a brief introduction to particle interactions[1]

8/2/2019 A Brief Introduction to Particle Interactions[1]

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arXiv:he

p-th/0511073v1

7Nov2005

A Brief Introduction to Particle Interactions 1

Ali H. Chamseddine

Center for Advanced Mathematical Sciences (CAMS)

and

Physics Department, American University of Beirut, Lebanon

ABSTRACTI give a brief introduction to particle interactions based on representations

of Poincare Lie algebra. This is later generalized to interactions based on repre-sentations of the supersymmetry Lie algebra. Globally supersymmetric modelswith internal symmetry and locally supersymmetric models leading to super-gravity theories are presented. I also discuss higher dimensional supergravitytheories and some of their applications.

1Series of lectures given at the Summer School Dirac Operators: Yesterday and Today

held at CAMS, American University of Beirut, August 27-September 7, 2001. Published by

International Press, 2005.

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Contents

1 Introduction 2

2 Poincare group 3

3 Poincare Lie algebra 4

4 Dynamical equations for free and interacting fields 7

4.1 Spin-0 : Klein-Gordon field . . . . . . . . . . . . . . . . . . . . . 74.2 Spin-12 : Dirac field . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3 Spin-1: Maxwell field . . . . . . . . . . . . . . . . . . . . . . . . . 84.4 Spin-32 : Rarita-Schwinger field . . . . . . . . . . . . . . . . . . . 104.5 Spin-2 : Gravitational field . . . . . . . . . . . . . . . . . . . . . . 104.6 The standard model . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Supersymmetry 15

5.1 Irreducible representations of supersymmetry . . . . . . . . . . . 165.2 Supersymmetric field theories . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Lagrangian for chiral multiplet . . . . . . . . . . . . . . . 19

5.2.2 Lagrangian for vector multiplet . . . . . . . . . . . . . . . 195.2.3 N=1 supergravity Lagrangian . . . . . . . . . . . . . . . . 20

6 Higher dimensional theories 21

6.1 N=1 eleven dimensional supergravity . . . . . . . . . . . . . . . . 23

6.2 N=1 ten-dimensional Supergravity and Super Yang-Mills . . . . 24

7 Further directions 25

7.1 Index of Dirac operators . . . . . . . . . . . . . . . . . . . . . . . 25

7.2 Holonomy and N=1 supersymmetry . . . . . . . . . . . . . . . . 277.3 Solutions of Einstein equations . . . . . . . . . . . . . . . . . . . 29

1 Introduction

Historically, the Dirac operator was put forward as a result of making theSchrodinger equation in quantum mechanics compatible with the special theoryof relativity. In modern physical theories, matter at very small distances, or

equivalenty, very high-energies, is made of elementary particles whose interac-tions are the four fundamental forces in nature: gravity, electromagnetic theory,weak-nuclear force and strong-nuclear force.

Locally the four-dimensional space-time is Minkowskian and physical statesare invariant under rotations (Lorentz transformations ) and translations. The

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physical states fall into representations of the Poincare algebra. Particles willbe characterized by their mass and spin. Matter particles have spins 0 and 12obeying respectively Klein-Gordon and Dirac equations. Particles with spin-1are the mediators of gauge interactions, e.g. photons mediate the electromag-netic interactions between electrons while all particles exchange gravitons forthe gravitational interactions.

The general plan is to first study representations of the Poincare algebra andplace particles in these representations, and to form physical states obeying thesesymmetries and write Lorentz-invariant interactions. The Poincare algebra isthen generalized to the supersymmetry algebra. We find representations ofthe supersymmetry algebra and construct interactions invariant under this newsymmetry. For more details the reader may consult the following references [1],[2] , [3] , [4], [5] , [6] , [7] , [8] .

2 Poincare group

Einsteins special theory of relativity states the equivalence of certain inertialframes of reference. The coordinates x in an inertial frame satisfy

ds2 = dx dx = dx dx ,

where x and x are the coordinates of the physical states in different framesand = diag (1, 1, 1, 1) is the Minkowski metric. The transformationsx x = a + x keeps the distance invariant provided that

= . (1)

Taking the determinant on both sides gives (det )2

= 1 or det =

1. In

particular 000000+i0ij j0 = 00, or 002 = 1 +i02 . This implies that00 1, or 00 1 and transformations are divided into four categories. Theentire set of comprises the homogeneous Lorentz group. The orthocronousLorentz transformations are charecterised by the condition 00 1 so that everypositive time-like vector transforms into positive time-like vector. If det =1 then this gives a group of restricted homogeneous Lorentz-transformations.These are the only ones that could be obtained by a continuous transformationof the identity. This is because it is not possible to continously transform stateswith det > 0 to states with det < 0, or states with 00 1 to stateswith 00 1. Therefore these states correspond to an SO(3, 1) group. Thetranslations a form a seperate group, an abelian one, defined as T(4) . Wewrite

x = x + a g (a, ) x,x = x + a = (x + a) + a = x + (a + a) .

But x = g (a, ) x = g (a, ) g (a, ) x g (a, ) x, which impliesg (a, ) g (a, ) = g (a + a, ) .

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The subgroup SO(3, 1) is defined by g (0, ) g (0, ) = g (0, ) and the abeliansubgroup T4 by g (a, 1) g (a, 1) = g (a + a, 1) . The inverse transformation isdetermined by the two conditions a + a = 0, and = 1 which implies

g1 (a, ) = g1a, 1 .

3 Poincare Lie algebra

We write = +

where

is infinitesimal. This equation implies that

= = is antisymmetric. Assume a is infinitesimal then we

need four essential parameters for translation and six essential parameters forrotations

U(a, ) = 1 + iaP i2

J.

The commutation relations of the Lie algebra can be obtained via the usualbasis of scalar functions

U(a, )f(x) = f

U1 (a, ) x

= f

1 (x a) .By going to the infinitesimal limit we get

1 + iaP i2

J

f

x

= f

x a x

= f

x a + x f(x) +

This implies that

P = i

x i,J = (xP x P) .

These generators obey the Lie algegra

[P, P ] = 0,

[J , P] = i ( P P ) ,[J , J] = i (J J J + J) .

By exponntiating, a general finite transformation is expressible in the formU(a, ) = exp

iaP i2 J

. The Casimir invariants (operators that

commute with the generators of the Lie algebra) are P2 = P P and W2 =

W

W

where W = 1

2 P

J

. One can easily verify thatP2, P

= 0,

P2, J

= 0,

W2, P

= 0,

W2, J

= 0.

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If P2 = 0 we can use the spin pseudo-vector S = 1P2

W. The Casimir

eigenvalues of P2 and W2 (or j (j + 1) associated with S2) are used to specifyirreducible representations:

P |p, j, = p |p, j, ,W2 |p, j, = W2 (j) |p, j, ,W0 |p, j, = W0 (j) |p, j, .

