Introduction to Supersymmetry

7.1. CHIRAL MULTIPLETS ONLY

45

where

i j = i

j pi

i

@ zj +

j

@ zi 21

@ zi@ zj

2

i jy =

i

j

p

j fi

i

i

@ zjy p

ifjy +

i

j @ zi

2

2

2

2

fifjy + 21 @ zi@ zjy

i

i @

j +

i

@ i

j

+

2

2

i

j

y

p

i

j

k

= 2

k

k@ zi 2i i j fky

i

i

k @ zj + j

+

2

i j ky ly

i

j

=

k

l

:

(7.9)

It is then easy to extract the D-term, i.e. the coe cient of

R d2 d2

K( i; iy) =

41 Ki@2zi 41 Ki@2ziy 41 Kij @ zi@ zj + h:c:

+ Kij fifjy + 21 @ zi@ zjy

i

i @

j +

i

@

i

j

2

2

+

i

Kijk

i

k@ zj +

j

k@ zi 2i i

j fky + h:c:

4

+

1 Kkl

i

j

:

k l

4

ij

(7.10)

Next note that

@ @ K(zi; zjy) = Ki@2zi + Ki@2ziy + 2Kij @ zjy@ zi

+ Kij @ zi@ zj + Kij

@ ziy@ zjy

(7.11)

so that we can rewrite (7.10) as

R d2 d2

K( i; iy) = Kij fifjy + @ zi@ zjy

i

i @

j +

i

@

i

j

2

2

+

i

Kijk

i

k@ zj +

j

k@ zi 2i i

j fky + h:c:

4

+ 41 Kijkl i j

k

l 41 @ @ K(zi; zjy) :

(7.12) where the last term is a total derivative and hence can be dropped from the Lagrangian.

Note that after discarding this total derivative, (7.12) no longer contains the \purely holomorphic" terms Kij or the \purely antiholomorphic" terms Kij .

45

46	CHAPTER 7. THE NON-LINEAR SIGMA MODEL

Only the mixed terms with at least one upper and one lower index remain. This shows that the transformation

K(z; zy) ! K(z; zy) + g(z) + g(zy)

(7.13)

does not a ect the Lagrangian. Moreover, the metric of the kinetic terms for the complex scalars is

	Kij =	@2	K(z; zy) :	(7.14)
		@zi@zjy

A metric like this obtained from a complex scalar function is called a K•ahler metric, and the scalar function K(z; zy) the K•ahler potential. The metric is invariant under K•ahler transformations (7.13) of this potential. Thus one is led to interpret the complex scalars zi as (local) complex coordinates on a K•ahler manifold, i.e. the target manifold of the sigma-model is K•ahler. The K•ahler invariance (7.13) actually generalises to the super eld level since

K( ; y) ! K( ; y) + g( ) + g( y)

(7.15)

does not a ect the resulting action because g( ) is again a chiral super eld

and its component is a total derivative, see (4.20), hence R d2 d2 g( ) = R d2 d2 g( y) = 0.

Once Kij is interpreted as a metric it is straightforward to compute the a ne connection and curvature tensor. However, in Riemannian geometry, indices are lowered and rised by the metric and its inverse, while here we used upper and lower indices to denote derivatives w.r.t. zi or zjy. To avoid confusion, we temporarily switch conventions, replacing zjy ! zj . Then Kij ! Kij so that

Kij

= K

ji ;

Kij = K

= 0

(7.16)

ij

and the inverse metric is Kij = Kji, Kij = Kij

= 0.

The a ne connection

is given as usual by abc

= 1 Gcd

(@aGbd + @bGad

@dGab) which for the K•ahler

@

2

@

metric simpli es since

Kjm

=

Kim

, etc. One nds

@zi

@zj

ijl

= Klm

Kijm

;

l

= KlmK

;

(7.17)

ijm

ij

all others with mixed indices like lij or lij vanish. The curvature tensor is given in general by

(Rab)cd = @a bdc @b adc + afc bdf bfc adf :

(7.18)

It is easy to see that in the K•ahler case the only nonvanishing components are

	ki)l j = @				Kijpk
(R			ijl = Klp			Kijm	KmnKnpk				(7.19)
		k

46

7.2. INCLUDING GAUGE FIELDS

47

and (Rik

)l j = (R

ki)l j , and similarly

= Klp Kipkj

:

l

l

l

K

(Rik

)

= (R

ki)

= @i

Kipm

mnKnkj

(7.20)

j

j

kj

Reverting to our previous notation, we write

! Kij ;

ijl ! ijl ;

! lij

Kij

l

; (R

ki)lj

! Rijkl;

(7.21)

ij

i.e.

