General Theory of Twisted Vertex Operators

9

General Theory of Twisted Vertex Operators

In this chapter we present analogues of the results of Chapter 8 for the twisted vertex operators studied in Chapter 3 and in Sections 7.3 and 7.4. We extend the twisted vertex operator representation to a representation of the full algebra of general vertex operators of the previous chapter, in a general sense involving the square roots of the formal variables. The important new feature of general twisted vertex operators is that the “naive” definition of general vertex operators analogous to their definition in Chapter 8 requires a correction. This modification is achieved by composing with the exponential of a certain quadratic differential operator. A similar expression turns out to have appeared in the formula for the socalled bosonic emission vertex, later interpreted as the H,-orbifold twist operator, in string theory (see the Introduction for more details on the connection with string theory and for the physics references). The resulting general vertex operators acting on the twisted space satisfy a Jacobi identity in the sense of Chapter 8. This fact will be important in the construction of the vertex operator algebra based on the moonshine module. The main results of this chapter were announced in [FLMS]. In Section 9.1, which is parallel to Section 8.4, we obtain a formula for the commutator of a pair of twisted vertex operators associated with arbitrary lattice elements satisfying natural conditions. This formula was

255

256

General Theory of Twisted Vertex Operators

found in [Lepowsky 41 in a slightly different form. We introduce general twisted vertex operators in Section 9.2 by analogy with the construction of Section 8.5 and parametrized by the untwisted space, but incorporating the modification mentioned above. Using the new definition we present the commutator formula in a form familiar from Chapter 8, but this time involving the square roots of the formal variables. In Section 9.3 we also extend the commutator formula to an arbitrary pair of general twisted vertex operators satisfying natural conditions, and we record some consequences and special cases. We relate the twisted construction of the Virasoro algebra and its commutation relations, from Chapter 1, with general twisted vertex operators in Section 9.4. In particular, the scalar adjustment required in the twisted case of the Virasoro algebra in Section 1.9 is now naturally determined by the exponential of the quadratic differential operator. Finally, in Section 9.5, we establish a Jacobi identity for twisted vertex operators, and from this we extract information about cross-brackets as in Section 8.9. Theorem 9.5.10 was announced in [FLMl]. The results of this chapter and of Chapter 8 are generalized to a setting based on an arbitrary lattice automorphism in [Lepowsky 61; see [Lepowsky 4,5] and [FLMS]. See also [Borcherds 41.

9.1. Commutators of Twisted Vertex Operators In this section, which is analogous to Section 8.4, our aim is to obtain a very general commutator formula for twisted vertex operators, removing the restriction on I (a, 6) I in Theorem 7.4.1. We recall the setting of Sections 7.3 and 7.4, where the twisted vertex operators X z + 1 / 2 ( az), , here abbreviated X(a, z ) (as in Chapter 7), were introduced. We have a nondegenerate lattice L ; 4 = L Oz IF; S(&+,/2) is -) is a central extension of L by ( K I K' = 1) with the fi[ - 11-moduleM(1); commutator map co; c(a,p) = W ~ O ( ~ . ' ) for a , p E L ; 8 E Aut(e; K , ( -, )) such that 0 = -1 on L and O2 = 1; T is an e-module such that K acts as o and for a E 8a = a as operators on T; V[ = S($;+1/2) 0 T;

(e,

-

e,

for

Q!

E

4; and for a E e with (&a)

E

h

(9.1.2)

257

9.1. Commutators of Twisted Vertex Operators

we have the vertex operator

(9.1.5) and

X*(a, z ) = X(a, z)

c

=

W a ,z )

(9.1.6)

x,'(n)z-".

(9.1.7)

n E (1/2)Z

(Of course, the assumption (9.1.2) is not necessary if IF contains appropriate roots of 2 , as we have commented in Remark 7.3.4.) Just as in Section 8.4, we change notation as follows: For a E 6, set a,,&)

=

C

n E Z+ 112

a(n)z-"-' = z-'a(z) (=z-'aOld(z)) (9.1.8)

and abbreviate this [as in (8.4.8)] by a(z): a(z) = anew(z) =

c+

nEZ

1/2

a(n)z-"-'

=

c

n E k + 1/2

a(n - l)z-". (9.1.9)

Then (9.1.1 0) Also introduce the notation change

a(z)' =

c

a(+n)zin-'

(9.1.11)

+ &I-.

(9.1 .1 2)

nEM+l/2

[cf. (8.4.10)], so that a(z) = a(z)' We set

a(n) = 0 for n E B , in particular,

a(0) = 0,

(9.1.13)

as in (7.4.5). Now in the twisted case there is only one natural notion of normal ordering-the one we have already used, denoted by : * : [recall Section 3.3 and (7.3.21)-(7.3.28)]. But to emphasize the parallels with the untwisted theory as treated in Chapter 8, we shall replace the normal ordering notation in the twisted case by open colons: 8 . 8 = :*:.

(9.I. 14)

258

General Theory of Twisted Vertex Operators

In the untwisted theory, the appropriate vertex operator Y(a,z) differs from the earlier one X(a, z) by a factor of z - ‘ ” ~ ) / [recall ~ (8.4.17)], and it turns out that this is the correct motivation for the definition of Y(a,z ) in the twisted case: For a E such that ( a , ii) E Z, define the new vertex operator by:

e

Y(a,Z ) = yz+1/2(a, 2 ) = 2-(”’) 8ela(z)8aZ-(P’P)/2 =

X(a, z ) z - ( ~ . ~ )=/ 2X(a, Z ) Z - W f t ( @ .

(9.1.15)

Then we have

X&)Y(a, Z)X = a!(z)-Y(a,z) + Y(a,z)a(z)’

(9.1 .16)

gy(a,Zl)y(b,Z z ) 8 = 2 - ( d * u ) - ( 6 .gel(u(z1)+6(Z3)8abz;(U,6)/2 6)

z 2-(6,6)/z

(9.1.17)

8Va, Zl)Y(b,Z2)8

=

til2‘;,)

(9.1.18)

40,G)gY(b,z ~ ) Y ( u21)s , 1/2

y(a, zl)Y(b,zz) = 8Y(a, zl)Y(b,zz)x

e

-

1/2

(d,b)

+ zz

(9.1.19)

for a! E b and a, b E satisfying the integrality condition (9.1.2). The last formula is a restatement of (7.3.28). The factor on the right is to be expanded in nonnegative integral powers of z;”, as the notation as usual suggests. We may also write (9.1.19) as follows:

Y(a,Zl)Y(b,zz) =

n

gY(a, Zl)Y(b,Z2)8

(2:’’

- (-1)PZz 112)((-1YU.h (9.1.20)

p=o,1

= SY(U,zl)Y(b,z2)8(z1- zz)(u’6)(z:/2 +

zi’2)-2(d*6).

(9.1.21)

Note also that

in contrast with (8.4.23), and that

~ ( az ), =

n

E

C

x,(n)z-n-(a~p)/2.

(9.1.23)

(1/2)2

By the properties of the involution 0 and its action on the module T , we have

Y(Oa,z) =

p 2

lim

+ -zl/2

Y(a,z) if

(a, a )

E

272,

(9.1.24)

259

9.1. Commutators of Twisted Vertex Operators

and more generally,

[see (7.4.16) and (9.1.15)]. We are ready to begin computing commutators. Let a, b E L and assume that (9.1.26) (a, a ) , (6,6>, (096) E Z,

(a, 6) I 0

(9.1.27)

and c(a, 6) = (- l p ?

(9.1.28)

Remark 9.2.2: The condition (9.1.27) is not a serious restriction because (9.1.24) or (9.1.25) can be combined with the present computation to find [Y(a,zl), Y(b,z2)]if (9.1.27) does not hold. Continuing with the computation, we would like to apply Proposition 8.2.2, but we must be careful about the fact that the exponents of z1 and z2 are half-integers [recall (9.1.17)). We start by observing that

6(z) = +(s(z1/2) + s(-z1/2)).

