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TO FERMILABPUB96/445T
INTRODUCTION SUPERSYMMETRY
arXiv:hepth/9612114v1 11 Dec 1996
JOSEPH D. LYKKEN Fermi National Accelerator Laboratory P.O. Box 500 Batavia, IL 60510
These lectures give a selfcontained introduction to supersymmetry from a modern perspective. Emphasis is placed on material essential to understanding duality. Topics include: central charges and BPSsaturated states, supersymmetric nonlinear sigma models, N=2 YangMills theory, holomorphy and the N=2 YangMills function, supersymmetry in 2, 6, 10, and 11 spacetime dimensions.
1
Introduction Never mind, lads. Same time tomorrow. We must get a winner one day. Peter Cook, as the doomsday prophet in The End of the World.
Supersymmetry, along with its monozygotic sibling superstring theory, has become the dominant framework for formulating physics beyond the standard model. This despite the fact that, as of this morning, there is no unambiguous experimental evidence for either idea. Theorists nd supersymmetry appealing for reasons which are both phenomenological and technical. In these lectures I will focus exclusively on the technical appeal. There are many good recent reviews of the phenomenology of supersymmetry. 1 Some good technical reviews are Wess and Bagger, 2 West, 3 and Sohnius. 4 The goal of these lectures is to provide the student with the technical background requisite for the recent applications of duality ideas to supersymmetric gauge theories and superstrings. More specically, if you absorb the material in these lectures, you will understand Section 2 of Seiberg and Witten, 5 and you will have a vague notion of why there might be such a thing as M theory. Beyond that, youre on your own. 2 2.1 Representations of Supersymmetry The general 4dimensional supersymmetry algebra
A symmetry of the Smatrix means that the symmetry transformations have the eect of merely reshuing the asymptotic single and multiparticle states. The known symmetries of the Smatrix in particle physics are: 1
Poincar invariance, the semidirect product of translations and Lorentz e rotations, with generators Pm , Mmn . Socalled internal global symmetries, related to conserved quantum numbers such as electric charge and isospin. The symmetry generators are Lorentz scalars and generate a Lie algebra,
j [B , Bk ] = iCk Bj j where the Ck are structure constants.
,
(1)
In 1967, Coleman and Mandula 6 provided a rigorous argument which proves that, given certain assumptions, the above are the only possible symmetries of the Smatrix. The reader is encouraged to study this classic paper and think about the physical and technical assumptions which are made there. The ColemanMandula theorem can be evaded by weakening one or more of its assumptions. In particular, the theorem assumes that the symmetry algebra of the Smatrix involves only commutators. Weakening this assumption to allow anticommuting generators as well as commuting generators leads to the possibility of supersymmetry. Supersymmetry (or SUSY for short) is dened as the introduction of anticommuting symmetry generators which transform in the ( 1 , 0) and (0, 1 ) (i.e. spinor) representations of the Lorentz group. Since 2 2 these new symmetry generators are spinors, not scalars, supersymmetry is not an internal symmetry. It is rather an extension of the Poincar spacetime e symmetries. Supersymmetry, dened as the extension of the Poincar symmee try algebra by anticommuting spinor generators, has an obvious extension to spacetime dimensions other than four; the ColemanMandula theorem, on the other hand, has no obvious extension beyond four dimensions. In 1975, Haag, Lopusza ski, and Sohnius 7 proved that supersymmetry n is the only additional symmetry of the Smatrix allowed by this weaker set of assumptions. Of course, one could imagine that a further weakening of assumptions might lead to more new symmetries, but to date no physically compelling examples have been exhibited. 8 This is the basis of the strong but not unreasonable assertion that: Supersymmetry is the only possible extension of the known spacetime symmetries of particle physics. In fourdimensional Weyl spinor notation (see the Appendix) the N supersymmetry generators are denoted by QA , A=1, . . .N. The most general four 2
Discrete symmetries: C, P, and T.
dimensional supersymmetry algebra is given in the Appendix; here we will be content with checking some of the features of this algebra. The anticommutator of the QA with their adjoints is: {QA , QB } =
m A 2 Pm B
.
(2)
1 To see this, note the righthand side of Eq. 2 must transform as ( 2 , 1 ) under 2 the Lorentz group. The most general such object that can be constructed out of Pm , Mmn , and B has the form: A m Pm CB
,
A where the CB are complex Lorentz scalar coecients. Taking the adjoint of the lefthand side of Eq. 2, using m m =
, , (3)
QA
= Q
A
A tells us that CB is a hermitian matrix. Furthermore, since {Q, Q} is a positive A denite operator, CB is a positive denite hermitian matrix. This means that A A we can always choose a basis for the QA such that CB is proportional to B . The factor of two in Eq. 2 is simply a convention. The SUSY generators QA commute with the translation generators:
[QA , Pm ] = [Q , Pm ] = 0
A
.
(4)
This is not obvious since the most general form consistent with Lorentz invariance is: QA , Pm [Q
A
=
A ZB m Q A ZB
B
, Pm ] =
QB m
,
(5)
A where the ZB are complex Lorentz scalar coecients. Note we have invoked here the Haag, Lopusza ski, Sohnius theorem which tells us that there are no n 1 ( 1 , 1) or (1, 2 ) symmetry generators. 2 A To see that the ZB all vanish, the rst step is to plug Eq. 5 into the Jacobi identity: (6) [QA , Pm ], Pn + (cyclic) = 0 .
Using Eq. 240 this yields:
4i (ZZ )B mn QB = 0 , A
(7)
3
Here we have used the fact that the rhs must transform as (0, 0) + (1, 0) under the Lorentz group. The spinor structure of the two terms on the rhs is antisymmetric/symmetric respectively under , so the complex Lorentz scalar matrices X AB and Y AB are also antisymmetric/symmetric respectively. Now we consider contracted on the Jacobi identity {QA , QB }, Pm + [Pm , QA ], QB [QB , Pm ], QA = 0 . (9)
which implies that the matrix ZZ vanishes. A This is not enough to conclude that ZB itself vanishes, but we can get more information by considering the most general form of the anticommutator of two Qs: mn {QA , QB } = X AB + Mmn Y AB . (8)
The complex Lorentz scalars X AB are called central charges; further manipulations with the Jacobi identities show that the X AB commute with the QA , QA , and in fact generate an Abelian invariant subalgebra of the compact Lie algebra generated by B . Thus we can write: X AB = aAB B , (12)
A and thus ZB is symmetric. Combined with ZZ =0 this means that ZZ =0, A which implies that ZB vanishes, giving Eq. 4. Having established Eq. 4, the symmetric part of the Jacobi identity Eq. 9 implies that Mmn Y AB commutes with Pm , which can only be true if Y AB vanishes. Thus: {QA , QB } = X AB . (11)
Since X AB commutes with Pm , and plugging in Eqs. 2,5,232 and 233, the above reduces to 4 Z AB Z BA Pm = 0 , (10)
where the complex coecients aAB obey the intertwining relation Eq. 264. 2.2 The 4dimensional N=1 supersymmetry algebra
The Appendix also contains the special case of the fourdimensional N=1 supersymmetry algebra. For N=1 the central charges X AB vanish by antisymmetry, and the coecients S are real. The Jacobi identity for [[Q, B ], B ] implies k that the structure constants Cm vanish, so the internal symmetry algebra is Abelian. Starting with [Q , B ] [Q , B ] = S Q = S Q 4 , (13)
it is clear that we can rescale the Abelian generators B and write: [Q , B ] = Q [Q , B ] = Q . (14)
Clearly only one independent combination of the Abelian generators actually has a nonzero commutator with Q and Q ; let us denote this U (1) generator by R: [Q , R] = Q [Q , R] = Q . (15)
Thus the N=1 SUSY algebra in general possesses an internal (global) U (1) symmetry known as R symmetry. Note that the SUSY generators have Rcharge +1 and 1, respectively. 2.3 SUSY Casimirs
Since we wish to characterize the irreducible representations of supersymmetry on asymptotic single particle states, we need to exhibit the Casimir operators. It suces to do this for the N=1 SUSY algebra, as the extension to N>1 is straightforward. Recall that the Poincar algebra has two Casimirs: the mass operator e P 2 = Pm P m , with eigenvalues m2 , and the square of the PauliLjubansk i vector 1 (16) Wm = mnpq P n M pq .
2
W has eigenvalues m s(s + 1), s=0, 1 , 1, . . . for massive states, and Wm = 2 Pm for massless states, where is the helicity. For N=1 SUSY, P 2 is still a Casimir (since P commutes with Q and Q), but W 2 is not (M does not commute with Q and Q). The actual Casimirs are P 2 and C 2 , where
2 2
C2 Cmn Bm
= Cmn C mn = Wm
1
, , . (17)
= Bm Pn Bn Pm
4
Q m Q
This is easily veried using the commutators:
[Q m Q , Q ] [Wm , Q ] = imn Q P n
, , (18)
= 2Pm Q + 5
4inm P n Q
which imply: [Cmn , Q ] = = 2.4 [Bm , Q ]Pn [Bn , Q ]Pm 0 .
(19)
Classication of SUSY irreps on single particle states
We now have enough machinery to construct all possible irreducible representations of supersymmetry on asymptotic (onshell) physical states. We begin with N=1 SUSY, treating the massive and massless states separately. Unlike the case of Poincar symmetry, we do not have to consider tachyons they are e forbidden by the fact that {Q, Q} is positive denite. N=1 SUSY, massive states We analyze massive states from the rest frame Pm = (m, 0). We can write: C2 Ji = 2m4 Ji J i , 1 Si Qi Q 4m
,
(20)
where Si is the spin operator and i is a spatial index: i = 1, 2, 3. Both Si and i obey the SU (2) algebra, so [Ji , Jj ] = iijk Jk , (21)
and J 2 has eigenvalues j (j + 1), j equal integers or halfintegers. The commutator of Ji with either Q or Q is proportional to P and thus vanishes since we are in the rest frame. Q , Q are in fact two pairs of creation/annihilation operators which ll out the N=1 massive SUSY irrep of xed m and j : 10 0 . (22) {Q , Q } = 2m = 2m 01 Given any state of denite m, j we can dene a new state Q1   = Q2  = Q1 Q2 m, j , 0. (23) =
Thus  is a Cliord vacuum state with respect to the fermionic annihilation operators Q1 , Q2 . Note that  has degeneracy 2j +1 since j3 takes values j, . . . j . 6
Acting on  , Ji reduces to just the spin operator Si , so  is actually an eigenstate of spin:  = m, s, s3 . (24) Thus we can characterize all the states in the SUSY irrep by mass and spin. It is convenient to dene conventionally normalized creation/annihilation operators: a1,2 a , 2 1 = = 1 Q1,2 2m 1 Q1,2 2m , . (25)
Then for a given  the full massive SUSY irrep is: a 1 a 2   (26) 1 = a a  21 2
 1 a1 a2  2
There are a total of 4(2j +1) states in the massive irrep. We compute the spin of these states by using the commutators: S3 , a 2 a 1 = 1 2 a 2 a 1 . (27)
1 Thus for  = m, j, j3 we get states of spin s3 = j3 , j3 2 , j3 + 1 , j3 . 2 As an example, consider the j =0 or fundamental N=1 massive irrep. Since  has spin zero there are a total of four states in the irrep, with spins 11 s3 = 0, 2 , 2 , and 0, respectively. Since the parity operation interchanges a1 with a2 , one of the spin zero states is a pseudoscalar. Thus these four states correspond to one massive Weyl fermion, one real scalar, and one real pseudoscalar.
N=1 SUSY, massless states We analyze massless states from the lightlike reference frame Pm = (E, 0, 0, E ). In this case C 2 = 2E 2 (B0 B3 )2 = E 2 Q2 Q2 Q2 Q2 = 0
2 1
.
(28)
7
Also we have: { Q1 , Q1 } { Q2 , Q2 } = 4E , =0.
(29)
We can dene a vacuum state  as in the massive case. However we notice from Eq. 29 that the creation operator Q2 makes states of zero norm:  Q2 Q2  = 0 . (30)
This means that we can set Q2 equal to zero in the operator sense. Eectively there is just one pair of creation/annihilation operators: 1 a = Q1 2E , 1 a = Q 1 2E . (31)
 is nondegenerate and has denite helicity . The creation operator a transforms like (0, 1 ) under the Lorentz group, thus it increases helicity by 2 1/2. The massless N=1 SUSY irreps each contain two states:  helicity , helicity +
1 2
a 
.