There are four distinct cases. i) P2 > 0, ii) P2 = 0, iii) P2 < 0, iv) P = 0.If P2 = 0 then we can take P1 = P2 = 0. The little group leaving this choiceinvariant contains J3 = J12, so we can associate with integer or

12 -integer

eigenvalues of J3 :J3 |p, j, = |p, j, ,

so that is always discrete.If P2 = m2 > 0 we choose the rest frame

P = (m, 0, 0, 0) ,W = m

0,

J

, Jij = ijk Jk,

S =

0,J

.

The little group is O(3) generated by [Ji, Jj ] = ijk Jk and is described by the

state vectors

|j, , = j, , j, j = 0, 12

, 1, where W2 = m2j(j + 1).

If P2 = 0, we can take P = (w, 0, 0, w), and W = w(J3, L1, L2, J3) whereL1 = J1 + K2, L2 = J2

K1 and J0i = K0i, i = 1, 2, 3. The generators L1 L2

and J3 satisfy the commutation relations

[L1, L2] = 0, [J3, L1] = iL2, [J3, L2] = iL1.The relations P2 = 0, W2 = 0 and PW = 0 imply that P

and W areparallel vectors and therefore we can write

W = P,

where is the helicity. This implies that W2 = w2L2 and unitary representa-tions are labelled by |l, .

For the Lorentz group we can write

Ni =1

2(Ji + iKi), N

i =

1

2(Ji iKi),

where these generators satisfy the commutation relationsNi, N

j

= 0,

[Ni, Nj ] = iijk Nk,

Ni , Nj

= iijk N

k.

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The Casimir operators are given, in terms of the generators Ni and Ni , by NiNi

and Ni Ni with eigenvalues n(n +1) and m(m +1) respectively. The Ni and N

i

obey independently Lie algebras of SU(2). We can label representations with(m, n) where m, n = 0, 12 , 1,

32 , are the separate SU(2) numbers. The two

SU(2)s are not independent. The spin representation are given by m + n. Asexamples we have:

(0, 0) is spin zero and is the scalar representation. (12 , 0) or (0, 12) is the spin- 12 representation for Weyl spinors, while Dirac

spinors are represented by (12 , 0) (0, 12). (12 , 12 ) = (12 , 0) (0, 12) is the spin-1 representation.We represent ( 12 , 0) and (0,

12) by two-component complex spinors and

.

which transform under Lorentz transformation as

= M

, .

=

M1 ..

.

This can be denoted by M and M1 . We identify generatorsJi with

12i and Ki with i2i. We can therefore write

M = ei2

i(wiivi),

M1

= ei2

i(wi+ivi).

We also introduce the 22 matrices ().

defined by

0

.

= I2, the identity

matrix and

i

.

= i are the Pauli matrices. The matrix P is defined

by P = P () and transforms under Lorentz transformations according to

P = M P M or equivalently P

PM

M. We also define

. =

..

.

.

These satisfy the proporties

( + )

= 2 ,

( + ).

.

= 2

.

.

.

From these matrices we can construct representations of the Lorentz algebra.They are given by

= ( ) ,

.

. = ( )

.

. .

A Dirac spinor takes the form

=

.

,

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and the Dirac gamma matrices are defined by

= 0 0

,which satisfy the anticommutation relations {, } = 2I4. This definesa Clifford algebra. The representation of the Lorentz algebra in terms of thegamma matrices is given by

S =i

4[, ] .

Let 12

= exp i

2 S

then 112

12

= where =

+

.

These proporties will be used in the next section when we derive the Diracequation.

4 Dynamical equations for free and interacting

fields

4.1 Spin-0 : Klein-Gordon field

The Hamiltonian in relativistic dynamics has eigenvalues E which can be de-termined from the relation PP

= m2. This relation implies that P20 =P .

P + m2 = E2, where we have set the velocity of light c to 1. Using the

quantum mechanical correspondence P =

i

xand setting = 1) and acting

with the operator

PP m2 on the scalar field gives

+ m

2

= 0,

which is the Klein-Gordon equation. This equation could be derived from avarriational principle using the action

I0 =

d4xL =

1

2

d4x

m22

.

The Euler-Lagrange equation obtained by demanding that I = 0, with the field vanishing at the boundary, is given by

L

=

L

,

and this implies the Klein-Gordon equation.

4.2 Spin- 12 : Dirac field

The Schrodinger equation is non-relativistic and is linear in time-derivatives butquadratic in space-derivatives

i

t=

2

2m

2.

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The relativistic equation should be linear in time and one must find a first orderdifferential equation of the form

i

t=c

ii

xi+ m,

where i and are 4 4 matrices. The square of this equation should yieldan equation of the Klein-Gordon type. This is possible if i and satisfy theproporties

ij + j i = 2ij, 2 = 1, i+ i = 0.

Multiplying both sides of the equation by and defining 0 = , i = i gives

(i m) = 0.The matrices satisfy the anticommutation relations {, } = 2 and canbe identified with the gamma matrices defined in the previous section. This isthe Dirac equation. One can easily verify that it is Lorentz covariant

(i m)

i1 m

12

1x

= 12

112

i1 m

1

2

1x

= 12

(i m)

1x

,

where we have used the property 112

12

= derived in the last section.

The action

I12

=

d4x(x) (i m) (x),

is Lorentz invariant because

11

2

. In the special case when m = 0, the

Dirac equation simplifies to0 i

i 0

LR

= 0,

and this splits into two independent equations, one for the left-handed compo-nent L and the other for the right handed component R

iL = 0, iR = 0.

4.3 Spin-1: Maxwell field

We are interested in massless vectors, in particular, the photon which obeys

Maxwells equations E = , E =

B

t,

B = 0, B = E

t+

J .

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The second and fourth equations are solved by

E = A0 At

, B = A ,

and are invariant under the gauge transformations

A =

A , A0 = A0 +

t.

Therefore one can impose a gauge choice, the Lorentz gauge, on the potentialA and A0

A + A0

t= 0.

In this gauge, Maxwell equations simplify to

A0 = , A = J ,which is a standard wave equation with a source. Maxwell equations could bewritten in a Lorentz covariant manner if we define the curvature

F = A Aso that

F = J .

We can replace the definition of F by the requirement

F + F + F = 0,

which, locally, can be solved by the above equation defining A. The curvatureF is invariant under the gauge transformations A

= A + . Maxwell

equations could be obtained from the action

I1 =

d4x

1

4F F

+ AJ

.

According to Noethers theorem, there is a conserved charged associated toevery symmetry of the action. The symmetry in this case gives electric chargeconservation. To find the interactions of electrons with photons, we note that theaction for electrons I1

2is invariant under the global symmetry (x) eie(x)

where is a constant phase. This global symmetry is lost if the symmetry ispromoted to a local symmetry by allowing to depend on x. In this case

(x)i(x) (x)i(x) + e(x)(x).To restore the symmetry, we add to the Dirac-action, the photon-electron inter-action term

e

d4x(x)(x)A,

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which transforms under the local symmetry according to

e(x)

(x)A e(x)

(x)A e(x)

(x).