(9.1.30)

Also, the operator d/dz may be expressed as a Laurent monomial in times the derivative with respect to the variable z1l2:

z”’

(9.1.3 1) as we see by formally using the chain rule or by applying both sides of (9.1.31) to a general formal series. More generally (although we shall

260

General Theory of Twisted Vertex Operators

not need this),

(9.1.32) for every m

E

IFx, a fact which can also be expressed in the natural form

d d z-& = nz"d(z")

(9.1.33)

for every n E F X . Actually, (9.1.31)-(9.1.33) have already arisen in the proof of Proposition 8.3.10. We now find using Proposition 8.2.2 and (8.3.11) that

[Y(a,211, Y @ ,2211 =

i p = O , l Res,,

~ , j ~ ~ ~ ) z ; zl)Y(b, ~ s Y ( aZ2)8(Z:/2 , + Z;/2)-2'u*''

. e-zdd/azdg(( - l)~z:/2/zi/2)

=

+ *

C

Res,,$*6)z;1

p=O,l

lim

z p -t ( - IYzy2

(sY(a,z 1 + zo)Y(b,Z2)S

((z1 + z0)lI2 + z21 / 2) - 2 ( a . 6 ) ) e - z o ( a / a t , ) g ( ( _ I)PZ:/~/Z;/~). (9.1.34)

Recalling that the exponent -2(a, 6) is a nonnegative integer, we have

( ( z , + zo)1/2+ p ) - 2 ( a J ) = ($2(1

+ z0/z1)'/2+ 2 ; / 2 ) - 2 ( a J )

, (9.1.35)

and the binomial series (1 + involves only nonnegative integral powers of z o / z l ,with constant term 1. Thus when we replace z:l2by -z:l2 in (9.1.35), the resulting formal power series in zo contains a factor of z;~('*'). It follows that the term corresponding to p = 1 in (9.1.34) contributes the residue of a formal Laurent series in zo in which the smallest exponent of zo is at least - (a, 6). This being nonnegative, the residue is 0, and so (9.1.34) gives just

[Y(a,211, Y(b,Z2)l = 3 Res,,z~P'6)z;'sY(a, 2 2 + 20) Y(b,22)s - ((z2 + zrJ)1/2+ 22112) - ~ ( a , 6 ) ~ - ~ , ( a / a z , ) g ( ~ : / 2 / ~ ; / 2 ) . (9. -36) This formula is the analogue of (8.4.35). There is an interesting analogue of Remark 8.4.3:

Remark 9.1.2: Formally, (9.1.21) gives [Y(a,211, Y(b,2211 =

Res,, z;'Y(a, z2 + zo)Y(b,~ ~ ) e - ~ ~ ( ~ / ~ ~ l ) B ( z(9.1.37) :'~/z~'~).

9.1. Commutators of Twisted Vertex Operators

1

- exp(ezo(d/dzd

261

)

a(z2)+ dzz abz,- ( u + 6 , u + 6 ) / 2

. zy.6)(l+ zo/z2)-(p.u)/2

(9.1.38)

This expression is not as simple as that in (8.4.32). We shall rectify the situation below by setting up general vertex operators appropriately, but for now we summarize what we have proved (cf. Theorem 8.4.2):

Theorem 9.1.3: Let a, b E

e and assume that

(a, a>, (6,6>,(it, 6) E z, (0,6> I 0

(9.1.39) (9.1.40)

262

General Theory of Twisted Vertex Operators

Formula (9.1.43) follows immediately from (9.1.38) and does not require condition (9.1.40) or (9.1.41). Note that the factor

is a formal power series in z0/z2with constant term 1. By expanding this expression explicitly, we could write down “concrete” expressions for the commutator in the spirit of (8.4.38)-(8.4.45), but it is better to wait until we have general vertex operators available, since we will then be able to exhibit a precise parallel between the untwisted and twisted theories. The discrepancy between (9.1.22) and its simpler-looking untwisted analogue (8.4.23) will be explained in the process.

Remark9.2.4: Theorem9.1.3 generalizesTheorem7.4.1 (cf. Remark8.4.5). Remark 9.2.5: In Section 7.4 and in the present section, we have assumed that 8a = a as operators on T for all a E When we investigate triality in Chapter 11, we will want to drop this assumption and to allow instead that for a particular a E either

e.

e,

8a = a as operators on

T

(9.1.45)

or 8a = - a

as operators on

T.

(9.1.46)

The only change required in the present section is that formulas (9.1.24) and (9.1.25) gain minus signs on the right-hand sides if (9.1.46) holds.

9.2. General Twisted Vertex Operators

263

9.2. General Twisted Vertex Operators In the last section we have computed the commutator of the twisted vertex operators Y(a, zl) and Y(b,z2)under certain natural hypotheses on a and b. The result, Theorem 9.1.3, seems more complicated than the corresponding result in the untwisted case (Theorem 8.4.2). But there are two reasons why we might expect that a proper definition of general twisted vertex operator will reveal the untwisted and twisted theories to be entirely analogous: The vertex operator constructions of the untwisted affine algebras a,, 8,, I?,, and of their twisted analogues given in Sections 7.2 and 7.4 are very similar, and so are the formal versions of the commutator result given in Remarks 8.4.3 and 9.1.2. This section, which is parallel to Section 8 . 5 , begins with an approach to the definition of general twisted vertex operator. We shall now assume for convenience that (@,a)E Z

for all a E L .

(9.2.1)

While this does not make L an integral lattice, condition (9.1.2) will always hold. Comparison of Corollary 8.5.1 and Theorem 9.1.3 leads us to try to define general twisted vertex operators Y(u,z ) = Yz+1/2(u, z ) for u E VL = S(6;) 0 IF[ LJ [the untwisted space; recall (7.1.18), (7.1.3 l)] having the following properties: The correspondence u Y(u, z) should be a linear map from V, to (End V[))(zJ(V; being the twisted space) such that when we extend the map canonically to VJzoJ,

-

for a,b

E

e. In this case, we would have

for a, b as in Theorem 9.1.3 [cf. (83.11)-(8.5.13)], and formally,

W ( a ,zo)r(b),22)

=

Y(a,22 + zo)Y(b,zz)

(9.2.4)

for all a, b E f, (cf. Remark 8.5.2). Of course, Y means Y z + , / 2 .[Note that we want YJa, zo), not Y(a,zO).]

264

General Theory of Twisted Vertex Operators

The "obvious" guess as to the right definition of Y(u, z), namely, the operator denoted Yo(u, z) in (9.2.23) below [formally the same as the general untwisted vertex operator given by (8.5.5)], certainly will not work.Instead, we shall construct a linear map of exponential form, exp(A,): V, -,vL(z1,

(9.2.5)

such that exP(Az)(Yz(a, zo)r(bN

[so that Yz(a,zo)r(b)is an "eigenvector"], and we shall take Y(u,z) Yo(exp(Az)v,z). This will be compatible with (9.2.2). Moreover, Az: V ,

-+

h(zJ

=

(9.2.7)

will be realized as a quadratic expression in the operators a(n) for [Y E 4 and E N-a "quadratic differential operator." We begin by noting that for a, b E [Y E b and n > 0, we have

n

e,

44

and

Yz(a, z 0 ) m = (a,a>zo"Yz(a,zo)r(b)

*

(9.2.8)

4) - Yz(G z 0 ) m = (a,a + 6>Y,(a, zo)r(b), (9.2.9) by (7.1.33), (7.1.46) and (8.4.17). Now let Ih,, ..., hi) be a basis of b and let lh; , ... h;) be its dual basis with respect to the form ( - , - ), as in (1.9.21)

and (8.7.2). Then for m,n > 0, I

C hi(m)hf(n) i= 1

*

&(a, z o ~ b =) (6,~ > G + " Y ~ zo)l(b), (u,

(9.2.10)

I

C hi(m)hf (0) Yz(~,Zo)r(b) = ( a + 6, a>Z,"Y,(a, *

i= 1

I

C h A W f( n )

i=1

I

C

i= 1

'