= (K 1)l

; ij

= (K 1)kKij ;

l

Kk

ij

k

ij

l

l

k

Rijkl = Kijkl Kijm(K 1)mn Knkl :

(7.22)

This allows us to rewrite various terms in the Lagrangian in a simpler and more geometric form.

De ne \K•ahler covariant" derivatives of the fermions as

D

i

= @

i + jki @ zj

k = @

i + (K 1)li Kjkl @ zj

k

(7.23)

D

j

= @

j + jki@ zky

i = @

j + (K 1)jl Klki@ zky

i :

The fermion bilinears in (7.12) then precisely are

i

Kij D i

j + h:c:. The four

2

fermion term is Kijkl

i j

k l. The full curvature tensor will appear after we

eliminate the auxiliary elds f i. To do this, we add the two pieces (7.12) and (7.6) of the Lagrangian to see that the auxiliary eld equations of motion are

fi = (K 1)ji wj	1	jki j k :	(7.24)
	2

Substituting back into the sum of (7.12) and (7.6) we nally get the Lagrangian R d4x hR d2 d2 K( ; y) + R d2 w( ) + R d2 [w( )]yi

= R d4xhKij @ zi@ zjy +

i

D

i

j

i

i D

j (K 1)ji wiwj

2

2

1

k

i j 1

ij

ij

k

1 kl i j

:

2

wij ij wk

2

w

k w

i j + 4 Rij

k li(7.25)

7.2Including gauge elds

The inclusion of gauge elds changes two things. First, the kinetic term K( ; y) has to be modi ed so that, among others, all derivatives @ are turned into gauge covariant derivatives as we did in section 4 when we replaced y by ye2gV .

47

g12 fab =

48	CHAPTER 7. THE NON-LINEAR SIGMA MODEL

Second, one has to add kinetic terms for the gauge multiplet V . In the spirit of the -model, one will allow a susy Lagrangian leading to terms of the form fab(z)F a F b etc.

Let's discuss the matter Lagrangian rst. Since
! ei ; y ! ye i y ; e2gV ! ei ye2gV e i	(7.26)
one sees that
ye2gV ! ye2gV e i :	(7.27)

Then the combination ye2gV i is gauge invariant and the same is true for

i

any real (globally) G-invariant function K( i; y) if the argument y is replaced

i i

by ye2gV . We conclude that if w( i) is a G-invariant function of the i, i.e. if

i

			wi(T a)i j j = 0			; a = 1; : : : dimG					(7.28)
then	d2 d2 K			i; ye2gV	i	+ Z	d2 w( i) + Z	d2 [w( i)]y			(7.29)
Lmatter = Z

is supersymmetric and gauge invariant.

To discuss the generalisation of the gauge kinetic Lagrangian (5.17), reall that W is de ned by (5.3) with V ! 2gV and in WZ gauge it reduces to (5.10) times 2g. Note that any power of W never containsR more than two derivatives, so we could consider a susy Lagrangian of the form d2 H( i; W ) with an arbitrary G-invariant function H. We will be slightly less general and take

Lgauge =	1	Z	d2 fab( i)W a W b + h:c:	(7.30)
	16g2

with fab = fba transforming under G as the symmetric product of the adjoint representation with itself. To get back the standard Lagrangian (5.17) one only needs to take 4 i Tr T aT b. Expanding (7.30) in components is straightforward and yields

Lgauge =	Refab(z) 41 F a F b i a D							b + 21 DaDb 41 Imfab(z)F a F b
+	41 fab;i(z) p		i	i aDb p		a iF b + a bfi + h:c:			e
		2			2
+	1 fab;ij (z) a b			i j + h:c:
	8

(7.31)

where F and D were de ned in (5.15) and fab;i = @z@i fab(z) etc.

