Edited by A. Garcia, C. Lammerzahl, A. Macias, T. Matos, D. Nu~nez; ISBN Science Network Publishing

Transcription

1 Lectures on Supersymmetry and Supergravity in 2+1 Dimensions and Regularization of Suspersymmetric Gauge Theories F. Ruiz Ruiz a;b and P. van Nieuwenhuizen c a Institut fur Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, D Heidelberg, Germany, ferruiz@eucmos.sim.ucm.es b Departamento de Fisica Teorica I, Facultad de Fisicas, Universidad Complutense de Madrid, Madrid, Spain. c Institute for Theoretical Physics, State University of New York State at Stony Brook, Stony Brook, NY , USA vannieu@insti.physics.sunysb.edu Abstract These notes are intended to provide an introduction in both x-space and superspace to N = 1; d = 2+1 rigid supersymmetry and supergravity. We give a detailed discussion at the classical level of various supersymmetric models, namely the Wess-umino model, Yang-Mills theory, Chern-Simons theory and supergravity. We also consider rigidly supersymmetric Yang-Mill-Chern-Simons theory at the quantum level and prove that the theory is ultraviolet nite to all loops. At the one, two and three-loop level in x-space, and at the one and two-loop level in superspace, certain diagrams are power-counting divergent. This raises the possibility that dierent regularization schemes may give nite but different results for the eective action. We consider the two most used schemes: ordinary dimensional regularization with d > 3 in x-space, and dimensional reduction with d < 3 in superspace. The well-known inconsistency of dimensional reduction (an ambiguity in the evaluation of the product of three epsilon tensors) is multiplied by d?3, so that it vanishes at d = 3. Using BRST Ward identities, supersymmetry Ward identities and general theorems of quantum eld theory, we show that both schemes yield the same eective action. Hence, for this model at least, the superspace approach respects gauge invariance. Recent Developments in Gravitation and Mathematical Physics. 1

2 Edited by A. Garcia, C. Lammerzahl, A. Macias, T. Matos, D. Nu~nez; ISBN Science Network Publishing

3 Contents 1 Classical rigid supersymmetry When and why supersymmetry? General properties of supersymmetric eld theories Spinors and Dirac matrices in three dimensions The simplest case: the Wess-umino model in x-space Supersymmetric Yang-Mills theory in x-space Supersymmetric Chern-Simons theory in x-space Three-dimensional rigid superspace The Wess-umino model in superspace The covariant approach to Yang-Mills theory The covariant approach to Chern-Simons theory Higher N models and gauge couplings to matter Quantum rigid supersymmetry: Yang-Mills-Chern-Simons theory Supersymmetric regularization of gauge theories Supersymmetric Yang-Mills-Chern-Simons theory Ward identities, dimensional regularization and regularization by dimensional reduction Perturbative niteness A BRST invariant and supersymmetric eective action Classical supergravity Supergravity in (2 + 1)-dimensional x-space Closure on the gravitino, the auxiliary eld S Supergravity in superspace Covariant derivatives A new basis for the gauge elds leading to vielbeins Constraints and Bianchi identities Action and eld equations References 68 3

4 1 Classical rigid supersymmetry 1.1 When and why supersymmetry? The assumption that eld theories have a Fermi-Bose symmetry leads to predictions which will be tested in the next decade, certainly at the LHC at CERN, and possibly earlier at the Tevatron at Fermilab. For example, in the minimal supersymmetric extension of the standard model, one needs two (instead of one) Higgs doublets, with one of the Higgs scalars classically lighter than the boson and quantum corrections being able to lift its mass at most to about 150 GeV. For some of the predicted suspersymmetric partners, upper and lower limits on their masses can be given, so that not nding these suspersymmetric particles will be a serious problem for SUSY. On the other hand, discovery of supersymmetric particles will rank, with quantum mechanics, special and general relativity and gauge theories, among the most important physical discoveries of our century. It is sometimes stated that there is not the slightest indication that nature is supersymmetric. This is not the whole story, though. The standard model becomes probably inconsistent at very high energies, of the order of GeV, due to what is called \triviality". This means that some extension or modication of the Standard Model is needed. SUSY is one such modication, perhaps the most consistent one available today. When combined with string theory, SUSY produces a theory of quantum gravity without innities. In addition, supersymmetric quantum gauge eld theories have duality symmetries which give detailed information on the nonperturbative sector of the corresponding eective actions, the hope being that also nonsupersymmetric gauge theories have similar features. In these notes we rst give an introduction to SUSY, both in x-space and in superspace. Then we discuss a fundamental problem with SUSY that has been around since its beginning: a regularization scheme that respects both SUSY and gauge invariance. In 2+1 dimensions we have found such a scheme, but we make no claims concerning 3+1 dimensions. We conclude with a detailed discussion of classical supergravity. Rather than giving a full account of the subject and the literature on it, something which would require much more space, we have opted for a more direct presentation which hardly requires any background on supersymmetry. 1.2 General properties of supersymmetric eld theories SUSY is a symmetry of certain actions with an anticommuting spinorial parameter, such that in the (anti)commutator of two supersymmetries one nds a translation. There are other symmetries with anticommuting parameters, BRST symmetry for example, which has an anticommuting constant parameter, but this parameter is a scalar instead of a spinor under Lorentz transformations. Suppose one has an action with some bosonic elds b(x) and fermionic elds f(x) which satisfy the usual spin-statistics connection. A crucial property from 4

