Fermionic construction of vertex operators for twisted affine algebras

Nuclear Physics B305 [FS23] (1988) 164-198 North-Holland, Amsterdam

FERMIONIC

CONSTRUCTION OF VERTEX OPERATORS TWISTED AFFINE ALGEBRAS L. FRAPPAT’s3,

2Dipartimento

A. SCIARRINO*,

FOR

P. SORBA’

‘L.A.P.P., B. P. 909, 74019 Annecy-le- Vieux, France di Scienze Fisiche, 80125 Napoli, Italy and I.N.F.N., Sezione de Napoli, Itu(v

Received

5 April 1988

We construct vertex operator representations of the twisted affine algebras in terms of fermionic (or parafermionic in some cases) elementary fields. The folding method applied to the extended Dynkin diagrams of the affine algebras allows us to determine explicitly these fermionic fields as vertex operators.

1. Introduction The Frenkel-Kac-Segal construction [l], via vertex operators, of level-one representations for untwisted affine algebras 9(l), relative to simple simply laced Lie algebras 9, constitute today an unavoidable section in any review of string theory as well as of Kac-Moody algebras. Since then, the case of vertex operators for non-simply-laced algebras has been considered [2,3] and also the case of twisted vertex operators [4,5]. This last study [5,6] is of first interest in the context of orbifold compactification and explicit breaking in superstring theory. Moreover, when the twist is associated to an outer automorphism of the root lattice A of the simply laced algebra 9, the developed method [5,6] gives rise to a direct construction of the vertex representation for the twisted Kac-Moody algebras (see also ref. 131). Hereafter we would like to present a quite different construction of level-one representations of twisted affine algebras. We will call our construction fermionic, since fermionic (or parafermionic in some cases) auxiliary fields will be explicitly introduced to compensate the fermionic nature of the “bare” vertex operators associated to the (affine) short roots. Such an approach has already been used in ref. [2] in order to build vertex operators relative to non-simply-laced algebras. Taking as an example the case of B, or SO(2I + 1) algebras with the set of roots { f e, k e,, +e,} with 1 I i #j I 1, one knows that the vertex representation of SO(21)“) can 3 Also at Universitt

de Savoie, 74000 Annecy,

France.

0169-6823/88/$03.50OElsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

L. Fruppat et al. / Twisted uffine algehrus

be written

in terms of bilinear

fermions

E”+‘J = \k’~(z)~(~), while for the short roots an auxiliary roots,

165

field r(z)

i#j,

allows one to recover,

for the short

the bosons E”=

W(z)I’(z),

with a conformal weight one and the moments of which satisfy the expected is a commutation relations of the SO(21+ 1)“) algebra. Actually, our construction sort of generalization, to the case of twisted affine algebras, of the method of ref. [2] for non-simply-laced algebras. Indeed, denoting (or, (Y~ and 01~+ $3 as the long, short and affine short root, respectively, of a twisted algebra ‘9’(*), one will have to introduce now two types of auxiliary fields T(z) and T’(z) such that E”s= !W(z)~(z), Ems+“*=

Ps(z)T’(z),

the number of different T(z) and r’(z) being equal to the number of orbits of the short roots under the Weyl group generated by the long ones. In our approach, symmetries of the Dynkin diagrams (or DD) relative to untwisted, affine, simply laced algebras 9(l) will be considered. Such symmetries are associated to outer automorphisms of 9(l) and the set of twisted Kac-Moody algebras A$‘), D/“), Ei2) and Di3) can be obtained as subalgebras of 9(l), invariant under such automorphisms. The aim of this paper is two fold. First, due to the fundamental role of fermions in the presently proposed unified string theories [7,8], one may think such a fermionic construction to be well adapted for a practical use in string physics. Moreover, we hope that the simple properties on twisted affine algebras developed hereafter, especially with the help of the symmetries of the extended Dynkin diagrams (or EDD), will lead to a better understanding on their structure and their possible relevance. The paper is organized as follows. After recalling some definitions and basic properties of the Kac-Moody algebras in sect. 2, we show in sect. 3 how special symmetries of the EDD of an affine algebra 9(i) easily allow us to obtain the twisted affine algebras as g(l) subalgebras left invariant under such discrete symmetries. This folding technique is then used in sect. 4 to build vertex operators for %crn) (m # 1) as bilinear in fermionic fields. Finally, for each type of twisted algebra, an explicit construction is performed in sect. 5 where, in particular, the cocycle problem is worked out in detail.

166

L. Frappat et al. / Twisted a/fine algebras

2. Reminder about Kac-Moody Let us briefly Kac-Moody

recall the definition

algebra

of untwisted

Y(l), constructed

algebras

and twisted affine algebras

from a simple

Lie algebra

[9]. A

9, is the loop

algebra W) = c( t, 1-l)

@ 9@ CC)

(2.1)

in which we denote by C(t, t-l) the algebra of Laurent polynomials in the complex variable t and by c the central extension term. Commutation relations among generators of 5??(l) are [tmc3ua,t”@b]

=tm+*c3

[a,b] +m(a,b)6,+,JJ,

(2.2)

where a, b are ?J generators and (a, b) denotes the usual Killing form on 9. A twisted affine algebra 5?(“‘) (m f 1) is defined with the help of an outer automorphism 7 of 9 of order m (i.e. m is the smallest positive integer such that 7 m = 1) in such a way that its elements (2.3) satisfy (2.4) Setting (2.5) where the quantities e2cnk/m (k = 0,1, .. . , m - 1) are the m eigenvalues has for ?J(“‘) the decomposition

of 7, one

m-1

g’(m) =

@

t*+k/mg,

gk

,

(2.6)

and for (2.7) the Z/mZ

gradation [gkk,g[]

c gk+l,

(modm).

(2.8)

Two remarks are in order. First 3(m) is actually a subalgebra of go) (this is obvious when replacing in eq. (2.2) the t variable by zm in order for the gcrn) generators to appear with integer powers in the complex variable). Second, if 7 were not an outer but an inner automorphism of 9, then +?(“‘I would be isomorphic to go) itself. By the same kind of arguments, affine algebras associated with a simple

L. Frappat et al. / Twisted affine algebras

Lie algebra

3 will be in one to one correspondence

where

Aut(Y)

tively

inner

W=Int(A,(B))

(respectively

Int(FQ)

with the quotient

161

Aut( g)/Int(

9)

the group of automorphisms (respecof 9. If A, denotes the root lattice of c?? and Weyl group, one has the well-known isomorphisms

automorphisms) its corresponding

denotes

Aut(Y)

Aut(A,(g))

-Int( S)

= Int(A,(g))

=F(‘)

(2.9)

where g(g) is the symmetry group of the Dynkin diagram of 9. In particular, since F(Y) is the identity when 9 is non-simply laced or equal to E, or E,, only untwisted affine algebras g(l) will be associated to such a 9, while the Z, = { 5 l} symmetry of the A,, D, (I # 4) or E, Dynkin diagrams leads to the existence of affine algebras A”‘), Dj”), E6”) with m = 1,2, and finally the S, symmetry group of D4 allows the construction of Dj”) with m = 1,2,3.

