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190 General Theory of Untwisted Vertex Operators 
8.4. Commutators of Untwisted Vertex Operators 
We have set up the machinery for computing commutators of vertex 
operators in great generality. We work in the setting of Section 7.1. In 
particular, L is a nondegenerate lattice; = L OZ 5; S(6;) is viewed as the 
&module M(1); (e, -) is a central extension of L by (K I K~ = 1) with 
commutator map co; x is a faithful character of (K) such that X(K) = o; 
F(L) = Ind?,, 5x = 5[L] (linearly); V, = S(@ 0 5ILI; 
for CY E ij; and 
(8.4.1) 
(8.4.2) 
(8.4.3) 
for a E L. 
Our main approach to commutators will be based on the additive trans- 
formation (8.3.3). It turns out to be natural and convenient to emphasize a 
new differentiation operator, a new notion of normal ordering and a new 
vertex operator. In place of the operator D = D,, we shall typically use 
d/dz. For a vector space V, we can define an operator 
on the space 
by the obvious formula 
(8.4.4) 
(8.4.5) 
(8.4.6) 
[cf. (2.2.20)-(2.2.22)]. 
We shall now make an important notation change which shall remain in 
CY,,&) = a(n)z-"-' = Z-'CY(Z) (=z-'aOld(z)) (8.4.7) 
force throughout the rest of this work. For CY E b, set 
neZ 
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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 
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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) 
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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) 
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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), 
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8.4. Commutators of Untwisted Vertex Operators 195 
which we may write, using normal ordering notation, as 
This argument is nothing but an elaborate form of (8.3.21). The expressions 
such as (z2 + y)" and (1 + yz;')" have unambiguous meanings as expan- 
sions in nonnegative integral powers of y, and all the necessary relations 
involving these expressions are easily proved. (Also recall that ;;6 = si + 6.) 
Although the variable y has played a special role, there is no harm in 
replacing y by a "generic variable" zo whose negative and nonintegral 
powers are allowed. We shall use the following residue notation: For a 
series 
f(Z0) = 1 vnzt 
"€F 
with coefficients in a vector space, we write 
Res,, f(zo) = u-~. (8.4.33) 
We have computed commutators of untwisted vertex operators in the 
following general form: 
Theorem 8.4.2: Let a, b E e and suppose that 
(si,L) c z, (6,L) c z, 
c(a, 6) = (-l)("J) 
[recall (7.1.21)]. Then 
(8.4.34) 
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196 General Theory of Untwisted Vertex Operators 
Moreover, 
Remark 8.4.3: A formal application of (8.4.25) shows that in a formal 
sense which we shall not rigorize here, 
[Va, Zlh Y(b, z2)l 
= Res,, z;’Y(a, z2 + zo)Y(b, z2)e-z~‘d’J2~)B(z1/z2). (8.4.37) 
Note the absence of normal ordering and of the quantity (a, 6). 
U! ing the polynomials Pk defined in Corollary 8.3.5, we find the follow- 
ing more “concrete” formulation of Theorem 8.4.2: 
(8.4.38) 
using prime notation for derivatives. That is, 
zi), Y(b, ~2)l 
Recalling the values of Pk tabulated in Remark 8.3.6, we find that the 
coefficients of the first few powers of zo in sY(a, z2 + zo)Y(b, Z2)8 are: 
2,”: Y(ab, z2) 
2;: 8d(Z2)Y(Ub, Z2)8 
(8.4.40) 
Zi : i8(a(Z2)2 + a(Z2)’) Y(Ub, 22)s 
2: : &8(6(22)3 + 31t(22)a(22)‘ + n(z2)”) Y(ab, 22)8. 
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8.4. Commutators of Untwisted Vertex Operators 197 
Thus from (8.4.39) we see that [Y(a, zl), Y(b, z2)], which we denote by Y, 
has the following values: If (a, 6) 2 0, 
Y=O. (8.4.41) 
If (a,6) = -1, 
If (li,6> = -2, 
Y = z,'Y(ab, z2)6(21/22). (8.4.42) 
Remark 8.4.5: A little checking using the notation-changes (8.4.7) and 
(8.4.17) shows that we have recovered and hence generalized Theorem 7.2.1 
for the case of an integral lattice. 