5 which SUSY springs is their mass dimension. Actions for bosonic elds contain two derivatives. Examples are the Maxwell and the Klein-Gordon actions, and also the spin 2 action of Fierz and Pauli which is the linearized limit of the Einstein-Hilbert action of general relativity. On the other hand, actions for fermions contain only one derivative. Quite familiar is the spin 1/2 Dirac action; but also the spin 3/2 Rarita-Schwinger action for \gravitinos", the supersymmetric partners of gravitons, has only one derivative. It follows that the sum of the mass dimensions of two bosonic elds and two derivatives is equal to the sum of the dimensions of two fermionic elds and one derivative. Equivalently, the dimensions of a Fermi and a Bose eld dier by one half the mass dimension of a derivative. If we dene the latter to be unity, [@ ] = 1, we nd [f]? [b] = 1=2 : (1) This leads to SUSY as we now show. Suppose that there are SUSY transformation rules which transform a boson into a fermion and vice-versa and which leave the action invariant. Then 1 b f and f b, with a Fermi eld. However, if b f contains no derivative, it follows from eq. (1) that [] =?1=2, so that in f b there is a gap of one unit of mass dimension. If there are no masses in the theory and we there are no dimensionful coupling constants, the only object that can ll this gap is a derivative. Hence b f : (2) (1) It is clear from the statistics of Bose and Fermi elds that must be anticommuting. Usually one chooses Grassmann variables which anticommute, so = 0. Recently, another choice for has been studied, namely Cliord variables Bouwknegt, McCarthy, and Nieuwenhuizen (1997). This seems to lead to quantum groups, and could lead to a completely dierent quantum superspace. (2) From the conservation of angular momenta and the integer/half-integer spin of bosons/fermions it follows that has half-integer spin. The simplest case is clearly spin 1/2. Spin 3/2 for leads to eld theories which have no positive denite Hilbert space in at spacetime. Thus, by angular momentum theory, bosons transform into fermions and fermions into bosons whose spins dier by 1/2. This means that the basic building blocks are Fermi-Bose doublets. (3) The commutator of SUSY transformations of b(x) leads indeed to a translation: 1 ( 2 b ) 1 (f 2 ) (@b 1 ) 2 1 (3) 1 To keep the notation simple, we denote SUSY variations by the letter without any subscript. 5

6 The of b(x) is over a distance 1 2 which is the same everywhere, since in rigid SUSY the 's are x-independent. For the fermion f(x) one nds a similar result 1 ( 2 f ) 1 (@b 2 ( 1 f) 2 1 : (4) (4) We shall see that matters are a bit more complicated than in point 3) above. In general, there are also \auxiliary elds" F which do not correspond to physical particles. Often they appear in the action as F 2 and have eld equations F = 0. In 2+1 dimensions they have mass dimension [F ] = 3=2, so that now one can ll the mass dimension gap also with an auxiliary eld b + F : (5) With auxiliary elds, the SUSY algebra \closes", meaning that the commutator of two supersymmetries leads to a sum of symmetries, namely a translation and sometimes a gauge transformation. Without auxiliary elds, the SUSY commutator of a fermionic eld contains in general fermionic eld equations. Their origin is clear: because F = 0 is a bosonic eld equation and, in general, eld equations rotate into eld equations under SUSY, the SUSY variation of F (x) is a fermionic eld equation. Hence, if the SUSY commutator in the theory with F closes, omitting F from the theory will introduce fermionic eld equations. 1.3 Spinors and Dirac matrices in three dimensions Before discussing the W model in 2+1 dimensions, we recall rst some simple facts about spinors and Dirac matrices in 3 dimensions. We shall exclusively discuss Minkowski spacetime and, in Section 1.7, its superspace. In Euclidean space, the treatment of spinors is dierent; for example, no real spinors exist in three-dimensional Euclidean space while in Minkowski spacetime they exist. One can go from Minkowski spacetime to Euclidean space by a Wick rotation, and construct in this way supersymmetric theories in Euclidean space from supersymmetric theories in Minkowski spacetime Nieuwenhuizen and Waldron (1996). We take the metric in at (2+1)-dimensional Minkowski spacetime to be = diag (?1; 1; 1) = 0; 1; 2 ; (6) while for the epsilon symbol we take 012 = +1. A spinor a (x) has two components ( = 1; 2) since the Dirac matrices f ; g = 2 (7) 6

7 are 22 matrices. If we choose the real representation 2 ( 0 ) =?i 2 ( 1 ) = 1 ( 2 ) a = 3 ; (8) where 1 = = 0?i i 0 3 = 1 0 0?1 (9) are the Pauli matrices, the Dirac is real and we can take (x) to be real. Then (x) describes one physical state. Compare with four dimensions, where one can also choose a real representation of the Dirac matrices, and where a real 4-component spinor describes 2 physical states, namely particles with helicities 1=2 which are their own antiparticles. An important dierence with four dimensions is that in three dimensions, and more generally in any odd number of dimensions, there is no 5 and hence no chiral spinors exist. This is due to the fact that in an odd number 2n + 1 of dimensions the product of all the Dirac matrices 0 1 : : : 2n+1 is proportional to the unit matrix. In three dimensions, the matrices f1l; g form a basis of the Cliord algebra, so that any 22 matrix will be a linear combination of them. These observations will be important when considering supersymmetric extensions of purely bosonic models. The real spinors we consider are a special case of Majorana spinors. In an arbitrary representation of the Dirac matrices, a Majorana spinor is a spinor for which its Dirac conjugate = y i 0 is equal to its Majorana conjugate = T C, where C is the charge conjugation matrix, C C?1 =?( ) T. For our real spinors, C equals i 0. By denition, we raise and lower spinor indices with the epsilon symbols and and the northwest-southeast convention: = and =. W= e dene 12 = +1. Raising the indices of as stated, = , shows that also 12 = 1. The Dirac matrices with both spinor indices down are given by = ( ) 0 0. It is straightforward to check that =?( 0 ), so that ( ) = ( 0 ), which leads to ( ) = f?1l;? 3 ; 1 g : (10) The Dirac matrices with both indices up are obtained analogously From these equations it follows that ( ) = f1l; 3 ;? 1 g : (11) ( ) ( ) =?2 ( ) ( ) =? ( + ) : (12) 2 In Euclidean space, one of the Dirac matrices is necessarily complex and no real spinors can be dened. 7

8 Lowering spinor indices with in the last equation, we obtain ( ) ( ) =? ( + ) : (13) It is useful to write vectors as bispinors v = ( ) v v =? 1 2 ( ) v (14) v = 0 : In this way both spinors and vectors can be described with one formalism (spinor formalism). In 4 dimensions one can distinguish between \dotted" and \undotted" indices, for spinors can be decomposed into left- and right-handed parts, but (again) in odd dimensions there exists no matrix 5 and hence no chiral spinors exist. The normalization of the action for real spinors is as usual L =? 1 ; (15) and is chosen such that the hamiltonian is positive denite H = = 1 2 dx 1 dx 2 H =? 1 2 = i 2 = X ~k dx 1 dx 2 k dx 1 dx 2 0 dx 1 dx 2 T _ E( ~ k) c y ( ~ k) c( ~ k)? 1 2 : (16) Two identities that the Dirac matrices satisfy in three dimensions and that will be used in the sequel are and =? [ ] (17) = [ ] + +? : (18) It is instructive to check them by taking particular values for ; and. Another two identities for real spinors and that we will repeatedly use are = =? : (19) To prove them, it is enough to use the denition of : = T 2. Note that the second equation in (19) ensures that the lagrangian L in (15) is not a total derivative. 8