We deduce that the action for the spin- 12 , spin-1 system governing the interactionof photons and electrons and invariant under the local gauge transformations(x) eie(x) and A A + , is given by

I( 12 ,1)=

d4x

1

4F F

+ (x) (iD m) (x)

,

where D = + ieA. Therefore, interaction terms are obtained by replacingordinary derivatives in the free Dirac action by covariant derivatives D.These covariant derivatives satisfy the commutation relations

[D, D ] = ieF.

4.4 Spin- 32

: Rarita-Schwinger field

The field that describes a spin- 32 particle is a vector-spinor where is aspace-time vector index and is a spinor index. The free field satisfies theRarita-Schwinger equation

= 0

where = , and

=1

3!( + + )

is completely antisymmetric in the indices . The product of spin-1 and spin- 12representations gives spin- 3

2

and spin- 1

2

representations1

2,

1

2

1

2, 0

=

1,

1

2

0,1

2

.

It will be seen that the Rarita-Schwinger equation is necessary to eliminate thespin-12 component from .

4.5 Spin-2 : Gravitational field

The massless spin-2 field is the graviton which is a symmetric traceless tensorh with gauge invariance

h h + + .

To see that this is none other than the linearized form of the metric tensor of theunderlying space-time manifold, let us assume that we would like to promotethe global Lorentz invariance of spinors to a local one:

(x) exp

i

4ab (x) ab

(x) ,

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where ab =12

ab ba

. To restore the invariance of the Dirac equation

under local Lorentz transformations, and in analogy with the electromagnetic

case, we introduce the covariant derivative

D = +1

4 ab ab.

The term (x) (iD m) (x) becomes invariant provided that

ab = ab + ac

bc bc ac .

The curvature tensor is defined by

[D, D ] =1

4R ab ab,

which can be easily evaluated to be

R ab = ab

ab + ac bc ac bc ,

and is covariant under local Lorentz transformations. To make contact with theRiemann curvature tensor, we introduce the soldering form ea satisfying

ea = ea ea + ab eb = 0,

where =12g

(g, + g, g,) is the Christoffel connection and g =eaabe

b . This equation implies that the covariant derivative of the metric under

coordinate transformations vanishes. The constraint equation can be solved for ab in terms ofe

a. Substituting the solution into R

ab () one can show that

R ab () eaeb = R(g),

is identical to the Riemann tensor as a function of the metric g.Equivalently, we can introduce gauge fields corresponding to both the Lorentz

generators Jab and translations Pa

D = + ab

Jab + eaPa,

then the curvature is

[D, D ] = Rab

Jab + Ta

Pa,

where R ab is the same as before while

Ta = ea ea + ab eb ab eb.

By setting the torsion Ta to zero we can uniquely solve for ab

in terms of ea

and substitute this expression into R ab to obtain the Riemann tensor.

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This last formulation is related to the Cartan structure equations

Ta = dea + ab eb,Rab = dab + ac bc .

By writing ea = eadx, ab = ab dx

, Rab = 12Rab

dx dx and Ta =

12T

adx

dx we can recover all the previous formulas.The Einstein-Hilbert action is given by

I2 =

d4xe

4ea e

b R

ab ,

where e = det

ea

. The dynamical equations governing the interaction of elec-trons and photons with gravity can be read from the equation

I( 12 ,1,2)= d4xei + eA + 1

4 ab ab m

d4xe

1

4F Fg

g +1

4ea e

b R

ab

.

4.6 The standard model

The principle of gauge invariance plays a fundamental role in physics. ThePoincare symmetry is a space-time symmetry effecting local space-time coordi-nates. On the other hand the known elementary particles in nature, fall intogroup representations, e.g. fermions have the local gauge invariance under the

transformations I

eiga

a(x)Ta

J

IJ where (T

a)J

I are the representations

of the gauge group. As in the Maxwell case, connections are introduced to insurelocal gauge invariance. Let

D = d + ig A, A = AaTadx,

so that the curvature of the connection A will be given by

D2 = igF = ig

dA + igA2

,

F =1

2Fa T

adx dx ,

so thatFa = A

a Aa + gfabcAbAc ,

where Ta, Tb = ifabcTc. Let us denote the quarks by q = uLdL , uR, dRwhich are in the representations

3, 2, 13

,

3, 1, 43

and (3, 1, 23) ofSU(3)c SU(2)w U(1)Y . The leptons are denoted by l =

eLL

, e+

Rwhich are in the

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(1, 2, 1) and (1, 1, 2) ofSU(3)c SU(2)w U(1)Y .. The Lagrangian that governall known interactions is given by

e1L = 142

R 14

Fi F

i + F F + B B

+ DHDH 2HH+ HH2

+ iq

+1

4 ab ab

i

2g2A

i6

g1B i2

g3Vi

i

q

+ idR

+

1

4 ab ab +

i

3g1B i

2g3V

i

i

dR

+ iuR

+

1

4 ab ab

2i

3g1B i

2g3V

i

i

uR

+ il +1

4

ab ab

i

2

g2A

+i

2

g1B l+ ieR

+

1

4 ab ab + ig1B

eR

+

kdqH dR + kuq2HuR + k

elH eR + h.c.

,

where the Higgs field H is in the (1, 2, 1) representation so that DH = Hi2g2A

i2g1BH and Vi, A and B are the gauge fields for the gauge Liealgebras SU(3)c, SU(2)w and U(1)Y . The k

d, ku and ke are 3 3 matricesmixing the three generations of particles. Minimizing the potential of the Higgsfields H gives a vacuum

H =

0v

,

which generates masses for the quarks and leptons, e.g. the leptons will acquirethe term

kelH eR = keveLeR.

Similarly the term DHDH will yield the mass terms

1

4v2g22

W+ W

+1

cos2 ZZ

,

where W =

A iA2

, Z =1

g21+g2

2

g2A

3 g1B

and sin2 =

g21

g21+g2

2

.

The symmetry of the Lagrangian is diffeomorphism local internal symmetry.All fermions are chiral (Weyl fermions) and acquire mass only after the symme-try is broken spontaneously from SU(3)c

SU(2)w

U(1)Y to SU(3)c

U(1)em.

At the quantum level chiral fermions break gauge invariance. The chirality con-dition is 5 = where 5 = i0123 so that 25 = 1. A Dirac spinoris decomposed according to = + + . The eigenvalue equation D = implies 5D = D so that the eigenvalue problem could not be set forchiral fermions alone without their partners except for zero eigevalues. After

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quantization one must sum over all states which would involve computing thedeterminat DDeD det D,and this involve the step of taking the trace over all states, which must now berestricted to the chiral ones

n| D |n+ .