Yz(~,ZoMb) = ( 0

+ 6, a)i$Yz(a,

hi(O)hf(O)' &(a, Zo)r(b) = (6 + 6, a

zo)r(b),

(9.2.11)

zo)r(b),

(9.2.12)

+ 6)Yz(a, zo)r(b). (9.2.13)

We shall take Az to be of the form (9.2.14)

9.2. General Twisted Vertex Operators

265

with cmnE F. This is a well-defined element of End(V,[z-']), that is, it preserves the space of polynomials in z-' with coefficients in V' . Of course, this operator can be canonically extended to the space VL[z-'](zo). Assuming that coo = 0, (9.2.15) we see from (9.2.10)-(9.2.12) that Az(Yz(a9 z 0 ) W

c

= ( < I I , O ) m.n+O

Cm,(zo/Z)m+n + (a, 6)

c

n>O

(cno

+ ~on)(zO~z~)

- Yz(a, z o ) m

(9.2.16)

Because of (9.2.15), exp(Az) is a well-defined element of End(V,[z-']), and from (9.2.16), exp(A*)(YZ(a,ZO)l(b)) = exp

((o,s> 1

m,nzO

Cmn(zo/Z)m+n+

( a , 6 > 1 ( ~ " 0+ cOn)(zo/z)"

- Yz(a, zo)@) = (ex,

1

m , n * O cmpj(zo/z)m+n)

- Yz(a, zo)r(b).

(8.0)

(exp

nsO

c

n>O

(cno

+ con)(4/z)n)

)

(u,6)

(9.2.17)

We want (9.2.17) to match (9.2.6). It is clear that we can arrange this with the constants cmndetermined by the formula

The c, are indeed well defined, and coo = 0. Wefix this choice of the cmn in the expression (9.2.14) for Az .

Remark 9.2.1: Of course, there is certain flexibility in the choice of the constants, but (9.2.18) will prove to be correct when we consider commutators of general twisted vertex operators.

-

Before defining these operators, we introduce operators denoted &(v, z). As in (83.1)-(8.5.6). we first observe that the correspondence i(a) Y(a, z)

266

for a

General Theory of Twisted Vertex Operators E

i extends uniquely to a well-defined linear map F(L] (End V[)[[z'l2, z - " ~ ] ] +

?J

&(v,

(9.2.19)

z),

by (7.1.20) and (7.3.19). In particular, we have Yo(@),z) = Y(a,z) for a E

L

(9.2.20)

and &(f(l), 2) = Y(1,z)

=

(9.2.21)

1.

More generally, let a

E

L,

a],.-.,(Yk E

b v n ] , ..., n k

E

and write

z+

u = ck!](-nl)' * * (Yk(-nk) ' f(a)E V,.

(9.2.22)

We define &(v,z) E (End V[)[[Z'/~, z-'"]]

by:

This gives us a well-defined linear map V'

+

u

(End V[)[[z'/', z - " ~ ] ] &(v,

z).

(9.2.24)

Recalling [from (8.5.1 l), for example] that

for a, b

E

e, we see that

where of course the map & is extended to VJz,,). We finally define the general twisted vertex operators Y(v,z): For v E V,, we set

Y(v, z ) =

YZ+I/&J,

z) = Yo(exp(A,)u, 21,

(9.2.27)

9.2. General Twisted Vertex Operators

267

well defined since exp(A,)u is a polynomial in z-' with coefficients in VL. We have a linear map (9.2.28) From (9.2.6) and (9.2.26) we find that we have achieved what we wanted: For a, b E

e,

[cf. (9.2.2)]. In particular (taking

Y(i(a),z)

=

Q

= l),

Y(a,z) for a

E

L,

(9.2.31)

Y M l ) ,2 ) = Y ( 1 , z ) = 1 , and we have: Corollary 9.2.2:

(9.2.30)

For a, b as in Theorem 9.1.3,

[ V a ,zl), Y(b,z2)l

=

t Res,, z;'Y(Y,(a, zo)r(b),z2)e-zo(J/Jz1)8(z:/2/z:/2).

(9.2.32)

Remark 9.2.3: Formally, Y(&(a, zo)db),22) = Y(u,22

+ z0) Y(b,z2)

(9.2.3 3)

for all a, b E L (cf. Remark 9.1.2). We recall from (8.9.20)-(8.9.24) the action of the involution 8 [recall (7.4.10), (7.4.11)] on VL: 8: & = ~(6;)

0F ~ L ) -+

x

0 r(a) -ex 0 r(ea)

(9.2.34)

for x E S(6;) and a E L , where 8 acts on S(6,) as the unique algebra automorphism such that ex =

-X

for

XE

6,.

(9.2.35)

General Theory of Twisted Vertex Operators

268

Recall that 0 is indeed well defined on ff(LJvia the formula

B(r(a))= r(0a) for a E

L

(9.2.36)

by the definition (7.1.18)and the fact that OK = K. With this action, we have the following generalization of the action (9.1.24)-(9.1.25)of 0 on the operators Y(a,z): Proposition 9.2.4:

If the lattice L is even, then for v

E V',

(9.2.37) More generally, for L satisfying (9.2.1) and v as in (9.2.22),

Proof: From (9.1.25)and the definition (9.2.23),we see easily that

for v as in (9.2.22). Also, applying (9.2.14)to u, we find that BOAz= A,oB

(9.2.39)

and hence

8 exp(A,) 0

= exp(Az) 0 0

(9.2.40)

on V,. The result now follows from the definition (9.2.27).

I

We would like to see what general twisted vertex operators look like in some special cases beyond (9.2.30).The expansion (9.2.18)begins as follows:

- log((l

+ x)1/2 + (1 + y)"') 2

\

= -$(x

+ y ) + &(xz + y Z ) + &xy +

I

** *

,

(9.2.41)

so that exp(A,) = 1 -

a

I

i= 1

(hi(l)hi(0)+ hi(0)hj(l))z-'

+ 6 1 (hi(2)hi(O)+ hi(O)h:(2))z-* + & c hi(l)h:(l)z-2 I

I

i= I

i= 1

(9.2.42)

9.2. General Twisted Vertex Operators

269

Hence we have exp(A,)a( - 1) = a( - l),

(9.2.43)

Y M - 11, z) = &(a(- I ) , z ) = a(z)

(9.2.44)

for a E f~[where a( - 1) of course denotes a( - 1) * i(1), as in (8.6.33)]; more generally, eXP(AzM4-1) * 4 ~ )=) 4 - 1 ) y(a(- 1) *

for a E

l(U), 2)

*

r(a) - +(aY, P ) z - ’ ~ ( u ) , (9.2.45)

= 8 ( a ( Z ) - * ( a , a)z-’)Y(a, Z)g

(9.2.46)

b and a E i [cf. (9.1.22)]; exp(A,)a( - 2) = a( - 2),

(9.2.47)

d Y(a(- 2), z ) = -a(z) dz

(9.2.48)

for a E b; and exp(A,)a(-

1)p(- 1)

Y(a(-l)b(-l),

=

a( - 1)p(- 1)

Z ) = Sa(Z)p(Z)8

+ + ( a ,p)z-’,

+ +(a,p)Z-’

(9.2.49) (9.2.50)

for a,p E 6. Recall the canonical quadratic element w from (8.7.2). Then by the last two formulas, exp(A,)o = w

+ &(dimfi)z-’,

(9.2.51)

+ c 8h:(z)hi(z)8+ *dim I

Y(w,z ) =

i= 1

b)z-’

(9.2.52)

[cf. (8.7.4)]. As in Section 8.5, we shall express our results in terms of the component operators u, E End V[ defined by the formula Y(U,Z) =

c

U,Z-n-*

ne9

= n

E

c

u,-’z-n

(9.2.53)

(1/Z)E

for u E V, . (The context should eliminate any confusion with the operators on the untwisted space VL denoted u, in Section 8.5.) From (9.1.23), we have

+ 1)