9 1.4 The simplest case: the Wess-umino model in x-space Suppose that we begin with a real scalar eld '(x) in (2+1)-dimensional Minkowski spacetime, and choose as its action the Klein-Gordon action S = d 3 x? 1 ' : (20) To make the system supersymmetric, we introduce a spin 1/2 fermion. We have already said that in 2+1 dimensions a spinor (x) has two components and describes one physical state. This gives equal numbers of states but not yet equal numbers of bosonic and fermionic eld components, since we have one bosonic eld component '(x) and two fermionic eld components (x). Thus we expect that we must add a bosonic auxiliary eld F (x). This suggests the free eld action S W = d 3 x? '? + 2 F 2 : (21) At this point it is not clear what the sign of the F 2 term is going to be, so we have introduced a coecient. From the previous section we are motivated to consider ' = F (22) + F ; with and constant coecients to be determined. We have scaled and F such that ' and F are normalized to respectively. Note that, since are real, the coecients and are real. We recall that in three dimensions there is no 5 and that the matrices f1l; g form a basis of the Cliord algebra. This accounts for the two terms in the SUSY transformation law for above, Lorentz covariance requiring the matrices in the rst term to be contracted Invariance of the action under the transformations (22) xes = 1 and =. Indeed, the variation of the Klein Gordon action gives ('), while the Dirac action varies into? and the F 2 term into the identities (19) implying then the thesis. Exercise 1: Show that the mass term S m = d 3 x m F '? 1 2 (23) is supersymmetric provided = 1 and = 1. Hence, invariance of both S W and S m requires that the F 2 term in (21) has positive sign, i.e. = 1. This has important consequences for SUSY breaking that we do not discuss in these notes. 9

10 Exercise 2: Show that the self-interaction term S g = d 3 x g (F ' 2? ') (24) is supersymmetric provided = 1 and = 1. To do this, note that the ( ) ( ) vanishes by itself, since ( ) is completely antisymmetric in all 3 's while there are only two independent 's. Thus, one can also use SUSY invariance of S W and S g to x = = = 1, so that ' = F (25) + F : (26) Exercise 3: Using eq. (22), derive the SUSY commutator [ 1 ; 2 ] ' = F? (1 $ 2) = 2 ( 2 1 ' (27) and show that exactly the same result holds for (x) and F (x) provided =. The elds '(x); (x) and F (x) ll up a real scalar supereld (x; ) living in superspace, (x; ) = '(x) + i (x) + i 2 F (x) ; (28) as we shall discuss in more detail in Section Supersymmetric Yang-Mills theory in x-space Next we consider Yang-Mills elds A a ; with a a gauge, Lie algebra index. Since A a describes one degree of freedom for xed index a, we add a real spinor eld a. Counting eld components shows that there are 2 fermionic eld components and 2 (and not 3) bosonic eld components. The reason is that gauge invariance can be used to gauge away one of the components of A a, for example A a 0 or any other combination. Hence, we do not need an auxiliary eld this time, and the action reads S YM = 1 m g 2 d 3 x? 14 F af a? 12 a (D= ) a ; (29) where m is a parameter with dimensions of mass, g is the dimensionless coupling constant, F a A A a + f abc A b Ac is the eld strength and D a c a c + f abc A b is the covariant derivative. The gauge transformations are g A a = (D ) a g a = f abc b c : (30) 10

11 It is straightforward to check that the gauge transforms of F a and (D ) a are g F a = f abc F b c and g (D ) a = f abc (D ) b c, from which the gauge invariance of the action follows. By writing in eq. (29) the overall factor 1=m, we have taken the coupling constant g to be dimensionless. In the literature one also nds the action (29) written without the factor 1=m. This corresponds to taking for g mass dimension [g] = 1=2. In either case, there is always a dimensionful parameter, which in our conventions is m. The parameter m can be used to ll the mass dimension gap between f and f discussed in Section 1.2, so that the SUSY transformation law for a may in principle have more terms than those given in eq. (2) for f. The most general SUSY transformation rules which are Lorentz covariant read A a = a a = F a A a + ma= a : (31) Note that a term F a in a is not independent because of the identity (17). The variation of the action (29) under (31) is given by S YM = 1 m g 2 d 3 x?f a (D A ) a? (D= ) a? 12 f abc a (A= b ) c : (32) We have used that ( a )(D= ) a is equal to a (D= ) a, as can easily be shown by partially integrating. The rst two terms in (32) are linear in and must cancel each other, while the last one is cubic in and must separately cancel. For the rst two, one nds after partial integration m (D F ) a a? 1 m a [ (D= F ) a? a? m (D= A ) a ] : (33) From this we already see that = = 0. We now use the identity (18). The term that arises with [ ] does not contribute due to the Bianchi identity? D[ F ] a = 0 ; (34) the term with does not contribute since F is antisymmetric, and the two remaining terms give each the same contribution? (D F ). Recalling that =?, we nd + 2 = 0. Hence, A a = a =? 2 F : (35) We still have to show that the third term in (14) cancels by itself. The same term appears in 4-, 6- and 10-dimensional supersymmetric theories, and always vanishes. In our case we must show that f abc ( a b ) ( c ) = 0 : (36) 11