(32)
However this is not a CPT eigenstate in general, requiring that we pair two 1 massless SUSY irreps to obtain four states with helicities , + 2 , 1 , and 2 . N>1 SUSY, no central charges, massless states Here we have N creation operators a . These generate a total of 2N states in A the SUSY irrep. The states have the form: 1 a 1 . . . a n  A A n! ,
N n
(33) . Denoting the helicity
with degeneracy given by the binomial coecient
1 of  by , the helicities in the irrep are , + 2 , . . . + N . This is not a CPT 2 eigenstate except in the special case = N/4. Examples of some of the more important irreps are given in Table 1.
8
Table 1: Examples of N>1 massless SUSY irreps (no central charge)
N=2 0 helicity no. of states 1 helicity no. of states 1 helicity no. of states N=4 1 N=8 2 helicity no. of states 2 1
3 2
0 1 1 1
1 2
1 1 0 1
1 2
2
1 2
0 and 1 together make one N=2 onshell vector multiplet.
2 0 2
2
1 2 1
A massless N=2 onshell hypermultiplet.
1
helicity no. of states
1 1
1 2
0 6
1 2
1 1
A massless N=4 onshell vector multiplet.
4
4
8
1 28
1 2
0 70
1 2
1 28
3 2
2 1
An N=8 gravity multiplet.
56
56
8
N>1 SUSY, no central charges, massive states In this case we have 2N creation operators (aA ) . There are 22N (2j + 1) states in a massive irrep. Consider, for example, the fundamental N=2 massive irrep: 0 : spin no. of spin irreps 0 5
1 2
1 1 3
4 8
total no. of states 5 9
There are a grand total of 16 states. Let us describe them in more detail: 1 state : 4 states : 6 states : 4 states : 1 state :
1 2 (aA1 ) (aA2 )
(aA )   

1 spin 0 state 4 spin 4 spin
1 2
states states
3 spin 1 and 3 spin 0 states
1 2
3 2 1 (aA1 ) (aA2 ) (aA3 ) 1 2 3 4 (aA1 ) (aA2 ) (aA3 ) (aA4 )

1 spin 0 state
The only counting which is not obvious is 6 = 3 spin 1 +3 spin 0; this can be veried by looking at the Lorentz group tensor products: (0, )1 (0, )2 (0, )1 (0, )2 =
2 2 2 2 1 1 1 1
(0, 0) + (0, 1) + (0, 0) + (0, 0) . The key point is that (aA ) (aA )  , by antisymmetry, only contains the sin glet. N>1 SUSY, with central charges In the presence of central charges QA , QA cannot be interpreted in terms of creation/annihilation operators without rediagonalizing the basis. Recall { QA , QB } = X AB =
XAB
, , (34)
{QA , QB }
where X AB is antisymmetric and, following Wess and Bagger, we impose the convention X AB =XAB . Since the central charges commute with all the other generators, we can choose any convenient basis to describe them. We will use Zuminos decomposition of a general complex antisymmetric matrix: 9
A X AB = UC X CD (U T )B D
,
(35)
where, for N even, X CD has the form (Z1 ab ) 0 ... 0 0 (Z2 ab ) . . . 0 . . .. . . . . . . . . 0 0 . . . (Z N ab )
2
,
(36)
10
where ab =i 2 . For N odd, there is an extra righthand column of zeroes and bottom row of zeroes. In this decomposition the eigenvalues Z1 , Z2 , . . . Z[ N ] 2 are real and nonnegative. Consider now the massive states in the rest frame. In the basis dened by Zuminos decomposition we have: {Q , Q }
aL 0 = 2m b M
, , , (37)
{QaL , QbM }
{QaL , QbM }
= ab LM ZM = ab LM ZM
where the internal indices A, B have now been replaced by the index pairs (a, L), (b, M ), with a, b = 1, 2 and L, M = 1, 2, . . . [ N ]. Here and in the 2 following, the repeated M index is not summed over. It is now apparent that there are 2N pairs of creation/annihilation operators: aL (aL ) bL (bL ) = = = = 1 2 1 2 1 2 1 2 Q1L + Q 2L 0 Q1L + 0 Q2L Q1L Q 2L 0 Q1L 0 Q2L
, , , . (38)
The Lorentz index structure here looks a little strange, but the important point is that Q transforms the same as Q under spatial rotations. Thus (aL ) , (bL ) create states of denite spin. The anticommutation relation are: {aL , (aM ) } = {bL , (bM ) } =
0 L (2m + ZM ) M 0 L (2m ZM ) M
, . (39)
This is easily veried from Eqs 37, 38 using the relations: 0 0
= = 11
0 0
, . (40)
BPSsaturated states Since {a, a } and {b, b } are positive denite operators, and since the ZM are nonnegative, we deduce the following: For all ZM in any SUSY irrep: ZM 2 m . (41)
When ZM < 2m the multiplicities of the massive irreps are the same as for the case of no central charges. The special case is when we saturate the bound, i.e. ZM = 2m for some or all ZM . If e.g. all the ZM saturate the bound, then all of the (bL ) are projections onto zero norm states; thus eectively we lose half of the creation operators. This implies that this massive SUSY irrep has only 2N (2j +1) states instead of 22N (2j +1) states. These reduced multiplicity massive multiplets are often called short multiplets. The states are often referred to as BPSsaturated states, because of the connection to BPS monopoles in supersymmetric gauge theories. 10 For example, let us compare the fundamental N=2 massive irreps. For N=2 there is only one central charge, Z . For Z < 2m we have the long multiplet already discussed: long : 0 spin no. of spin irreps total no. of states 0 5 5
1 2
1 1 3
4 8
There are a grand total of 16 states. For Z = 2m we have BPSsaturated states in a short multiplet: short : 0 spin no. of spin irreps total no. of states 0 2 2
1 2
1 0 0
1 2
There are a grand total of 4 states. Note that the spins and number of states of this BPSsaturated massive multiplet match those of the N=2 massless hypermultiplet in Table 1. 12
1 Let us also compare the j = 2 N=2 massive irreps. For Z < 2m we have a long multiplet with 32 states:
long : 1
2
spin no. of spin irreps
0 4
1 2
1 4
3 2
6
1 4
total no. of states 4
12 12
For Z = 2m we have a short multiplet with 8 states: short : 1
2
spin no. of spin irreps total no. of states
0 1 1
1 2
1 1 3
3 2
2 4
0 0
Note that the spins and number of states of this BPSsaturated massive multiplet match those of the N=2 massless vector multiplet (allowing for the fact that a massless vector eats a scalar in becoming massive). Automorphisms of the supersymmetry algebra In the absence of central charges, the general 4dimensional SUSY algebra has an obvious U (N ) automorphism symmetry: QA U A B QB , QA QB U B A , (42)
where U A B is a unitary matrix. SUSY irreps on asymptotic single particle states will automatically carry a representation of the automorphism group. For massless irreps U (N ) is the largest automorphism symmetry which respects helicity. For massive irreps, we have already noted that Q and Q transform the same way under spatial rotations. Assembling these into a 2N component object, one nds that the largest automorphism group which respects spin is U Sp(2N ), the unitary symplectic group of rank N. 11 In the presence of central charges, the automorphism group is still U Sp(2N ) provided that none of the central charges saturates the BPS bound; this follows from our ability to make the basis change Eqs. 38, 39. When one central charge saturates the BPS bound, the automorphism group is reduced to U Sp(N ) for N even, U Sp(N +1) for N odd. The automorphism symmetries give us constraints on the internal symmetry group generated by the B . In the case of no central charges U (N ) is 13
the largest possible internal symmetry group which can act nontrivially on the Qs. With a single central charge, the intertwining relation Eq. 264 implies that U Sp(N ) is the largest such group. Supersymmetry represented on quantum elds So far we have only discussed representations of SUSY on asymptotic states, not on quantum elds. Q , Q can be represented as superspace dierential operators acting on elds. The Cliord vacuum condition Eq. 23 becomes a commutation condition: (43) [Q , (x)] = 0 . For onshell elds, the construction of SUSY irreps proceeds as before, with the following exception. If (x) is a real scalar eld, then the adjoint of Eq. 43 is [Q , (x)] = 0 . (44) In that case, Eqs. 43, 44 together with the Jacobi identity for {[(x), Q], Q} implies that (x) is a constant. Thus we conclude that (x) must be a complex scalar eld. This has the eect that some SUSY onshell irreps on elds have twice as many eld components as the corresponding irreps for onshell states. Because we already paired up most SUSY irreps on states to get CPT eigenstates, this doubling really only eects the SUSY irreps based on the special case , = N/4. The rst example is the massless N=2 hypermultiplet. On asymptotic single particle states this irrep consists of 4 states (see Table 1); the massless N=2 hypermultiplet on elds, however, has 8 real components. 2.5 N=1 rigid superspace
Relativistic quantum eld theory relies upon the fact that the spacetime coordinates xm parametrize the coset space dened as the Poincar group modded e out by the Lorentz group. Clearly it is desirable to nd a similar coordinatization for supersymmetric eld theory. For simplicity we will discuss the case of N=1 SUSY, deferring N>1 SUSY until Section 5. The rst step is to rewrite the N=1 SUSY algebra as a Lie algebra. This requires that we introduce constant Grassmann spinors , : { , } = { , } = { , } = 0 . (45)
This allows us to replace the anticommutators in the N=1 SUSY algebra with commutators: [ Q, Q ] = 2 m Pm 14 ,
[ Q, Q ] = 0 [ Q, Q ] = 0
, .
(46)
Note we have now begun to employ the spinor summation convention discussed in the Appendix. Given a Lie algebra we can exponentiate to get the general group element: G(x, , , ) = ei[x
m i Pm + Q+ Q] 2 mn Mmn
e
,
(47)
where the minus sign in front of xm is a convention. Note that this form of the general N=1 superPoincar group element is unitary since (Q) = Q. e From Eq. 47 it is clear that (xm , , ) parametrizes a 4+4 dimensional coset space: N=1 superPoincar mod Lorentz. This coset space is more come monly known as N=1 rigid superspace; rigid refers to the fact that we are discussing global supersymmetry. There are great advantages to constructing supersymmetric eld theories in the superspace/supereld formalism, just as there are great advantages to constructing relativistic quantum eld theories in a manifestly Lorentz covariant formalism. Our rather long technical detour into superspace and supereld constructions will pay o nicely when we begin the construction of supersymmetric actions. Superspace derivatives Here we collect the basic notation and properties of N=1 superspace derivatives.
= =
,
, ,
= = = =
=
, ,
=
, , , , (48)
=
, ,
= =
() ()
2
=
= , ( ) = 2 = 2 ,
=4
,
( ) = 4 . 15
2
Superspace integration We begin with the Berezin integral for a single Grassmann parameter : d d d f () =1 =0 = f1 , , , (49)
where we have used the fact that an arbitrary function of a single Grassmann parameter has the Taylor series expansion f () = f0 + f1 . We note three facts which follow from the denitions of Eq. 49. Berezin integration is translationally invariant: d( + ) f ( + ) = d d f () = d d f () 0. , (50)
Berezin integration is equivalent to dierentiation: d f () = f1 = d d f () . (51)
We can dene a Grassmann delta function by () . (52) These results are easily generalized to the case of the N=1 superspace coordinates , . The important notational conventions are: d2 d2 d4 = d d
= d d 4 1 1
, , (53)
= d d2
4 2
.
Using this notation and the spinor summation convention, we have the following identities: d2 d2 = = 16 1 1 , . (54)
Superspace covariant derivatives If we wanted to treat a general curved N=1 superspace, we would have to introduce a 4+4=8dimensional vielbein and spin connection. Using M to denote an 8dimensional superspace index, and A to denote an 8dimensional supertangent space index, we can write the vielbein and spin connection as A mn EM and WA respectively. The general form of a covariant derivative in such a space is thus 1 A mn DM = EM (A + WA Mmn ) , (55)
2
where A =(m , , ). Naively one might expect that DM reduces to M for N=1 rigid superspace, since the rigid superspace has zero curvature. However it is possible to show 3 that N=1 rigid superspace has nonzero torsion, and thus that the vielbein is nontrivial. The covariant derivatives for N=1 rigid superspace are given by: Dm D D D D 3 3.1 N=1 Superelds The general N=1 scalar supereld
= = = = =
m
,
m + i m
, , , (56)
m i m m i m
+ i m m
.