(9.2.54)

deg r(a), = n - wt r(a) + 1

(9.2.55)

r(a), = x,(n

for a E

-

+(a, 0 )

e and n E Q , as in (8.5.16), and so

270

General Theory of Twisted Vertex Operators

by (7.3.18). Note that l(l)n

=

We claim that deg u,

=

(9.2.56)

dn,-l*

n - wt u

+1

(9.2.5 7)

for every nonzero homogeneous element u E I$ and n E Q [cf. (8.5.22)]. To see this, first observe that if we define operators u), by

y,(u,z) =

C

uAz-,-l,

(9.2.5 8)

- wt u + 1

(9.2.5 9)

as9

then

deg u; = n

just as in (8.5.21) and (8.5.22). Now note that if A is a monomial in the operators hi(m)h:(n)z-"-"making up Az [see (9.2.14)] and if we define operators 4 by

Y,(Au,z) = then we still have

1

(9.2.60)

u;fZ-n-l,

ns4

(9.2.61) degut = n - wtu + 1. This proves the claim. The component operator form of Corollary 9.2.2 (the commutator result) is similar to Corollary 8.5.4:

Corollary 9.2.5: In the notation of Theorem 9.1.3, let

m ,n E +Z. Then as operators on VT, (9.2.62)

(finite sum). (On the right-hand side, i(a)i is an operator on V' .)

The result follows readily, as in Corollary 8.5.3.

I

9.2. General Twisted Vertex Operators

271

Hence we have: Corollary 9.2.6:

In the same notation,

Now for every homogeneous element u E &, set

X ( u , z ) = X Z + 1 / 2 ( V , z) = Y(u,z)zWtU =

c

-n

n E (1/2)2

~fl+wtu-lz

(9.2.65)

and define x,(n) E End VF for n E tZ by

X(u,z) = Y(u,z ) =

c c

n E (1/2)h

n E (1/2)Z

x,(n)z-" Xu(n)z-n-w'u

(9.2.66)

as in (8.5.27), (8.5.28). Then

xu(n) =

and

u,+,tu-l

degx,(n)

=

n

[dm u , z)]

=

-DX(v, 2 )

(9.2.67)

(9.2.68)

[cf. (7.3.17), (7.3.18)]. We define X(u, z) for arbitrary elements of V , by linearity:

v

-

m u , z).

(9.2.69)

Using the first formula in (9.2.66) to define x,(n) for arbitrary u E V , , we see that (9.2.68) is valid for all such u. We have

X ( W , z ) = X(a, z ) for a E

e and n E iZ,and

xL@)(n)= xu(n) X(a( - 11, z) = za(z)

xa(-l)(n)= for a E

b and n E

[see (9.1.8), (9.1.9)], as in (8.5.33), (8.5.34).

(9.2.70)

(9.2.71)

212

General Theory of Twisted Vertex Operators

Proposition 9.2.4 gives the following generalization of (7.4.16) and (7.4.17) [which we first check for u as in (9.2.22)]:

For all u E V',

Proposition 9.2.7:

In particular, xdn) =

xu@)

if

-xu(n) if

n n

E

Z

E

h

(9.2.73)

+ +.

There is an interesting formulation of X(u,z) in terms of binomial coefficients, analogous to (8.5.38). For every homogeneous element u E VL, we define

Xo(v, 2 ) = &(u, z)zWfU,

(9.2.74)

and we extend to all u E V , by linearity. Then for u homogeneous, Xo(exp(AzIz=

z ) = Yo(exp(Az)u,z)zw"',

(9.2.75)

since for every monomial A in the operators hi(rn)h~(n)z-"-"making up AZ , &(A lz= 1 u, z) = &(A u, Z)ZWtu. (9.2.76) Thus from the definitions, X(u, Z) = Xo(exp(AzIz=l)u,Z )

(9.2.77)

for u homogeneous and hence arbitrary in V'. But we also note that for u as in (9.2.22),

[recall (8.5.35)-(8.5.38)], and so for arbitrary u, X(u, z) can be expressed via (9.2.77) in terms of these binomial coefficients. For example, as in (9.2.45)-(9.2.50) we see that

-

X(d-1) [(a),z ) =

for a E fi and a E

e,

S(ZCY(Z) -

+
X(cY(-2),z) = ( D - l)(za(z))

(9.2.79) (9.2.80)

273

9.2. General Twisted Vertex Operators

for a E

b, and X ( a ( - l ) P ( - l ) , ~ ) = g ( ~ a ( ~ ) ) ( ~ P ( z+) Q )g

(9.2.81)

for a,P E b. This last formula should be compared with (8.5.39); note the added constant. As in (8.5.40) and (8.5.41), it is convenient to define a&?) = x a c - l ) p ( - l , ( n )for

n E $Z (actually, n

E

(9.2.82)

Z),

UP takes place in S2(b)[cf. (9.2.71)]; then ~a(z)P(z):+ +(a,P > z - ~= Y(CZ(-I)P(-I), z) = C aP(n)z-n-2.

where the product

neZ

(9.2.83)

Recalling (9.2.51), we see that for the canonical quadratic element o,

Just as in Corollary 8.5.6, we find from Corollary 9.2.5:

Corollary 9.2.8: In the notation of Theorem 9.1.3, let

m,n

E

+Z.

Then as operators on V:,

(finite sum). (On the right-hand side, x J i ) is an operator on V, .)

r

With the aid of (8.5.44), (9.2.70) and (9.2.71), we see that this result gives Theorem 7.4.1 for an integral lattice, as expected: if [~u(m),xb(n)l = S x d m + n) if +ii(m + n) + $mdm+n,Oif

(a,6) = 0 (a, 6) = -1 ( a , ~ =) 2, b

=

a-'. (9.2.86)

We shall comment on the action of the involution 0 on various structures. We have already defined an action of 0 on the untwisted space V, in (8.9.22) and (9.2.34), and we have discussed some properties in (8.9.26)-(8.9.29).

274

General Theory of Twisted Vertex Operators

Now we define an action of 8 on the twisted space V: as follows: 8: :v =

~(fi;+~/~)8

T-,

vf

0 r -Ox 0 ( - r )

x

= -Ox

0r

(9.2.87)

for x E S(fi;+l/2)and r E T , where 0 acts on S(g;+ ,,2) as the unique algebra automorphism such that

ex = - X

for x E fi;+1/2.

(9.2.88)

Then it is easy to see from the definitions (9.1.15), (9.2.23) and (9.2.34) [keeping in mind (7.4.14)] that

ea(n)e-'

= -(Y(~I) for

eY(a,z)e-'

=

eu,(v, 2)e-I

= Y,(BU, Z )

y(&,

Z)

E rj,

t~ E

(11

a. + +,

for a E L , for u E

(9.2.89) (9.2.90)

v,,

(9.2.91)

and it follows from (9.2.40) and (9.2.91) that

e w , 2)e-l = r(eU, Z )

for

u

E

v,.

(9.2.92)

In particular, in the notation of (8.9.25), if

u E V l then

0 commutes with

Y(V,Z)

(9.2.93)

and in this case, the component operators of Y(u,z ) preserve the f l-eigenspaces (v,')'~ of e in v;T:

(9.2.94)

:v

=

(vfy

@

(v:)-e.

(9.2.95)

From (9.2.87) and (9.2.88) it is clear that these two subspaces of Vf are distinguished by their gradings:

(9.2.96)

[recall (1.9.53) and (7.3.911.

Remark 9.2.9: Continuing with the generalization discussed in Remark 9.1.5, we see that the only changes needed in the present section are the following: If the element a E J? entering into the vector u in (9.2.22) satisfies

275

9.3. Commutators of General Twisted Vertex Operators

(9.1.46), then (9.2.37), (9.2.38), (9.2.72), (9.2.73) and (9.2.90)-(9.2.92) all acquire sign changes, and (9.2.93) is replaced by: if

then

u E V;'

B commutes with

Y(u,z).