3. Folding and symmetries of extended Dynkin diagrams 3.1. FOLDING OF EXTENDED DYNKIN DIAGRAMS

Actually, the Dynkin diagrams relative to twisted affine algebras can be obtained by folding of the EDD of an untwisted affine one. Let us be precise about how it works in the general case. We consider an affine Kac-Moody algebra g(l) with its EDD associated to a simple root system R R=

{cY~=~-(Y,_(Y~,

l
(3.1)

(Ye is the affine root, 6 is the isotropic direction and 0~~is the highest root of 9. If A,, is the root system for the horizontal algebra 9 and A the root system for the affine algebra ?Y(l), one has A=

{A,+m8,nS,mEZ,nEZ*}.

(3.2)

We assume that the EDD of 9 (l) has a symmetry, which can be related to an outer automorphism r of F?(l). If p is the order of the automorphism r (i.e. the smallest integer such that Tp = l), the set of roots of the affine algebra invariant under 7 is given by

(3.3)

Let us note gCrn) this invariant algebra. The simple root system of @-cm, is given by the invariant combinations of simple roots of g(l) under 7 (m is an integer equal to

L. Fruppat et al. / Twlsted

168

affinealgebras

1 or p, see later)

iI=

1++~(a)+ ...

BtRj,

++(p)),

and the associated DD is obtained by folding of the DD of 9(i) according symmetry corresponding to 7. At this point, one has to consider two different

(3.4) to the cases:

(1) The automorphism r has a trivial action on the affine root (Y,, (i.e. CQ is invariant under T). In this case, the automorphism r is actually an outer automorphism of the horizontal algebra 9. One obtains an affine invariant algebra S?(l), the horizontal part being a non-simply-laced algebra. (2) The automorphism r has a non-trivial action on the affine root (i.e. it mixes the affine direction and the horizontal roots of 9). In this case, r is no more an outer automorphism of 9. One obtains as invariant algebra under 7, a twisted affine algebra g(P), where p is the order of r. Notice however, that if there is no outer automorphism of the corresponding Lie algebra isomorphic to the affine algebra S?(l). In the following, we will write the commutation the Cartan-Weyl basis, i.e.

5?, the twisted relations

algebra

g(P)

is

of the affine algebra

in

[HA, HJ] = m&,,+n6,j

(3.5)

[H;,

(34

,l,n = cu’Ea m+n) ’

[EZ,@]

=

+,P)E::fl,

if

a+p

(Y.Hmlm+m&+,,,

if

p=

1 0,

isaroot, -(Y,

(3.7)

otherwise.

algebra. As mentioned above, Case I. Take for 9?(l) a simply laced Kac-Moody if CQ is invariant under r, the problem is equivalent to the folding of a simply laced Lie algebra. The folding leads in this case to a non-simply-laced algebra. Although this has been already done [2], let us be reminded, in a few lines, of the philosophy of the method (for simplicity, we treat only the case of an automorphism of order 2). One can always divide the root system A, of 9 into two separate subsets Ai and A: such that

AL,={P~AolP=~(P)}~

(3.8)

As,= {PEA~IP-(P)=O},

(3.9)

are such (i.e. we assume that the Dynkin diagrams considered in this construction T). If that two connected roots cannot be related to each other by the automorphism

L. Frappntet al. / Twisteduffineulgehrus j3 is a root

consider Now algebra

of 9, we set p= p + m8 (m E 2)

that the step operator the automorphism go)

Et corresponds

7 is extended

169

and p is a root of Y(l). One can to this root p.

from the root system

A of Y(l)

to the

by T( E,P) = E:,‘@ )

x-T(H,) the 2-cocycle

being chosen invariant

(3.10)

=7(x)4,,

under

(3.11)

7

4LP2) = +(PlL a>)) which is still possible,

(3.12)

since /? = 7( fi) or fi. T( /3) = 0. Then taking

y invariant

under

T, one has E,p+E;@)]

[y.H,,

[ d, Ei + E:@‘]

= tv.(P+~(P))(E,P+E~(P)),

(3.13)

= m( EE + ELcp)) ,

(3.14)

(d is the derivation

operator) which means $(p + T(P)) of C??(l) at level m is

that the generator

attached

to the root

if b=T(p),

E,P

fiCE; + E;(p))

if

p-~(j3)

(3.15) =O.

(3.16)

Case 2. We suppose for simplicity that T is of order 2. As before, one starts from an affine algebra 9 (l) . Now , the automorphism T does not leave the affine root invariant: T(CQ) = y where y is a simple root of 3. Notice that 9 is not necessarily a simply laced algebra. If A is the root system for the affine algebra 9 cl), the root system of the affine algebra gCrn), invariant under 7, is (3.17)

d”= {:@+r(@))lPEA}. if R = {(Ye, q,. . . , a,} is the simple

In particular, root system

root system

of ~??o), the simple

of C!YCrn) is d=

and the affine R-basis

{cx:=:(~+T((Y~)),

O
root is (~6= $(a, + y). The roots of Y(l) can be decomposed

j7=

2 r=O

mp,,

(3.18) in the

(3.19)

L. Frappat et al. / Twisted affine algebras

170

whereas

the roots of @-cm, are decomposed

in the k-basis

I’ = rank @ ,

(3.20)

(g’ is the invariant integral subalgebra of $ cm)). Using these decompositions, one can divide the root system d” of the affine algebra gCrn) invariant under 7 into three subsets: @= which corresponds a;=

to the long roots of 8’, appearing {:(p+-(p))&r(p)

which corresponds AT=

(3.21)

{i(P+7(P))lP=r(P)},

=o,

at each integral

0 m’=O

to the short roots of ‘@, appearing {:(p+r(p))ljLr(p)

=o,

level;

(mod 2) } , at each integral

0 m’=l

(3.22)

level; and (3.23)

[mod 21) ,

which corresponds to the roots of the twisted part (#“’ modules), appearing at each half-integral level. One obtains, therefore, in this way a twisted affine algebra C&(2) (i.e. m = 2)*. As before, the automorphism r is extended from the root system to the algebra ?Y(‘) by r( Et)

= E;(p),

(3.24)

x~(H,)=r(x).H,,

the 2-cocycle being chosen invariant taking y invariant under 7, one has [yeH,,E,p+E;@)]

1d,E;+E;‘B’] For the consistency

under

(3.25) 7: c(&, p2) = ~(r(/?t),

(3.26)

=m(E,P+E;@)).

(3.27)

of these relations,

fi(Ei+ l

if

EAta))

one has to take m E Z for f(fi + r(p))

/3=7(p), if

Then

= :Y.(P+~(~))(E,P+E~(~)),

but m E Z + i for $(fi + r(p)) E A’. The step operators $( /3 + r(p)) of CL@(~) are therefore EL

T(&)).

p.r(p)

associated

(3.28)

(mEZ), =O,

(mEZ+

E A’

to the root

$).