12 Using that a = T i 0, that ( 0 ) = ( ) and eq. (13), we have for the left-hand side?f abc a b c ( + ) ; (37) which vanishes by (anti)symmetrization. For the SUSY commutator acting on A a we nd [ 1 ; 2 ] A a =? F a? (1 $ 2) : (38) We now use the identities (18) and (19), and obtain for the right-hand side? F a. Hence [ 1 ; 2 ] A a =?2 2 ( 2 1 A A a + f abc A b A c : (39) The SUSY commutator acting on the gauge eld gives thus a covariant translation. To interpret its meaning we split o the ordinary translation term A a and write the two remaining terms as a covariant derivative [ 1 ; 2 ] A a = 2 2 ( 2 1 A a? 2 2 ( 2 1 ) (D A ) a : (40) We have found a translation and a gauge transformation, the latter with parameter a =?2 2 ( 2 1 )A a. Thus, the algebra closes, but not only on translations: it also produces gauge transformations. To nd the same translation as for the W model we need 2 = 1, which we assume from now on. Thus, = 1. We choose the sign of a such that = 1. In other words, if =?1, we redene a as? a and get the same action and SUSY transformation laws as for = 1. Hence A a = a =? 1 2 F : (41) Note that one can not set = 1 by rescaling, since one could put together the W model and supersymmetric Yang-Mills theory and was already rescaled to normalize the SUSY transformation ' of the scalar led ' in the W model to. Coming back to eq. (40), in superspace one nds only a translation. However, if one chooses a so-called Wess-umino gauge, one needs to add compensating gauge transformations to the ordinary SUSY transformations in order to stay in this gauge, and these produce then the terms with gauge transformations in the SUSY commutator. Exercise 4: Check that for a one nds the same result as in (25), namely a covariant translation with (D ) a. Exercise 5: Suppose one were to add a mass term S m = d 3 x c 1 m (A a ) 2 + c 2 a a (42) 12

13 to the action S YM. Show by counting states that one would need another real physical spinor eld. Counting eld components, show that one would need one real auxiliary bosonic eld. All these elds ll up a real spinor supereld A a (x; ) = a (x) + H a (x) + V a (x) + i 2 1 a (x)? a (x) (43) where is essentially and V a is essentially A a, as we shall see in Sections 1.9 and The mass term (42) breaks gauge invariance, since (A a ) 2 is not gauge invariant. In three dimensions, however, it is possible to give a mass to Yang-Mills elds without breaking invariance under innitesimal gauge transformations by adding to the Yang-Mills action a Chern-Simons term Jackiw and Templeton (1981), Schonfeld (1981). We see this in the next section. 1.6 Supersymmetric Chern-Simons theory in x-space In three dimensions, out of the gauge eld A a, the and a dimensionless coupling constant g, one can construct the following local action invariant under gauge transformations (30) S CS = 1 g 2 d 3 1 x 2 A a f abc A a A b A c : (44) Here, as is usual in quantum eld theory, local means polynomial in the eld A a and its derivatives. This action is known as the Chern-Simons action and has eld equation F a = 0 : (45) As opposed to the Yang-Mills action, the Chern-Simons action is only invariant under innitesimal gauge transformations (30). Suppose that the gauge group is SU(N) and consider nite gauge transformations A! A h = h + g h?1 A h, where A = A a T a ; with T a antihermitean generators of the gauge Lie algebra. Then S CS is not invariant under large gauge transformations. Only the quantity e SCS is invariant, provided 4=g 2 is an integer Deser, Jackiw, and Templeton (1982). In these notes, however, we are concerned with perturbative quantization, so that we are only interested in innitesimal gauge transformations, for which there is no restriction on g. In what follows, unless stated otherwise, we will refer to innitesimal gauge transformations as gauge transformations. The same counting of states and components as for Yang-Mills theory implies that the supersymmetric extension of the Chern-Simons action will involve the ; 13

14 eld A a and the spinor eld a. In this case, it is straightforward to verify that the action S CS = 1 g 2 d 3 1 x 2 A a f abc A a Ab Ac? 12 a a ; (46) is gauge invariant and supersymmetric. Absence of dimensionful parameters and locality imply that the fermion a can only enter the action as a term a a, but does not x the coecient. Invariance under the SUSY transformations (41) xes the value of the coecient. Exercise 6: Show that indeed SUSY requires the coecient of the term a a to be -1/2. One can combine the Yang-Mills and Chern-Simons actions into one single action S YMCS = S YM + S CS : (47) The resulting theory is called Yang-Mills-Chern-Simons theory or topologically massive Yang-Mills theory. The action S YMCS is gauge invariant and gives, after gauge xing, a massive propagator for the eld A a. To see this, let us consider the nonsupersymmetric theory and work in the ordinary Landau A a = 0. The Faddeev-Popov procedure gives then for the gauge-xed classical action where the gauge-xing term S GF reads S GF = S = S YM + S CS + S GF ; (48) d 3 x [?b A a? (@ ^c a ) ( D c) a ] ; (49) b a is a Lagrange multiplier eld imposing the A a = 0, and ^c a and c a are the Faddeev-Popov antighost and ghost elds. The part of the gauge-xed action quadratic in A a and b a has in momentum space the form where? 1 2 d 3 p (2) 3 h A a (p) K (p) A a (?p) + b a (p) p A a (?p) K (p) =? p + i m i ; (50)? p 2? p p : (51) This denes the kinetic matrix of A a and b a as K T (p) = (p)?p p 0 : (52) 14

15 The propagator matrix (p) = (p) (p) (?p) 0 (53) is the result of inverting T (p): T (p) (p) = : (54) To nd (p), we write for (p) and (p) the most general expressions compatible with Lorentz covariance, (p) = f 1 p + f 2 + f 3 p p (p) = f 4 p ; (55) with f 1 ; : : : ; f 4 functions of p 2 and m to be determined, and impose eq. (54). One thus nds and (p) =? mg 2 p 2 (p 2 +m 2 m p + i p 2? i p p?io) (56) (p) = p p 2 : (57) We see that the propagator (56) has a pole at p 2 =?m 2, which shows that the gauge eld has a mass. In Section 2.1 we will consider a supersymmetric gauge in which the propagator of the gauge eld is the same as in (56). The propagator (56) has been obtained in three dimensions. When we dene dimensional regularization, we will use the original 't Hooft-Veltman prescription, Hooft and Veltman (1972) and Breitenlohner and Maison (1977), for in n dimensions, which is the only algebraically consistent one known to date. We will see that this prescription introduces in the n-dimensional propagator extra terms which vanish for n=3 and which loosely speaking can be regarded as proportional to n? Three-dimensional rigid superspace Having a symmetry between bosonic and fermionic elds suggests also to consider a symmetry between bosonic coordinates x and new fermionic coordinates. The simplest choice are \spin 1/2 coordinates", with = 1; 2. Since x are real, we take also real. According to eqs. (1) and (2), under SUSY, x should vary into. Hence [] =?1=2, just as [] =?1=2. The reverse law would x (q 1 + q 2 ), with q 1 and q 2 constants, but x =, this simplies to. Hence with p and q real constants. x = p = q ; (58) 15