This is not invariant under the chiral rotation n+ ei5n+. To see thisexplicitely we have

T r(F(D)) =

n

n| F(D) |n+

n

ei

aTan

F(D)ei

aTan+

.

Since F(D) depends only on geometrical quantities, one can use the heat kernelexpansion to find the lowest terms

T r(F(D)) 2iaT r 5Ta Fb TbFcTc= 2iaFb F

cT r

Ta

Tb, Tc

.

We deduce that gauge invariance is not destroyed by chirality if

T r

Ta

Tb, Tc

= 0.

If we include gravitational terms then the lowest order term is

2iaT r

5T

a R cd Rcd

,

which vanishes ifT r (Ta) = 0,

insuring the absence of gravitational anomalies. Both conditions on the repre-sentations of the chiral fermions in the standared model are satisfied. These arevery strong constraints on possible representations of the fermions in a realisticmodel and lead to the conclusion that these are not satisfied by accident butare derivable from higher principles. One such explanation is unification wherethe chiral fermionic representations

3, 2, 13

3, 1,2

3

3, 1, 43

(1, 2, 1) (1, 1, 2)

could be obtained from the 5 10 representations of SU(5) to be broken spon-taneously into two stages

SU(5) SU(3)c SU(2)w U(1)Y SU(3)c U(1)em.

This is not the unique possibility but the simplest.

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5 Supersymmetry

In nature there are particles of different spins, in particular integral and half-integral spins. In the ultimate theory there must be a symmetry that placesparticles with different spin in the same multiplet. Until 1974 all internal sym-metries were studied and there was a theorem by Coleman and Mandula statingthat, under certain physical assumptions, it is impossible to have a space-timesymmetry larger than the Poincare symmetry. This obstruction was overcomeby the simple extension of considering Z2 graded algebras whose generators areclassified into two classes, even (bosonic) and odd (fermionic) and obey

[even,even] = even, [even,odd] = odd, {odd, odd} = even.

With every generator A in the graded Lie algebra we associate a number a =

0 if A is even

1 if A is oddand define the graded commutator by

[A, B} = AB (1)ab BA,

which satisfies the graded Jacobi identity

[A, [B, C}} + (1)a(b+c) [B, [C, A}} + (1)c(a+b) [C, [A, B}} = 0.

Denote by Qi the fermionic generators where the index transforms underthe Lorentz group and the index i transforms under an internal algebra withgenerators Tr

[Tr, Ts] = frstTt, [P, Tr] = 0, [J , Tr] = 0,

Qi, J = (b) Qi, Qi, P = (c) Qi,Qi, Q

j

= r (C) P

ij + s ( C) J ij

+ C Uij + (5C) V

ij + (5C) Lij .

By the Coleman-Mandula theorem we can set Lij to zero because it carries a

Lorentz vector index. Without any loss in generality we can assume that Qisatisfy the Majorana condition Qi = C Q

i.

Using the Jacobi identityJ ,

J, Q

i

+

J,

Qi

+

Qi, [J , J]

= 0,

we deduce that (b )

should form a representation of the Lorentz group and

must be identified with(b )

=

i

2( )

From the identityP,

P , Q

i

+

P ,

Qi, P

+

Qi, [P, P ]

= 0,

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one deduces that [c, c ] = 0. But it is always possible to change (c)

to the

form c = c ()

and this implies that c = 0. From the identityQi,

Qj , Q

k

+

Qj ,

Qk, Qi

+

Qk,

Qi, Qj

= 0,

we deduce that Uij and Vij commute with all the generators. Finally from theidentity

P,

Qi, Qj

+

Qi,

Qj , P

Qj,

P, Qi

= 0,

we deduce that s = 0. We shall rescale the parameter r to 2. Summarizing theresults we have the most general graded algebra which is an extension of thePoincare symmetery and consistent with unitarity:

Qi, J = i2 ( ) Qi, Qi, P = 0,Qi, Q

j

= 2 (C) Pij + C Uij + (5C) Vij .

5.1 Irreducible representations of supersymmetry

The Casimir operators are P2 and W2 whereW = P J + Qi5Qi.

Since P2 is a Casimir, all particles belonging to the same multiplet must have thesame mass. Now introduce the fermion number (1)NF which acts on bosonicand fermionic states as follows

(1)NF |boson = |boson ,(1)NF |fermion = |fermion ,

which implies that

(1)NF Qi |. = Qi (1)NF |. .

For any finite dimensional representation we have

T r

(1)NF

Qi, Qi

= T r

(1)NF (QiQi

+ Qi

Qi

= T r(1)

NF () P

= T rQi (1)NF Qi + Qi (1)NF Qi = 0.

From this we deduce thatT r

(1)NF

= 0,

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implying that the number of fermionic and bosonic degrees of freedom are iden-tical.

The energy in supersymmetric theories is non-negative. To see this consider

0

i

Qi

Qi

+

Qi

Qi

=

i

Qi

Q

i0

+

Qi0

Qi

= T r

i

Qi,

Q

i0

= 2N T r (P0) = 8N P0.

Therefore E 0.We would like to characterize the massless representations of supersymmetry:

P2 = 0, and we can choose P = (w, 0, 0, w) . In this frame, and after settingthe central charges Uij and Vij to zero, we get

Qi, Q

j

= 2ij (C) P = 2w

ij ((0 + 3) C)

= 4wij 0 0 0 10 0 0 00 0 0 0

1 0 0 0

.We therefore have one independent relation

Qi1, Q

j4

= 4E ij. From the

relation Qi = CQi

we have

Qi3 = Qi2 , Q

i4 = Qi1

so we can write

Qi1, Qj1

= 4wij. Normalizing the Qi1 we can finally write

Qi1, Q

j1

= ij .

The subgroup of the Lorentz group leaving the state (w, 0, 0, w) invariant is03 = 0, 10 = 13, 20 = 23 implying that the generators are

T1 = J10 + J13, T2 = J20 + J23, J = J12.

These obey the commutation relations

[J, T1] = T2, [J, T2] = T1, [T1, T2] = 0,J, Qi1

=

1

2Qi1,

J, Q

i

1

= 1

2Q

i

1,

Qi1, Qj1

= ij .

Since Qi1 and Qi1 form a Clifford algebra, they correspond to a set of N fermi

creation operators Qi1 and N fermi annihilation operators Qi1. The vacuum

state is defined by |

Qi1 | = 0, J| = | ,

Qi1 | = 12 , i

, Qi1 Q

j1 | = | 1, [ij] ,

Q11 Q21 QN1 | =

N2

.

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5.2 Supersymmetric field theories

5.2.1 Lagrangian for chiral multiplet

The Lagrangian for N = 1 chiral multiplet (A,B,) is given by

L =1

2

A

A + BB + i

.