(3.29)

This discussion can be done actually for an automorphism 7 of order p in the same way. One finds then that the twisted affine algebra obtained is 59 -CP). The case p = 3 will be treated as an example in subsect. 5.4 where we discuss the folding Eb’) + Di3’.

L. Frappat et al. / Twisted affine algehrus 3.2. SYMMETRY

GROUP OF EXTENDED DYNKIN DIAGRAMS

It might be worthwhile

to emphasize

$-(g(l))

the role of the symmetry

=

diagram

of 9(l),

group

Aut A,( @) W(@l’)

of the Dynkin

171

where A,(9(‘))

’ is the root lattice

of 59(l) and

W(9(‘)) its Weyl group - also called affine Weyl group. Denoting Z(G) the center of the universal covering group G of G [lo], the Lie algebra of which is 9, itself isomorphic to the quotient A,($)/A.( 9) of the weight lattice by the root lattice of ‘9 [ll], one has the isomorphism

‘F(S(l))

eg) (see table algebra).

1 which

contains

= Z(G))

the list of these

finite

(3.30)

groups

for each

simple

Lie

Note that the symmetries of the 99(l) Dynkin diagrams have already been used, for example in the reduction of Toda field equations [lo] and more recently in the construction of modular-invariant partition functions for strings [12,13]. Let us add that elements of 9(9(l)) acting non-trivially on the affine root (~a (i.e. r((~a) # (wO) induce a changing from a homogeneous gradation into another one [3,12]. A homogeneous gradation corresponds to defining an euclidean hyperplane orthogonal to the isotropic root 6 in which lies the root system of ‘9. The number of homogeneous gradations of CC+‘(‘) is therefore equal to the order of Z(G).

4. Vertex operators for twisted affine algebras Now we will consider the construction of vertex operators for twisted affine algebras, using the folding method of sect. 3. The possibility of writing the generators of a folded algebra as linear combinations of the generators of the non-folded algebra allows us to construct the corresponding vertex operators for the twisted affine algebra, once the vertex operators for untwisted affine algebras are known. As in sect. 3, we will consider separately the cases where the automorphism r acts trivially or not on the affine root. Case 1. The outer automorphism 7 acts trivially on the affine root. The folding of the affine algebra 9 (l) leads to another affine algebra g(l), such that C!? is a non-simply-laced algebra. The vertex operators are constructed in the following way: If Q*(z) are Fubini-Veneziano fields in number rank 9

L. Frappat et al. / Twisted affine algebras

172

Symmetries

TABLE 1 of extended Dynkin

diagrams

Automorphism

Affine algebra g(l)

Dynkin

diagram

group 9(W))

Automorphism Center Z(G)

group S(9)

D /+1

Z ,+1

z2

z2

z2

1

z2

z2

1

z2

z2

1

D4

z2xz2

z2

(I even) z4

D4 (I

1

x

2

z2

odd)

1 z2xz2

s,

o-c-L3

s,

z3

Z2

z2

z2

1

1

1

1

1

1

1

1

1

1

1

s4

1

1

2

1

2

3

2

1

1234321

24654321

?-Y-P?+

173

L. Frappat et al. / Twisted affine algebras TABLE 1 (continued)

Automorphism

Twisted algebra g(m)

Dynkin

group F(@““)

diagram

We note that Z,, is the cyclic group of order n, 5, the permutation group of )I objects and D,, the dihedral group with 2n elements [lo]. Algebras labelled by the index I have DD with I + 1 vertices.

c

Q’(z)=q’-ip’lnz+i

mzo we define

the vertex operator

?z--;

U(j3, z) where /? is a root of 99 by

u(P, z) =zfl*/2:exp(ip)Q(z):. The momenta step operators

(4.2)

p

are belonging to the weight lattice of the Lie algebra 9’. Then, the associated to the folded roots can be written as follows.

The algebra ‘#i” generated by the long roots of @-(I) is a subalgebra vertex operators corresponding to the long roots are thus

of Y(l). The

E(P>4 = U(P, Z)CB. The step operator at level m for the short corresponding vertex operator is

roots

(4.3) being

fi(EL

+ E;‘p)),

the

L. Frappat et al. / Twisted affine algebras

174

where y&= +(p -t r(p)). Since /3 . r(p) = 0 in this case, one has y: = y 2 = 1 and y+y_= 0, and the vertex operator part factorizes as

w,

4 = U(Y,>

Z)U(YL

WPL 4 = U(Y+,

Z>WYL

z), 4.

(4.5)

We have also to deal with the factorization properties of the cocycle. More precisely, we would like to write the same factorization relation as above, i.e. $3 =

cy,cy-

cT(P)= cy+ccy_ .

)

(4.6)

To do that, one has to extend cy with y E AR(S) to cy+ with y + = :( y + T(Y)) E A +. The construction of the cocycle operator on the lattice A + will be worked out m each case. Notice however that the root lattice A n( gL) of the algebra generated by the long roots of @ is a sublattice of A+. It follows that the cocycle operator in eq. (4.3) can also be constructed as a cocycle operator on the lattice A +. Assuming for the moment that the cocycle can be factorized as eq. (4.6) one obtains for the step operator

with

r,-(z) is an auxiliary field associated to the short root y,. One can verify that U( y+, z) and yy_(z) have the conformal weight 4, which gives the right conformal weight 1 for the generator E(y+, z). Notice that, in general, the same auxiliary field can be related to different short roots. Actually, all the short roots related to each other by a Weyl reflection with respect to a long root have the same auxiliary field T(z). In other words, if the set of short roots of @ is divided into orbits D under the Weyl group generated by the long roots, one associates one auxiliary field m(z) for each orbit 9. These auxiliary fields are not necessarily independent of each other. More precisely, if yP E a, yL E 3’, one can have y-y’=

kl

and

y_+yL

isalongroot,

Y-Y’

= + $

and

y_T y’

is a short root belonging

r,(z)rP(w) (c( y-, yr)

= ccY_, y:)z1/2(z being a two-cocycle

- w)-‘/2r,,,(z) constructed

to the orbit a”,

+ regular terms in z - w,

from the cocycle operator

(4.9)

cym appearing

L. Frappat et al. / Twisted affine ulgebrm

175

in eq. (4.8)) y-y’=0 Case 2. assume

and

r,(~)r~(w)isaregularfunctionofz--w.

The outer automorphism

r does not act trivially

on the affine root. We

that

T is of order 2. The folding of the affine algebra c!?@) leads to the to the generators twisted affine algebra 9-(*) . The vertex operators corresponding with integral moments (invariant integral subalgebra g’,“) of @-(‘)) are constructed as for case 1. The main problem is to construct vertex operators associated to the roots of AT at half-integral levels, i.e. vertex operators with half-integral moments, instead of integral moments as for the untwisted Kac-Moody algebras. To do this, consider the invariant integral subalgebra of gC2) we have denoted by @,“) above. Let A be the weight lattice of @, the horizontal algebra of g-,“). One extends this lattice to the lorentzian lattice 2, by adding to the lattice A the isotropic direction 6; if (~!)i ~; ~ dimA is a basis of A, one has a2=0

and

Se,=O.