Remark 8.4.6: These formulas could have been derived from formula 
(8.2.18) (with T, = d/azl) for low values of n, as in the proof of Theorem 
7.2.1. 
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198 General Theory of Untwisted Vertex Operators 
8.5. General Vertex Operators 
The similarity between the exponential expression in (8.4.36) and the 
expression for the vertex operator (cf. Remark 8.3.7) can be exploited in an 
interesting way: We shall define "general vertex operators" parametrized 
by the vector space yL in such a way that the vertex operator Y(a, z) for 
a E L corresponds to [(a) = a 0 1 E F(L] [recall (7.1.19)]. Observe in fact 
that the correspondence r(a) - Y(a, z) for a E e extends uniquely to a well- 
defined linear map 
(8.5.1) 
by (7.1.20) and (7.1.50). [The only linear relations among the elements r(a) 
are satisfied by the operators Y(a,z).] Note that we are allowing the 
notation 
Y(z(a), z) = ~(a, z) for a E L. (8.5.2) 
For instance, 
Y(r(l),z) = Y(1,z) = 1. 
[recall the optional notation Y, from (8.4.17)] as follows: 
(8.5.3) 
(8.5.4) 
(8.5.5) 
the normal ordered product being defined in an obvious way extending the 
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8.5. General Vertex Operators 199 
rules already given. The result is a well-defined linear map 
VL = S(&) 0 FILI -, (End VL)M 
(8.5.6) 
extending (8.5.1). The expressions Y(u, z) are called general (untwisted) 
vertex operators (or for brevity, vertex operators). Note that if 
n, = ... = nk = 1 and a = 1, 
then 
Y(u, 2) = xcr,(z) *** a&): (8.5.7) 
[cf. (3.3.8), (3.3.9)]. 
It is easy to check from the definition (8.5.5) that for all u E V,, 
lim Y(u, z)r(l) = u. 
z-0 
(8.5.8) 
That is, Y(u,z) "creates the vector u." Here of course the limit has 
algebraic meaning because Y( u, z)r( 1) involves only nonnegative integral 
powers of z. 
Let us replace z by z2 in (8.5.6): 
and then extend this linear map canonically to V,(z,): 
That is, if 
(u, E V,) then 
Now let a, b E e and consider 
(8.5.9) 
(8.5.10) 
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200 General Theory of Untwisted Vertex Operators 
in VL[zo). Then we have 
= 8Y(a, 22 + Zo)Y(b, Z2)8Zhp'6) (8.5.12) 
[see (8.4.36)], and Theorem 8.4.2 asserts: 
Corollary 8.5.1: In the notation of Theorem 8.4.2, 
[Y(a, ZI), Y(b, z2)l 
Remark 8.5.2: It is interesting to compare this corollary with Remark 
8.4.3. In view of (8.5.12), a formal application of (8.4.25) gives 
Y(Y(a, ZoMb), z2) = Y(a, z2 + zo)Y(b, 22). (8.5.14) 
This suggestive formula is not rigorous in the present setting, although at 
the end of Chapter 8 we shall develop an approach in which it can be 
interpreted rigorously. 
We would also like to express Corollary 8.5.1 in terms of the component 
operators u, E End 5 defined by the formula 
Y(u,z) = c U,z-n-l = c u,-Iz-n (8.5.15) 
neQ neQ 
for u E VL. Note that by (8.4.24), for a E e and u = r(a) the operators u, are 
re-indexings of the operators x,(n): 
r(a), = x,(n - +(d,it) + 1) (8.5.16) 
for n E Q. In particular, 
deg ~(a), = n - *<a, a) + 1 = n - wt ~(a) + 1 (8.5.17) 
[see (7.1.49) and recall Remark 7.1.31. We have 
411, = an, -1 (8.5.18) 
for n E Q (cf. Remark 7.1.1). The case u = a(-1) = a(-1) 0 i(1) for 
a E ij gives 
Y(a(-1), z) = a(z) = c a(r7)z-n-l (8.5.19) 
ncZ 
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8.5. General Vertex Operators 201 
[see (8.4.8), (8.5.7)], so that 
a(-l), = a(n) 
deg a( - I), = n 
for n E Q. More generally, for u as in (8.5.4) and n E Q, 
(8.5.20) 
deg u, = n - n, - - nk - +<o, O) + 1. (8.5.21) 
It follows that for all homogeneous elements u, w E VL and n E Q, 
deg u, = n - wt u + 1 
wt(u,. w) = wtu + wt w - n - 1. 