The general scalar supereld (x, , ) is just a scalar function in N=1 rigid superspace. It has a nite Taylor expansion in powers of , ; this is known as the component expansion of the supereld: (x, , ) = f (x) + (x) + (x) + m(x) + n(x) m (x) + () (x) + ()( )d(x) + vm (x) + ()
.
(57)
The component elds in Eq 57 are complex; redundant terms like m vm have already been removed using the Fierz identities listed in the Appendix. The fermionic component elds (x), (x), (x), and (x) are Grassmann odd, i.e. they anticommute with each other and with , . To compute the eect of an innitesimal N=1 SUSY transformation on a general scalar supereld, we need the explicit representation of Q, Q as 17
superspace dierential operators. Recall that for ordinary scalar elds the translation generator Pm is represented (with our conventions) by the dierential operator im . Let be a constant Grassmann complex Weyl spinor, and consider the eect of left multiplication by a supertranslation generator G(y, ) on an arbitrary coset element (x, , ): G(y, )(x, , ) = ei[y
m m m
Pm +Q+ Q] i[xm Pm +Q+Q]
e
= ei[(x +y )Pm +( + )Q +( + )Q + 2 [Q,Q]+ 2 [Q,Q] , = (xm + y m i m + i m ), + , +
i
i
(58)
where, to obtain the last expression, we have used the commutators: [ Q, Q] = 2 m Pm , [ Q, Q] = 2 m Pm .
(59)
From Eq. 58 we see that, with our conventions, Pm , Q, and Q have the following representation as superspace dierential operators: Pm Q Q : im ,
m : i m
m : i m
, .
(60)
It is now a trivial matter to compute the innitesimal variation of the general scalar supereld Eq.57 under an N=1 SUSY transformation: (x, , ) = (Q + Q)(x, , ) + i m m f + 2m + m vm i m m f = + +2 n + m vm + i( m )m + ()( ) i( m )m
m m
(61) ) + 2( )( ) +()( ) i m + i m + 2()( i m ()m m + i m n m vn + 2()( )d + i m ( )m n m n m + i m ( )m . i m vn + 2()( )d i ()m
Using the Fierz identities, we then have that the component elds of transform as follows: f
= + = =
,
m 2 m + [im f + vm ] , m 2 n + [im f vm ]
,
18
m n vm d
i = m m 2 i = + m m 2 = m + m +
, , i i m m , 2 2 (62)
i = 2 d + m vm + i( m ) m m , 2 i = 2 d m vm + i( m ) m n , 2 i . m m + m = 2
Note the important fact that the complex scalar component eld d(x) transforms by a total derivative. We have thus demonstrated that the general scalar supereld forms a basis for an (oshell) linear representation of N=1 supersymmetry. However this representation is reducible. To see this, suppose we impose the following constraints on the component elds of : (x) n(x) vm (x) (x) (x) d(x) =0 , (63) =0, = im f (x) , i = m m , 2 =0, = 2f (x)
4 1
.
It is easy to verify that the N=1 SUSY component eld transformations Eq. 62 respect these constraints. Thus the constrained supereld by itself denes an oshell linear representation of N=1 SUSY (in fact, an irreducible representation). This suces to prove that representation dened by is reducible. In fact there are several ways of extracting irreps by constraining , however the general scalar supereld is not fully reducible, i.e. the reducible representation is not a direct sum of irreducible representations. We can also use to demonstrate the importance of the superspace covariant derivatives D , D . Consider (x, , ): this has fewer component elds than since, for example, there is no ()( ) term in its component expansion. However the commutator of with Q is nonvanishing:
m [ , Q ] = i m
,
(64)
19
and this implies that an N=1 SUSY transformation generates a ()( ) term. Thus (x, , ) is not a true supereld in the sense of providing a basis for a linear representation of supersymmetry. The superspace covariant derivatives, on the other hand, anticommute with Q and Q: { D , Q } { D , Q } = { D , Q } = { D , Q } =0 , =0 , (65)
Thus if is a general scalar supereld, then m , D , and D are also superelds. 3.2 N=1 chiral superelds
An N=1 chiral supereld is obtained by the constraints Eq. 63 imposed on a general scalar supereld. A more elegant and useful denition comes from realizing that Eq. 63 is equivalent to the following covariant constraint: D = 0 . (66)
Covariant constraints are constraints which involve only superelds (and covariant derivatives of superelds, since these are also superelds). It is a plausible but nonobvious fact that the superelds which dene irreducible oshell linear representations of supersymmetry can always be obtained by imposing covariant constraints on unconstrained superelds. Let us nd the most general solution to the covariant constraint Eq. 66. Dene new bosonic coordinates y m in N=1 rigid superspace: y m = xm + i m . (67)
We note in passing that the funny minus sign convention in Eq. 47 is tied the fact that sign in Eq. 67 above is plus. Since D y m D = = 0 0 , ,
(68)
it is clear that any function (y, ) of y m and (but not ) satises D (y, ) = 0 . (69)
It is easy to see that, since D obeys the chain rule, this is not just a particular solution of Eq. 66 but is in fact the most general solution. 20
Thus we may write the most general N=1 chiral supereld as: (y, ) = A(y ) + 2 (y ) + F (y ) , (70)
where A(y ), F (y ) are complex scalar elds, while (y ) is a complex left handed Weyl spinor. The 2 is a convention. There are 4+4 = 8 real oshell eld components; this is twice the number in the onshell fundamental N=1 massive irrep. The full , component expansion is obtained by using the Fierz identity Eq. 247. The result is: (y, ) = A(x) + 2 (x) + F (x) i 1 m +i m A(x) + ()m (x) m ()( )2A(x) 4 2
.
(71)
An innitesimal N=1 SUSY transformation on the chiral supereld yields: A = F = = 2 , 2F + 2i m m A m 2im .
,
(72)
Note that F (x) is a total derivative. Antichiral superelds, i.e. righthanded chiral superelds, are dened in the obvious way. In particular, if (y, ) is a chiral supereld, then is an antichiral supereld; it satises D = = 0, (y , );
y = xm i m
.
(73)
Since D and D obey the chain rule, any product of chiral superelds is also a chiral supereld, while any product of antichiral superelds is also an antichiral supereld. However it is also clear that if (y, ) is a chiral supereld, the following are not chiral superelds: , .
+
For future reference, let us write down the expressions for the covariant derivatives acting on functions of (y, , ): 21
D D D D
m = + 2i m
, , (74)
= =
,
= 2i m m
,
where of course here m is a partial derivative with respect to y m rather than xm . 3.3 N=1 vector superelds
Vector superelds are dened from the general scalar supereld by imposing a covariant reality constraint: V (x, , ) = V (x, , ) , or, in components: f m vm = f = = n , , , (76) (75)
= vm , = ,
d = d
.
Thus in components we have 4 real scalars, 2 complex Weyl spinors (equivalently, 2 Majorana spinors), and 1 real vector. The 8+8 = 16 real components in this oshell irrep are twice the number in the onshell 1 massive irrep. The presence of a real vector eld in the N=1 vector multiplet suggests we use vector superelds to construct supersymmetric gauge theories. But rst we must deduce the supereld generalization of gauge transformations. WessZumino gauge If (y, ) is a chiral supereld, then + is a special case of a vector supereld. In components: + = (A + A ) + 2 + 2 + F + F + i m m (A A ) i i 1 + () m m + ( ) m m ()( )2(A + A ) . (77) 4 2 2 22
2
From this we see that we can dene the supereld analog of an innitesimal abelian gauge transformation to be V V + + , (78)
since this denition gives the correct innitesimal transformation for the vector component: vm vm + m = i(A A ) ; . (79)
The meaning of the bigger supereld transformation Eq. 78 is that any supereld action invariant under abelian gauge transformations will also be independent of several component elds of V (x, , ). More precisely, notice that the rst 5 component elds of + in Eq. 77 are completely unconstrained. This means that without loss of generality we can decompose any vector supereld as follows: V (x, , ) = VW Z + + where VW Z only has 4 component elds instead of 9:
1 VW Z = m vm + i() i() + ()( )D 2
,
(80)
,
(81)
where, to conform with Wess and Bagger, I have changed notation slightly: vm vm , i , d
1 2
D
.
VW Z is known as the WessZumino gaugexed supereld. This decomposition is unambiguous except for the remaining freedom to shift part of vm into the corresponding component of + , i.e. vm vm im (AA ). Thus xing WessZumino gauge does not x the abelian gauge freedom. 3.4 The supersymmetric eld strength
Note that the supersymmetry transformations do not respect the WessZumino gaugexing decomposition. This is somewhat disappointing since it means that a supereld formulation in terms of V (x, , ) necessarily carries around 23
a number of superuous elds. We can however dene a dierent supereld which has the property that it only contains the WessZumino gaugexed component elds vm (x), (x), and D(x). We dene left and righthanded spinor superelds W , W = (W ) : W W = (DD)D V (x, , )
4 1 4 1
, (82)
= (DD)D V (x, , ) .
An equivalent denition, which we will need when we go from the abelian to the nonabelian case, is: W W = = 1 (DD)e2V D e2V , 8 1 (DD)e2V D e2V . 8
(83)
W is a chiral supereld: D W = D (D D )D V (x, , ) = 0 ,
4 1
(84)
where we have used the fact that since the Ds anticommute and have only 2 components, (D)3 = 0. W is an antichiral supereld. W is not a general chiral spinor supereld, because W and W are related by an additional covariant constraint: D W = D W This constraint follows trivially from Eq. 82: D W =
D W = D (DD)D V 1 4 1 4
.
(85)
= 1 (DD)(DD)V 4 = D W
= D (DD)D V .
(86)
W and W are both invariant under the transformation Eq. 78. Let us prove this for W : W (DD)D (V + + ) ,
4 1
= = =
W (D D)D
4 1 4
1
,
(since D = 0) , (since D = 0) (87)
W + D {D , D } W , 24
where in the last step we have used: { D , D }
m = 2 Pm
, (88)
[ D , Pm ] = 0
.
Since W and W are both invariant under Eq. 78, there is no loss of generality in computing their components in WessZumino gauge, i.e. write W W = = (DD)D VW Z (x, , ) , (DD)D VW Z (x, , ) .
4 4 1 1
(89)
Since W = W (y, ) we write VW Z (x, , ) = VW Z (y i , , ) and expand W in component elds which are functions of y : W = i i (y ) + D(y ) ( m n ) (m vn n vm )(y ) 2 m +() m (y ) , i i (y ) + D(y ) + ( m n ) (m vn n vm )(y ) 2 m () m (y ) . , D, fmn m vn n vm . (90)
(91)
W
=
So indeed W , W contain only the component elds
This is an irreducible oshell multiplet known as the curl multiplet or eld strength multiplet; it has 4+1+3 = 8 real components. Nonabelian generalization We can exponentiate the innitesimal abelian transformation Eq. 78 to obtain the nite transformation eV ei eV ei
,
(92)
where, to conform with the standard notation of Ferrara and Zumino, 12 we now denote the chiral superelds of Eq. 78 by:
i 25
i ,
.
To obtain the nonabelian generalization we write V [T ,T ] tr T T
a b a b a Tij Va , a Tij a ;
(93) ,
=
if
abc
T ,
c
=
ab
a where the Tij are the hermitian generators of some Lie algebra. The form of the nonabelian transformation is then the same as Eq. 92. To nd the innitesimal nonabelian transformation, we can apply the BakerCampbellHausdor formula to Eq. 92. One can show that, to rst order in , Eq. 92 reduces to: 13
V = iLV /2 ( + ) + iLV /2 cothLV /2 ( ) where the operation LX Y denotes the Lie derivative: LX Y (LX ) Y
2
,
(94)
= = etc.