(4.10)

If the Q’( z)‘s are the Fubini-Veneziano fields introduced in eq. (4.1) we extend the number of components of the oscillators & and pi from dim A to dim x with the following conditions &Y~=O,

(m#O)

We denote by Qi( z) the obtained the extended vertex operators

and

&,=Sp=l.

Fubini-Veneziano

(4.11)

fields. Then if fi belongs

to A,

(4.12)

U(P+ f6,z)=zBZ’2:exp[i(p+

(4.13)

iS)]Q(z):,

have series expansion either in integral or in half-integral powers of z, but are always of opposite Ramond or Neveu-Schwarz character. More precisely, if fi2 = 1, U( p, z) is of NS character and U( /3 + ?S, z) of R character if the momentum p has only integral components, and U(p, z) is of R character and U(fi + :S, z) of NS character if the momentum p has only half-integral components. Now we are in position to write a vertex operator for the roots of A,

~‘(u,,z)=~(U(P+:s,z)c,+ where

y*=

$(P * r(p))

as before.

Since Dr(p)

u(T(p)+ ~s,z)c,(p)), = 0, one has /37(p)

(4.14)

= 0 and thus

L. Frappat et al. / Twisted affine algebras

176 y+y_

=

0. The vertex operator

part factorizes,

therefore,

as in the untwisted

case

U(p+:s,z)=U(Y+,z)U(Y_+js,z), U(r(P)

+ $8, z) = U(y+,

“>U( -y_+

:s, z).

(4.15)

Moreover, the factorization properties of the cocycle are unchanged in comparison to the untwisted case, since it depends only on the root lattice of the horizontal algebra. One has, consequently,

CL7 = CY+CY’ The vertex operator fermionic character

can therefore

E(Y+,

CT(P)=

be written

cy+c-y_.

as a product

z) = WY,,

(4.16)

of two vertex operators

of

(4.17)

4cy+q_(z),

with T,_(z)=

fgu(y_+js,z)c,_+

q-y_+

:s,z)c_,_).

(4.18)

r;_(z) is an auxiliary field associated to the short root y+. One can show, as in the untwisted case, that all the short roots related to y+ by a Weyl transformation with respect to the long roots of @r(‘) have the same auxiliary fermionic field r,_(z). Therefore rr_(z) is associated to the orbit Q of the short roots under the Weyl group generated by the long roots of @r, which y+ belongs to: r,_(z) = rcCv+,(z). Remark. The short roots c~s of $(*I appear both with integral and half-integral levels, associated with the generators E(%

z) = U(%

z)C,S&YS)(z)>

(4.19)

Because of the structure of the twisted algebra gC2), the auxiliary fields roCaS,(z) and r’oCa,)(z) are in general not independent of each other but satisfy the following O.P.E.‘s: if y E L?, y’ E fi’ and yy’ = f $ such that either y + y’ or y - y' belongs to some orbit r,(z)r,,(w)

a”, one has = e(y, y’)~l/~(z

ra(z)r;f(w) =+,f)~~/*

~(y, y’) being

the two-cocycle

+ regular terms in z - w,

(4.21)

z) + regular terms in z - w,

(4.22)

- w)-“‘r,,,(z) (Z - w)-1’21’&(

defined

in eq. (4.9).

L. Fruppat et al. / Twisted ajjine algebras 5. 5.1. FOLDING

Consider

177

Explicit constructions of the vertex operators

B(” + Dj*)

the non-simply-laced

symmetry

affine algebra

Bi’), whose extended

DD has a Z,

c::>---yy

The outer automorphism

of order 2 related

to this symmetry

is defined

by

(5.1) The root system 6=(&e,+ which corresponds

of the folded algebra

is

e, + m8, k e, + +m8,

(5.2)

to the folded DD

with a6 = $(a,, + (or) = ~$6- e2. One algebra Dy), whose invariant integral consequently: for the invariant part integral level)

obtains therefore the twisted Kac-Moody subalgebra is B{?,. The vertex operators are (corresponding to long and short roots at

(5.3) c being the cocycle operator B,, and

defined

on the Z”

E(~ei,z)=U(~ej,z)C+.T(z),

sublattice

of the Z’ root lattice of

(5.4)

T(z) being an auxiliary fermionic field (which can be constructed for instance by folding of the affine algebra D/y,); for the twisted part (corresponding to short roots at half-integral level) one has following the previous general discussion

L. Frappat et al. / Twisted

178

the cocycle

being

defined

on the 2’ C

&

uffinealgehrm

lattice of B,, one can factorize

e, f e,

=C

* e,C*.,

it immediately (5.6)

3

and one obtains E’( *e,,

2) = U( *e,,

z)c+.,P(z),

(5.7)

where r’(z)

= fi(

U( e, + $8, z)ccl + u( -e,

+

$3, Z)C_,))

(5.8)

is another auxiliary fermionic field, independent of F(z) and of opposite Neveu-Schwarz or Ramond character. c + e, is a cocycle operator defined on the Z’-’ root lattice of B,_ 1 and c + eI is a cocycle operator defined on the Z lattice orthogonal to the previous one.

Consider

the affine algebra

DC) with the EDD

aO

%?c

al

c121-1

The outer automorphism

of order 2 which defines

+,>

=

a/,

7(ai)=aZI_,

the folding

.

is

(OIiSl-1).

(5.9)

The simple root system of the folded algebra is given in terms of the resealed roots qi = fi(ei - e */+1-i) (1 zz i I 1) by

which corresponds

folded

to the folded DD I aO

One obtains therefore subalgebra C,.

the twisted

affine

algebra

A(i/)_r with

invariant

integral

179

L. Frapput et al. / Twisted affine algebras

The root system of A($_,

is

Construction of the vertex operators For the long roots of the invariant +2~jJ&=

k(e,-e,,+,_,)=

and the long roots at integral Therefore

the vertex operator q

+29,/k%

is as follows. part, one has f(q+

level correspond

...

(5.12)

+qPj),

to roots of Di:’ invariant

under

7.

for the long roots is given by z) = u( *2%/m

+i2-o,,~ (5.13)

= ~(+(ei-e2,+l-i)~~)~i~e,~e2,+,-,~~ For the short roots of the invariant fi(?l,-~j)=$(e--e

I

part, one has -e

2/+1-r

,

+e2,+,_,)=

$(fi+T(P)),

(5.14)

with /3 = e; - ej and T(P) = e2,+ I _i - e2,+ I _ ,. Therefore

E( fi(v;

- ~~1, z) = /F( u(e,

- e,, iIce,-.,

+ U(ezr+lp, Similarly,

-

e21+1-,,

4Ce*,_l_,-P*,+,_,)~

(5.15)

one has ~(4i+9,)=:(e,-e,,+,-,+e

with p = e, - e2/+i_, E[fi(V,

and r(p)

I -e2,+,-j)=:(B+~(P))9 = e, - ++i_i.