16 We denote the derivative with respect to : Noting satises f@ ; g =, it is clear that (@ ) y Similarly, from [@ ; x ] = it follows that (@ ) y =?@. Since both x can be written as bispinors by means of eq. (14), x = ( ) x = ( and since the matrices ( ) are real, we also have (@ ) y =?@. Fields (x; ) dened on superspace are called superelds and are functions of both coordinates x and. A supereld will have an expansion in powers of, with terms of order 0, 1 and 2 in. This is so since the coordinates anticommute and there are two such coordinates ( = 1; 2), so that one can have at most products =?2 1 2 of two 's. For example, for a scalar supereld (x; ), one has (x; ) = '(x) + i (x) + i 2 F (x) ; (59) where the coecients '; and F are elds dened on x-space, usually called component elds. The SUSY transformations (58) can be viewed as a translation in superspace. Superelds (x; ) will then transform with respect to SUSY as scalars, i.e. only with orbital parts but not with spin parts. In other words, 0 (x 0 ; 0 ) = (x; ), where x 0 = x + p and 0 = + q. The SUSY generator Q, called supercharge, will therefore be such that (x; ) = Q (x; ) : (60) Note that Q must be a spinor operator, for SUSY transformations are linear in. In order that the commutator of two SUSY transformation gives a translation, we claim that we need Q : (61) To see this, let us take a scalar supereld (x; ). Using the expansion (59) and acting with on it, we get on the one hand (x; ) = ) (x; ) = i? '(x) + i F (x)? 1 2 ; (62) and on the other (x; ) = '(x) + i (x) + i 2 2 F (x) : (63) In deriving (62), we have used that =? 1 2 2, where 2 denotes 2. Comparing eqs. (62) and (63), we have ' = i = (?i ) = (64) ' + F, + F (65) F =?i = ; (66) 16

17 in accordance with eq. (25). From these transformation laws, and using eqs. (19), the SUSY commutators [ 1 ; 2 ] 8 < : ' F 9 = ; = 2 ( < : follow. The observant reader may notice that ' F 9 = ; (67) [ 1 Q ; 2 Q ] = 2 1 fq ; Q g = 2 1 (?2i@ ) =?2 2 (68) has opposite sign. The reason is that eq. (61) gives a representation for the supercharge Q as a Lie derivative and the generator P of translations is represented also by the Lie and minus the Lie derivatives form (on general coset manifolds) a representation of the algebra. For example, f@? g = 2i ( ) (?@ ): (69) From either (68) or (69) it follows that in superspace the commutator of two SUSY transformations yields only a translation and no gauge transformation. As always in eld theory, it is useful to introduce the notion of covariant derivatives. Here this means derivatives, denoted by D and D, which (anti)commute with the Lie derivatives Q It is very easy to nd that they are given by D + D : (70) [The theory of coset manifolds can be applied to the coset (P + Q + M)=M, where M is the Lorentz subalgebra, nding that the Lorentz connections on Q ; P ; D and D all vanish]. Summarizing so far: Superspace is parameterized by coordinates x and, superelds (x; ) transform as (x; ) = Q (x; ), where Q is the supercharge, and there exist covariant derivatives D + and D such that fd ; Q g = 0 : (71) Hence D = Q (D ) = D ( Q ). Furthermore, since ( a ) y =, (@ ) y and (@ ) y =?@, one has (D ) y = D. It is clear that fd ; D g = 2i@ [D ; D ] =? D 2 [D ] = 0 ; (72) where D 2 D D. Three-dimensional N = 1 superspace is much simpler than four-dimensional N = 1 superspace. There are no chiral superelds, and hence no representation \preserving constraints". We recall that, as already mentioned, the notion of chirality does not exist in an odd number of dimensions. Imagine one were nevertheless to dene a chiral supereld by the condition D 1 = 0. Then 17

18 D 1 D 1 = i@ 11 = 0, 11 = =?(@ 0 1 ). This restricts the x-dependence of, which is inadmissible. Another simplication in three dimensions is due to the simple fact that objects with three spinor indices which are totally antisymmetric vanish. Namely, for any object O one has the identity O + O + O? O? O? O = 0 : (73) Although this follows trivially from the observation that spinors in three dimension have only two indices, it leads to many simplications. For example, taking O = D D D and contracting with, we nd D D D + D D D + D D D = 0 : (74) If one next writes D D =?D D + fd ; D g in the rst term and D D =?D D + fd ; D g in the second term, the two terms with an anticommutator cancel each other, fd ; D gd + D fd ; D g = [fd ; D g; D ] = 2i [@ ; D ] = 0, and one is left with?d D D? D D D + D D D = 3D D D = 0. Hence From this fundamental identity, others follow; e.g. D D D = 0 : (75) D D 2 + D 2 D = 0 : (76) The measure in superspace is d 3 xd 2, where d 3 x is real and has mass dimension [d 3 x] =?3 while d 2?2d 1 d 2 is imaginary and has mass dimension [d 2 ] = 1. The normalization factor?2 in the denition of d 2 has been introduced for convenience (see below). Integration over Grassmann variables is dened by d = 0 d = 1 : (77) In the case we are considering here of two Grassmann coordinates, we have d a = 0 Thus, in an integral d a = ) d 2 = 2 : (78) d 3 x d 2 F (x; ), integration over d 2 picks the term in F (x; ) quadratic in 's. This coincides precisely with the result of acting with D 2 on F (x; ) and taking afterwards = 0, the reason for this being that D 2 ( ) = 2. Hence one has d 3 x d 2 F (x; ) = d 3 x D 2 F (x; ) ; (79) 18