The action is invariant under the supersymmetry transformations

A = , B = 5,

= (iA + B5) ,as can be easily verified. The supersymmetry parameter is independent of thecoordinates x and supersymmetry is a global symmetry.

5.2.2 Lagrangian for vector multiplet

The vecor multiplet is given by (A, ) where A = AaT

a and = aTa which

are Lie algebra valued:

Ta, Tb

= ifabcTc. The Lagrangian which is valid indimensions D = 4, 6 and 10 is given by

L = T r

1

4F F

+i

2D

,

where F = dA + A2 = 12F dx dx , F = Fa Ta and D = +

ig [A, ] = DaTa. The action is invariant under the supersymmetry trans-

formations

A = i,

=1

2F

.

To verify supersymmetry one has to use Fierz identities for the fermions whichare valid only in dimensions 4, 6 and 10.

One can couple vector multiplets to chiral multiplets and have a generaliza-tion of the standard model. In the supersymmetric standard model it turned outthat all known fields must have partners, e.g., the electron must have a bosonicpartner, the s-electron, while gauge fields A will have as partners the gauginos. As we have seen, the fermions in the standard model are chiral. The chiralitycondition cannot be satisfied for theories with more than N = 1 supersymmetry.

In other words, for supersymmetry to be of any relevance, the Lagrangian atlow energies must have N = 1 supersymmetry. However, in the real world su-persymmetry is not present because no partners for the observed particles withthe same mass are found, so it must be broken. One of the main advantages ofsupersymmetry is that it has good quantum behaviour, where divergencies inthe renormalization of the bosonic fields are cancelled by those coming from the

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fermions. To preserve good quantum behaviour, the symmetry must be brokenspontaneously. The Higgs mechanism which is essential in the bosonic case must

also be present here. The Goldstone phenomena where the scalar fields associ-ated with the broken generators are absorbed by the massless vector fields tobecome massive must be generalized. In this case the field associated with thebroken supersymmetry generator, the Goldstino, must be absorbed by a mass-less vector-spinor field. In the discussion in the last section we have seen thatthis is the Rartia-Schwinger field. This implies that the supergravity multipletmust be coupled to the matter Lagrangian of the standard model in such a waythat supersymmetry is broken spontaneously, so that the gravitino absorb onefermionic field to become massive. In supergravity the gravitational multipletplays a role in the breaking of supersymmetry and this can be used to inducethe electroweak breaking in the standard model. This is a novel phenomena asthis implies that physics of the extremely weak gravitational field influences thelow-energy sector.

5.2.3 N=1 supergravity Lagrangian

The easiest way to construct the supergravity Lagrangian is to generalize theconstruction of the Einstein-Hilbert action based on gauging the Poincare alge-bra. Here we gauge the supersymmetry algebra instead. Let

D = + ab

Jab + eaPa +

Q,

be the connection associated with the supersymmetry algebra. The curvatureis easily computed to be

[D, D ] = Rab

Jab + Ta

Pa + Q,

where

R ab = ab

ab + ac bc ac bc ,Ta = e

a ea + ab eb ab eb + a ,

=

+

1

4 ab ab

+

1

4 ab ab

,

and the action of Jab on the spinor is represented by14ab. To proceed one

first impose the torsion free constraint

Ta = 0,

which can be solved to express ab

in terms of ea

and

to obtain

ab = ab

(e) +1

4

ab ba + a

b

where ab (e) is the same expression we had in the non-supersymmetric case,and indices are changed from flat to curved by using the vierbein ea and its

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inverse ea . The supersymmetry transformations are given by

ea = a,

ab = 0,

=

+

1

4 ab ab

.

The torsion constraint is not preserved under these transformations. We canpreserve the torsion constraint by modifying the ab accordingly. The resultis

ab = 12

eaeb ( + ) .

The Lagrangian invariant under the supersymmetry transformations is given by

e1LSG = 1

4 R + + 14 ab ab

The proof of invariance is simplified by using that eR = 12abcde

ae

b R

cd

and by writing the varriation with respect to ab in the formL

ab ab . Since

ab appears linearly and quadratically one can easily see that, after integrating

by parts, its varriation L ab

vanishes if the torsion vanishes and therefore there

is no need to substitute the very complicated expression for ab . This methodis known as the 1.5 formalism and was used widely in the early formulationsof supergravity because it drastically simplifies the proof of the supersymmetryinvariance of the proposed actions.

6 Higher dimensional theories

By matching the bosonic and fermionic degrees of freedom we shall determinewhat are the possible supersymmetric multiplets in all possible dimensions. Firsta massless vector field in D dimensions have D 2 degrees of freedom as the A0component does not propagate because it has no time derivatives in the actionand one of the transverse components of Ai can be gauged away. For fermions

a Dirac spinor has 2[D2 ] components, but in certain cases could be subjected

to either the Weyl chirality condition or to the Majorana condition or both.The first step is to define the Clifford algebra in dimensions D. The Dirac

gammas matrices M where M = 0, 1, , D 1 will be 2[D2 ] 2[D2 ] matricess.In even dimensions we can define the gamma matrix D+1 =

01 D1

where (D+1)2

= 1 so that 2

= (1)s

t

= 1 (mod8), where s is the number ofspace-like coordinates and t the number of time like coordinates. For our con-siderations we will always take s = D 1 and t = 1. Therefore (D+1)2 = 1 andone can always impose the Weyl condition D+1 = . No Weyl fermionscan exist in odd dimensions. The Majorana condition = C requires theexistence of imaginary representations of the gamma matrices in D dimensions

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so that M

= CMC1. This is only possible in D = 2, 3, 4 (mod8) . It isalso possible to impose both the Weyl condition and the Majorana condition

simultaneously only in D = 2 (mod8) . Therefore a spinor have 22 12 = 2components in 4 dimensions as one imposes the Majorana condition. (Super-symmetric multiplets are always defined with respect to Majorana spinors sothe Weyl condition can only be imposed in dimensions 2 (mod8). The numberof components for in 5 dimensions is 4 while in 6 dimensions it is 23 12 = 4components. In dimensions 7 and 8 it is 8 components. In dimension 9 it is16 (although in this case it is possible to modify the Majorana condition toreduce it to 8 components). In dimension 10 one can impose simultaneously theWeyl and Majorana conditions so that the number of independent componentsof is 8. We notice that the number of independent components of AM and match in D = 3, 4, 6 and 10 and this explains why the super Yang-Mills actionis invariant under supersymmetry transformations only in those dimensions. Indimensions higher than 10 the number of independent fermionic components in-crease much faster than the number of independent components for the vectorsAM and supersymmetric actions for the pure vector multiplet do not exist. Thisis to be expected because when these higher dimensional theories are analyzedas viewd in four-dimensions, they will correspond to vector multiplets with su-persymmetry higher than N = 4 which is not possible from our analysis of thesupersymmetry representations.