+ 17,), Z) = @(U(e,

+ If we define

the vectors

5, = K(e;

(5.16)

Therefore -

e2/+l-j>

uCe,-

+ e2,+i_,),

z)Ce,-eZ,+l_,

e2t+l-i3

Z)Ce,-ez,+, ~)_

orthogonal

to the

(5.17)

qj’s, one can

L. Frappat et al. / Twisted uffine algebras

180

factorize

easily the vertex operator

part

Now we have to deal with the factorization of the cocycle operator. To perform this factorization, one must extend cy with y E A R(D21) to cy+ where y * = 4( y f 7(y)). be a basis of y E A.(D,,).

Let (c,)r.i,2,

A lattice vector is defined

by

21 y=

xy,e,

with

FZ,EZ

and

CniE2Z.

(5.19)

i=l

A direct calculation

shows that

i g n,(e,-

y+=

e2,+1_i) = fi

i=l

y_=

i f

i

p,~i

with

Pi E

with

qi E Z,

Z,

cpi

E 22,

(5.20)

zqiE

22.

(5.21)

I=1

n,(e, +

i=l

i=l

This implies that y+ and y_ belong to two orthogonal lattices A +, each of them being isomorphic to the root lattice of D, resealed by 6. The cocycle operator factorizes then as

cy= cy+cy’

CT(Y) =

cy+c-y_>

(5.22)

once one is able to define the cocycle operator on A i. To do that, it is necessary and sufficient to define a symmetry factor S(x, y) on the lattice EA a(D,). Let ( yr, . . . , y,) be a basis of simple roots of D,, resealed by the factor

6,

such that y,’ = 1. Then, on this basis we define the symmetry

factor S

by

SjjE

i

S(Y,?Y,)

=I,

s(y,,y,)

=

(-l)“‘j,

S( y,, y,) = - S( y,, y,) = einy~y~,

i +j,

Y,Y, E Z, Y,YjEz+

+3

i
(5.23)

L. Frappat et al. / Twisted uffine algebras

Then,

if x = Cx,y, and y = Cy,y, are some points

S(x, y) =

1X1

of KAR(DI),

one sets

nsy .

(5.24)

i, J It is obvious

to verify that S possesses

the properties

of a symmetry

S(x,x)=l,

SC-x, YP(Y,

x) = I,

s(x, y + z) =

factor:

qx, YNXY 4.

(5.25)

This allows us to construct the corresponding cocycle ~(x, y) and the corresponding cocycle operator on the lattice fina ( see ref. [2] for e.g.). Now the factorization of the vertex operator for the short roots is complete and one can write finally

E(J(~17;+9j)‘z)=U(I/‘:(+9,f9,)~z)c(+,,_,,),~T,,(z)~ (5.26) with

(5.27) rij(z) is an auxiliary fermionic short root belongs to

field, which depends

on the orbit

fin,, to which the

There is :/(I - 1) such orbits for A$, and therefore one needs :1(1- 1) auxiliary fields q,(z) to construct the vertex operators associated to the short roots at integral level. For the twisted part the short roots fi( +n, * 17,) appear also at half-integral levels (recall that the twisted subalgebra). One has

part forms a representation

of the invariant

integral

:6-~(1),+17j)=js+:(-e,+e2,+,-,-e,+e2i+l_,)=:(P+7(P)), (5.29) with p = 6 - ei - e, and r(p)

= e2,+ 1_, + e2,+ 1_ ;. Therefore

+ U(e~+i-,

+ e2/+lP, + $6, z)c~,~,+,_,+~~,~~~,)~

(5.30)

L. Frapput et al. / Twisted affine algebras

182

according

to the general

form of the twisted vertex operator.

+8-~(7j,-nj)=j8-kL(-e

2

with fi = 6 - ej - e,,+,_j

I

Similarly,

+e,,+,_,+e.-e,,+,_j)=~(~+7(~)) J

one has 9

(531) .

and ~(/3) = ej + e2,+1_i. Therefore

E’(~(-$,+gj),z)=~(U(:S-ej-e2/+l~j,Z)C-,,-,1,+I~,

+ 49 The vertex operator UC’

21+1-j

+

e21+1-i

part factorizes

+ ej+ e21+lPij z)G,+~~,+~J.

(5.32)

as follows

+~s~z)=u(~(-S,-,j),z)‘(~(I,+~j)+:~,z),

U(-ei-ej+j8,z)=U(~(-~i-7jJ),z)U(-~(~i+~j)+~S,z),

u(

-ej

+

e2t+1-i

u( -e, -

+

is, z) = u(fi( -

Q/+1-j,

9r+IIJ)>z)u(fi(ti+tj)+

is,‘),

Z)~U(~(~~~+~~)~z)U(~J:(~~+~j)+~s~z)~ (5.33)

The previous discussion on the cocycle shows that it factorizes as in the untwisted sector. It follows that the vertex operator for the twisted part can be written as E’(~(+Pi+9j),Z)=U(~(+lli~?Ij),Z)C(iq,+.,,/~S’,(Z)~

(5’34)

with

+

u(-fi(Si +

tj)

+

(5.35) is,z)c-(~,+<,)/@).

The number of auxiliary fermionic “twisted” fields c’,(z) is still given by the number of orbits Gjj of the short roots under the Weyl group generated by the long ones. One needs also $1(Z - 1) auxiliary fermionic twisted fields I;)(z) for the twisted part of A(;LI)_i. In summary, one has for AT/)_i: Invariant part: generators associated

to the long roots at integral

E( +2n/‘fi,

Z) = U( *2ni/“fi, = U(+ (ei -

level

Z)c*2vl/fi e2,+1-r),

z)cice,-eZ,+,_,);

(5.36)

L. Frappai et al. / Twisted affine algebras

generators

associated

to the short roots at integral

183

level

with

c-(E,-~,vfi i . (5.38) Twisted generators

part: associated

to the short roots at half-integral

level

U(~(+9ifII,),z)c(.,,.I,),ilr~:tZ)~ (5.39)

E’(j:(,lli*?+)= with

+

u(- fi(& +E,)+ :s,Z)c-(t,+t,)/fij.

The auxiliary fields cj(z) and c$(z) are not independent respect the Z, grading of the twisted algebra, i.e.

(5.40)

in the sense they must

~,(~)~~(~)=(~-~~-~‘~(~~(~)+regulartermsin

(5.41)

z-w),

(5.42)

~j(z)~k(w)=(z-~))“2(~~(~)+regulartermsinz-w), c)(z)Tl,(w)

= (z - w)-“2(&(w)

+ regular

(5.43)

terms in z - w).