19 where the vertical bar denotes restriction to = 0. With another choice of normalization for d 2, this identity would have to be modied accordingly. Let us consider the action S = d 3 x d 2 L(; D ; D D ; : : : ) ; (80) with L a lagrangian that does not depend explicitly on coordinates. Under a SUSY transformation, the variation of L is L = Q L. The L that arises from in Q is made of terms which are order zero and one in and which, therefore, vanish upon integration over d 2. Similarly, the term L that arises from taking in Q gives rise to a total spacetime derivative which can be ignored. Having L = 0, one concludes that the action S is supersymmetric: S = The Wess-umino model in superspace Since actions are dimensionless (we set ~ = 1) and d 3 x d 2 has mass dimension?2, to obtain the superspace action for the W multiplet, we need a lagrangian L W with mass dimension 2. The scalar supereld (x; ) in eq. (59) has two scalars, ' and F, and one spinor,. In three dimensions, and assuming that there are no dimensionful parameters, a scalar eld has mass dimension 1/2, and a spinor eld has mass dimension 1. This and the fact that [ ] = 1=2 forces us to take ' as the scalar with mass dimension 1/2, since only then a has mass dimension 1. Thus [] = ['] = 1=2. Recalling tha= t [D ] = 1=2, we see that L W = (D )(D ) has the correct mass dimension. Furthermore, because L W is a function of and D, the argument given at the end of the last section implies that S W = 1 8 d 3 x d 2 (D ) (D ) (81) is supersymmetric, where the factor 1/8 has been introduced for convenience. We can also add a mass term L m = m 2 and a self-coupling L g = g 3. Note that [m] = 1 but [g] = 1 2. To obtain the component action from the supereld action (81), we use eq. (79): S W = 1 4 d 3 x (D ) (D 2 D )? (D D ) (D D ) : (82) If we write D 2 D in the rst term as D 2 D = D (?D D + fd ; D g) = recast D D in the second term as D D = 1 2 fd ; D g [D ; D ] = 1 2 D 2 and note that ' = =?id F = i 2 D2 ; (83) 19

20 we obtain S W = d 3 x? 1 4 (@ ') (@ ')? i F 2 : (84) This is precisely the W action (21). Note that the F 2 term comes out indeed with a positive sign. 1.9 The covariant approach to Yang-Mills theory To describe Yang-Mills theory in superspace, we need a supereld with a spin 1 eld. The real scalar supereld (x; ) in (59) can therefore not be used. The spinor supereld A(x; ) = (x) + H(x) + V (x) + i 2 1 (x)? (x) ; (85) contains a vector V a and hence can be taken as starting point. Because we want to construct covariant derivatives and D, we consider A as the spinor part of a vector superconnection A M = fa ; A g A = ( ) A : (86) The connections are Lie algebra valued A M = A a M T a ; (87) with T a the antihermitean generators of the gauge Lie algebra T cy =?T c [T a ; T b ] = f abc T c : (88) A = A : (89) In order that the vector eld V a be real, Aa must be real. Then also a ; H a and a are real elds. Once we have a superconnection, we dene a gauge covariant superderivative and use it to construct gauge transformations. Since D is real, as we already saw, we dene the spinor part r of the gauge covariant superderivative r M by r D + ia : (90) Note that the i in front of A is needed because A is Lie algebra valued and the generators T a are antihermitean. We dene the vector part of the gauge covariant superderivative by + A : (91) 20

21 is imaginary, A a imaginary; note that there is no i in front of A. Gauge transformations are dened by g (ia ) = r = D + i [A ; ] (92) g A = r + [A ; ] ; (93) where = a T a with a real. The covariant derivatives themselves transform covariantly g r = [r ; ] g r = [r ; ] : (94) In general, given a covariant derivative r M, the supertorsion T MN P and the group supercurvature F MN are dened by [r M ; r N g = T MN P r P + F MN ; (95) where [a; bg is the graded commutator, equal to fa; bg if both a and b are fermionic, and equal to [a; b] otherwise. Explicit evaluation gives that only T ; is nonvanishing and yields with fr ; r g = 2ir + F ; (96) F = id A + id A? fia ; A g? 2iA : (97) The unusual term?2ia ensures that F transforms covariantly under gauge transformations. Indeed, under a gauge transformation (92)-(93), some straightforward algebra shows that g F = [F ; ]. The presence of A in A can be understood by noting that rigid superspace, though at, has a nontrivial spin connection. The inverse rigid vielbeins E M (0) and E M (0) follow from D = E M M ; D = E M M, and read o E M (0) = n ; i 0 ( ) 0 E M (0) = f0 ; g : (98) If one changes the basis from fa M g to f AM ~ g, with A = E M (0) AM ~ and A = E M (0) AM ~, the curvature takes on the usual Yang-Mills form, as one may check. The connection in r is A, but one may always add a tensor O that transforms covariantly under gauge transformations, since the new connection A 0 = A + O will also transform as g A 0 = r0. If we go back to the beginning and start with the modied connection A 0 = A + 1 2i F, we then end up with fr ; r g = 2 = ir 0. Thus, by a redenition of the vector connection we have obtained F = 0, and from F = 0 we have th= at A 0 = 1 h i D A + D A + i fa ; A g : (99) 2 21

22 Hence we have imposed the conventional constraint F = 0, which is simply an allowed redenition of A. From now on we drop primes. Only A is left as an independent eld, while A is expressed in terms of A by A = 1 h i D A + D A + i fa ; A g : (100) 2 Clearly, A a is real. Next we study the Bianchi identities We rst look at the identity [ r M ; [ r N ; r L g g + cyclic = 0 : (101) [r ; fr ; r g ] + cyclic = 0 : (102) From this equation, the anticommutator fr ; r g = 2ir and [r ; r ] = F ;, we get F ; + F ; + F ; = 0 : (103) This, the decomposition F ; = 1 h i ( F ; + F ; + F ; ) + ( F ;? F ; ) + ( F ;? F ; ) 3 (104) and F ;? F ; = F ; (105) implies that F ; = 1 3 h i (?F ;) + (?F ;) : (106) The object F ; is the basic supereld strength in the theory. For reasons to become clear, we normalize it as Then we have [ r ; r ] = F ; =? 3 2 W : (107) [r ; r ] = F ; = 1 2 W W : (108) The eld strength is thus given by a (graded) commutator of two covariant derivatives, as in ordinary Yang-Mills theory, but not by fr ; r g, which only yields a torsion term, but rather by [r ; r ]. The third commutator that can be formed with the covariant derivatives, namely [r ; r ], gives the derivative of the eld strength, as we show below. Note that W is real because r is real 22