[X, Y ] , [X, [X, Y ]] , . (95)
Eq. 94 is meant to be evaluated by its Taylor series expansion, using x cothx = 1 + x4 x2 + ... 3 45 . (96)
This becomes much more illuminating if we x the nonabelian equivalent of WessZumino gauge. Unlike the abelian case, the relationship between the component elds of V (x, , ) and (y, ) in the WessZumino gauge xing is nonlinear, due to the complicated form of Eq. 94. However the end result is the same: VW Z (x, , ) is as given in Eq. 81. Furthermore, as in the abelian case, the WessZumino decomposition does not x the freedom to perform gauge transformations parametrized by the scalar component of i(+ ). Consider then the transformation Eq. 94 with V replaced by VW Z , and with only the scalar component of + nonvanishing (which also implies that only the component of is nonvanishing). Clearly only the rst term in the Taylor series expansion of the hyperbolic cotangent remains, since 3 the next higher order term gives something proportional to 3 . Thus having xed WessZumino gauge the innitesimal nonabelian gauge transformation is just i V = i( ) [( + ), V ] . (97) 2 26
This implies the usual nonabelian gauge transformations for the component elds vm (x), (x), and D(x) (vm (x) is the nonabelian gauge eld while (x) and D(x) are matter elds in the adjoint representation). W and W are given by Eq. 83 in the nonabelian case. Let us com pute how W and W transform under Eq. 92. First notice that, under the transformation Eq. 92: e2V D e2V ei2 e2V D e2V ei2 + ei2 D ei2 , (98)
which follows from the fact that D = 0. Thus, using also the fact that D commutes with , we see that 1 W ei2 W ei2 ei2 (D D)D ei2 8 . (99)
Furthermore the second term vanishes, just as in Eq. 87, using the identities Eq. 88. So our nal result is that W and W transform covariantly in the nonabelian case: W ei2 W ei2
i2
, . (100)
W e
W e
i2
Let us be more explicit in the nonabelian case about the derivation of the component expansion for W . There is no loss of generality in computing this in WessZumino gauge. From the denition Eq. 83 we have the explicit expression: W 1 = (D D)e2VW Z D e2VW Z 8
1 1 4 2
,
1 4
(101)
2 = (DD)D VW Z + (D D)VW Z D VW Z (DD)D VW Z
where we have used our knowledge (see Eq. 81) of the component expansion for VW Z (y i , , ): VW Z (y i , , ) = + ()( ) (D(y ) + i m vm (y ))
2
m vm (y ) + i() (y ) i( )(y )
1
.
(102)
Using the form Eq. 75 for D acting on functions of (y, , ), we have:
m D VW Z = vm (y ) + 2i (y ) i( ) (y )
+ ( ) (D(y ) + i m vm (y )) i()( m n ) m vn (y ) m +()( ) m .
(103)
27
A little more straightforward computation gives:
2 D VW Z = ( )v m vm
,
(104)
as well as: VW Z D VW Z =
1 2 n ( )v m vm (y ) + m ( )[vm , vn ] 4 1
1 m i()( ) [vm , ] . 2
(105)
Putting it all together, we have:
mn m W = i (y ) + D(y ) Fmn (y ) + () m (y )
,
(106)
where Fmn m = = m vn n vm + i[vm , vn ] , m + i[vm , ] ;
(107)
Fmn is the YangMills eld strength, while m is the YangMills gauge covariant derivative. We also need 1 W = (DD)e2V D e2V ; (108) 8 raising the index on Eq. 106 and Fierzing, we get:
mn W = i (y ) + D(y ) + Fmn (y ) () m m (y ) . (109)
3.5
N=1 linear multiplet
In the previous subsection we obtained the eld strength multiplet by starting with the chiral spinor supereld W , and imposing the additional covariant constraint Eq. 85. Let us again start with a chiral spinor supereld , = ( ) , and construct a new supereld L(x, , ) as follows: L(x, , ) = i D + D The supereld L(x, , ) is real, since D
.
(110)
= D = D 28
;
(111)
so L(x, , ) is a vector supereld which satises two additional covariant constraints: (DD)L = (D D)L = 0 . (112) These constraints follow trivially from the fact that is chiral, and (D)3 = (D)3 = 0. The component elds of L(x, , ) comprise the linear multiplet. These are a real scalar C (x), a complex lefthanded Weyl spinor , and a real divergenceless vector eld Am , m Am = 0. Thus the linear multiplet has 1+4+3 = 8 real components. 4 N=1 Globally Supersymmetric Actions
Recall from the previous section that both the F component of a chiral supereld and the D component of a vector supereld transform by a total derivative under an N=1 supersymmetry transformation. Thus we immediately deduce two classes of N=1 globally supersymmetric actions: d4 x d2 (y, ) + d2 (y , ) (113)
is an invariant real action for any chiral supereld (y, ), while d4 x d4 V (x, , ) (114)
is an invariant real action for any vector supereld V (x, , ). 4.1 Chiral supereld actions
The WessZumino model 14 is the simplest (sensible) N=1 SUSY model in four dimensions. The action is d4 x d4 d4 x d2 ( m2 + g 3 ) + h.c.
2 3 1 1
,
(115)
where is a chiral supereld. Let us work out the part of this action containing bosonic component elds. The bosonic components of and are: (y, ) = A(x) + F (x) + i m m A(x) ()( )2A(x)
4 1 1
, (116)
(y , ) = A (x) + F (x) i m m A (x) ()( )2A (x)
4
.
29
Thus:
= 2A A A 2A + F F + m A m A ,
4 4 2
1
1
1
(117)
where to obtain the last term we have used the Fierz identity Eq. 247. We also have
1 2
m 2 + g 3
3
1
= mAF + gA2 F
,
(118)
so the part of the WessZumino action containing only bosonic elds is: d4 x m A m A + F F (mAF + gA2 F + h.c.) (119)
We immediately notice that this action contains no derivatives acting on F (x), i.e. F (x) is an auxiliary eld which can be eliminated by solving its equations of motion: L F L F = = F mA gA2 = 0 , F mA g (A )2 = 0 . (120)
This means we can write the bosonic part of the WessZumino action as just d4 x [ m A m A V (A, A )] where the scalar potential V (A, A ) is given by: V (A, A ) = F 2 = [mA + g (A )2 ][mA + gA2 ] . More generally we could write d4 x d4 d4 x d2 W () + h.c. , (123) (122) , (121)
where the superpotential W () is a holomorphic function of , i.e. a functional only of , not . In this more general case the scalar potential is VF (A, A ) = F 2 = W
2 =A
.
(124)
Note that the scalar potential is obviously positive denite. Since a cubic superpotential leads to a quartic scalar potential, we also see that the WessZumino model is the most general unitary, renormalizable fourdimensional SUSY action for a single chiral supereld. 30
An even more general construction than Eq. 123 is L=
d4 K (i , j )
d2 W (i ) + h.c.
,
(125)
where K (i , j ) is called the Khler potential, and we now have an arbia trary number of chiral superelds i . The Khler potential is a vector supera eld; unlike the superpotential it is obviously not a holomorphic function of the i . From the component expansion Eq. 71 it is clear that the Khler potential a produces kinetic terms with no more than two spacetime derivatives. If we replace some of the i by covariant derivatives of superelds, we will either obtain a higher derivative theory, or a theory which can be collapsed back to the form Eq.125. Thus if we exclude higher derivative theories Eq.125 is the most general action for (not necessarily renormalizable) N=1 SUSY models constructed from chiral superelds. 4.2 N=1 supersymmetric nonlinear sigma models
Bosonic nonlinear sigma models in Ddimensional spacetime have an action of the form:
1 2
dD x gij (A) m Ai (x)m Aj (x)
,
(126)
where the Ai (x) are real scalar elds. The functional gij (A) can be thought of as the metric of a target space Riemannian manifold with line element ds2 = gij dAi dAj . (127)
Nonlinear sigma models are not in general renormalizable, except in the case D = 2 with gij the metric of a symmetric space. 15 The general chiral supereld action Eq.125 denes the supersymmetrized version of 4dimensional nonlinear sigma models. 16 To see this, note that Eq. 117 implies that the kinetic term for the complex scalar components Ai (x) is gij m Ai (x) m Aj , (128) where: 2 K (Ai , Aj ) Ai Aj 31
gij
.
(129)
Since the Khler potential is real, the target space metric gij is hermitian. a To obtain a correct sign kinetic term for every nonauxiliary scalar eld, we must also require that gij is positive denite and nonsingular; this implies (mild) restrictions on the choice of the Khler potential. a A complex Riemannian manifold possessing a positive denite nonsingular hermitian metric which can be written (locally) as the second derivative of a scalar function is called a Khler manifold. Thus Eq.125 denes supera symmetric generalized nonlinear sigma models whose target spaces are Khler a manifolds. This is a rather powerful observation, since it implies that models with horrendously complicated component eld Lagrangians can be characterized by the algebraic geometry of the target space. As an example, we will discuss the possible holonomy groups of sigma model target spaces. Consider the parallel transport of a vector around a contractible closed loop using the Riemannian connection in a Ddimensional Riemannian space. The transported vector is related to the original vector by some SO(D) rotation. The SO(D) matrices obtained this way form a group, the local holonomy group of the manifold. Obviously the holonomy group is either SO(D) itself or a subgroup of it. Four important examples are given below (we use the convention that Sp(2D) is the symplectic group of rank D): Manifold Maximum Holonomy Group ....... SO(D)
General Riemannian space with real dimension D:
Khler manifold with complex dimension D, a real dimension 2D: . . . . . . . . . . . . . . . . . . . . . . U (D) HyperKhler manifold with real dimension 4D: a Quaternionic manifold with real dimension 4D: ........ .... Sp(2D)
Sp(2D) Sp(2)
Note that the Khler structure Eq. 129 (and thus also the action) is ina variant under a Khler transformation: a K (Ai , Aj ) K (Ai , Aj ) + (Ai ) + (Aj ) . (130)
It is also clear that both the Khler structure Eq. 129 and the Riemannian a structure Eq. 127 are preserved by arbitrary holomorphic transformations of the target space coordinates Ai . 32
4.3
N=1 supersymmetric YangMills theory
We recall that W is a chiral spinor supereld and that a gauge transformation on the vector component of W is induced by the supereld transformation W ei2 W e+i2
1 2
.
(131)
It follows that a gauge invariant supersymmetric action is d4 x d2 tr W W (132) ,
=
i 1 1 d4 x tr Fmn F mn Fmn F mn i m m + D2 4 4 2
where we have used the explicit component expansions Eqs. 106,109. The dual eld strength is dened as:
1 F mn mnpq Fpq 2
.
(133)
This action is not real and lacks any dependence upon the YangMills gauge coupling g . The dualityfriendly way to remedy these deciencies is by introducing a complex gauge coupling : = 4i YM +2 2 g , (134)
where YM is the YangMills theta parameter. The N=1 YangMills action we want is then 1 Im d4 x d2 tr W W 8 1 1 1 = 2 d4 x tr Fmn F mn i m m + D2 g 4 2 YM d4 x tr Fmn F mn . 32 2 (135) (136)
The minus sign in front of the YM term is correct given the minus sign convention of Eq. 47. Under a gauge transformation, chiral superelds in the adjoint representation transform as:
ei2 e 33
, . (137)
i2
Thus tr is not gauge invariant. However from Eq. 97 we see that the following is a gauge invariant kinetic term for chiral superelds: tr e2V . (138)
In fact this is gauge invariant for in an arbitrary representation R, not just the adjoint. In this case =ta a , where the ta are matrices in the represenij ij tation R. Thus Eq. 138 is still gauge invariant that provided all the tensor products contain the singlet. This is indeed true because the tensor product of R with its conjugate R contains both the singlet and adjoint representations, while every term in the series expansion of exp(2V ) also contains either the singlet or the adjoint (or both). Thus, supposing I have chiral superelds i transforming in representations Ri , the gauged version of the WessZumino action is: tr d4 x tr d4 i e2V i d4 x d2 ( mij i j + gijk i j k ) + h.c.
2 3 1 1
.
(139)
Note that by gauge invariance mij can only be nonvanishing if Ri = Rj . (140)
Similarly, gijk can only be nonvanishing if Ri Rj Rk contains the singlet. The gauged kinetic term Eq. 138 contains a Dterm A DA. The only other dependence on the auxiliary eld D is the term D2 /2g 2 in the YangMills action. Thus when we eliminate this auxiliary eld by its equation of motion we nd Da = g 2 Ab A tr (T a T b T c ) c
1 2
= ig 2 f abc Ab A c
,
(141)
where the second line follows from the fact that the adjoint representation is always anomalyfree. Thus: D = T a Da = g 2 [A, A ] .
2 1
(142)
This implies that in the coupled YangMillsWessZumino model there is a new contribution to the scalar potential: VD = g2 12 [Ai , Ai ] D= 2 2g 8 34
2
.