(9.2.97)

9.3. Commutators of General Twisted Vertex Operators The stage is now set for computing commutators in general. Following the method of Section 8.6, we shall suitably extend Corollaries 9.2.2, 9.2.5 and 9.2.8 to arbitrary elements of V' in place of [(a) and r(b),thus exhibiting the span of the operators u, or x,(n) as a Lie algebra with precisely known structure. Most important is the remarkable similarity between the Lie algebra structures in the untwisted and twisted settings. Continuing to assume (9.2.1), we start with a result which, as one easily sees, generalizes Corollary 9.2.2: Theorem 9.3.1: Let a, b E t and suppose that

( 8 , b ) E Z and

c(8, 6) = (-l)('*').

(9.3.1)

Let u', u' E S(6;) and set

u

= uf

u

= U'

0 r(a) = uo - r(a) E V, 0 [(b)= u0 * f ( b )E V,.

(9.3.2)

Then

for all u, u E VL. Proof: Let k , I I1 and let a , , ..., a k ,b l , .. ., b, E ?, be as in (8.6.4). Define A and B as in (8.6.5). Then as we saw in the proof of Theorem 8.6.1, the coefficients in the formal power series A and B span S(&) 0 r(a) and

276

General Theory of Twisted Vertex Operators

0 z(b), respectively, and so it suffices to prove the Theorem with u and u replaced by A and B, respectively. Recall the expressions for A and B given in (8.6.6)-(8.6.8). Now we want the analogue of (8.6.9). Using (9.2.23) and (8.3.3), we have S(&

(9.3.5) Just as in (9.1.38), this expression has extra factors, compared with its untwisted analogue (8.6.9). But what we really want is Y(A,z), not &(A,z). Recalling (9.2.27), we see that we need to determine exp(A,)A, and this computation will be similar to (9.2.17). From (8.6.6), A = Yz(a1, w1) ' * ' Yz(ak, w k ) l ( l )

n

lsi
(w; -

w j ) - ( u i ' " J ) .(9.3.6)

277

9.3. Commutators of General Twisted Vertex Operators

As in (9.2.8)-(9.2.13), we find that for a E

b and m, n

L 0,

c (a, k

a(n) * A

=

iii)W,"A

i=l

(9.3.7) (9.3.8)

[here I is as in (9.2.10)-(9.2.14)]. Thus with Az as in (9.2.14), k

Az(A) =

C

i,j= 1

(08,

c

m.n 2 0

Cm,(wi/z)"(wj/z)"A,

(9.3.9)

and if coo= 0,

cm,(wi/z)"(wj/z)" i,j= 1

(9.3.10)

m,na0

[cf. (9.2.15)-(9.2.17)]. Now take the official choice of constants cm, given by (9.2.18). Then

(9.3.11) and from (9.2.27) and (9.3.5) we find that

Note that this last factor is to be understood as a formal power series in the w's, which is where it came from:

278

General Theory of Twisted Vertex Operators

Also note that the coefficient of each monomial in the w’s is a Laurent monomial in z. We now have the analogue of Remark 8.6.2:

Remark 9.3.2: A formal application of (9.1.21) to (9.3.12) [see also (9.3.6)] yields that for all a , , ...,ak E I!,,

This formally generalizes Remark 9.2.3. It also justifies the precise choice of constants c, in (9.2.18); recall Remark 9.2.1. Of course, all these considerations for A also apply to B . For instance,

(9.3.15)

(9.3.16)

and lim xY(A, z,)Y(B,Z 2 ) 8

exists,

(9.3.1 8)

21 - 2 2

that is, if X(zl, z2) denotes the coefficient of any fixed monomial in w l , ..., x1 in 8Y(A,zl)Y(B,Z 2 ) 8 , then lim X(zl,zz) exists. 21-22

(9.3.19)

9.3. Commutators of General Twisted Vertex Operators

279

Moreover, from (9.1.21) we see that y(A9Zl)y(B,22) = Sy(A, Zl)Y(B, Z2)8

(2, -

z2 + wi - xj)(uivhj)

1 sjs%

fl

*

fl

lsis

((z, +

+ (22 + X j ) 112)-2(ai*'j),

(9.3.20)

Isis

Isjx%

where the two products over i and j are of course to be expanded in nonnegative integral powers of (z2 + xj)lI2,wi and x j . Now consider the factor

g ( z l ,z2) = ((zl + w i ) 1 / 2+ (z2 + xj)1'2)-2(uis'j) =

C

m,n E M

grnn(zl9 z2)wi"xj"

(9.3.21)

of (9.3.20). If the exponent -2(ai, 6,) (EZ) is negative, then lim g ( z l ,z2) does not exist,

(9.3.22)

21 - 2 2

since for instance lim goo(zl,z2) does not exist

(9.3.23)

ZI-'22

[cf. (9.3.13)!]. Thus we multiply and divide (9.3.20) by a suitable factor: Choose M E N such that

M 2 2(ai, 6,) for all i, j.

(9.3.24)

Then

Y ( A ,ZI)Y(B, 22)

=

XY(A, zl)Y(B,Z2)8 *

n

lsis

(zl - z2 + wi - x ~ ) ( ' ~ * ' ~ )(9.3.25) -~G~,

1 sjs%

where GM =

n

(((21

1 sis

+

+ (22 + x j ) 1 / 2)- 2 ( a , . b , ) + M

1 sjs?

*

((~1

and Ghf E

so that

+

- (22

w2,z1

1/2

+ xj)112)M

, z y 2 ,z,1/2]"wI,..., x,]],

lim GM exists. 21 - 2 2

19

(9.3.26) (9.3.27) (9.3.28)

280

General Theory of Twisted Vertex Operators

If the roles of A and B and of z1 and z2 are switched in Gw, the result is ( - l)k'wGw.

(9.3.29)

Fix a monomial P in w l , . ..,x, as in (8.6.13), and choose N E N so large that the coefficient of P and of each monomial of lower total degree than P in FN

=

(z, - z2)N

n

(2, -

~2

1S;S

+ w i-

~,)'"i*'j)"'''

(9.3.30)

1sjs4

is a polynomial in z1 - z2 [cf. (8.6.14)]. Denote by Y p ( z l ,z2)the coefficient of P in

Then and lim Yp(zI, z2) exists.

(9.3.33)

z1-22

The coefficient of P in Y(A,zI)Y(B,z2)is YP(Z1

9

z2)(21

(9.3.34)

- 22rN,

where (zl - z2)-Nis to be expanded in nonnegative integral powers of z 2 . Similarly, reversing the roles of A and B and of z , and z2, and using (9.3.16) and (9.3.29), we find that Y(B,Zz)y(A,Zl) = 8Y(A,zl)Y(B,Z 2) ( - 1 ) ( a

9

6 )+ klM

where n ( z 2 - z1 + xi - ~ ~ ) ( " i * ' j ) -is~ now to be expanded in nonnegative integral powers of zl, wi and xj [cf. (9.3.20)]. Moreover, the coefficient of P and of each monomial of lower total degree than P in (zl - z2)N( - l)(a.b)+klMn ( z 2 - z1 + xj -

wi)(aiv6,)-M

is a polynomial in z1 - z2 which agrees with the same for FN [see (9.3.30)]. Hence the coefficient of P in Y(B,z2)Y(A,z,) is YP(21,z2)(21 - z21-Y

(9.3.35)

9.3. Commutators of General Twisted Vertex Operators

281

where this time the last factor is to be expanded in nonnegative integral powers of z 1 [cf. (9.3.34)]. Exactly as in (8.6.20)-(8.6.21), and using (9.1.30)-(9.1.31), Proposition 8.2.2 and (8.3.1 1). we see that the coefficient of P in [Y(A,z l ) ,Y(B,z2)] is the coefficient of z0N-l in 2 ;

yP(z , z2)e-zo(J/Jz~)d(z /z2) =

+

C

z ; ' y P ( z I ,z ~ ~ ~ - ~ o ( ~ / 1/2~ / ~z 21/2~ 1) ~ ~ ~ - ~ ) P z ~

p=O,I

We now examine the limit for p = 0: lim YP(Z1 + zo, 22) = YP(22 + zo, z21,

(9.3.37)

21-22

which from (9.3.18),(9.3.26), (9.3.28), (9.3.30) and (9.3.31) is the coefficient of P in

SY(A, Z 2 *

+ Zo)Y(B,Z 2 ) S Z r

n

1 n i,n

n

Isin I njnf

(Zo

+ Wi - Xj)("'*b')-M.