5.3. FOLDING EC’ + EL”

Consider symmetry

the

case of the exceptional

Lie algebra

E,

whose

EDD

has

a Z,

L. Fruppat et al. / Twisted affine algehrcrs

184

with the simple

root system ao=8+e7-es, q=$(e,--e2-e3-e4-e5-e6-e7+e,),

a2=e,+e,, cq=

-e,+e,,

a4=

-e,+e,,

a5=

-e,+e,,

(5.44)

a,+ -e4+e,, -e,+e,.

a,=

The outer automorphism

+I)

7 of order 2 which defines

=

the folding

is +x4)

a6,

=

a4.

(5.45)

One obtains

the folded EDD c, a;

with the corresponding

a:

(y.‘s

ai

a’2

simple root system

(5.46)

where the q,‘s are the resealed

roots

171 =

+( e5

-

e7 +

eS>

q2

$( e,

+ e2 + e3 +

e4),

=

e6 -

7j3 =

+( -e, - e2 +

q4=

$(-e,+e,-e3+e4).

)

e3 +

e,) , (5.47)

L. Frappat et al. / Twisted affine algebras

The folded

root system, expressed

185

in terms of the vi’s is

lSi#j<4,mEZ,nEZ*}. It appears

that the invariant

(5.48) integral

subalgebra

of Eg6) is the Lie algebra

F4, and

that the twisted part is generated by the short roots of F4. Construction of the vertex operator is as follows. For the invariant integral algebra, since the invariant integral subalgebra of Ef) is F4, we will find the generators associated to the long and short roots of F4 with integral moments, the short roots splitting into three different orbits under the Weyl group generated by the long roots iI,=

{-tq,

lris4},

a2=

{:(f~l+~2+~3+q4),

evennumberof

L?,=

{f(~~l+q2+q3~q4),

oddnumberof

+

signs},

+ signs}.

(5.49)

Vertex operators associated to the long roots. The long roots of F4 are roots of E, invariant under the automorphism 7. One has simply (5.50) with fn, + qj expressed in terms of invariant roots of E,. Since the long roots hqI, & n, constitute the root system of D,, it is obvious to construct a cocycle operator c + 9rk ‘I on the root lattice of D,. Vertex operators associated to theshort root. The main problem is to construct a cocycle operator on the lattices A += { :( y f r(y)) Iy E An(E,)}, factorization of the cocycle be possible

cy= cy+cy9

Let y be a lattice

vector of As(E,).

y+=’

2

4

Cp,q,

CT(v)= cy,ccy_ .

A direct calculation

with

PiEZ>

such

that

the

(5.51)

shows that

CPtE2Zy

(5.52)

i=l

with r=l

q,EZ,

cq,=O,

(5.53)

L. Fruppat et al. / Twisted affine algebras

186

where the Ei’s are linear combinations of the e,‘s, orthogonal to the TJ~‘s. This implies that y+ belongs to the lattice A += +A R(D4) and y_ belongs to the lattice A _ = fin

a(A 3), the two lattices

A + being orthogonal.

To construct the cocycle cy+ on the lattice A+, it is necessary and sufficient to define the corresponding symmetry factor S(x, y). Let (yi, y2, y3, y4) be a basis of simple roots of D4, resealed by the factor 4, such that y,* = $. Since a point x E A+ is defined

by x = Cx,y,

with

xi E 2

and Cx; E 22,

the basis

vectors

y, do not

belong to the lattice A+, but the vectors 2y, do. It follows that the symmetry S(x, y) should satisfy on the basis (y,)

(5.54)

s(2Y,,2Y,)=l~s(Y,,Y,)E{+1,+i}.

One is led to define

the symmetry

factor

factor S on the basis (yi) by

S(Y,,Y;) = -i, s,j=

s(Y,,Yj)= (-Q"",

i S( y,, y,) = S( y,, y,)-’

= eiaygyl,

Y,YjEz*

i#j,

yiyjEZ+$

or

Z+

:,

i
Therefore,

if x = Cxiy, and y = Cyiyi are two points S(x,

This symmetry

of iA.(

.

y) = ns;, i. j

one has (5.56)

factor satisfies the usual properties S(x,x) s(x,

=l,

S(x,

Y)S(Y,

Y + z> = s(x,

Y)S(X,

x) = 11 z>.

(5.57)

For the construction of the cocycle operator on the lattice A_= fina( see ref. [2]. One finds finally the expression of the vertex operators for the short roots associated to the three different orbits E(U,, with

z> = W%>

+,,I+)>

(5.58)

L. Frappat ei al. / Twisted affine algebras

(even number

of + signs), with

=

(odd number

187

(5.62)

u(:(+~~+~2+~3~~4),z)c(i~,~g,i~,~’14),2r3(z),

of + signs), with

Actually, this can be easily understood when invoking the “triality”. In fact, the subalgebra corresponding to the long roots of F4 is D, which possess the property of triality. The three different orbits Q, of the short roots under the Weyl group generated by the long ones are related to the three representations 8, (orbit L?,), 8, (orbit Q,) and 8; (orbit Q,) of D,. The triality implies, therefore, some relations when computing the O.P.E.‘s between the auxiliary fields T,(z). For the twisted part, the twisted part forms a representation of the invariant subalgebra F4, thus one finds at each half-integral level the short roots of F4, associated to the twisted generators, which split again into the three different orbits 52, (i = 1,2,3). One has E’(&n,,

(5.64)

z) = q&77,> Z)Cf,,NZ)~

with r~(z)=~(vi~(~,-El)+1s~z)C~~,~f,l/~

+ U(&(&

- &) + :s, Z)%,-r,,,,li)

’

(5.65)

E’(t(-tlll$_112~173f174),z) = U(:(+111f~2+93~~14),Z)C(i11+szi9,iqq),2r~(Z)7 (even number

(5.66)

of + signs), with

(5.67)

(5.68)

L. Frappat et al. / Twisted affine algebras

188

(odd number

of + signs), with

+ 5.4. FOLDING

qTcs4- b> + 3%Z)‘(&E,),fi).

(5.69)

ES’ --* DA”

Let us now examine the case of an automorphism r of order 3. This case arises for the affine algebra EL’) whose EDD exhibits a Z, symmetry

with the simple

root system

‘a,=6-

~(e,+e,+e,+e,+e,-ee,-e,+e,),

al=i(el-e,a2

R=

=

e, + e2

ag=

-e,+e,,

ad=

-e,+e,,

a5=

-e,+e,,

a6=

-ee,+e5.

The outer automorphism +X0)

e3 - e4 - e5 - e6 - e, + es), (5.70)

7 of order 3 associated

= +,>

= (Yg,

+X2)

to this symmetry

= r(cQ)

= 015,

is defined

T( (Y4) = (Yq.

by (5.71)

If p is a root of the horizontal algebra E,, we set p= j? + m8, which is a root of the affine algebra E,(‘) . The root system of the affine algebra invariant under 7 is A=

{f(j?+r(@+r2(p)),j%A},

where A is the root system of E&l). The folded EDD is

(5.72)

L. Frappat

with the corresponding

et al. /

Twisted crffine crlgehras

189

simple root system

( a;,

ii=

=

;a

-

$(v*

ai=

:(2772-771-7J3),

a; =

T3

-

+

773 -

2771)

9

(5.73)

772 >

q4 = e,, n, = e, (i f 1,4). One obtains with n1 = -e4, Di3). If we write an affine root p of E&l) as

the twisted

affine

algebra

P= i,,,,.