To obtain the component expansion of the matter Lagrangian (7.29) is a bit lengthy. The computation parallels the one leading to (7.12) but paying

48

7.2. INCLUDING GAUGE FIELDS

49

attention to the gauge eld terms. The result can be read from (7.12) by gauge covariantising all derivatives and adding (7.6). Furthermore, it is clear that one also obtains the Yukawa interactions that already appeared in (5.27) with the K•ahler metric appropriately inserted. Note also that the term gzyDz now is replaced by gziyDKi. Taking all this into account it is easy to see that one obtains

Lmatter =

Kij hfifjy + (D z)i(D z)jy

i

i D

i

D

i

j i

j +

2

2

+

1 Kk i

jfy

+ h:c: +

1 Kkl

i

j

f

f

k l

2

ij

k

4 ij

(7.32)

wifi + 21 wij

i

j + h:c:

p

i

y

j

p

i

j

y

i

i z

;

+ i 2gKj

zi

i 2gKj

+ gzi DK

zj

T azj

a

before gauge indices have been suppressed, e.g.

where as

j

N

a

a M

i

i

R

a M

( i)M (TR) N (z )

where (TR)

N

are the matrices of the representation carried

by the matter elds (zj )N and (

i)N . The derivatives D

acting on the fermions

are gauge and K•ahler covariant, i.e.

f

D

i = @

i

igvaTRa

i + jki @ zj

k

f

a

a

ki

y

(7.33)

@

j

T

@ z

:

D

j =

igv

R j + j

k i

f

The full Lagrangian is given by L = Lgauge + Lmatter. The auxiliary eld equations of motion are

fi

(K 1)ji wj 21 Kklj k l 41 (fab;j )y

a

b

=

Da

(Ref)ab1 gziyT bKi +

2pi

fbc;i i c

2pi

(fbc;i)y

i

c

(7.34)

=

:

2

2

It is straightforward to substitute this into the Lagrangian L and we will not write the result explicitly. Let us only note that the scalar potential is given by

V (z; zy) = (K 1)i wiwj +	g2	(Ref) 1	(zyT aKi)(zyT bKj ) :		(7.35)
j	2	ab	i	j

49

50	CHAPTER 7. THE NON-LINEAR SIGMA MODEL

50

Chapter 8

N = 2 susy gauge theory

The N = 2 multiplets with helicities not exceeding one are the massless N = 2 vector multiplet and the hypermultiplet. The former contains an N = 1 vector multiplet and an N = 1 chiral multiplet, alltogether a gauge boson, two Weyl fermions and a complex scalar, while the hypermultiplet contains two N = 1 chiral multiplets. The N = 2 vector multiplet is necessarily massless while the hypermultiplet can be massless or be a short (BPS) massive multiplet. Here we will concentrate on the N = 2 vector multiplet.

8.1N = 2 super Yang-Mills

Given the decomposition of the N = 2 vector multiplet into N = 1 multiplets, we start with a Lagrangian being the sum of the N = 1 gauge and matter Lagrangians (5.17) and (5.25). At present, however, all elds are in the same N = 2 multiplet and hence must be in the same representation of the gauge group, namely the adjoint representation. The N = 1 matter Lagrangian (5.25) then becomes, after rescaling V ! 2gV ,

LmatterN =1 = R d2 d2

Tr ye2gV

Tr h(D z)yD z i D

+ fyf

=

ip

gzyf ; g ip

gf

;

gz + gD[z; zy]i

+

2

2

(8.1)

where now

z = zaT a ; =

aT a ; f = faT a ; a = 1; : : : dimG

(8.2)

in addition to = aT a; D = DaT a; v = vaT a. The commutators or anticommutators arise since the generators in the adjoint representation are given

51

52

CHAPTER 8.

N = 2 SUSY GAUGE THEORY

by

(Tada )bc = ifabc

(8.3)

and we normalise the generators by

Tr T aT b = ab

(8.4)

so that

zy a

izy af

c = izyf

zy

!