(((z2 + zo + wi)1'2+ (z2 + x j ) 1 / 2 ) - 2 ( " i * 6 j ) + M

l5,A

*

((22

+ 20 + w i y 2 - (22 + X j ) 1/2 )M).

(9.3.38)

Here it is important to note that even though lim F N does not exist, we are justified in replacing z I by z2 + zo in F N because we are looking only at the coefficient of P in (9.3.31) [cf. (8.6.23)]. Also, the factors

(z2 + zo +

wi)'I2

+ (z2 + x ~ ) " ~and

(z2 + zo + w ~ ) ' ' ~ (z2 + x j ) ' I 2 (9.3.39)

are of course to be expanded in nonnegative integral powers of z o , wi and

x j , and they may be raised to negative powers. Thus (9.3.38) simplifies to

282

General Theory of Twisted Vertex Operators

lsjsl

(9.3.41)

e-zo(a/az1)8(z:/2/z:/2).

Next we determine the limit f o r p = 1 in (9.3.36). First we observe that in the notation of (9.3.31),

= y(eA, z ~ ) Y ( Bz2)(z1 , -

z2Y

(9.3.42)

by Proposition 9.2.4 and the fact that ( z , - z ~involves ) ~ only integral powers of z l . To obtain BA from A , one replaces each ai by Oai and each Oi by -ai [recall (8.6.6)]. Now without affecting the argument so far, we increase M if necessary so that

M 2 -2(ni, E j ) [cf. (9.3.24)] and then we increase N

n

for all i , j

if necessary so that i f we define

~ ) (zl ~ - z2 + wi - ~ ~ ) - ( ‘ i * ’ j ) - ,~

F i = (zl - z

(9.3.43)

(9.3.44)

1s;s 1S J A

then the coefficient of P and of each monomial of lower total degree than P in F i is a polynomial in z 1 - z2 [cf. (9.3.30)]. Then as in (9.3.31), the expression (9.3.42) equals

sY(eA, zl)Y(B,z2)8FiG&,

(9.3.45)

where GL denotes the expression (9.3.26) with Di replaced by -ai. Moreover, denoting by Y,!(zl,z2)the coefficient of P in (9.3.42), we see that as in (9.3.33), Iim Y,“(z,,z 2 ) exists, (9.3.46) 21-22

and that this limit may be computed using (9.3.45) by the methods above. Returning to the limit for p = 1 in (9.3.36), we now have 4/2,

lim

-4/2

y P ( z Iz2) , = (- ~)(‘”’)Y,!(z~, z2)

(9.3.47)

9.3. Commutators of General Twisted Vertex Operators

283

by (9.3.42), and hence

- ( _ l ) ( u , a yPe( Z 1 + zo zz) 9

since 1 Thus

(9.3.48)

+ zo/zl and its powers are unaffected by the change of sign of z:”.

-

(-~)“*”Y,B(Z~ + zo, z2).

(9.3.49)

[note the sign changes from (9.3.41)]. All that is left is to express the commutator in terms of twisted vertex operators. First, Y & l , zo)B is given by (8.6.25). Thus by (9.3.12) with A replaced by Y z ( A ,zo)B,

284

General Theory of Twisted Vertex Operators

Comparing with (9.3.17), we now see that

and in conjunction with (9.3.41) and (9.3.50), this proves the Theorem. I As in Corollary 8.6.3, we find:

Corollary 9.3.3: In the notation of Theorem 9.3.1, [u0, Y(V,z)] =

3p =Co , 1 (-i)p(dgn)’r((epU)o- V , z),

and if L is even and (9.3.1) holds for all a, b E [u0,Y(V, z)] = +Y(@

Remark 9.3.4:

e, then

+ eu), - U, z).

Formally combining (9.3.20) and (9.3.51), we obtain Y(Yz(u, ZO)U,

22) =

Y(u, 2 2 + ZO)Y(U, Z 2 )

(9.3.52)

for all u, v E V,, and in particular, in the setting of Theorem 9.3.1, WU,

Zl), Y(V, 2211 =

3 C

p=o.1

(-l)p(a*a)

Res,, z;’Y(Bpu,z2 + zO)Y(u,Z ~ ) ~ - ~ O ( ~ -/ l)p~i/2/~:/2). ~~~)B(( (9.3.53) (Compare with Remark 8.6.4.) As in Corollaries 8.6.5-8.6.7, 9.2.5 and 9.2.8, we can express Theorem 9.3.1 in terms of component operators as follows:

Corollary 9.3.5: In the notation of Theorem 9.3.1, let

m, n E @, Then as operators on VT,

(finite sum).(On the right-hand side, (8pu)iis an operator on the untwisted space VL.)

9.3. Commutators of General Twisted Vertex Operators

285

Hence: Corollary 9.3.6:

In the same notation,

In the same notation, suppose that the element u is homogeneous. Then as operators on VT, Corollary 9.3.7:

(finite sum). (On the right-hand side, x o p u ( iis ) an operator on V' .) We call attention again to the similarity between these results and the' corresponding results in the untwisted case-Theorem 8.6.1 and Corollaries 8.6.5-8.6.7. It is important to note that this similarity increases when we assume that the element u is fixed by the action of 9 on V' . With the help of (9.2.38), we have: Corollary 9.3.8:

suppose that

In the notation of Theorem 9.3.1 and Corollary 9.3.5, (ti,6) E 2z

(9.3.56)

+ eu.

(9.3.57)

and set w=u

Then Y(w,z)=

c

wnz+l,

n E Z

(9.3.58)

that is, Y(w,z ) involves only integral powers of z, and [Y(w,z , ) , Y(u,z,)]

= Res,,

z;'Y(YZ(w, zo)u, z2)e-Z~(d'dz1)6(zl/z2).

(9.3.59)

Moreover, i f m E Z, then (9.3.60)

and

if in addition

w is homogeneous,

286

General Theory of Twisted Vertex Operators

Corollary 9.3.9: In the same notation,

and

Remark 9.3.20: The operators X’(a, z) discussed in (7.4.18)-(7.4.28) (or rather their variants Y +(a,z) = Y(a,z) + Y(ea,z ) ) are examples of operators Y(w,z) discussed in Corollary 9.3.8 (assuming that (a, a ) E 2Z). In particular, the commutation relation (7.4.28) follows from the corollary.

[cf. (7.3.15)-(7.3.16)],

(9.3.65)

9.4. The Virasoro Algebra: Twisted Construction Revisited

287

(9.3.71) where ~ ~ ( - ~ ) ~ (+- ,n) ) (is1given by (9.2.66), (9.2.77) and (9.2.78).

Remark 9.3.11: In the more general setting of Remark 9.1.5 (see also Remark 9.2.9), the only necessary changes in the present section are as follows: if the element Q E entering into the element u in (9.3.2) satisfies (9.1.46), then the sign is to be changed in (9.3.42), (9.3.45) and (9.3.47)(9.3.50), and correspondingly, 13is to be replaced by - 0 in Theorem 9.3.1 [(9.3.3) and (9.3.4)], Corollary 9.3.3, Remark 9.3.4 (9.3.53) and Corollaries 9.3.5-9.3.9.

e

Remark 9.3.22: The proof of Theorem 9.3.1 shows more generally that Y(U, zl)Y(v, z2) - (- l)(u*6)c(c7, 6)Y(v, z2)Y(u, z , ) is given by the right-hand side of (9.3.3) even if c(a, 6) # (-l)(u*6); similarly for (9.3.4) and the corollaries (cf. Remark 8.6.9).

