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208 General Theory of Untwisted Vertex Operators 
Continuing with the proof, we adopt the obvious definition of the normal 
ordered product of Y(A, zl) and Y(B, z2) and we have 
8Y(A9 Zl)Y(B, Z2)8 = (-l)'d'6'8Y(B, z~)Y(A, 21)s (8.6.11) 
using (8.6.1) [cf. (8.4.21)]. Also, from (8.4.25) and (8.6.9), 
Y(A, Zl)Y(B, z2) 
= SY(A9 tl)Y(B, Z2)8 n (21 - ~2 + Wi - Xj)('i*'J), (8.6.12) 
lsisk 
lsjs/ 
where n(zl - z2 + wi - xj)("iP6j) is understood as an expansion in non- 
negative integral powers of z2, wi and xj; it may be computed using the 
binomial expansion of 
(Ui. bj) (~1 - (22 - wi + xj)) 
or two successive binomial expansions of 
((Zl - 22) + (Wi - Xj))('iJj). 
Fix a monomial 
P = n w,'i n xp (ri,sj 1 0) (8.6.13) 
in the wi and xi. We may and do choose N 2 0 so large that the coefficient 
of P and of each monomial of lower total degree than P in 
lsisk lsjsl 
N FN = (Zl - 22) n (zl - ~2 + wi - ~j)('i'~j) (8.6.14) 
lsisk 
1 sjsl 
is a polynomial in z1 - z2. Let Yp(zI, z2) denote the coefficient of P in 
y(A, Zi)Y(B, Z2)(Zl - ZZ)~ = SY(A, Zl)Y(B, 22)8F~. (8.6.15) 
Since 
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8.6. Commutators of General Vertex Operators 209 
(8.6.16) 
lim Wp(zl, z2) exists. 
z1-22 
Moreover, the coefficient of P in Y(A, zl)Y(B, z2) is 
where (zl - z2)-N is to be expanded in nonnegative integral powers of z2. 
Similarly, reversing the roles of A and B and of z1 and z2, and taking into 
account (8.6.1 l), we see that 
Y(B, Z2)y(A, Zl) 8Y(B, Z2)y(A, Z1)8 n (Z2 - Z1 + Xj - Wi)(ai'6j) 
lsisk 
1 sjsl 
= 8y(A, Zl)Y(B, 22)8(-l)'a'.6' n (z2 - z1 + xj - Wi)(PiJj) 
lsisk 
Isjsl 
(8.6.18) 
where n(z2 - z1 + xj - wi)(ais6j) is now to be expanded in nonnegative 
integral powers of zI , wi and xj. Moreover, the coefficient of P and of each 
monomial of lower total degree than P in 
(zl - ~~)~(-1)(~*~) n (z, - z1 + xj - wi)(ai*bj) 
is a polynomial in zI - z2 which agrees with the same for FN. Thus the 
coefficient of P in Y(B, z2)Y(A, z,) is 
Yp(zI, z2)(z, - z2)-N = Yp(zl, Z,)(-Z~~(I - 3) , (8.6.19) 
where (z, - z~)-~ is to be expanded in nonnegative integral powers of zl. 
-N 
z2 
It follows that the coefficient of P in [Y(A, zl), Y(B, z2)] is 
(8.6.20) 
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210 General Theory of Untwisted Vertex Operators 
[cf. (8.4.26)]. As in (8.4.27)-(8.4.30) we find that (8.6.20) is the coefficient 
of in 
z2- 'yP(z1 , z2)e-Z~(a/az~)~(z1 /z2). (8.6.21) 
By (8.6.16), we may apply Proposition 8.3.12, and (8.6.21) becomes 
22- 'yp(z2 + zo , z2)e-zo(a/azl)((z1 /z,)'s(z /z2)) (8.6.22) 
[cf. (8.4.31)]. But by (8.6.14) and (8.6.19, Yp(z2 + zo, z2) is the coefficient 
of P in 
8Y(A, 22 + Zo)Y(B, Z2)8Z: n (Zo + Wi - Xj)(dt'6~), (8.6.23) 
Isisk 
1 sjsl 
which is understood as an expansion in nonnegative integral powers of wi 
and x,; 8Y(A, 22 + zo)Y(B, Z2)8 is also an expansion in nonnegative integral 
powers of zo. We conclude that the coefficient of P in [Y(A, z,), Y(B, z2)] is 
the coefficient of z,'P in 
z2-I 8 Y(A , Z2 20) Y(B, Zz) 8 
* n (zo + wi - (8.6.24) 
and therefore that [Y(A, zl), Y(B, z2)] is the coefficient of z;' in (8.6.24). 