23 and r is imaginary. Hence W a is imaginary. Explicit evaluation using (107) and the relation in eq. (100) yields W =?D D A? i [A ; D A ] [A ; fa ; A g] : (109) Another expression for W is W = i r r r : (110) To derive it, apply eq. (73) to O = r r r to nd and use this in the denition of W r r r + r r r + r r r = 0 ; (111) W = i 3 [r ; fr ; r g] : (112) Note that the right-hand side in eq. (110) denes a function in superspace, not an operator. This is so since, as a result of the basic identity D D D = 0, no free derivatives are left in r r r. The easiest way to check this is to rst act with r r r on a function, and then show that the expression r r r contains no derivatives of. From eq. (94) it follows that W is covariant since it transforms covariantly under gauge transformations g W = [W ; ] : (113) In x-space, the variation (any variation, not necessarily a gauge variation) of a curvature is the covariant derivative of the variation: F = D (A )? D (A ). The same holds in superspace: W a =?r r (A ). This follows easily from (107) if one uses that A = 1 2 (r A + r A ), which in turn arises from (100). The next Bianchi identity we study is fr ; [r ; r ]g + [r ; fr ; r g]? fr ; [r ; r ]g = 0 : (114) It can be used to express F ; 1 2i [fr ; r g; r ] = ( ) ( ) A + [A ; A ] ) (115) in terms of W. We begin by decomposing the curvature F ; into the sum of terms symmetric in ; and terms antisymmetric in ;. From the denition of F ; in eq. (115) it follows that F ; = cf ; =?F ;, which in turn implies that the terms in F ; symmetric (respectively antisymmetric) in ; are antisymmetric (respectively symmetric) in ;. This allows us to write without loss of generality F ; = f + f ; (116) 23

24 with f symmetric in its indices. We could have decomposed F ; using other pairs of indices, with the rst index in f; g and the second index in f; g. For example, we could have decomposed in ; and written F ; = f + f : (117) Tracing eq. (114) with and using eq. (116) yields Recalling now eq. (108) we nd Hence fr ; [r ; r ]g? fr ; [r ; r ]g =?4if : (118)?2 r W r W =?4if : (119) r W = 0 f = 1 2i r (W ) : (120) Exercise 7: Verify that the remaining Bianchi identities [r ; [r ; r ] ] + cyclic = 0 [r ; [r ; r ] ] + cyclic = 0 (121) give no further information. Hint: substitute (116) and (108) and then use that r W = 2if. Let us now obtain the gauge action. Recalling that g is in our conventions dimensionless and that W a has mass dimension 1/2, an action which is gauge and super Poincare invariant and has the correct mass dimension is given by Gates Jr., Grisaru, Rocek, and Siegel (1983) and Siegel (1979b) S YM = c g 2 d 3 x d 2 W a W a ; (122) with c a constant. Using W =?r r A and integrating by parts, the eld equations are found to be given by r r W = 2ir W = 0 : (123) To nd the component content of S YM, we use again d 2 = D 2, but we may replace D 2 by r 2 = r r since the action is gauge invariant. In other words, for a gauge invariant action, d 3 x d 2 L = d 3 x D 2 L = d 3 x r 2 L : (124) We obtain then S YM = 2c d 3 x W a r 2 W a? (r W a ) (r W a ) : (125) 24

25 Using r W = 0 and the identity (73), one gets for r 2 W a in the rst term r 2 W + 2ir W = 0 : (126) From this and eq. (120) it follows S YM = 2c? d 3 x?2i W a r W a + 4f a f a : (127) Noting the relations in eq. (13) for ( ), we obtain F ; F ; = ( ) ( ) F ( ) ( ) F = 4 F F = ( f + f ) ( f + f ) = 4 f f : (128) Recalling that W a is imaginary and noting eq. (115), we dene a = i 2 W a A a = A a : (129) Finally, using that i a ( ) b = a b and making the choice c =?1=32 g 2, we obtain S YM = 1 mg 2 d 3 x? 14 F a F a? 12 a (D= ) a : (130) This is indeed the component action of eq. (29). The last subject we wish to study in Yang-Mills theory are the SUSY transformation laws. The fact that f a and W a transform covariantly under gauge transformations suggests to use covariant derivatives r for the SUSY transformations. Thus we write 0 = r. The result of acting with SUSY on any gauge covariant quantity consists of the sum of the usual SUSY transformation going with D (whose commutator yields an ordinary translation) plus a gauge transformation (which leads to terms quadratic in superelds). The invariance of the action S YM in eq. (122) under 0 follows from its gauge invariance and the fact that ( 0? )W a = g W a. Using 0 we have or in vector notation 0 a = i 2 r W a =?i f a ; (131) 0 a =? 1 2 F a : (132) To nd the SUSY transformation law for the eld A a we note that the action of A on any supereld, with an arbitrary variation, is given by [A ; ]. This and the identities [A ; ] = (r )? r () r r? r r = [r ; r ] = 1 2 ( W + W ) (133) 25