9.4. The Virasoro Algebra: Twisted Construction Revisited We recall the structure of the Virasoro algebra o, given by (1.9.9) or (8.7.1). In Section 1.9 we constructed u in both untwisted and twisted settings, and in Section 8.7 we reinterpreted the first case in terms of general untwisted vertex operators. Here we do the same for the twisted case, having already noted in Remark 7.3.3 that u acts on our space VT. In the untwisted construction, u was realized using the canonical quadratic element o of V' defined in (8.7.2); see (8.7.4)-(8.7.10). In the present context, set I

U z ) = &+&)

=

Y ( o , z )=

i= 1

shj(z)hi(z)s + #dimb)z-'

[see (9.2.52)], the optional subscript Z

+

(9.4.1)

1/2 designating the twisted

General Theory of Twisted Vertex Operators

288

construction as usual. Also, for n E t Z set I

L(n) = x,(n) = I

=

t1

i= 1

hi hi(n) shj(n - k)hi(k)s + &&,dimb

i= 1 k ~Z+1/2

(9.4.2)

[recall (9.2.66) and (9.2.82)]. Then

L(n) = 0 unless n E B . Also,

(9.4.3) (9.4.4)

by (9.2.66). The operators L(n) agree with those in (1.9.23) for 2 = Z + 1/2 (recall Remark 3.3.1). We already know the relation between L(0) and the degree operator d on the space VF [recall (7.3.9), (7.3.10)]: L(0) = - d

+ kdimb

(9.4.5)

[see (1.9.49), (1.9.50) and (1.9.54)]. For a homogeneous element u E VT,

L(0)u = (wt u)u.

(9.4.6)

Remark 9.4.1: The scalar &dimQ in (9.4.2), introduced in (1.9.24) to “make the algebra fit together,” is now understood from avery general viewpoint. It comes from the action of the natural operator exp(A,) on the natural quadratic element w [recall (9.2.51)]. In particular, the operator exp(A,) determines the correct difference in the degree-shifts (1.9.51) and (1.9.53) between the untwisted and twisted cases. Since the scalar & dim b is precisely the right one to give us the commutation relations (8.7.1) (as we shall see again below), exp(A,) can be thought of as determining these relations. The best way to use the general theory to show that the L(n) satisfy (8.7.1) is to establish some general principles first rather than to invoke (9.3.70) [cf. (8.7.7)]. By analogy with (8.7.24), we show: Proposition 9.4.2:

For u E VL, d Y(L,( - l)u, z) = - Y(u,z), dz

(9.4.7)

where Lz(-l) is the operator L(-1) on V,, not V;. (We shall use the subscript Z to indicate operators on the untwisted space.)

289

9.4. The Virasoro Algebra: Twisted Construction Revisited

Proof: First note that a special case of this already follows from (8.7.25), (9.1.22) and (9.2.46):

To establish (9.4.7) in general, it is most natural to refer to the following consequence of (8.7.24): er&z( - 1) v =

Ym(v,20) * 41)

(9.4.9)

(see Proposition 8.7.4). On the other hand, as operators on V',

Y(YZ(U,zo) 41), 2 ) = Y(v,z

+ 20)

(9.4.10)

by (9.3.51) and the first paragraph of the proof of Theorem 9.3.1 [cf. (9.3.52)]. Hence from (8.3.3) and (9.4.9),

ezo(d/dz)y(u,z) = Y ( V , Z + zo) = Y(ez&z(-') v , z), and (9.4.7) follows by extracting the coefficient of zo.

(9.4.1 1)

I

Using (9.4.6) and (9.4.7) it is now easy to continue the analogy with the basic results of Section 8.7. By Corollary 9.3.8, we have just as in (8.7.33) the formula [L(zl),Y(v,z2)1 = Res,, z;'Y(Lm(zo)v,z2)e-z0(a/az1)6(z1/z2)

+ Res,,z;'

Y(Lz(n)v,z2)z0-,-2 e -zoca/az 45(z1/z2)

(9.4.12)

fl>O

for all v E V,. Taking u = o gives us precisely the formula (8.7.7) for [L(z,),L(z2)].Hence we also have (8.7.8) and (8.7.9). That is,

[L(m), L(n)] = (rn - n)L(rn + n) + $rn3 - rn)(dim b)6,+,,o (9.4.13) for m,n E Z, and we obtain as in Proposition 8.7.1 the result of Theorem 1.9.6 for Z = Z + 1/2: Proposition 9.4.3: The operators L(n) provide a representation of the Virasoro algebra 0 [see (8.7.1)] on VF with

L, c

-

L(n) for n E Z dimb.

(9.4.14)

290

General Theory of Twisted Vertex Operators

To continue with other consequences of (9.4.6), (9.4.7) and (9.4.12), formulas (8.7.1 1)-(8.7.13) hold in the present setting. In particular,

d

+ (rn + 1)z?)a(z2)

for a E fi,

m E Z. (9.4.15)

Also, (8.7.14)-(8.7.16) become formulas involving Y ( ~ r ( - l ) ~z2) , in place of 8 ~ Y ( Z 2 ) ~ 8so , that for example

[wo, Y M - 112, -

2211

( z ? + l L + 2(m dz2

1 + 1)z?)Y(a(-1)2,z2) + -(m3 6

- m ) ( a ,a)z?-2 (9.4.16)

for a E b and m E Z.As in (8.7.17)-(8.7.19),

for a E and m E Z. By analogy with Proposition 8.7.5, we find (using Corollaries 9.3.3 and 9.3.6): Proposition 9.4.4:

For u E V,,

d [ U - l ) , Y(u, z)l = Y(L,(-l)v, z ) = dz Y(u, z)

(9.4.18)

[,!,(-I), u,] = (Lz(-l)u),, for n E +h.

(9.4.19)

Just as in Proposition 8.7.6, we can iterate (9.4.18) to obtain [see also (9.4.1 l)]:

From (9.4.7) and (9.4.12) we have the analogue of one direction of Proposition 8.7.7:

9.4. The Virasoro Algebra: Twisted Construction Revisited

291

Proposition 9.4.6: If v E Vr is a lowest weight vector for D with weight h, then

or [L(m),x,(n)]= (hm - m - n)x,(m

+ n)

for m E Z, n

E

iB. (9.4.23)

Remark 9.4.7: The step missing in the converse argument would say that Y(L(O)u,z) = hY(u,z ) would imply that L(0)u = hu. The results from (8.7.40)through Proposition 8.7.9now hold, except for the converse assertion in Proposition 8.7.9. For instance: If u E V, is a lowest weight vector for D with weight h, then

for n E Z, with z1 as in (8.7.43)[cf. (8.7.44)]. Proposition 9.4.8: If u E V, is a lowest weight vector for span ( L - , , L o , L , ) with weight h [see (8.7.47)], then

[L(n),Y(u,z)] = (z""

f o r n = 0, h l , with

(: :)

=

d

+ h(n + l)zn

Y(u,z )

(a - C Z ~ ) ~ " Y ( U2,, )

(9.4.25)

(9.4.26)

and z1 as in (8.7.50)-(8.7.52).

Remark 9.4.9: Themoregeneralsetting of Remarks9.1.5,9.2.9and9.3.11 entails no changes in the present section.

292

General Theory of Twisted Vertex Operators

9.5. The Jacobi Identity and Cross-Brackets: Twisted Case Here we shall develop twisted analogues of the results of Sections 8.8 and 8.9. For convenience, we shall assume that the lattice L is even and that (9.3.1) holds for all a, b E Then in particular, (9.3.4) holds. (See also the more general assertions in Theorem 9.5.3.) As above, we use the notation Y(u,z ) for the twisted operators Yz+I/2(u, z) and the notation Yz(u,z) for the untwisted operators. Recall that

e.