(143)
So altogether the complete scalar potential is the sum of positive denite F and Dterm contributions: V (Ai , Ai ) = VF + VD = F 2 +
12 D 2g 2
(144)
If we forget about renormalizablity we can write a very general N=1 action by gauging Eq. 125: L= + d4 K (i , j e2V ) 1 Im 8
d2 W (i ) + h.c. , (145)
d4 x d2 tr f (i )W W
where f (i ) is a new holomorphic function called the gauge kinetic function. Note that every term in f (i ) must transform like a representation which is contained in the tensor product of two adjoints. 5 5.1 N=2 Globally Supersymmetric Actions N=2 superspace
There are several dierent ways to extend our treatment of N=1 rigid superspace to the case of N=2 rigid superspace. 17 Some methods, e.g. harmonic superspace, build in the SU (2) automorphism symmetry of the N=2 generators Q1 , Q2 . We will make do with the most naive extension of N=1 to N=2 superspace parameterizations: , D D d2 , , , D , D D , D (146) d2 d2
If we want to restore the SU (2) global R symmetry, we should think of ( , ) etc. as SU (2) doublets. 5.2 N=2 chiral superelds
An N=2 chiral supereld (x, , , , ) is dened as an N=2 scalar supereld which is a singlet under the global SU (2) and which satises the covariant 35
constraints D (x, , , , ) = 0 D (x, , , , ) = 0 , . (147)
It is convenient to introduce new bosonic coordinates y m = xm + i m + i m , which obviously satisfy D y m = D y m = 0 . (149) (148)
If we expand an N=2 chiral supereld in powers of , the components are N=1 chiral superelds. Thus: y = (, ) + i 2 W (, ) + G(, ) , y y (150) where (y +i m , ) and G(y +i m , ) are N=1 chiral superelds (note the eective y coordinate is shifted by i m ), and W (y +i m , ) is an N=1 chiral spinor supereld. Since is an SU (2) singlet, while ( , ) is an SU (2) doublet, it follows that the fermionic components of and of W also form an SU (2) doublet. On the other hand the bosonic component elds A of and vm of W are SU (2) singlets. 5.3 N=2 supersymmetric YangMills theory
a (, ) Tij a (, ) . y y
Suppose we write (151) (152) Then, since 2
= W W
+2G
the obvious form for N=2 YangMills theory is: 1 Im 4 d4 x d2 d2 tr 2
2 1
.
(153)
This clearly describes an N=1 YangMills theory coupled to chiral superelds in the adjoint representation. Unfortunately something is wrong, since the second term in Eq. 152 is not a sensible Lagrangian for chiral superelds. 36
Clearly what we want is to be able to regard G(, ) as an auxiliary supereld, y and thus eliminate it in favor of and V , reproducing (at least) the N=1 gauge invariant kinetic term Eq. 138. Thus, while Eq. 153 is the correct action for N=2 YangMills theory, we must impose additional covariant constraints on the N=2 chiral supereld . The correct constraints turn out to be:
b (Da D ) = (D D ) a b
,
(154)
where a, b are global SU (2) indices: Da D
a
= (D, D) = (D, D)
, . (155)
Rather than solve these constraints directly, it is much easier to simply assume that G can be eliminated in favor of and V , then deduce the correct expression from the requirement of gauge invariance (i.e. gauge invariance in the N=1 sense). Roughly speaking, we need something like G(, ) e2V y . (156)
However, while the righthand side transforms correctly under gauge transformations, it is clearly not an N=1 chiral supereld. So consider instead the more sophisticated expression: G(, ) = y
y y d2 ( i , )e2V (i,,)
,
(157)
where the integral is meant to be performed for xed y . The result of the integral is obviously a function only of y and , so G(, ) y thus dened is an N=1 chiral supereld, as required. Under the N=1 supereld transformation which induces a gauge transformation, the integrand of Eq. 157 transforms as: e2V
y e2V ei2(i+i,) y = e2V ei2(,) ,
(158)
so we can pull the exp(i2(, ) factor out of the integral. Thus y
y G(, ) G(, )ei2(,) y y
,
(159)
as required for gauge invariance. 37
The overall coecient of 1 in Eq. 157 is xed by the global SU (2) symmetry. As we noted above the fermionic components of and of W form an SU (2) doublet. Thus the relative coecient of the kinetic terms for in G and in W W must be equal. The resulting N=2 YangMills theory is thus equivalent to N=1 YangMills coupled to matter elds in the adjoint representation. There is no superpotential, but there is a scalar potential coming from the Dterm. The nonauxiliary elds form an oshell N=2 vector multiplet: vm , A, and the global SU (2) doublet (, ). Onshell this multiplet gives 4+4 = 8 real eld components, which of course agrees with the counting for the massless N=2 vector multiplet of single particle states. 5.4 The N=2 prepotential
If we forget about renormalizability, we can write a much more general action for N=2 chiral superelds satisfying the covariant constraint Eq. 154: 1 Im 4 d4 x d2 d2 tr F () , (160)
where the holomorphic functional F () is called the N=2 prepotential. Obviously 1 F () = 2 (161)
2
gives back the classical N=2 YangMills action of Eq. 153. Let us dene Fa () = Fab () = F () , a 2 F () . a b
(162)
Then the general Lagrangian can be written in terms of N=1 superelds as follows: 1 Im 4
b d2 Fab ()W a W + 2 1
d4 ( e2V )a Fa ()
.
(163)
Thus from the N=1 point of view we have a special case of Eq. 145: the superpotential vanishes, the Khler potential is a K = Im ( e2V )a Fa () 38 , (164)
and the gauge kinetic function is f () = Fab T a T b . (165)
Notice that in this more general N=2 action the scalar elds Aa describe a nonlinear sigma model whose target space Khler potential has the special a form above, i.e. it can be written in terms of a derivative of a holomorphic function. The target space is a special Khler manifold known as the special a Khler manifold. 18 a 5.5 N=2 hypermultiplets
While was assumed to be a singlet under the global SU (2) symmetry, we can also consider a general N=2 scalar supereld which is an SU (2) doublet: a (x, , , , ) An N=2 hypermultiplet supereld is then dened by the covariant constraints
a D b
= =
1 ab 2 1 ab 2
c D c
, .
a
D b
a
D c
c
(166)
a These constraints simply remove the isotriplet parts of D b and D b :
[ ] + [ ] = [0]antisymm. + [1]symm.
2 2
1
1
.
(167)
The independent component elds are: Aa (x) , , complex scalar isodoublet two isosinglet spinors complex auxiliary scalar isodoublet
(x), (x) F a (x) ,
Onshell this implies 4+4 = 8 real components, which as we have already noted is twice the number in the massless N=2 hypermultiplet of single particle states. A free superspace action for an N=2 hypermultiplet supereld a is
c b d4 x Da D a Dc D b
.
(168)
With more diculty, we can couple N=2 hypermultiplets to N=2 YangMills; the details are not particularly illuminating. 39
Note that there can be no renormalizable selfinteraction for a since there is no cubic SU (2) invariant. We can construct N=2 generalizations of nonlinear sigma models out of the hypermultiplets. It is easiest to start with the N=1 case in components gij m Ai (x) m Aj + . . . , (169)
then impose the extra constraints of N=2 supersymmetry. The end result 19 is that the target spaces of N=2 hypermultiplet nonlinear sigma models are hyperKhler manifolds. a 6 Supergravity
So far we have only considered global supersymmetry, generated by Q + Q with , constant Grassmann parameters. If we want local supersymmetry, we should promote these parameters to functions of spacetime: , (x), (x) . (170)
Rigid superspace then becomes curved superspace. From the superspace A mn vielbein EM and spin connection WA we can construct the superspace curvature and torsion: RMN A B , TMN A . Recall that N=1 rigid superspace has already nonzero torsion, so we cannot constrain all components of the curved superspace torsion to vanish as we do in general relativity. On the other hand the superspace vielbein and connection have too many independent components to dene a sensible theory. Thus the main diculty in constructing supergravity theories is nding and solving an appropriate set of covariant constraints. This gets very complicated, 20 and is beyond the scope of these lectures. Let us instead quote results. One can construct an oshell supergravity multiplet with the following eld content: ea , m
m
, ba ,
vectorspinor spin 3/2 gravitino auxiliary real vector eld auxiliary complex scalar eld 40
vierbein spin 2 graviton
M,
From these elds we can construct a supergravity Lagrangian: 8GL = eR eM 2 + eba ba
2 3 3 1 1 1
+ e
2
1
klmn
k l Dm n k l Dm n
(171)
where: G e R Dm = Newtons constant, = det ea , m = Ricci scalar curvature, = covariant derivative for spin 3/2 elds.
The action is invariant under general coordinate transformations, local Lorentz transformations, local N=1 supersymmetry. Lets count the oshell degrees of freedom of N=1 supergravity. Because the action is invariant under three types of local symmetries we should only count gauge invariant degrees of freedom: ea : m 4 general coordinate m 6 local Lorentz ab 4 4 = 16
= 6 bosonic real components 4 4 = 16 4 local N = 1 SUSY
m :
= 12 fermionic real components real vector = 4 bosonic real components complex scalar = 2 bosonic real components 41
ba :
M:
Thus we have a total of 12 bosonic and 12 fermionic real components in the oshell N=1 supergravity multiplet. Onshell we have only 2 + 2 components, corresponding to a massless spin 2 graviton and a massless spin 3/2 gravitino. Of course, we really want to be able to couple N=1 supergravity to N=1 supersymmetric YangMills and N=1 chiral supereld matter, all in a way which is consistent with local supersymmetry. This is again a complicated problem and the nal result is not particularly intuitive. 2 We can also extend 4dimensional N=1 supergravity to N=2, 3, 4, or 8 supergravity. These extended supergravities automatically couple gravity to gauge elds and matter elds in a way consistent with local supersymmetry, just as N=2 YangMills couples gauge elds to matter in a way consistent with global supersymmetry. Extended supergravities are easier to construct and understand if we use dimensional reduction. For example, 4dimensional N=8 supergravity can be obtained by dimensionally reducing 11dimensional N=1 supergravity, which is a rather simple theory to describe. We will return to this fact when we discuss supersymmetry in higher dimensions. 7 Renormalization of N=1 SUSY Theories
Consider again the WessZumino model: d4 x d4 d4 x d2 ( m2 + g 3 ) + h.c.
2 3 1 1
,
(172)
We would like to work out the supereld Feynman rules of this theory. However we encounter an immediate diculty which is that is not a general scalar supereld, but rather a constrained supereld. Thus in computing perturbative diagrams with chiral superelds we must deal with the occurence of integrals d4 x d2 over only part of the full N=1 rigid superspace. This diculty is overcome by introducing a projection operator for chiral superelds. The projection operator we need is 1 2 D D2 . (173) 162 This operator clearly has the property that it takes a general scalar supereld to a chiral supereld, i.e. P+ D P+ (x, , ) = 0 (174)
follows trivially from the fact that (D)3 = 0. To prove that P+ is in fact a projection operator, we must also show that (P+ )2 = P+ 42 .
We use the following identity (which can be veried by brute force): [ D , D2 ] = 8i(D m D)m 162 . Thus: (P+ )2 = = = = = 1 1 2 2 D D2 D D2 162 162 2 1 2 2 D D2 D D2 162 1 162 1 162 P+ ,
2 2
(175)
D D2 (D2 D + 8i(D m D)m 162)
2
2
2
(176)
D D2 (162) (177)
2
where in the fourth line we used (D)4 =(D)3 =0. Similarly the projection operator for antichiral superelds is P 1 2 D2 D 162 . (178)
We can now deal with the occurence of integrals d4 x d2 over only part of the full N=1 rigid superspace. The judicious use of projection operator insertions allow us to convert these into integrals over the entire superspace. For example: d2 m = = d2 m P+ 4 d4 m
2
1 D2 162
,
(179)
where in the last line we used the fact that D = 0 and that, modulo surface terms, d4 x D 4
2
d4 x
d2
.