(5.74)

I=0 and the folded

root p of D,j3’ as

~‘=~(~+r(~)+r’(~))= one can divide a” into three different

a;= {:(p+r(p) which

corresponds

appearing

roots

L’= {:(p+r(p)

+ 72(P))IP=Gq} of the

invariant

roots

(5.76)

7

integral

-t~‘(~))~~~~(~)=O,rn~=O

to the short

at each integral

(5.75)

subsets

a”;= {;(p+r(p) which corresponds to the long appearing at each integral level.

trn:ai, r=O

of the invariant

[mod3]}, integral

subalgebra

G,,

(5.77) G,,

level.

lt~~(~))~~~~(~)=O,rn~fO

[mod3]},

which corresponds to the roots of the twisted part, which appear Z + $. The root system of Di’) is therefore

12Li#j#kI3,mEZ,nEZ*}. The automorphism by the relations

subalgebra

r is now extended

from the root system

r( E,P) = &‘fl’,

xr(H,)

=r-‘(x)H,,

(5.78)

at level Z + $ and

(5.79) A to the affine algebra (5.80) (5 21)

190

L. Frappat et al. / Twisted affine algebras

the 2-cocycle has

being chosen invariant

under

r. Now taking

y invariant

under

r, one

yff EP +E’(b) m1=L n,m m+E”(b)

3y(B+7(P)+72(P))(E~+E:(P)+E~2(P)), (5.82)

d,

E,p + ET’81 + E”t@ m

“1

1 ( =

m

Et + E;(a) + E”(P)

The step operator attached to the root algebra at level m is therefore

$(p + 7(p) + r’(p))

E!i Et

+ E;(P) +

(5.83)

m

E;‘(P)

of the folded

affine

if P=T(~) (m~z),

(5.84)

if

(5.85)

@r(p)

=O.

In this last case, m E Z + $I where m(, = p [mod 31. To construct the corresponding vertex operators, we use the same trick as in the case of twisted algebras 9-@). One extends the weight lattice A of the invariant integral subalgebra G, of the folded algebra to a lorentzian lattice n. The vertex operators for the roots of a”: are E(P, the vertex operators

(5.86)

z) = u(P> z)c,q,

for the roots of a”‘, are

+ and finally

the vertex operators

u( TV> z)c,?(~)) > (5.87)

for the roots of AT are

+ U( T’( P) + $3, z) ~~2~~))

if

rn& = 1

[mod 31 (5.88)

+U(r2(p)+

:S,z)c7zCpj)

if

mb=2

[mod3], (5.89)

L. Frappat et al. / Twisted affine algebras cp

is here non-folded

the cocycle operator constructed on the root Lie algebra. Construction of the vertex operators

For the invariant part, the long roots of G, at integral Es) under the automorphism 7. Therefore E(+(Y+)=

191

lattice A.(E,) is as follows.

of the

level are invariant

roots of

(5.90)

U(f(4;-~,):z)C+(rl,-1),).

The short roots of G, at integral level divide into two distinct under the Weyl group generated by the long roots Q,=

The vertex operator

E(as,

lIi#j#kI3}.

- 7, - %J>

part factorizes

fi2, and

9,

lIi#j#k53},

{i(29,-11,-%)>

9, = { - :@L

orbits

(5.91)

easily as (Y~E Qi,

(5.92)

z) + u(‘&, z) + U(&> Z))>

(5.93)

z) = u(as,

z)&(z)

if

with Ti(Z)

= E(U(&,

T*(z) = &(U(-5i,

z) + U(-E2,

z) + u(-‘&,z)>>

(5.94)

where

(5.95) However, one must also examine the factorization properties of the cocycle! Actually, the cocycle problem in this case has been studied in refs. [2,3]. Let /3 be a long root of the underlying D, in E,. Then, the short roots & at integral level can be written

as

&=~(P+T(P)+T*(/?)), For QL,p E A.(D,),

prootof

D,cE,,

with

p~(p)=O.

(5.96)

one has (_I)“P=

(_I)3%fiS_

(5.97)

It follows from this that one can construct a suitable cocycle operator cp, for the short roots by using a cocycle operator of D4 associated to the corresponding root j3 of D4, related by eq. (5.96). In other words, the auxiliary fields ri(z) and T,(z) do

192

L. Frapput et al. / Twisted uffine algehrcrs

not contain cocycle operators in their expression (or it reduces to the trivial unity operator). Therefore, the vertex operator associated to the short roots can be written as

E(% 4 = ub,, z)c,T,(z), if cxs E 52, and (us = ~(CI + r(a) + ~~(a)). For the twisted part, it is constituted by the generators

(5.98)

associated

to the short

roots of G,, appearing at level Z + : and Z + f. As above, the short roots decouple in the two orbits 9, and J&. The vertex operators are determined by the usual method. One has to introduce vertex operators with moments at level Z + f and Z + $. This is achieved by considering the vertex operators of the form U( CX~ + $8, z) and U(a, + $6, z). Therefore, the vertex operators are: for the short roots of 9, at level Z+ ;

E’bs, z) = WY,, +,&w

>

(5.99)

with

T;(z) = ~(u(s; +

;s, z) + U(& + ;s, z) + U(& + $8, z)) ;

(5.100)

for the short roots of Q2, at level Z-t <

E%s, z) = U(%,z>cJ;‘(z),

(5.101)

with I’;‘(z)

= fi(U(t;‘+

26, z) + U(&‘+

26,~)

+ U([;‘+

26, z));

(5.102)

for the short roots of 0, at level Z+ ;

E’bs, z) = w+,,z)c&‘(z),

(5.103)

with

r;‘(z)

=

/gu(

-2y

+ ;s, z) + u( -E;‘+

$6, z) + u( -,l”

+ ;s, z)) ; (5.104)

for the short roots of 9, at level Z+ 5

E”bs,

z>=w+.‘+J;(z>>

(5.105)

with

G(z) = &(u(

-8

+ $6, z) + u( -E;

+ $8, z) + U( -5;

+ :S, z)),

(5.106)

L. Frupput et d. / Twisted

affineulgehrus

193

where E;=

i(-711-172-

7?3 -

3774- 3715+ 3176+ 3% - 3%) 2

c;= +(-vl-v2-v3+3

?jd- 3775- 3q, - 3v, + 3178))

E;

=

j(171+ 92

+

7J3 +

3%) 9

E;’

=

f(Q

+

773 -

3%).

+

v,

(5.107)

Notice that the different auxiliary fields c(z), T:(z) and q”(z) are not independent since Ej + [: -t- [1’= 0 for i = 1,2,3. These auxiliary fields, together with the elementary vertex operators associated to the short roots of Dj’), are no longer fermionic fields since the short roots (us have squared length $ and the vectors E,, t(, [:’ have squared length t. Therefore, the corresponding vertex operators U(cus, z) have conformal weight ; and U(.$,, z), U(<:, z), U([,!‘, z) have conformal weight $, the O.P.E.‘s between these fields having branching points rather than poles. One can say that one obtains, in this case, a parafermionic construction of the vertex operators of the twisted algebra Dj3’.