This last assertion is independent of P and N. 
lsisk 
lsjsl 
On the other hand, 
y(A, zolB = 8Y(al, Zo + w1) "' Y(a,, z0 + wk) 
' Y(b19 Xi) **. Y(b1, X/)8l(l) n (20 + Wi - Xj)('f*6J), (8.6.25) 
lsisk 
lsjsl 
using (8.6.6) and (8.6.9). Thus by (8.6.9) with A replaced by Y(A, zo)B, 
W(A, zdB, 22) = sY(a1, z2 + 20 + w1) *** Y(ak I z2 + zo + w,) 
In particular, 
y(y(A,Zo)B,Z2) = 8Y(A,Z2 + Zo)Y(B, Z2)8 n 
Isisk 
lsjsl 
(20 + wi - x.)('i'6J) J 
(8.6.27) 
where of course (zo + wi - xj)(d~s6j) represents an expansion in nonnegative 
integral powers of wi and xi. This proves the Theorem. I 
===== PDFPAGE 265 =====
8.6. Commutators of General Vertex Operators 211 
Using Remark 8.5.3 and the fact that the residue of a derivative is 0, we 
find: 
Corollary 8.6.3: In the notation of Theorem 8.6.1, if (a, L) c Z, then 
[u,, Y(v, 211 = Y(u0 * u, 2). 
Remark 8.6.4: A formal combination of (8.6.12) and (8.6.27) gives us 
[even without the assumption (8.6. l)] a second formal but suggestive 
generalization of Remark 8.5.2 (cf. the last remark): 
Y(Y(U, Z0)h z2) = Y(u, 22 + ZO)Y(V, 22) (8.6.28) 
for all u, u E VL. In particular, for u and u as in Theorem 8.6.1, 
Mu, Zl), Y(hZ2)l 
= Res,, z;'Y(u, z2 + zo)Y(u, Z~)~~~~~~'~~~~((Z~/Z~)~~(Z~/~~)). (8.6.29) 
This formally generalizes Remark 8.4.3. If we write z1 for z2 + zo in (8.6.28) 
and use the expansion (8.5.15), we obtain: 
Y(u, 21) Y(u, 22) = Y(Y(u, 21 - z2)u, 22) = c Y(u, * u, 22)(21 - Z2)-n-1, 
nEZ 
assuming for convenience that (a, L) C Z. This is an example of an 
"operator product expansion" in the sense of quantum field theory. 
Although not strictly rigorous in the present setting, it is very useful. For 
example, from the right-hand side we can read off the commutator formula 
(8.6.3) simply by replacing (zl - z2)-"-' by ST'(( - 1)"/n!)(~/~z,)"~(z,/z2) 
for n 1 0. Here it is illuminating to observe that for n 2 0, 
as rational functions [cf. (8.4.27)-(8.4.29)]. Note that only the finitely many 
"singular terms" in the operator product expansion contribute to the com- 
mutator. In Section 8.10, we shall interpret these and related matters in a 
deeper way. 
Using the component operators u, defined in (8.5.15), we have 
Y(Y(u, zo)u, z2) = c (Ui * u)jz;i-lz;j-l (8.6.30) 
i,j E 0 
for u, u E V', as in (8.5.24). Thus Corollary 8.5.4 generalizes to: 
===== PDFPAGE 266 =====
212 General Theory of Untwisted Vertex Operators 
Corollary 8.6.5: In the notation of Theorem 8.6.1, if (a, L) c Z then 
(8.6.31) 
(finite sum) for m E Z, n E Q. More generally, let M be a subset of L such 
that 
(a,M) C Z + m, 
for fixed m, E Q. (If L is integral, we may take m, = 0.) For 
mEZ-mm,, nEQ, 
formula (8.6.31) holds on the space V, [recall the notation (7.1.67)]. 
Hence: 
Corollary 8.6.6: In the same notation, if (a, L) C Z, then 
bo, 41 = (uo * u), * 
In particular, if L is integral and (8.6.1) holds for all a, b E e, then the 
operators uo for u E V, form a Lie algebra: 
[u,, uo] = (u, * u), for u, u E V,. 
We also generalize Corollary 8.5.6, using the component operators xu@) 
defined in (8.5.28). Changing indices in Corollary 8.6.5 and keeping in 
mind (8.5.29) and (8.5.30), we find: 
Corollary 8.6.7: In the notation of Theorem 8.6.1 and Corollary 8.6.5, 
suppose that the element u is homogeneous, and let 
m E Z - m, - +(a, a) = Z - mo - wt u, n E Q. 
Then 
on V, (finite sum). If the lattice L is even, we take m, i E Z. 
We list some special cases of Theorem 8.6.1 and Corollaries 8.6.5 and 
8.6.7. Any of these formulas could be proved more directly (cf. Remark 
8.4.6 and Section 8.7 below). We start with the case 
u = a(- 1) = a(-1) - i(1) = a(-I) @ r(1) for a E 8. (8.6.33) 