To analyse supergravity theories in higher dimensions we must first deter-mine the number of independent degrees of freedom for the graviton. This is

essentially a symmetric metric with D(D+1)2 components where the componentsg0i do not propagate because they do not acquire second time derivatives. Fromthe other components gij the D diffeomorphism parameters could be used to

reduce the components to D(D3)2 =(D2)(D1)

2 1 so essentially the physicaldegrees are those of a symmetric traceless metric in (D 2) dimensions. Fromthe Rarita-Schwinger equation for a free field, MN PNP = 0, we can easilyshow by first contracting this equation with M and making the gauge choiceMM = 0 that

NNM = 0 and MM = 0. Therefore M behaves like a

massless vector with D 2 spinor components and the condition NNM = 0eliminates one more spinor component to give finally (D 3) 2[D2 ] r wherethe reduction factor r is 12 if the Majorana condition is imposed, r is

14 if the

Majorana-Weyl condition is imposed and r = 1 if no conditions are imposed.From this analysis it is easy to see that the supergravity multiplet in D = 4 isthe metric (or vierbein) ea and the gravitino as we have seen before. In fivedimensions, the metric has 5 degrees of freedom while the gravitino has 8 . A vec-tor field has 3 degrees of freedom so the number of fermionic and bosonic degreesmatch if we take the multiplet ea, A, . We can classify all possible theoriesin various dimensions, but it is easier to start with maximal supergravity whichis N = 8.

6.1 N=1 eleven dimensional supergravity

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A spinor in 11 dimensions have 2[112 ] 12 = 16 degrees of freedom because the

Majorana condition could be imposed. As viewed from 4 dimensions wherea spinor have 2 degrees of freedom, this corresponds to a spinor with SO(8)symmetry, and according to the classification discussed before, this correspondsto a theory with N = 8 supersymmetry in four-dimensions. Therefore D = 11 isthe highest dimensions where a supergravity theory could be constructed. Alsoin higher dimensions the number of fermionic degrees increase much faster thanthose of the metric and other bosonic fields. To determine the field content ofthe supergravity multiplet in D = 1 we note that the metric has 44 degrees offreedom while the gravitino has 128. The difference is 84 degrees which must bepresent in other bosonic configuration. An antisymmetric tensor of rank p (a

p-form ) has

D 2

p

degrees of freedon, which is equal to 84 when p = 3.

We deduce that in the multiplet

eAM, AMN P, M

the bosonic and fermionic

degrees of freedom match. This is a necessary but not sufficient condition tohave a supersymmetric multiplet and construct a supersymmetric action. Thisis indeed possible and the full supersymmetric Lagrangian is

e1L = 14

R i2

MMN PDN

+

2

P 1

48FM N P QF

M N P Q

+2e1

124M1M8P QRFM1M4FM5M8AP QR

+1

192

M

MNPQRSN + 12P

QRS

FPQRS + FPQRS ,where

FM N P Q =1

4

(MAN P Q

NAP QM + PAQNM

QAMN P) ,

ABM = ABM (e) + 14 MAB MBA + AMB ,FPQRS = FPQRS 3[P QR S].The action is invariant under the supersymmetry transformations

eAM = iAM,

M =

M +

1

4 ABM AB + i144 PQRSM 8PMQRS FPQRS

,

AMN P =3

2[MN P].

A theory in higher dimensions can produce more complicated content in lowerdimensions either by compactification or by dimensional reduction. In dimen-sional reduction one assumes that the fields are independent of the coordinatesin a certain number of internal directions. The action then reduces to that of alower dimensional theory with more fields. As an exmaple a metric in D dimen-

sions reduces in d dimensions to a metric, (D d) vectors and (Dd)(Dd+1)2

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scalars. A vector in D dimensions reduces in d dimensions to a vector and(D

d) scalars.

6.2 N=1 ten-dimensional Supergravity and Super Yang-

Mills

The super Yang-Mills action can be reduced from 10 dimensions to 4. Thevector fields AaM reduce in 4 dimensions to the vector A

a and the 6 scalars A

ai ,

Bai = Aai+3, i = 4, 5, 6. The spinors

a in 10 dimensions reduce to four spinorsin 4 dimensions K , K = 1, , 4. The reduced action takes the form

I =

d4xT r

1

4Fa F

a +1

2DA

ai D

Aai +1

2DB

ai D

Bai

+

g

2 K iKL Ai + i5iKLBi, L+

g2

4

[Ai, Aj ]

2+ [Bi, Bj ]

2+ 2 [Ai, Bj ]

2

,

where iKL and iKL are 6 real antisymmetric 4 4 matrices obeying SU(2)

SU(2) algebra.Of particular importance is N = 1 supergravity in 10 dimensions. The su-

permultiplet consists of the vielbein eAM, an antisymmetric field BMN, a dilatonfield , a gravitino M and a spinor . The supergravity action in ten dimensionscan be obtained from the eleven dimensional action by dimensional reductionand consistent truncation. The eleven dimensional vierbein gives in ten dimen-sions a vielbein, a vector and a scalar dilaton field. The antisymmetric field

AMN P gives in ten dimensions antisymmetric tensors of ranks three and two.The gravitino in eleven dimensions give in ten dimensions two gravitinos andtwo spinors. A consistent truncation is to set the vector, the rank three an-tisymmetric tensor, one of the gravitinos and one of the spinors to zero. Onecan show that one of the two supersymmetries will be preserved by this trunca-tion leaving a theory with N = 1 supersymmetry. The untruncated theory hasN = 2 supersymmetry in ten dimensions. The N = 1 supergravity Lagrangianis given by

e1L = 14

R +1

12e2HMN PHMN P +

1

2M

M

i2

MMN P

DN +

DN

P +

i

2M

DN +

DN

+1

22 MNMN + DN+

i

48

M

MNPQRR + 6N

PQ i

2MN P QM

HNP Q + HNP Q ,

where HMN P = 3[M B NP]. This action is invariant under the supersymmetry

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transformations

eAM = i

A

M,

= 12

,

BMN = e

i[M N]

,

M = DM 248

e

NP QM 9NMP Q HN P Q,

=i2

MDM + i12

2

eN P QFNP Q.The main advantage of the N = 1 supergravity Lagrangian in ten dimensionsis that it can be coupled to the ten-dimensional super Yang-Mills theory withan arbitrary gauge group. The main modification to the pure supergravity

Lagrangian is that the field HMN P is replaced withHMN P = HMN p MN P,MN P = T r

A[M NA P] +

g

3A[M ANA P]

,

where MN P is the Chern-Simons form over the gauge Lie algebra. Whenhigher derivative terms are allowed in the supergravity Lagrangian then themodifications to HMN P will also include the Chern-Simons gravitational term.The field HMN P satisfies the equation

dH = tr (F F + R R) ,where

H is a three-form, F is the gauge field strength two-form and R is the

curvature tensor two-form.