26 implies that 0 A a =? 1 2 ( ) 0 A a = i ( ) a = a : (134) Eqs. (132) and (134) are the x-space transformation rules of (41) The covariant approach to Chern-Simons theory The non-supersymmetric Chern-Simons action (44) can also be written as S CS = 1 d 3 x A a F a 4? 1 3 f abc A b Ac : (135) In superspace we therefore expect an action of the form S CS = i g 2 d 3 x d 2 A a c 1 W a + ic 2 f abc A b (D A c ) + c 3 f abc f cde A b A d Ae ; (136) with c 1 ; c 2 and c 3 real coecients. Invariance under gauge transformations g (ia a ) = (r ) a requires c 2 = c 1 =3 and c 3 =?c 1 =6, which gives Gates Jr. et al. (1983), Siegel (1979b) S CS = ic 1 g 2 d 3 x d 2 (D A a )(D A a ) + 2i? 1 6 f abc f cde A a A b A d A e 3 f abc A a A b (D A c ) : (137) Another way to obtain this expression is the following. We expect the eld equation F a = 0 for the nonsupersymmetric theory to generalize to W a = 0. Any action which under an arbitrary variation yields S CS d 3 x d 2 W a A a (138) will be gauge invariant, since g (ia a ) = (r ) a and r W a = 0. Hence it is enough to construct an action of the form (136) whose variation is (138). The answer is eq. (137). To nd the component expression for the action (137),we make use of the fact that the action is gauge invariant to set A j = 0 D A a j = 0 ; (139) which denes a Wess-umino gauge. The point is that these two conditions can be imposed by suitably choosing the components D a j and D 2 a j of the supereld a in g (ia ) = r a, while leaving the component a j arbitrary, which is the only one that enters the gauge transformation laws of the physical elds A a and a. Indeed, from g (ia a )j = D a j+if abc A b c j it follows that it 26

27 is enough to take D a j = 0 in order to have A a j = 0. Similarly, onc= e we have A a j = 0 and D a j = 0, it follows from g (id A a )j = D 2 a j+if abc D (A b c ) j that, to have D A a j = 0, it is enough to take D2 a j = 0. Note, however, that no restriction has been imposed on a j, which according to eqs. (93) and (113) is the only component of a that enters in g A a and g a. Then, in the Wess- umino gauge (139), A a = A a = D ( A a ) and the action becomes S CS = 2ic 1 g 2 d 3 x a = i 2 W a =? i 2 D D A a (D 2 D A a ) (D A a ) + (D D A a ) (D D A a )? 2i 3 f abc (D A a ) (D A b ) (D A a ) Furthermore, using eqs. (75) and (76) to derive and noting : (140) (141) D 2 D A = 2i@ D A D 2 A = 2D D A? A ; (142) D D A = i@ A? 1 2 D 2 A D A = D ( A )? 1 2 D A ; we have S CS = 2ic 1 g 2 d 3 x 2i (@ A a ) A a + (D D A a ) (D D A a ) + 2i 3 f abc A a A b A c : Finally, recalling eq. (140) and taking c 1 = 1=16, we arrive at S CS = 1 g 2 d 3 x 1 2 A a f abc A a A b A c? 12 a a (143) (144) : (145) This is the component action of eq. (46). Here we have used the Wess-umino gauge (139) to derive the component form of action from the supereld form (137). We must emphasize, though, that the same component action is obtained if one does not make assumptions about the components of the supereld A a. To prove this, one directly integrates (137) over d 2 using d 2 = D 2 j and expresses everything in terms of A and W. a It is very important to keep this in mind since in Section 2.2 we will work in a supersymmetric gauge which imposes dierent conditions on A a j and Da A a j. One may dene the components of the supereld A a by a = A a V a = D ( A a ) H a = 1 2 D A a a = i 2 D D A a : (146) 27

28 Note that up to this moment we have not used any explicit form for the spinor supereld A a as an expansion in powers of a, but only the fact that it contains a vector V a. The denition of the components given here reproduces the - expansion in eq. (85). Using the expressions of A a and W a in eqs. (100) and (109), the physical elds A a and a can be written in terms of a ; H a ; V a and a as A a = V a f abc b c (147) a = a f abc H b c? 1 2 f abc A= b c f abc f cde b ( d e (148) ) The transformation laws of a ; H a ; V a and a under SUSY as given by = Q are linear in elds and read a = V= a? H a H a =? a V a = ( a ) a a V a : (149) From these and the expressions in eqs. (147) and (148), we get the transformations rules A a = a + (D ) a a =? 1 2 F a + f abc b ( c ) (150) By subtracting a gauge transformation with parameter a, we obtain the usual x-space rules (41) for A a and a. The same result is obtained in superspace if one adds a compensating gauge transformation which keeps one in the Wess- umino gauge a = H a = Higher N models and gauge couplings to matter One can construct rigidly supersymmetric models with N 8 SUSY. One way to obtain them is by dimensional reduction from the d = models where rigid SUSY exists for N 4. For example, the N = 2 Wess-umino model in d = corresponds to the N = 1 model in d = and contains two real spinors, two real scalars and two auxiliary elds. It can clearly be written in complex notation as a model with one complex scalar, one complex spinor and one complex auxiliary eld. The reader may check that is invariant under SW = d 3 x?(@ ' y ) (@ + F y F (151) ' = + F F : (152) 28

29 One may consider and as independent parameters, and consider separately the variations of the action with and. For example, for the variation with, one nds (')? + F )? F (@ ) ; (153) which clearly cancels after partial integration. Similarly, one can write the action for the N = 2 Yang-Mills and Chern- Simons models. In this case, the N =2 multiplet consists of the gauge eld A a ; two real spinors a i (i = 1; 2) and two real auxiliary elds C a and D a, and the actions have the form S YM = 1 mg 2 d 3 x? 1 4 F a F a? 1 2 a 1(D= i ) a? 1 2 (D C) a (D C) a (Da ) 2? 1 2 f abc ij a i b j Cc (154) and S CS = 1 g 2 d 3 x 1 2 A a f abc A a Ab Ac? 12 a a + C a D a ; (155) where 12 = 1. The SUSY transformation rules that leave these action invariant are A a = i a i a i =? 1 2 F a i + ij D a j + ij D= C a j C a =? ij i a j (156) D a =? ij i D= a j + f abc i b i Cc : It is also possible to set a supereld formalism for N =2 SUSY, Aragone (1983) and Ivanov (1991), but we will not discuss this here. The N = 2 actions S YM and S CS can also be obtained from a truncation of corresponding N =3 actions in 2+1 dimensions Kao, Lee, and Lee (1996). Exercise 8: Verify that S Y M and S CS in (154) and (155) are invariant under the transformations (156). So far we have discussed supersymmetric models for scalar elds and for gauge elds. It possible to construct supersymmetric models for matter elds coupled to gauge elds. Although this subject lies outside the scope of these notes, let us briey mention how to couple the N = 1 Wess-umino model for scalars to gauge elds while preserving SUSY. To do this, one puts the scalars in a particular representation R with generators (T a ) i j and replaces in x-space 29