the following cancellations the X X terms cancel: the terms with the stress tensor T αβ (X) in the first line of (A.7) cancel the terms with the stress tensor T αβ (X) in the second line of (A.8). All F -dependent terms cancel. These terms are: Variation of Term III: e ɛρ χf 2 + F 2 2 ɛρα [D α (ω 0 )ψ ρ β ( D β X)χ α F χ α ] (A.9) Rest of variations: e 2 F ɛρα D α (ω 0 )ψ + e 2 (χ αρ β ρ α ɛ)f ( D β X) (A.0) The F 2 terms in (A.9) clearly cancel, and also the terms linear in F cancel in the sum. 2

13 The remaining total set of variations is given by e 4 (ɛρα χ β )(ψρ β D α (ω 0 )ψ) e(ɛρ 4 χ)(ψρα D α (ω 0 )ψ) e 2 (ψρα ρ β D α (ω 0 )ɛ) β X + e 4 ψρα ρ β D α (ω 0 )[(χ β ψ)ɛ] + e 2 (ψρα ρ β D α ( ω)ɛ) β X e 4 (ψρα ρ β D α ( ω)ɛ)(χ β ψ) (A.) + e 4 (χ βρ α ρ β ψ)(ψd α ( ωɛ) + e 4 (χ βρ α ρ β ψ)(ɛd α ( ω)ψ) The two terms in the middle on the left-hand side cancel each other up to χ 2 terms from the torsion. There remain 6 terms; if we partially integrate the third of these 6 terms, we find 2 terms with D α (ɛ) and 4 terms with D α ψ e 4 (ɛρα χ β )(ψρ β D α (ω 0 )ψ) e 4 (ψρα ρ β D α ( ω)ɛ)(χ β ψ) e(ɛρ 4 χ)(ψρα D α (ω 0 )ψ) + e(χ 4 βρ α ρ β ψ)ψd α ( ω)ɛ (A.2) e 4 (ɛρβ ρ α D α (ω 0 )ψ)(χ β ψ) + e 4 (χ βρ α ρ β ψ)(ɛd α ( ω)ψ) If we bring in the last two terms the spinors ψ and D α ψ together by a Fierz rearrangement, all ψdψ terms cancel up to χ 3 terms. Also the 2 terms with D α ɛ cancel. Hence, up to total derivatives, and up to χ 2 and χ 3 variations, the action is invariant under local susy transformations. The complete set of total derivatives can be collected from the previous results. There was one total derivative containing δx in (A.5), and one total derivative containing δψ in (A.6). Then there was the total derivative due to partially integrating the third of the 6 terms in (A.); this term partially cancels part of the total derivative with δψ. The final result reads δs = π dt dσ α k α 0 (A.3) where k α = e 2 (ɛψ)hαβ β X + e 4 ɛ( / X)ρ α ψ + e 4 (ɛρα ψ)f e 8 (ɛρβ ρ α ψ)(χ β ψ) (A.4) In the main text it is shown that all boundary variations cancel due to suitable boundary conditions. References [] P. van Nieuwenhuizen, Springer Lecture Notes in Physics 6 (980) 276. [2] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D 3 (976) 32 and S. Deser and B. Zumino, Phys. Lett. B 62 (976)

14 [3] P.A. Collins, R.W. Tucker, Phys. Lett. B 64 (976) 207 constructed the Hamiltonian for the spinning string using Dirac s canonical formalism, but they only obtained a complicated action which they did not exhibit; L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. B 65 (976) 47 and S. Deser and B. Zumino, Phys. Lett. B 65 (976) 369. [4] M. Fierz, Z. f. Physik 04 (937)