7 Further directions

The topics considered so far in these lectures would enable us to study somevery interesting questions in physics using results from mathematics. For lackof space, these directions will only be described briefly. The interested readercan pursue these problems in the literature.

7.1 Index of Dirac operators

Consider Dirac equation for a massless spinor in ten-dimensions

iD10 = 0 = iMDM = 0,

= i (D4 + DK ) ,

where K is the internal 6 dimensional manifold. Let (4) = i0123 and

(K) = i45 9 and (10) = 01 9. This implies that

(4)2

=

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(K)

2

=

(10)

2

= 1 and (10) = (4)(K). The Weyl condition on fermions

in ten-dimensions (10) = is equivalent to (4) = (K). Defining D4 =(4)D4 and DK = (4)DK it is easy to see that one can diagonalize simultane-ously D4 and DK . Let H = (iDK )2 and let be an eigenstate with energy E,H = E. As H commutes with (K) we have

H(iDK ) = E(iDK) ,

so that and DK are degenerate in energy. On the other hand

DK

(K)

= (K)DK ,

so that and DK have opposite chiralities with respect to (K) and are

therefore linearly independent, unless DK = 0. For every state with (K) = 1

there is a state with (K) =

1 except for zero eigenvalues which do not haveto be paired. The index of the Dirac operator is then defined by

index (iDK ) = n+ nwhere n+ and n are, respectively, the number of zero eigenvalues with eigenval-ues (K) = 1 and (K) = 1. In the absence of gauge fields the Dirac operatorsatisfies the properties

(DK )

= DK ,

(K)DK

= (K)DK .

This is so because in six-dimensions the gamma matrices are real and (K) ispurely imaginary , and therefore complex conjugation exchanges n+ and n, sothat n+ = n

and the index is zero. In presence of gauge fields, and for spinors

in some representation Q of the gauge group we have

indexQ (iDK ) = indexQ (iDK )because complex conjugation exchanges the representation Q with its complexconjugate Q, and this implies that the index is zero if the representation Q isreal or pseudoreal. If Q is complex then the index is not necessarily zero. Interms of the gauge fields and the spin-connection of the underlying manifold theindex of the Dirac operator can be computed to be

indexQ (iDK ) =1

48(2)3

trQ (F F F) 1

8trQ (F) tr (R R)

.

One can apply this property to explain physical phenomena. We have seen thatin the standard model of particle physics, there are three families of particles.In higher dimensional theories it is assumed that particles are in some largerepresentation of a gauge group which is spontaneously broken and the fourdimensional theory is obtained by compactification from the higher dimensionalone. From the higher dimensional point of view all these particles are massless

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as they acquire their small masses (compared with the high energy scale ofcompactification) at a later stage. The number of massless families is then

linked to the index of the Dirac operator on the internal manifold. Thereforewe can write

indexQ (iDK) =1

2 (K)

where (K) is the Euler number of the internal manifold. To build a realisticmodel with three families one must choose the Euler number of the internalmanifold to be 6.

7.2 Holonomy and N=1 supersymmetry

At low-energies it is desirable to have a field theory with N = 1 supersymme-try in four-dimensions. To have some unbroken supersymmetry, let Q be thesupersymmetry generator annihilating the vacuum

|0

. This implies that

0| {Q, A} |0 = 0,for all operators A. If A is bosonic then {Q, A} is fermionic, but 0| fermionic|0 = 0 as a fermionic state has no expectation value. If A is fermionic then{Q, A} = A and the above relation implies that A = 0. In supergravitytheories we have seen that the supersymmetry transformations of the gravitinoand the other fermions are proportional to the other bosonic fields and thesupersymmetry parmeter. For example, assuming that bosonic fields, exceptfor the metric, are set to zero, the gravitino transformation is given by

M = DM = 0.

But this is a Killing spinor equation and is a covariantly constant spinor. Thiscan be satisfied provided Killing spinors exist on the manifold. This also implyconsistency conditions

[DM, DN] = 0,

or equivalently RM N P QP Q = 0. Multiplying by N and using the property

NP Q = NP Q + gNP Q gNQ P,where NP Q is completely antisymmetric in the three indices, and using theRiemann identity RM[NP Q] = o, we deduce that

RMQ Q = 0.

Let us impose the constraint that the vacuum state of the ten-dimensional space

is of the form M4 K where M4 is a maximally symmetric four-dimensionalspace and K is a compact six-dimensional manifold. In this case the consistencycondition on the Riemann tensor does not admit deSitter or anti deSitter spacesas solutions because in this case

R =r

12(gg gg)

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which togother with the consistency condition implies that r = 0.Upon parallel transport around a contractible closed curve , a field f is

transformed into U f where

U = P exp

dx

where is the spin-connection of the n-dimensional internal manifold K. TheSO(n) matrices U obtained this way always form a group H called the holonomygroup. We can then ask : Under what conditions the manifold K admits aspinor field obeying Di = 0 ? A covariantly constant spinor keeps itsoriginal value under parallel transport: U = . As an example, let us considercompactifying ten-dimensional supergrravity to four dimensions. In this case theinternal manifold K is 6 dimensional. The subgroup of the SO(6) is isomorphicto SU(4) and the left-handed spinor obeying the above condition will be in the

fundamental 4-representation. By an SU(4) rotation we can always transform to the form

=

0000

and this is left invariant by the SU(3) subgroup. We can write the followingdecomposition

SO(1, 9) SO(1, 3) SO(6) SO(1, 3) SU(4),16 = (2, 4) (2, 4)

where we have decomposed a 16 dimensional spinor of SO(1, 9) (the Lorentz

group in ten dimensions) in terms of the four-dimensional spinors of SO(1, 3)and the 4 and 4 spinor representations of SO(6). From these considerations weconclude that by requiring the manifold K to have SU(3) holonomy, the com-pactified theory will have N = 1 supersymmetry in four-dimensions. Thereforethe question to ask now is: What sort of manifold K admits a metric withSU(3) holonomy?

If is a covariantly constant spinor, then so would be the Kahlerian formkij = ij , the complex structure J

ij = g

ilklj and the volume form ijk =

Tijk . In six-dimensions, if the holonomy group is not SO(6) but U(3) then atany point in K tangent vectors which transform as vectors of SO(6) decomposeas 3 + 3 under U(3). There is a unique matrix Jij acting on tangent vectors

in the 3 and 3 representations. J defines an almost complex structure and is

invariant under the holonomy group. Therefore it is covariantly constant andits derivatives vanish. A manifold of U(N) holonomy is complex and is alsoKahler. If we ask whether it is possible to find a Kahler manifold with SU(N)holonomy instead of U(N) holonomy, then the answer lies in the Calabi-Yauconjecture. This states that a Kahler manifold of vanishing first Chern classadmits a Kahler metric of SU(N) holonomy and this metric is unique.