(180)
A similar diculty occurs for the cubic interaction term g 3 . These vertices correspond to functional derivatives with respect to chiral sources J (y, ). These functional derivatives produce superspace delta functions 4 (x1 x2 ) 2 (1 2 ) 43
whereas what we want (for internal lines anyway) are delta functions for the full superspace: 4 (x1 x2 ) 4 (1 2 ) = 4 (x1 x2 ) 2 (1 2 ) 2 (1 2 ) = 4 (x1 x2 )(1 2 )2 (1 2 )2
(181)
This is easily remedied by using the identity 4 (x1 x2 ) 2 (1 2 ) = D 4 (x1 x2 ) 4 (1 2 )
4 2 1 2
.
(182)
In loop graphs, one factor of D from each vertex will get used up converting 3 an d2 to an d4 . Of course we also have a similar trick for the g vertices. We can now compute the superspace propagator of the WessZumino model by performing the functional integral of the quadratic part of the action, written in the form: d4 x d4
1 2
( )
1m 2 4 D
1
1m 2 4 D
1
(183)
+( )
1 4 D J
2
2
1 4 D J
.
7.1
Nonrenormalization
Without further ado we can now summarize the superspace dependence of the resulting Feynman rules for 1PI diagrams: There is an d4 for each vertex.
For a 3 vertex n of whose lines are external, there are 2n factors of 2 3 D . For a vertex, there are 2n factors of D2 . There is a Grassmann delta function 4 (1 2 ) = (1 2 )2 (1 2 )2 for each propagator. There is a factor of D2 for each  propagator, and a factor of D for each  propagator. 44
2
(184)
Consider now an arbitrary loop graph. Integrating the various D2 and D factors by parts, we can perform all but one of the d4 integrations using the delta functions. Let d4 denote the nal integration, and 4 ( 2 ) be the one remaining Grassmann delta function. This delta function is already supposed to be evaluated at = 2 , due to the 2 integral already performed. So the graph vanishes unless there is precisely one factor of D2 and one factor 2 of D acting on the nal delta function: D2 D 4 ( 2 ) = 16
2
2
.
The only remaining dependence comes from the external lines. Thus an arbitrary term in the eective action can be reduced to the form: d4 d4 x1 d4 xn F1 (x1 , , ) Fn (xn , , )G(x1 , . . . xn ) , (185)
where the F s are superelds and covariant derivatives of superelds, and all the spacetime structure is swept into the translationally invariant function G(x1 , . . . xn ). This result is called the N=1 Nonrenormalization Theorem. It generalizes to N=1 actions containing arbitrary numbers of chiral and vector superelds. An important consequence is that if all of the external lines are chiral, or if all of the external lines are antichiral, the expression above vanishes. Thus: The superpotential is not renormalized at any order in perturbation theory. Another important result of the above analysis is that all vacuum diagrams and tadpoles vanish. This is consistent with the fact that the vacuum energy is precisely zero in any globally supersymmetric theory. Note our derivation implicitly assumed that the spacetime loop integrals are regulated in a way which is consistent with supersymmetry. This is not the case if we employ dimensional regularization, since the numbers of fermions and bosons vary dierently as you vary the dimension. Supersymmetric loop diagrams are usually evaluated using a regularization called dimensional reduction, where the spinor algebra is xed at four dimensions while momentum integrals are performed in 4 2 dimensions. This is not a completely satisfactory procedure either. 3
45
7.2
Renormalization
What renormalization do we have to perform in an N=1 SUSY model with chiral superelds i and a vector supereld V ? We have wave function renormalization: i 0 V0 = = Z 1/2 j
1/2 ZV ij
, , (186)
V
and we also have gauge coupling renormalization: g 0 = Zg g . (187)
Even better, if we compute using the background eld method, the background eld gauge invariance implies the relation: 3 Zg ZV
1/2
=1 .
(188)
The end result is that we can characterize the renormalized theory in terms of two objects: The beta function: g
(g ) =
;
The anomalous dimensions matrix of the i : ij = Z 1/2 8
ik
1/2 kj Z
.
Holomorphy and the N=2 YangMills Beta Function
In this section we will review some of Seibergs original arguments about the N=2 supersymmetric SU (2) YangMills beta function. 21 This type of argumentation deals with the eective infrared (i.e. low energy) limit of the theory, described by the Wilsonian action. 22 The form of this eective action will be constrained by three kinds of considerations: holomorphy, global symmetries, the existence of a nonsingular weak coupling limit. 46
Let us begin by listing the global symmetries of the classical N=2 supersymmetric SU (2) YangMills Lagrangian. These are: the global SU (2) R symmetry arising from the automorphism of the N=2 algebra, and an additional U (1) R symmetry dened by: ei , ei , e2i
(189) . (190)
R There is an axial current jm corresponding to the U (1) R symmetry. Since both fermions and have Rcharge +1, there is a nonvanishing ABJ triangle anomaly. We write the anomalous divergence of the R current, remembering that the fermions are in the adjoint representation: R m jm = 2C2 (G)
g2 g2 Fmn F mn = Fmn F mn 16 2 4 2
.
(191)
Next we deduce the moduli space of gauge inequivalent classical vacua for N=2 supersymmetric SU (2) YangMills. The theory contains an SU (2) triplet complex scalar eld A(x) whose scalar potential is (see Eq. 143): V (A) = 12 g2 2 D= ([A, A ]) 2g 2 8 . (192)
Unbroken supersymmetry requires that V (A) = 0 in the vacuum. Up to a gauge transformation, the most general solution to this requirement is: A(x) = a 3
2 1
,
(193)
where 3 is the Pauli matrix and a is a complex constant. The parameter a does not quite give only gauge inequivalent vacua, since by Weyl symmetry vacua labeled a and a are gauge equivalent. So the classical moduli space is described by a complex parameter u, with u = a2 = tr A2
2 1
.
(194)
For a generic nonvanishing value of u, the SU (2) gauge symmetry is broken down to a U (1). Since N=2 SUSY is still in force, masses are generated not just for two components of the SU (2) gauge eld, but also for their N=2 superpartners. Thus the remaining light elds consist of a U (1) gauge boson, a massless uncharged complex scalar, and two massless uncharged fermions. 47
The infrared eective action clearly exists in this case. The point u = 0, on the other hand, appears singular. The form of the infrared eective action is severely constrained by N=2 supersymmetry. The part of this action which contains no more than two spacetime derivatives and interactions of no more than four fermions must have the same form as the classical action of the ultraviolet theory: 1 Im 4 d4 x d2 d2 tr Fe () , (195)
Because of the anomaly, the eective action is not invariant under a U (1)R transformation. Instead, the AdlerBardeen theorem tells us that the eective action is shifted by = d4 x 1 Fmn F mn 4 2 d4 x d2 d2 1 2 2 2 . (196)
Im
Neglecting (for the moment) instanton eects, the eective action is still constrained by U (1)R invariance modulo this shift, which is manifestly a oneloop eect. The only holomorphic functional of with this property is F1 = i 1 2 2 ln 2 2 , (197)
where is a dynamically generated scale. Actually, since we can absorb the treelevel F into the denition of , F1 is the full eective prepotential to all orders in perturbation theory. From the shift Eq. 196 we see that a single instanton violates R charge conservation by 8 units, breaking the global U (1)R symmetry down to Z8 (in fact, since u carries R charge 4, there is a further breaking down to Z4 ). This suggests that the complete nonperturbative eective prepotential has the form: F =i 1 2 2 ln 2 + 2
k=1
Fk
4k
2
,
(198)
where the Fk are numerical coecients, and the k th term arises as a contribution of k instantons. Returning to the all orders perturbative result, we can deduce the eective Wilsonian gauge coupling gW (u) from the gauge kinetic function: f () = 2 F () 48 ; (199)
thus:
1 3g 2 g2 1 u = 2 1 + 0 + 0 ln 2 gW (u) g0 4 4 2 g3 4 2
.
(200)
This gives us the allorders perturbative beta function: (g ) = . (201)
We could try to make an analogous derivation in the case of N=1 SUSY YangMills. However in the N=1 case there is a separate wave function renormalization of the Dterm of the chiral supereld action. Because of this gW =ge and Eq. 201 is not valid. From the expression for the eective gauge coupling we see that the u limit is the weak coupling limit. This explains why we did not include negative k contributions to Eq. 198, i.e. why the sum extends only over positive k . Terms with negative k blow up like a power as u , behavior which is inconsistent with the existence of a nonsingular weak coupling limit. 9 9.1 Supersymmetry in spacetime dimensions 2, 6, 10, and 11 Spinors in arbitrary spacetime dimensions
The dimension of Dirac spinors in d spacetime dimensions can be deduced by constructing the Dirac gamma matrices obeying the Cliord algebra { m , n } = 2 mn The result is d = 2d/2 2(d1)/2 d even, d odd. (203) . (202)
Starting with Dirac spinors, we can investigate whether it is possible to impose Weyl, Majorana, or simultaneously Weyl and Majorana conditions on these spinors. For Weyl spinors, we need to generalize the notion of the chirality operator 5 . Recall that in four dimensions, CPT conjugate spinors have opposite chirality, implying that there are no gravitational anomalies. 23 This is related to the fact that 5 , dened as 5 = 0123 has eigenvalues i, since ( 5 )2 = I 49 . (204) (205)
Table 2: Properties of spinors in spacetime dimensions 2 to 12. d is the dimension of Dirac gamma matrices.
d d minimum spinor dim. Weyl? Majorana? MajoranaWeyl? gravitational anomalies?
2 2 1 X X X X
3 2 2
4 4 4 X
5 4 8
6 8 8 X
7 8 16
8 16 16 X X
9 16 16
10 32 16 X
11 32 32
12 64 64 X
X
X
X
X X
X
X
X
X
For any spacetime dimension d = 4k , k = 1, 2, . . ., we can dene 5 in an exactly analogous way: 5 = 0 1 4k1 , (206) and Eq. 205 still holds. Thus in d = 4k dimensions Weyl spinors exist and gravitational anomalies are absent. In d = 4k +2 dimensions ( 5 )2 = I ,
(207)
implying that CPT conjugate spinors have the same chirality. Thus Weyl spinors exist and gravitational anomalies are possible. In odd dimensions 5 dened as above is the identity; there is no chirality operator and thus no Weyl spinors. The analysis of Majorana and MajoranaWeyl conditions in arbitrary dimensions is more involved; a good reference is Sohnius. 4 The results are summarized in Table 2. 9.2 Supersymmetry in arbitrary spacetime dimensions
To discuss supersymmetry in spacetime dimensions other than four, we need an improved notation for keeping track of the number of independent supersymmetry generators. In four dimensions N=1 SUSY means that there are 50
four independent SUSY generators: Q1 , Q2 , Q1 , Q2 .
This is of course the minimum number of supersymmetries in four dimensions since the minimum spinor dimension is four. Let us refer to this as N=(1)4 supersymmetry. More generally, N=(p)pnmin denotes p nmin supersymmetries, where nmin is the minimum spinor dimension. In this new notation the possible global supersymmetries in four dimensions are: N N N N = = = (1)4 (2)8 (4)16
= (8)32 ...