5.5. FOLDING A’,:‘, 1 --+A$+)

Finally, we will study the case where one can exploit the symmetry a twisted affine algebra. One starts from the EDD of A(ij+t

of the EDD of

(:I%--~+,~ associated

to the simple root system

(5.108) This diagram

has a Z, symmetry

defined

by the automorphism

r (5.109)

194

L. Frappat et al. / Twisted affine algebras

The simple

root system of the folded algebra

is

R=(a;,=;s-fin2,

which corresponds

to the folded DD W

----@S

I

aO

I

a'2

cl;+,

ae

One obtains therefore the twisted affine algebra A($ with invariant bra C,. The root system of A(;] is

integral

2~i+j1f+l,mEZ,nEZ*).

subalge-

(5.111)

Construction of the vertex operators is as follows. Notice first that A$_, is a regular subalgebra of A($]. Therefore, the corresponding vertex operators are those which were constructed in subsect. 5.2 in the case of A($]- 1. For the invariant part; the invariant p = r(p). It corresponds to the roots

which appear

where 5.2.)

T;,(z)

at each integral

part

is given

level. The vertex operators

E( *217/a,

z) = u( *27&z

is a fermionic

auxiliary

by the roots

/I such

are (5.113)

+*2q,,fi

field depending

on the orbit

L?,, (see subsect.

(5.115)

%,=(/GWJi,>. The twisted part at level Z+ $ is constituted short roots of A,,_,. The vertex operators are

P(fi(+q,*g,),z)

that

by the generators

associated

= U(~(,9ifII,)7Z)c(.,,.,,,~r,‘,(z).

with

(5.116)

L. Frappat et al. / Twisted affine algebras

Folding

where l)>(z) is a fermionic Ramond or Neveu-Schwarz The “very

short”

+ a. The vertex

TABLE 2 schemes for affine and twisted algebras

auxiliary field depending on the orbit a,,, character than cj(z) (see subsect. 5.2).

roots are the roots

operators

195

+ ~$7;

which appear

on opposite

at level Z+

i and Z

for these roots at level Z + i are

+ U( - &?jl

- 6/4, z)l;:(z)j.

(5.117)

L. Frappat et al. / Twsted affine algebras

196

TABLE 2 (continued)

(2) E6

(1)

DA

The generators

associated

by action of the generators on the generators

associated

to the bosonic associated

roots

f gq,

at level 2-t

to the short roots fi(

to the bosonic

roots

+ 6%

i are obtained

f-q, f qj) at level Z + $ at level Z + :. One finds

+u(-fig,-S/~,Z)~‘;(Z)).

(5.118)

L. Frupprrr et ai. / Twisted uffine dgehrus

The auxiliary

fields are defined

by (see subsect.

5.2)

+u(-J:(t,+sJ +~/2.4-(&+[,),&

+~(-~(E,-5,)+S7zjC-(C,~E,)/\IZ). The properties

of I’i,(z)

and T;,(z)

insures

197

(5.120)

(5.121)

that one has

[gl, 31 c g,+,, [mod4],

(5.122)

where 3, represents the set of generators at level Z + ii (i mod 4). For the cocycle operator, one can construct it as a cocycle operator on the resealed root lattice iA.( since the invariant horizontal algebra of A(;] is B,. This construction is very similar to those explained in the A$_r case where the cocycle operator on the lattice :As(D,) was constructed. Remark. We remind that the twisted algebra A (24’is actually isomorphic to A($ [I41.

6. Conclusion In the method we have presented to construct the vertex operators for twisted affirm algebras CY(m) (m # l), the property of a twisted algebra to show up as a subalgebra of an untwisted one is widely used. One could ponder on the relevance of such I@ subalgebras associated to outer automorphisms of 3(l) in physics and in particular in string theories where the notion of twist, relative to inner and outer c?? automorphisms, appears today as a basic tool (cf. orbifold compactification). Let us remark that maximal simple affine subalgebras of !Z?(‘) with B compact can be classified in three different classes: the regular (respectively singular) subalgebras Z(l) with 2 being a regular (respectively singular) maximal ?? subalgebra, and the twisted subalgebras %(“‘). This last class of subalgebras - when it exists ~ is at the finite level. directly due to the affine structure of g(l) and has no counterpart

L. Frappat et al. / Twisted affine algebras

198

Concerning

the above vertex construction

being conceptually

simple, its fundamental

itself, let us emphasize feature

standing

its property

in the adjunction

of

of (in

general dependent) auxiliary (para)fermionic fields to short-root and also to affine short-root operators. The folding of EDD allows a rather elegant approach, more tedious may appear the cocycle construction which has to be done separately in each case. Finally, let us note that the same type of method can be used to construct vertex operators for untwisted and twisted affine superalgebras [15].

References [l] LB. Frenkel and V.G. Kac, Invent. Math. 62 (1980); G. Segal, Commun. Math. Phys. 80 (1981) 301 [2] P. Goddard, W. Nahm, D.I. Olive and A. Schwimmer, Commun. Math. Phys. 107 (1986) 179 [3] D. Bernard and .I. Thierry-Mieg, Commun. Math. Phys. 111 (1987) 181 [4] V.G. Kac and D.H. Peterson, Proc. Conf. on Anomaly, geometry and topology, Argonne, 1985, ed. A. White (World Scientific, Singapore) [5] J. Lepowsky, Proc. Nat. Acad. Sci. USA 82 (1985) 8295 [6] P. Sorba and B. Torresani, Int. J. Mod. Phys. A3, (1988). 1451 and references therein [7] D. Gross, J. Harvey, E. Martinet and R. Rohm, Nucl. Phys. B256 (1985) 253; B267 (1986) 75 [8] H. Kawai, D.C. Lewellen and S.H.H. Tye, Nucl. Phys. B288 (1987) 1; I. Antoniadis, C.P. Bachas and C. Kounnas, Nucl. Phys. B289 (1987) 87 [9] V.G. Kac, Ad. Math. 30 (1978) 85; V.G. Kac, Infinite dimensional Lie algebras, (Cambridge University Press, Cambridge, 1985) [lo] D. Olive and N. Turok, Nucl. Phys. B215 (1983) 47 [ll] N. Bourbaki, Groupes et algebres de Lie (Hermann, Paris, 1968) chap. 4, 5, 6 [12] D. Bernard, Nucl. Phys. B288 (1987) 628 [13] D. Altschtiler, J. Lacki and Ph. Zaugg, Phys. Lett. B205 (1988) 281 [14] A. Feingold and I.B. Frenkel, Ad. Math. 56 (1985) 117 [15] L. Frappat, A. Sciarrino and P. Sorba, LAPP TH 216/88