(T a )

bc

c =

abc

bac

a c

b

ad

b

b

(8.5)

= zy a cTr T b[T a; T c] = Tr zy

f

;

g

b

and

zyDz ! zbyDa (Tada )bc zc = ifabczbyDazc = Tr D[zy; z] = Tr D[z; zy] : (8.6)

We now add (8.1) to the N = 1 gauge lagrangian LNgauge=1 (5.17) and obtain

LNYM=2 = 321 Im ( R d2 Tr W W ) + R d2 d2 Tr ye2gV

= Tr 14 F F i D i D + (D z)yD z

+		g2 TrF	F	+ 1 D2	+ fyf	(8.7)
	32 2
			e	2

p p

+i 2gzyf ; g i 2gf ; gz + gD[z; zy] :

A necessary and su cient condition for N = 2 susy is the existence of an SU(2)R symmetry that rotates the two supersymmetry generators Q1 and Q2 into each other. As follows from the construction of the supermultiplet in section 2, the same symmetry must act between the two fermionic elds and . Now the relative coe cients of LNgauge=1 and LNmatter=1 in (8.7) have been chosen precisely in

such a way to have this SU(2)R symmetry: the and	kinetic terms have the
same coe cient, and the Yukawa couplings zyf ; g and f
		; gz also exhibit this
symmetry. The Lagrangian (8.7) is indeed N = 2 supersymmetric.

Note that we have not added a superpotential. Such a term (unless linear in

) would break the SU(2)R invariance and not lead to an N = 2 theory.
The auxiliary eld equations of motion are simply
fa	=	0			(8.8)
Da	=		g [z; zy]a
leading to a scalar potential
V (z; zy) = 2g2 Tr [z; zy]				:	(8.9)
	1		2

This scalar potential is xed and a consequence solely of the auxiliary D- eld of the N = 1 gauge multiplet.

52

8.2. EFFECTIVE N = 2 GAUGE THEORIES

53

8.2E ective N = 2 gauge theories

As for the non-linear -model, if one considers e ective theories, disregarding renormalisability, one may allow more general gauge and matter kinetic terms and start with an appropriate sum of (7.29) (with w( i) = 0) and (7.30). It is clear however that the functions fab cannot be independent from the K•ahler potential K. Indeed, the SU(2)R symmetry equates Refab with the K•ahler metric Kab. It turns out that this requires the following identi cation

16 2 fab(z) (2g)

16 2 K(z; zy)

(2g)

= i@z@a2@zb F(z) iFab(z)

=

i

zy

@

(z) + h:c:

i

zy

(z) +

i

[

Fa

(z)]y z

(8.10)

a

2

2

a @za F

aFa

2

where the holomorphic function F(z) is called the N = 2 prepotential. We have

pulled out a factor 16 2 for later convenience. Also, we again absorb the factor

(2g)

2g into the normalisation of the eld. This makes sense since ImFab will play the role of an e ective generalised coupling. Hence we set

2g = 1:

(8.11)

Then the full general N = 2 Lagrangian is

Le

= h

R

d2 Fab( )W a W +

R

ye

Fa( )i + h:c:

d d

64 i

32 i

N =2

1

b 1

2 2

a

V

a

=

1

Im h21 R d2 Fab( )W a W b + R d2 d2

yeV

Fa( )i

:

16

(8.12) Note that with the K•ahler potential K given by (8.10), the K•ahler metric is proportional to ImFab as required by SU(2)R :

Kab =	1	ImFab =	1		Fab Faby :	(8.13)
	16		32 i

The component expansion follows from the results of the previous section on the non-linear -model, using the identi cations (8.10) and (8.13), and taking vanishing superpotential w( ). In particular, the scalar potential is given by (cf (7.35))

V (z; zy) =	1	(ImF)ab1 [zy; Fc(z)T c]a [zy; Fd(z)T d]b :	(8.14)
	2

Let us insist that the full e ective N = 2 action written in (8.12) is determined by a single holomorphic function F(z). Holomorphicity will turn out to be a very strong requirement. Finally note that F(z) = 12 Tr z2 gives back the standard Yang-Mills Lagrangian (8.7).

53

54	CHAPTER 8. N = 2 SUSY GAUGE THEORY

54