In 4k +2 dimensions we can have independent chiral and antichiral SUSY generators, since CPT conjugates have the same chirality. Thus we need a notation which distinguishes chiral from antichiral SUSY generators: N = (p, q )(p+q)nmin where p, q are the number of chiral/antichiral SUSY generators, respectively. From Table 2 it is clear that in any spacetime dimension there is a minimum number of supersymmetries (other than zero). Thus: As few as 4 supersymmetries can only occur for: . . . . . . . . . d 4 As few as 8 supersymmetries can only occur for: . . . . . . . . . d 6 As few as 16 supersymmetries can only occur for: . . . . . . . . . d 10 As few as 32 supersymmetries can only occur for: . . . . . . . . . d 11 Furthermore, in any spacetime dimension, the maximum number of supersymmetries of physical interest is always 32 or less. This is because for more than 32 supersymmetries all massless multiplets contain unphysical higher spin particles, i.e. particles with spin greater than that of the ddimensional graviton. 51
Clearly we expect that many SUSY theories in dierent spacetime dimensions but with the same number of supersymmetries can be related, presumably through some form of dimensional reduction or truncation. Indeed this is true as we will see in several examples. Of particular interest is the possibility of relating models with extended supersymmetry in four dimensions to simpler models in the mother dimensions 6, 10, and 11. 9.3 Supersymmetry in 2 dimensions
In two dimensions we have (p, q ) type superalgebras. I will briey describe some examples. (1, 0)1 supersymmetry: here we have a single lefthanded MajoranaWeyl spinor: Q+ { Q+ , Q+ } [ Q+ , Pz ] = Q + , (208) (209)
= 2iPz , = [ Q+ , Pz ] = 0 ,
where the antihermitian generators Pz , Pz generate left and rightmoving translations in the twodimensional spacetime parameterized by coordinates z , z = x0 x1 . The minimal SUSY multiplet has just two states: one leftmoving real scalar (a chiral boson), and one leftmoving MajoranaWeyl fermion. (1, 1)2 supersymmetry: here we can construct a (1, 1) supersymmetric nonlinear sigma model by supersymmetrizing dzdz gij (A)z Ai z Aj . (210)
The target spaces of such models are general Riemannian manifolds. (2, 2)4 supersymmetry: here again we can construct a supersymmetric nonlinear sigma model. The target spaces are Khler manifolds. Note a that this agrees with the fourdimensional N=(1)4 case, which has the same number of supersymmetries. (4, 4)8 supersymmetry: here we have supersymmetric nonlinear sigma models whose target spaces are hyperKhler manifolds. Again this agrees a with the fourdimensional N=(2)8 case, which has the same number of supersymmetries. 52
Twodimensional supersymmetry has important applications to superstring theory, where it is interpreted as worldsheet supersymmetry rather than spacetime SUSY. For more details see Hirosi Ooguris lectures in this volume. 9.4 Supersymmetry in 6 dimensions
In six dimensions the minimal case is N=(1,0)8 or (0,1)8 . The SUSY generators can be expressed as a single complex Weyl spinor: Qa { Qa , Q }
b
, =
1 2
a = 1, . . . 8; (1 + 7 )c ( M )b PM a c , (211)
where we use capital Roman letters to denote 6dimensional spacetime indices, and 7 is the chirality operator, i.e. the 6dimensional version of 5 . In six dimensions massless particles are labelled by irreps of the little group Spin(4) SU (2)SU (2). Let us describe the possible massless irreps of (0,1)8 supersymmetry in terms of their SU (2)SU (2) helicity states: hypermultiplet: 2( 1 , 0) + 4(0, 0), i.e. one complex Weyl fermion and 2 two complex scalars, for a total of 4+4 = 8 real components.
1 1 vector multiplet: ( 2 , 1 ) + 2(0, 2 ), i.e. a massless vector and one com2 plex antiWeyl fermion, for a total of 4+4 = 8 real components. 1 gravity multiplet: (1, 1) + 2( 2 , 1) + (0, 1), i.e. a graviton, two gravitini, and one selfdual 2nd rank antisymmetric tensor, for a total of 9+12+3 = 24 real components. 1 tensor multiplet: (1, 0)+2( 2 , 0)+(0, 0), i.e. one antiselfdual 2nd rank antisymmetric tensor, one complex Weyl fermion, and one real scalar dilaton, for a total of 3+4+1 = 8 real components.
Dimensional reduction 6 4 Consider sixdimensional N=(0,1)8 supersymmetric YangMills theory. We write the action in terms of elds describing the onshell massless vector multiplet described above. The action is: d6 x FMN F MN + i M M
4 2 1 1
,
(212)
where M is a gauge covariant derivative. 53
Now we imagine compactifying this theory on a torus. Let x4 , x5 be the compactied coordinates. Then the 6dimensional gauge eld AM breaks up into a 4dimensional gauge eld Am and two real scalars A4 , A5 . The complex antiWeyl fermion breaks up into two 4dimensional complex Weyl fermions. In addition we will have an innite tower of massive KaluzaKlein states, corresponding to the Fourier decomposition (xm , x4 , x5 ) =
n4 ,n5
e[in4 m4 x
4
in5 m5 x5 ]
n4 n5 (xm )
,
(213)
where (xm , x4 , x5 ) denotes any 6dimensional eld component, and m4 , m5 are inversely related to the compactication radii. The SUSY generator Qa splits up into two Q s, implying that the 4dimensional theory has N=2 supersymmetry. The 6dimensional translation generator PM splits into Pm , P4 , P5 . From the 6dimensional SUSY algebra we see that P4 and P5 commute with the Q s and with Pm . They also appear on the righthand side of {Q, Q}. Clearly what we have here are two real = one complex central charge: X ab (P4 + iP5 )ab .
Multiplying the two Q s by a phase to convert to Zuminos basis (see Eq. 35), this corresponds to a single real central charge Z=
2 2 P4 + P5
.
(214)
Thus the dimensionally reduced theory consists of 4dimensional N=2 supersymmetric YangMills with central charges and an innite number of massive multiplets. Now suppose we repeat the above excercise, but rst adding some massless d = 6 hypermultiplets (A, B, ) to our N=(0,1)8 supersymmetric YangMills theory. Each d = 6 hypermultiplet will give one d = 4 multiplet of elds with the counting of the d = 4 N=2 massless hypermultiplet, plus a tower of additional massive multiplets. The 6dimensional onshell condition for the complex scalars A and B M M A = M M B = 0 xes the masses of the 4dimensional hypermultiplet scalars: 4 m2 = Z 2 . (216) (215)
Thus the dimensionally reduced theory contains massive BPSsaturated N=2 short multiplets, which as already noted do indeed have the same counting as the d = 4 N=2 massless hypermultiplet. 54
Anomaly cancellation A striking feature of the 6dimensional gravity multiplet is that it contains the selfdual part of a 2nd rank antisymmetric tensor, without the antiselfdual part. As is also the case in four dimensions, it is impossible to write a Lorentz covariant Lagrangian formulation of just the selfdual antisymmetric tensor eld. However one can write Lorentz covariant equations of motion, and it appears that the corresponding eld theory exists and is Lorentz invariant, despite the lack of a manifestly Lorentz invariant action principle. If ng = nt = 1 , (217)
where ng , nt are the number of gravity and tensor multiplets, then of course we can write a Lorentz covariant Lagrangian. Thus we have the interesting result that every 6dimensional supergravity theory either contains a dilaton eld or has no manifestly Lorentz invariant action principle. In six dimensions we have both gravitational anomalies and mixed gaugegravitational anomalies. Anomaly cancellation is a severe constraint on the particle content, and in particular on which combinations of SUSY multiplets yield anomalyfree theories. For example, when ng =nt =1, the necessary and sucient condition for anomaly cancellation is: nh = nv + 244 , (218) where nh , nv are the number of hypermultiplets and vector multiplets. Thus we always need a remarkably large number of hypermultiplets to cancel anomalies. Lets look at two examples of anomalyfree 6dimensional N=(0,1)8 supersymmetric supergravityYangMillsmatter theories. Gauge group E8 E7 , with 10 massless hypermultiplets in the (1, 56) of E8 E7 , and 65 singlet hypermultiplets. Thus: nv nh which satises Eq. 218. Gauge group SO(28) SU (2), with 10 massless hypermultiplets in the (28, 2) of SO(28) SU (2), and 65 singlet hypermultiplets. Thus: nv nh which satises Eq. 218. 55 = 378 + 3 = 381 ; = 560 + 65 = 625 ; (220) = = 248 + 133 = 381 ; 560 + 65 = 625 ; (219)
These examples arise as compactications of the tendimensional E8 E8 and SO(32) heterotic strings, respectively, onto the complex dimension 2 Khler manifold K3. a 9.5 Supersymmetry in 11 dimensions
Eleven is the maximum dimension in which we can have as few as 32 supersymmetries. Thus d = 11 is the maximum dimension of interest to supersymmetry theorists, unless one is willing to make some drastic assumptions such as altering the Minkowski signature of spacetime. Futhermore, there is only one sensible supersymmetric theory in eleven dimensions: N=1 supergravity. The N=(1)32 SUSY algebra is generated by a single Majorana spinor Qa , a = 1, . . . 32. Here is the eld content of the onshell massless d = 11 N=1 gravity multiplet, characterized by irreps of the little group SO(9): eA , the 11dimensional vielbein. Onshell this constitutes a 44 of SO(9). M
a M , a = 1, . . . 32, an 11dimensional massless vectorspinor, i.e. the gravitino eld. Onshell this is a 128 of SO(9).
AMN P , a 3rd rank antisymmetric tensor. Onshell this is an 84 of SO(9). The Lagrangian of 11dimensional supergravity, in terms of these component elds of the onshell multiplet, is rather simple. 24 The rst three terms are: where: e R MN P DN FMN P Q = = = = = 11dimensional gravitational coupling det eA M , Ricci scalar , ,
1 1 1 eR eM MN P DN P eFMN P Q F MN P Q 2 2 2 48
,
(221)
e M e N e P [A B C ] , ABC Lorentz covariant derivative for 11dim. vectorspinors , [M AN P Q] , i.e. a eld strength ,
and the square brackets denote antisymmetrization. Elevendimensional supergravity is the eld theory limit of Mtheory, which is in turn a strong coupling limit of superstring theory. 56
Dimensional reductions of d = 11 supergravity We can truncate d = 1 supergravity down to a 4dimensional theory by simply suppressing the dependence on x4 x10 . Since the resulting 4dimensional theory still has 32 supersymmetries, we obviously must get N=(8)32 extended supergravity. Provided one is satised with onshell formulations, this is a simpler way of deriving the rather complicated 4dimensional theory. Another useful example is to truncate d = 11 to d = 10. The Majorana spinor Qa (32 components) splits into two 10dimensional MajoranaWeyl spinors Q , Q (16 components each) with opposite chirality. Thus the truncated theory is a nonchiral d = 10 N=(1,1)32 supergravity, commonly known as Type IIA supergravity. 25 The components of the 11dimensional vielbein break up as follows: eA M eA M 0 AM , (222)
where we have set the 1 10 block on the lower left to zero using the freedom of those local Lorentz transformations which mix the x11 direction with the other ten. The 10dimensional massless vector AM arises from the e11 components M of the vielbein, while the real scalar (the 10dimensional dilaton) arises from the e11 component. 11 The 11dimensional antisymmetric tensor eld AMN P splits into a 10dimensional 3rd rank antisymmetric tensor AMN P and a 2nd rank antisyma metric tensor BMN (from AMN 11 ). The 11dimensional gravitino eld M becomes: a M M , M , , ; (223) giving two 10dimensional MajoranaWeyl vectorspinors of opposite chirality, and two 10dimensional MajoranaWeyl spinors of opposite chirality. Since Type IIA supergravity is vectorlike it is trivially free of gravitational anomalies. It is the eld theory limit of the Type IIA superstring. 9.6 More on supersymmetry in 10 dimensions
Type IIA supergravity has N=(1,1)32 supersymmetry. Since 16 is the minimum spinor dimension in ten dimensions, we can in principle construct a chiral N=(1,0)16 supergravity theory as well; however such a theory has gravitational anomalies. The only other possibility in ten dimensions is a chiral N=(2,0)32 supergravity. This theory, known as Type IIB supergravity, turns out to be anomalyfree, due to highly nontrivial cancellations. Type IIB cannot be obtained by dimensional reduction of 11dimensional supergravity. 57
Table 3: Comparison of Type II supergravities.
Type IIA:
1 8c
28 BMN 8s 28 AMN 8s
35v eA M 56s 35v eA M 56s
8v AM 56c 1 A 56s
56v AMN P
Type IIB:
1 A 8s
28 MN A
35c AMN P Q
N=(2,0)32 supersymmetry is generated by two MajoranaWeyl spinors Q , Q of the same chirality. The onshell Type IIB supergravity multiplet has the following eld content: eA , the 10dimensional vielbein. M AMN P Q , a selfdual 4th rank antisymmetric tensor. AMN , AMN , 2nd rank antisymmetric tensors. A, A, real scalars.
M , M , MajoranaWeyl vectorspinors of the same chirality. , , MajoranaWeyl spinors of the same chirality.