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7.3 Solutions of Einstein equations

There are advantages to deal with a supersymmetric system when solving Ein-stein equations resulting from a coupled gravitational system. This is doneby embedding the bosonic system in a supergravity theory, whenever this ispossible. One first starts with an ansatz for the metric dictated by the symme-try of the problem. The supersymmetry transformations depend on first orderderivatives of the fields. Setting the supersymmetric transformations to zeroand requiring that the solution preserve all or part of the supersymmetry, re-sults in a system of first order differential equations, in contrast to the secondorder differential equations which one have in the original bosonic system. Thismethod has been very successful in finding new solutions for Einstein-coupledsystems. To illustrate the power of this method we consider the follwing model.

Consider the coupling of gravity to a dilaton field and an SU(2) Yang-Millsfield Aa. described by the action

S =

14

R +1

2

14

e2 Fa Fa +

1

8e2

g d4x,where Fa = A

a Aa + abcAbAc , and the dilaton potential can be viewed

as an effective negative, position-dependent cosmological term () = 14e2.We shall consider static, spherically symmetric, purely magnetic configura-

tions of the bosonic fields, and for this we parameterize the fields as follows:

ds2 = N 2dt2 dr2

N r2(d2 + sin2 d2),

aAadx = w (2 d + 1 sin d) + 3 cos d,

where N, , w, as well as the dilaton , are functions of the radial coordinater. The field equations, following from the action, read

(rN) + r2N 2 + U + r2() = 1,N r2

= (U r2()),

r2

N e2 w

= e2 w(w2 1), = (r 2 + 2e2 w 2/r),

where U = 2e2

N w 2 + (w2 1)2/2r2. This is a system of second orderdifferential equations which could not be solved. We now adopt the strategyof embedding this action into a supersymmetric action and use proporties of

supersymmetry to find a solution.The action of the N=4 gauged SU(2)SU(2) supergravity includes a vierbein

e , four Majorana spin-3/2 fields I (I = 1, . . . 4), vector and pseudovec-tor non-Abelian gauge fields A

(1) a and A

(2) a with independent gauge coupling

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constants g1 and g2, respectively, four Majorana spin-1/2 fields I, theaxion a and the dilaton . The bosonic part of the action reads

S =

1

4R +

1

2

+1

2e4a a 1

4e2

2s=1

F(s) a F(s) a

12

a

2s=1

F(s) a F(s)a +g2

8e2

g d4x.

Here g2 = g21+g22 , the gauge field tensor F

(s)a = A

(s) a A(s) a +gC abc A(s) b A(s) c

(there is no summation over s = 1, 2), and F(s) a is the dual tensor. The dilatonpotential can be viewed as an effective negative, position-dependent cosmolog-ical term () = 18 g2 e2. The ungauged version of the theory correspondsto the case where g1 = g2 = 0. We consider the truncated theory specified by

the conditions g2 = Aa(2) = 0. In addition, we require the vector field Aa to be

purely magnetic, which allows us to set the axion to zero.For a purely bosonic configuration, the supersymmetry transformation laws

are

= i2

12

e aFa +

1

4e ,

=

1

2mn

mn +1

2aAa

1

2

2e aFa

+i

4

2e ,

the variations of the bosonic fields being zero. In these formulas, I are fourMajorana spinor supersymmetry parameters, a aIJ are the SU(2) gaugegroup generators, and mn is the tetrad connection.

The field configuration is supersymmetric, provided that there are non-trivial

supersymmetry Killing spinors for which the variations of the fermion fieldsvanish. Specifying to the above configuration and putting = = 0, thesupersymmetry constraints become a system of equations for the four spinors I.The procedure which solves these equations is rather involved. The consistencyof the algebraic constraints requires that the determinants of the correspondingcoefficient matrices vanish and that the matrices commute with each other.These consistency conditions can be expressed by the following relations for thebackground:

N 2 = e2(0),

N =1 + w2

2+ e2

(w2 1)22r2

+r2

8e2,

r =r2

8Ne21 4e4 (w

2 1)2r4 ,

rw = 2w r2

8Ne2

1 + 2e2

w2 1r2

,

with constant 0. Under these conditions, the solution of the algebraic con-straints yields in terms of only two independent functions ofr. The remaining

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differential constraint then uniquely specify these two functions up to two inte-gration constants, which finally corresponds to two unbroken supersymmetries.

Introducing the new variables x = w2 and R2 = 12r2e2, the above equationsbecome equivalent to one first order differential equation

2xR (R2 + x 1) dRdx

+ (x + 1) R2 + (x 1)2 = 0.

IfR(x) is known, the radial dependence of the functions, x(r) and R(r), can bedetermined. Define the following substitution:

x = 2 e(), R2 = d()d

2 e() 1,

where () is obtained from

d2()d2

= 2 e().

The most general (up to reparametrizations) solution of this equation whichensures that R2 > 0 is () = 2 lnsinh( 0). This gives us the generalsolution. The metric is non-singular at the origin if only 0 = 0, in which case

R2() = 2 coth 2

sinh2 1,

one has R2() = 2 + O(4) as 0, and R2() = 2 + O(1) as . Thelast step is to obtain r(s), which finally gives us a family of completely regularsolutions of the Bogomolny equations:

ds2 = a2sinh

R()

dt2 d2 R2()(d2 + sin2 d2) ,

w = sinh

, e2 = a2sinh

2 R(),

where 0 < , and we have chosen 20 = ln 2. The geometry describedby the above line element is everywhere regular, the coordinates covering thewhole space whose topology is R4. The geometry becomes flat at the origin, butasymptotically it is not flat, even though the cosmological term () vanishes atinfinity. In the asymptotic region all curvature invariants tend to zero, however,not fast enough. The Schwarzschild metric functions for r are N ln r andN 2

r2/4 ln r, the non-vanishing Weyl tensor invariant being 2

1/6r2.

References

[1] S. Weinberg, Quantum Field Theory, volume 1, Cambridge University Press,1998.

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[2] P. West, Introduction to Supersymmetry, second edition, World Scientific,1992.

[3] J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton Univer-sity Press, 1992.

[4] M. Sohnius, Introducing Supersymmetry, Phys. Rep. 128 (1985) 39.

[5] P. van Nieuwenhuizen, Supergravity, Phys. Rep. 68 (1981) 189.

[6] J. Scherk, Extended Supersymmetries and Extended Supergravity Theories,Cargese Summer Institute on Recent Developments in the Theory of Grav-itation, Plenum Press, 1979.

[7] M. Duff, B. Nilsson and C. Pope, Kaluza-Klein Supergravity Phys. Rep. 130(1986) 1.

[8] M. Green, J. Schwartz and E. Witten, Superstring Theory, volume 2, Cam-bridge University Press, 1987.

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