===== PDFPAGE 245 =====
8.4. Commutators of Untwisted Vertex Operators 191 
and by abuse of notation write 
This notation conflicts with the original notation a(z) introduced in (4.2. I), 
but no confusion should arise as the reader becomes aware of the signifi- 
cance of the change. Note that 
and we have a new way of writing an important ingredient of the vertex 
operator. Also introduce the notation change [cf. (4.2. l)] 
a(z)+ = C a(n)z-"-' 
a(z)- = 1 a(n)z-"-', 
ntO 
(8.4.10) 
n<O 
so that 
a(z) = a(z)+ + a(z)- (8.4.11) 
[cf. (4.2.2)]. 
For j E N, note that 
(dy - a(z) = (d>i - a(z) + + (dy - 421- 
dz dz dz (8.4.12) 
and that (d/dzya(z) * involves only negative (respectively, nonnegative) 
powers of z. 
Our new notion of normal ordering will be denoted by open colons. In 
place of (7.1 S4)-(7.1.56), we set 
8a(z)as = xacu(z)x = a&) (8.4.13) 
sa(O)as = saa(O)s = aa(0) (8.4.14) 
&a8 = 8aza8 = aza (8.4.15) 
for (Y E 4. a E e. That is, the operators a(n) for all n E h including n = 0 
are to be placed to the right of the operator a before the multiplication is 
performed, in contrast with the convention motivated in Remark 4.2.1. For 
an expression built entirely from operators a(n) and series a(z), the new 
normal ordered product shall agree with the old one (see the beginning of 
===== PDFPAGE 246 =====
192 General Theory of Untwisted Vertex Operators 
Section 4.2). For instance, 
for al, a2 E 6. 
Recall from Remark 7.1.3 the notion of weight of a homogeneous 
element of V,. For a E e, our new vertex operator is the normal ordered 
expression 
Y(a, Z) = Yz(a, Z) = gel(P(z)-d(O)z-’)aZp8 
- - 8 el(d(Z) -d(O)Z- I) azd 
- - X(a, Z)Z-(P,d)/Z = x(~ I Z)z-WfdU) (8.4.17) 
[cf. (8.4.2)]. In the context of the motivational Remark 4.2.1, formula 
(4.2.12) would be replaced by the simple formula 
Y(a, Z) = xe1a(z)s. (8.4.18) 
Formulas (7.157)-(7.1.60) have the following analogues: Define 
Sa!(Z)Y(a, Z)8 = Sy(a, Z)a(Z)S = a!(z)-Y(a, Z) + Y(a, z)a(z)+ (8.4.19) 
8 Y(a, Z l)y( b, Z2) 8 = 8 el(u(z I)-d(o)ZT ’) +1(6(zd-6(o)z~ 8 ab? 12 2 (8.4.20) 
for a! E 6 and a, b E e. Then 
8y(a, Zl)Y(b, 2218 = C(a, 6)8y(b, Zz)Y(U, Z1)8 
Y(a, Zi)Y(b, Z2) = Sy(a, Zl)Y(b, Z2)8Z~””(1 - Z~/Z~)‘”~’ 
(8.4.21) 
(8.4.22) 
[recall the notation c from (7.1.21)]. There are obvious analogues of 
formulas (7.1.61)-(7.1.64) too. For instance, 
d - y(4, Z) = 8n(Z)Y(U, Z)8. 
dz (8.4.23) 
Note that 
Y(a,z) = C Xa(n)Z-n-(d*d)/2 = C xa(n - 2 n))z-“. (8.4.24) 
ne4 noQ 
Remark 8.4.2: 
symmetrical form: 
Formula (8.4.22) can be written in the following more 
Y(a, Zl)Y(b, Z2) = 8Y(a, Zl)Y(b, Z2)8(Z1 - ZZ)‘~’~), (8.4.25) 
===== PDFPAGE 247 =====
8.4. Commutators of Untwisted Vertex Operators 193 
where (zl - z2)(d*6) is understood as the binomial expansion in non- 
negative integral powers of z2 : 
if (a, 6) I 0 fo 
(8.4.28) 
===== PDFPAGE 248 =====
194 General Theory of Untwisted Vertex Operators 
and this is the coefficient of yN in 
22 - I e -Y(a/azl)~(Z1/Z2), (8.4.29) 
for y as in (8.2.9). It follows that [Y(a, zl), Y(b, z2)] is the coefficient of yN 
in 
22-l 8 Y(U, Zl) Y(b, 22) 8 e -y(J/az')6(Z 1/22) (8.4.30) 
(whether or not N 2 0). 
applies to (8.4.30), which equals 
By the integrality assumption on (a, L) and (6, L), Proposition 8.2.2 
Z2-18Y(a, 22 + J')Y(b, Z2)8e-Y(a/az')B(Z~/Z2). (8.4.31) 
Here we have also used the change-of-variable formula (8.3.11). To 
compute (8.4.31), we use (8.3.3). From (8.4.20), 