Because the eld content includes the selfdual part of a 4th rank antisymmetric tensor, Type IIB supergravity does not have a Lorentz covariant Lagrangian formulation. The little group in 10dimensions is SO(8). Because of the special automorphism symmetry of the Lie algebra D4 , all of the dimension< 224 irreps except the adjoint occur as triplets of irreps with the same dimension and index. These irreps are distinguished by subscripts v , s, and c. Thus we can gain more information about the dierences between Type IIA and Type IIB supergravity by listing the SO(8) irreps corresponding to each component eld. Note that both theories have 128+128 = 256 real eld components. 58
Let us consider again 10dimensional N=(1,0)16 supergravity, also known as Type I supergravity. As we have noted, this theory is anomalous. Remarkably though, by coupling this theory to the 10dimensional supersymmetric YangMills multiplet, we can in certain cases obtain theories free of both gravitational and mixed gaugegravitational anomalies. The eld content of the chiral supergravity is just a truncation of the Type IIA elds: eA , BMN , M M , ; ;
for a total of 64+64 = 128 real eld components. The 10dimensional onshell N=(1,0)16 supersymmetric YangMills multiplet consists of a massless vector and a massless MajoranaWeyl spinor AM , ,
in the 8v and 8c irreps of the little group, and the adjoint representation of some gauge group. The details 26 of how to couple N=(1,0) supergravity to N=(1,0) YangMills, consistent with both local supersymmetry and gauge invariance, is beyond the scope of these lectures. We merely note that one surprising result is that the gauge invariant eld strength of the massless antisymmetric tensor eld BMN has an extra contribution which is a functional of the YangMills gauge eld AM . Then, even though BMN is a gauge singlet, gauge invariance requires that BMN transforms nontrivially under YangMills gauge transformations. This strange fact makes possible the GreenSchwarz anomaly cancellation mechanism. The coupled d = 10 supergravityYangMills theory is anomalyfree only for the following choices of gauge group: SO(32), E8 E8 , E8 [U (1)]248 , [U (1)]496 .
The rst two choices correspond to the eld theory limits of the SO(32) and E8 E8 heterotic strings; the other two choices can probably also be connected to superstring theory using Dbrane arguments. 10 Conclusion
It is a remarkable fact that many technical SUSY topics, thought until recently to be of purely academic interest, have turned out to be crucial to obtaining deep insights about the physics of stronglycoupled nonabelian gauge theories 59
and stronglycoupled string theory. Even if there is no weak scale SUSY in the real world even if there is no SUSY at all supersymmetry has earned its place in the pantheon of Really Good Ideas. Furthermore, we are encouraged to continue to develop and expand the technical frontiers of supersymmetry, condent both that there is still much to learn, and that this new knowledge will nd application to important physical problems. Acknowledgments The author is grateful to the TASI organizers for scheduling his lectures at 9 a.m, to Je Harvey for keeping me apprised of the Bulls playo schedule, and to Persis Drell for explaining to me what a minimum bias event is. Fermilab is operated by the Universities Research Association, Inc., under contract DEAC0276CH03000 with the U.S. Department of Energy. Appendix Notation and conventions My notation and conventions in these lectures conforms with Wess and Bagger 2 with the following exceptions: I use the standard West Coast metric: 1 1 mn = 1
1
(224)
This is the standard metric convention of particle physics. Wess and Bagger use the East Coast metric still popular with relativists and other benighted souls. Changes which follow from my dierent choice of metric: 1. My 0 and 0 are the 2x2 identity matrix instead of minus it. 2. I dene the SL(2,C) generators with an extra factor of i:
mn =
i m n n m 4
.
(225)
3. I dene the LeviCivita tensor density 0123 = +1 instead of minus one. 60
4. As a result of this, mn is selfdual and mn is antiselfdual, instead of viceversa. 5. In the full component expansion of the chiral supereld (x, , ), ) term comes in with the opposite sign. the ()( Spinor denitions and identities
Irreps of SL(2,C) SO(1,3): ( , 0) = lefthanded 2 component Weyl spinor
2 1
(0, )
2
1
= righthanded 2 component Weyl spinor
1 In Van der Waerden notation, undotted = ( 1 , 0), dotted = (0, 2 ): 2
( , 0)
2
1
: :
, .
(0, )
2
1
( ) ; ( )
Also: ( ) . (226)
We raise and lower spinor indices with the 2dimensional LeviCivita symbols: = =
= =
0 1 0 1
1 0 1 0
; = i 2 . (227)
Thus:
Note also:
= =
; ;
= =
; . (228)
61
= = = =
, ,
(229) , .
Pauli Matrices: 1 = 01 10
2 =
0 i
i 0
3 =
10 0 1
(230)
From which we dene: m m = (I, ) = m = (I, ) = m , ,
(231)
where I denotes the 2x2 identity matrix. Note that in these denitions bar does not indicate complex conjugation.
m m has undotteddotted indices:
m has dottedundotted indices: m We also have the completeness relations: tr m n m m = = 2 mn , 2 .
(232)
m and m are related by the LeviCivita symbols:
m m = m = m
;
.
(233)
It is occasionally convenient to do a fake conversion of an undotted to a dotted index or vice versa using the fact that 0 and 0 are just the identity matrix: = ( ) 0 ; = ( ) 0 . (234) 62
Because the Pauli matrices anticommute, i.e. { i , j } = ij I we have the relations: ( m n + n m ) ( m n +
n m )
i, j = 1, 2, 3
(235)
= =
2 mn 2 mn
, . (236)
Spinor Summation Convention: = = = = = = = =
(237)
Note that these quantities are Lorentz scalars. We also have: ( ) ( m ) = ( ) = = = m = Lorentz vector
(238)
Other useful relations are: = = = =
2 1
, , , (239)
2 1
2
1
.
2
1
The SL(2,C) generators are dened as i m n n m 4 i m n n m 4 63
mn
= =
mn
(240)
We have: mnpq pq mnpq pq = 2i mn = 2i mn ; ; self dual (1, 0) anti self dual (0, 1)
(241)
And thus the trace relation: tr [ mn pq ] =
1
i ( mp nq mq np ) + mnpq 2 2
.
(242)
Fierz identities
()( )
=
()( )
2 1 ( )( ) 2
1
(243) (244) (245) (246) (247) (248) (249) (250)
( )( ) = m m = =
m m
1 mn 2
( m )( n ) = ( m ) ( n ) = ()( ) = ( )() =
()( ) () i( mn ) ( )
1 mn 2 1 2 1 2
( m )(m ) ( m )(m ) = ()( )
64
General 4dimensional SUSY algebra
{QA , QB } { QA , QB } {QA , QB } [QA , Pm ] [QA , Mmn ] [QA , Mmn ] [Pm , Pn ] [Mmn , Pp ] [Mmn , Mpq ]
= = = = = = = = 0
A m 2 Pm B
(251) (252) (253) 0 (254) (255) (256) (257) (258)
aAB B a B AB [Q , Pm ]
A
=
mn QA
mn QA
i(np Pm mp Pn )
= i mp Mnq mq Mnp
np Mmq + nq Mmp
A S B Q B SA B QB
(259) (260) (261) (262)
[QA , B ] [QA , B ] [B , Bk ] [Pm , B ]
= = = =
iCk j Bj [Mmn , B ] = 0
(263)
Where the a are antisymmetric matrices, and S , a must satisfy the intertwining relation:
A S C aCBk
= 65
aACk SC
B
(264)
Note also the perverse but essential convention implicit in Wess and Bagger: aAB N=1 SUSY algebra in 4 dimensions = a AB (265)
{Q , Q } { Q , Q } [Q , Pm ] [Q , Mmn ] [Q , Mmn ] [Pm , Pn ] [Mmn , Pp ] [Mmn , Mpq ] [Q , R] [Q , R] [Pm , R] References
= = = = = = = = =
m 2 Pm
(266) = 0 0 (267) (268) (269) (270) (271) (272) (274) (275)
{Q , Q }
[Q , Pm ] = mn Q
mn Q
= i (mp Mnq mq Mnp np Mmq + nq Mmp )(273) RQ RQ
0 i(np Pm mp Pn )
=
[Mmn , R]
=
0
(276)
1. J. Amundson et al, Report of the Supersymmetry Theory Subgroup, hepph/9609374; S. Dawson, SUSY and Such, hepph/9612229; M. Drees, An Introduction to Supersymmetry, hepph/9611409; J. Bagger, Weak Scale Supersymmetry: Theory and Practice, Lectures at TASI 95, hepph/9604232; X. Tata, Supersymmetry: Where it is and How to Find it, Talk at TASI 95, hepph/9510287; H. Baer et al, Lowenergy Supersymmetry Phenomenology, hepph/9503479. 2. Supersymmetry and Supergravity, J. Wess and J. Bagger (2nd edition, Princeton University Press, Princeton NJ, 1992.) 3. Introduction to Supersymmetry and Supergravity, Peter West (2nd edition, World Scientic, Singapore, 1990.) 4. M. Sohnius, Introducing Supersymmetry, Phys. Rep. 128, 39 (1985). 5. N. Seiberg and E. Witten, Nucl. Phys. B 426, 19 (1994). 6. S. Coleman and J. Mandula, Phys. Rev. 159, 1251 (1967). 7. R. Haag, J. Lopusza ski, and M. Sohnius, Nucl. Phys. B 88, 257 (1975). n 66
8. For an interesting attempt, see H. van Dam and L. Biedenharn, Phys. Lett. B 81, 313 (1979). 9. B. Zumino, J. Math. Phys. 3, 1055 (1962). 10. See Je Harveys lectures in this volume. 11. S. Ferrara, C. Savoy, and B. Zumino, Phys. Lett. B 100, 393 (1981). 12. S. Ferrara and B. Zumino, Nucl. Phys. B 79, 413 (1974). 13. M. Grisaru, M. Rocek, and W. Siegel, Nucl. Phys. B 159, 429 (1979). 14. J. Wess and B. Zumino, Nucl. Phys. B 70, 39 (1974). 15. D. Friedan, Phys. Rev. Lett. 45, 1057 (1980); M. Rocek, Physica D 15, 75 (1985). 16. B. Zumino, Phys. Lett. B 87, 203 (1979). 17. R. Grimm, M. Sohnius, and J. Wess, Nucl. Phys. B 133, 275 (1978); A. Galperin, E. Ivanov, S Kalitzin, V. Ogievetsky, and E. Sokatchev, Class. Quant. Grav. 1, 469 (1984); N. Ohta, H. Sugata, and H. Yamaguchi, Annals of Physics 172, 26 (1986); P. Howe, Twistors and Supersymmetry, hepth/9512066. 18. A. Strominger, Commun. Math. Phys. 133, 163 (1990). 19. L. AlvarezGaume and D. Freedman, Commun. Math. Phys. 80, 443 (1981). 20. P. van Nieuwenhuizen, Supergravity, Phys. Rep. 68, 189 (1981). 21. N. Seiberg, Phys. Lett. B 206, 75 (1988). 22. M. Shifman and A. Vainshtein, Nucl. Phys. B 277, 456 (1986). 23. L. AlvarezGaume and E. Witten, Nucl. Phys. B 234, 269 (1983). 24. E. Cremmer, B. Julia, and J. Scherk, Phys. Lett. B 76, 409 (1978). 25. Superstring Theory, M. Green, J. Schwarz, and E. Witten (Cambridge University Press, Cambridge, 1987.) 26. G. Chapline and N. Manton, Phys. Lett. B 120, 105 (1983).
67
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Concordia Canada  COMM  308
Print Last Name:Print First Name:ID Number:COURSE FINANCENUMBER COMM 308/4SECTIONS: ( Circle your section) CC, DD, H, I, J, K, L, M TIME 3 hours # OF PAGES 18 including coverEXAMINATION DATE Final Exam April 20, 2009 VERSION BLUE INSTRUCTOR: ( Under
Concordia Canada  COMM  308
Print Last Name:Print First Name:ID Number:COURSE FINANCENUMBER COMM 308/4SECTIONS: ( Circle your section) CC, DD, H, I, J, K, L, M TIME 3 hours # OF PAGES 18 including coverEXAMINATION DATE Final Exam April 20, 2009 VERSION BLUE INSTRUCTOR: ( Under
Concordia Canada  COMM  308
Print Last Name:Print First Name:ID Number:COURSE FINANCENUMBER COMM 308/4SECTIONS: ( Circle your section) G, H, I, J, K, M, CC, DD TIME 3 hours # OF PAGES 20 including cover/cribEXAMINATION DATE Final Exam APRIL 16, 2008 VERSION BLUE INSTRUCTOR: (
Concordia Canada  COMM  308
Print Last Name:Print First Name:ID Number:COURSE FINANCENUMBER COMM 308/4SECTIONS: ( Circle your section) G, H, I, J, K, M, CC, DD TIME 3 hours # OF PAGES 20 including cover/cribEXAMINATION DATE Final Exam APRIL 16, 2008 VERSION BLUE INSTRUCTOR: (
Concordia Canada  COMM  308