Constructions N = 2 superconformal algebra

1 constructions

1.1 free field construction
1.2 su(2) supersymmetric coset construction
1.3 kazama–suzuki supersymmetric coset construction

constructions
free field construction

green, schwarz & witten (1988) give construction using 2 commuting real bosonic fields



(

a

n


)


{\displaystyle (a_{n})}

,



(

b

n


)


{\displaystyle (b_{n})}

[

a

m


,

a

n


]
=


m
2



δ

m
+
n
,
0


,




[

b

m


,

b

n


]
=


m
2



δ

m
+
n
,
0



,





a

n


∗


=

a

−
n


,





b

n


∗


=

b

−
n





{\displaystyle \displaystyle {[a_{m},a_{n}]={m \over 2}\delta _{m+n,0},\,\,\,\,[b_{m},b_{n}]={m \over 2}\delta _{m+n,0}},\,\,\,\,a_{n}^{*}=a_{-n},\,\,\,\,b_{n}^{*}=b_{-n}}

and complex fermionic field



(

e

r


)


{\displaystyle (e_{r})}

{

e

r


,

e

s


∗


}
=

δ

r
,
s


,




{

e

r


,

e

s


}
=
0.




{\displaystyle \displaystyle {\{e_{r},e_{s}^{*}\}=\delta _{r,s},\,\,\,\,\{e_{r},e_{s}\}=0.}}

l

n




{\displaystyle l_{n}}

defined sum of virasoro operators naturally associated each of 3 systems

l

n


=

∑

m


:

a

−
m
+
n



a

m


:
+

∑

m


:

b

−
m
+
n



b

m


:
+

∑

r


(
r
+


n
2


)
:

e

r


∗



e

n
+
r


:


{\displaystyle l_{n}=\sum _{m}:a_{-m+n}a_{m}:+\sum _{m}:b_{-m+n}b_{m}:+\sum _{r}(r+{n \over 2}):e_{r}^{*}e_{n+r}:}

where normal ordering has been used bosons , fermions.

the current operator




j

n




{\displaystyle j_{n}}

defined standard construction fermions

j

n


=

∑

r


:

e

r


∗



e

n
+
r


:


{\displaystyle j_{n}=\sum _{r}:e_{r}^{*}e_{n+r}:}

and 2 supersymmetric operators




g

r


±




{\displaystyle g_{r}^{\pm }}

by

g

r


+


=
∑
(

a

−
m


+
i

b

−
m


)
⋅

e

r
+
m


,





g

r


−


=
∑
(

a

r
+
m


−
i

b

r
+
m


)
⋅

e

m


∗




{\displaystyle g_{r}^{+}=\sum (a_{-m}+ib_{-m})\cdot e_{r+m},\,\,\,\,g_{r}^{-}=\sum (a_{r+m}-ib_{r+m})\cdot e_{m}^{*}}

this yields n = 2 neveu–schwarz algebra with c = 3.

su(2) supersymmetric coset construction

di vecchia et al. (1986) gave coset construction of n = 2 superconformal algebras, generalizing coset constructions of goddard, kent & olive (1986) discrete series representations of virasoro , super virasoro algebra. given representation of affine kac–moody algebra of su(2) @ level



ℓ


{\displaystyle \ell }

basis




e

n


,

f

n


,

h

n




{\displaystyle e_{n},f_{n},h_{n}}

satisfying

[

h

m


,

h

n


]
=
2
m
ℓ

δ

n
+
m
,
0


,


{\displaystyle [h_{m},h_{n}]=2m\ell \delta _{n+m,0},}






[

e

m


,

f

n


]
=

h

m
+
n


+
m
ℓ

δ

m
+
n
,
0


,


{\displaystyle [e_{m},f_{n}]=h_{m+n}+m\ell \delta _{m+n,0},}








[

h

m


,

e

n


]
=
2

e

m
+
n


,




{\displaystyle \displaystyle {[h_{m},e_{n}]=2e_{m+n},}}








[

h

m


,

f

n


]
=
−
2

f

m
+
n


,




{\displaystyle \displaystyle {[h_{m},f_{n}]=-2f_{m+n},}}

the supersymmetric generators defined by

g

r


+


=
(
ℓ

/

2
+
1

)

−
1

/

2


∑

e

−
m


⋅

e

m
+
r


,




g

r


−


=
(
ℓ

/

2
+
1

)

−
1

/

2


∑

f

r
+
m


⋅

e

m


∗


.




{\displaystyle \displaystyle {g_{r}^{+}=(\ell /2+1)^{-1/2}\sum e_{-m}\cdot e_{m+r},\,\,\,g_{r}^{-}=(\ell /2+1)^{-1/2}\sum f_{r+m}\cdot e_{m}^{*}.}}

this yields n=2 superconformal algebra with

c
=
3
ℓ

/

(
ℓ
+
2
)


{\displaystyle c=3\ell /(\ell +2)}

.

the algebra commutes bosonic operators

x

n


=

h

n


−
2

∑

r


:

e

r


∗



e

n
+
r


:
.


{\displaystyle x_{n}=h_{n}-2\sum _{r}:e_{r}^{*}e_{n+r}:.}

the space of physical states consists of eigenvectors of




x

0




{\displaystyle x_{0}}

simultaneously annihilated




x

n




{\displaystyle x_{n}}

s positive



n


{\displaystyle n}

, supercharge operator

q
=

g

1

/

2


+


+

g

−
1

/

2


−




{\displaystyle q=g_{1/2}^{+}+g_{-1/2}^{-}}

(neveu–schwarz)




q
=

g

0


+


+

g

0


−


.


{\displaystyle q=g_{0}^{+}+g_{0}^{-}.}

(ramond)

the supercharge operator commutes action of affine weyl group , physical states lie in single orbit of group, fact implies weyl-kac character formula.

kazama–suzuki supersymmetric coset construction

kazama & suzuki (1989) generalized su(2) coset construction pair consisting of simple compact lie group



g


{\displaystyle g}

, closed subgroup



h


{\displaystyle h}

of maximal rank, i.e. containing maximal torus



t


{\displaystyle t}

of



g


{\displaystyle g}

, additional condition that

the dimension of centre of



h


{\displaystyle h}

non-zero. in case compact hermitian symmetric space



g

/

h


{\displaystyle g/h}

kähler manifold, example when



h
=
t


{\displaystyle h=t}

. physical states lie in single orbit of affine weyl group, again implies weyl–kac character formula affine kac–moody algebra of



g


{\displaystyle g}

.