&

Y ( * , z ) is a linear map

+

(End V:)[[z'l2,

z - " ~ ] ] (9.5.1)

[see (9.2.28)],

Y(z(l),2 ) = 1

(9.5.2)

[see (9.2.31)] and = Y(ez&z(-l)u,z ) = Y(U,z e@(-')Y(u, z)e-z&(L(-l)

+ zo)

(9.5.3)

for u E VL [see (9.4.20)]. If u E VL [recall the notation (8.9.25)] then

Y(u,z) =

c U,z-n-',

(9.5.4)

nsh

i.e., Y(u, z) involves only integral powers of z , and if Bu = - u then

Y(u,z) =

c

(9.5.5)

U,Z+l,

n E 2+1/2

as we see from (9.2.37) [cf. (9.3.58)]. From (8.8.7) and (9.5.3) we obtain:

Proposition 9.5.1:

For u, u E VL,

Remark 9.5.2: Combining this formally with (9.3.52), we find a formal "commutativity" relation for operators on V: Y(U,2 2 + zo)Y(u, 2 2 ) = Y(u, z2)Y(u,22

+ 20)

(9.5.7)

(cf. Remark 8.8.4). Now for u,u E V, and n E Z define the following alternating or commutative generalization of Lie bracket (the case n = 0) as in (8.8.13):

[Y(u,21)

Xn

Y ( u ,22)1

= (Zl - Z2)"Y(U,Zl)Y(u, ~

2) (-22

+ z ~ ) " Y ( uzZ)Y(U, , ~ 1 ) .

(9.5.8)

9.5. The Jacobi Identity and Cross-Brackets: Twisted Case

293

As before we call [ * x,.] the cross-bracket. Also define the expansion coefficients [u x I v],, and [xux I x v ] ( mn) , as in (8.8.15) and (8.8.16). Then (8.8.17)-(8.8.24) hold here. Also form the generating function

[Y(u,2 , ) xz, Y(u, 22)I =

c [Y(u,

2,)

n G H

xn Y(v,z ~ ) I z ~ ~ - ' . (9.5.9)

Then (8.8.26)-(8.8.28) remain valid here. In particular,

(9.5.10) We can now state the Jacobi identity for twisted operators:

Theorem 9.5.3: For u, v E V, (or more generally, fully general setting of Theorem 9.3.I ) ,

=

+

c

Z;'d(c-lY

(2, - 20)1/2

p=o.1

Equivalently, for n [Y(u,21)

Xn

E

z;/2

if (a, a ) E 2 2

y(yp(epu, z0)v, z2).

in the

(9.5.1 1)

72,

Y(v, 2211

= p=O.I

Res,,z~z;1Y(Yz(8Pu, zo)v, z2)e-Z0(a'azi)6(( -l ) p ~ ~ / ~ / z ~ ' ~ ) (9.5.12)

(In the greater generality of Theorem 9.3.I and Remarks 9.3.I I and 9.3.12, (is to be inserted after the summation sign; if a E J? in (9.3.2)

satisfies (9.1.46), then 8 is to be replaced by -8; and (-l)(o*b)c(i?,6) is to be inserted in the second term on the left in (9.5.I I).)

Proof: The equivalence is clear. To prove (9.5.12), we use the proof of

Theorem 9.3.1 except for the following modifications: By (9.3.34), the coefficient of P in (zl - z2)"Y(A,zl)Y(B,z2)is YP(ZI

9

z2)(21 - 22)

and similarly, the coefficient of P in ( -z2 Yp(21, 22)(-22

-(N-n)

+ zl)"Y(B,z2)Y(A,z , ) is

+ Z1)-(N-").

294

General Theory of Twisted Vertex Operators

Thus the coefficient of P in [Y(A,z l ) x, Y(B,z2)] is the coefficient of zf-"-' in (9.3.36), and it follows that [Y(A,z l ) x, Y(B,z2)] is the coefficient of zgn-' instead of z'; in the expression on the right-hand side in formula (9.3.4). I If we restrict u to lie in V:, the Jacobi identity looks exactly like (8.8.29):

Corollary 9.5.4:

For u E V: and u E V',

(9.5.13)

Remark 9.5.5: By taking Res,, of the Jacobi identity, we of course recover formula (9.3.4). As in Remarks 8.8.11 and 8.8.12, we can take Res,, and Res,, instead. If u E V: we find that Y(Yz(u,zo)u, z2) is given by (8.8.31) and that Y(Yz(u,zo)u, z1 - zo) is given by (8.8.32). Remark 9.5.6: Formula (9.5.13) can be rewritten as in Remark 8.8.16 [see (8.8.38)-(8.8.40)].

Remark 9.5.7: Formula (9.5.13) can be expressed in component forms exactly as in Corollaries 8.8.17-8.8.19. Remark 9.5.8: The vertex operators Yz(u,z) and Yz+1,2(v,z) that we have been studying are all parametrized by u E V,. It is possible to develop a theory of operators Y(u,z) for u E V[ by starting with (8.8.7) to define the action of such operators on V'. We shall not pursue this direction in the present work; cf. [Frenkel-Huang-Lepowsky] . Remark 9.5.9: For u, u E V i of weight 1, the discussion of (8.9.2)(8.9.12) holds here, with the operators acting on V[. Note that 8 is an automorphism of the Lie algebra g of (8.9.6) and that the form ( * , - ) of (8.9.8) is &invariant. Let g(o) denote the Lie subalgebra of &fixed elements of g: B(0)

=g

n v:.

(9.5.14)

Then VF is a g(o,-module under the action u

-

uo,

(9.5.15)

295

9.5. The Jacobi Identity and Cross-Brackets: Twisted Case

and VE is in fact a (g,))^-module, as we already knew from Theorem 7.4.10. Of course, we could also recover the full action of i[O] on VF given in Theorem 7.4.10. Now suppose that

u, v E

v - , wt u = wt v = 2.

(9.5.16)

Then the discussion at the end of Section 8.9 applies, with the operators acting on VE. Suppose that L is positive definite and that L2 = 0, as in (8.9.30). Recall that the space 1 consisting of the elements of V: of weight 2 (8.9.32) is a commutative nonassociative algebra under the product u x v = u1 v and that the form ( u , v ) = u3 v is symmetric and associative. As in Proposition 8.9.5 we have a graded representation of the commutative affinization f on VT by cross-bracket:

-

Theorem 9.5.10:

The space VF is a graded f-module under the action R:

defined by

n: u 0 t" R:

e

-

-

+

End VT

xu@) for

(9.5.17)

u E f, n E Z

(9.5.18)

1.

Let p be the space of elements of VF of weight 2:

P

= (~LT(dirntj)/24-2

(9.5.19)

Then the identity element 30 of € (recall Remarks 8.9.2 and 8.9.4) acts as the identity operator on p , by (9.4.6): (+o),* w = w

for

w E p.

(9.5.20)

Unless dim b is chosen carefully, p = 0 [recall (1.9.57)]. Let us assume that dim4 = 24,

(9.5.21)

the case that we shall be most interested in. Then

P=bOT,

(9.5.22)

where we make the identification Ij =

4

@

t-'/2

(9.5.23)

296

General Theory of Twisted Vertex Operators

in the notation (1.7.12), (1.7.15) [recall (7.3.8)]. With the notation of Remark 8.9.7, we can express the action of € on p explicitly as follows:

- (h 0 7 ) = +(h - 2(0, h ) a ) 0 a '5 k2)1- (h 0 7 ) = ((8,h)g + +
(x,'),

(9.5.24) (9.5.25)

for a e L 4 , g , h E b , 7 e T, using (7.3.4), (7.4.14), (9.1.15), (9.2.50) and (9.2.53).

Remark 9.5.21: Later, for the case in which L is the Leech lattice and where L and T are chosen specially, we shall make f @ p a commutative nonassociative algebra with a nonsingular associative symmetric form-the Griess algebra. This has mostly been accomplished already. We shall also extend Theorems 8.9.5 and 9.5.10 by constructing a representation of the commutative affinization of this algebra by cross-bracket on V: @ (VLTe for a suitable choice of &the moonshine module [recall the notation (9.2.94)]. The Monster will act compatibly on both the algebra and the module.