From (8.5.8) and (8.5.15) we see that 
u-] * r(1) = u 
for all u E Vr. 
(8.5.22) 
(8.5.23) 
Remark 8.5.3: For u E VL , 
uo = Res, Y(u, z). 
Now in the notation of Corollary 8.5.1, 
and 
and so Corollary 8.5.1 can be reformulated as follows (where we note that 
r(a); * i(b) = 0 for i sufficiently large): 
Corollary 8.5.4: In the notation of Theorem 8.4.2, let 
m, n E Z. 
Then 
(8.5.26) 
the sum being finite. 
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202 General Theory of Untwisted Vertex Operators 
In particular we have: 
Corollary 8.5.5: In the same notation, 
We also want to express Corollary 8.5.1 in terms of the original operators 
x,(n). For every homogeneous element u E V,, set 
xu, z) = Xdu, z) = Y(u, z)zWtU 
(8.5.27) 
[cf. (8.4.17)1, and define xu@) E End V, for n E Q by the expansions 
X(u,z) = c x"(n)z-" 
neQ 
(8.5.28) 
Y(v,z) = c xu(n)z-n-wtu. 
ne9 
Then for n E Q, 
and 
xm = un+wtu-1 (8.5.29) 
degx,(n) = n 
(8.5.30) 
[cf. (7.1.48), (7.1.49)]. By (8.5.23) and (8.5.29), 
xu(-wt V)' r(1) = u. (8.5.31) 
Extend the correspondence v - X(u, z) from homogeneous elements to 
arbitrary elements of V' by linearity: 
(8.5.32) 
and use the first formula in (8.5.28) to define operators x,(n) in general. 
Then (8.5.30) holds for all u E V,. 
For the case u = (a), a E e, we have 
(8.5.33) 
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8.5. General Vertex Operators 203 
For u = a(-1), a E b, 
X(a( - l), Z) = za(z) 
(8.5.34) 
xa(-])(n) = a(n) for n E Q 
[recall (8.4.7), (8.4.8); za(z) is the expression previously denoted a(~)]. 
More generally, for u as in (8.5.4), 
X(v, 2) 
[see (8.5.5)]. But 
using the usual binomial coefficient notation 
T(T - l)-..(T - m + 1) 
(z) = m! 
(8.5.36) 
(8.5.37) 
for an element T of an associative algebra. Hence (8.5.35) becomes 
For example, 
X(al(-l) .*. ok(-l), Z) = 8(Zal(Z)) *.* (Zak(Z))8 (8.5.3 9) 
[cf. (8.5.7)]. In this connection we generalize the notation a(n) [cf. (8.5.34)] 
by setting 
al ... = xa,(- l)...uk(- for n E Q. (8.5.40) 
Here the product a, . . ak is of course to be viewed as an element of the kth 
symmetric power Sk(h), as in (3.3.9). We have 
8al(Z) **a ak(Z)X = 1 a1 c~k(n)Z-~-~. (8.5.41) 
nez 
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204 General Theory of Untwisted Vertex Operators 
Changing indices in Corollary 8.5.4, we find: 
Corollary 8.5.6: In the notation of Theorem 8.4.2, let 
m E h - +<a, a), n E Z - +(5,6). 
Then 
(finite sum). 
To help in the use of this formula, we record that 
using (8.5.38). 
For example, 
if (a, 6) L 0 
[xa(m), xb(n)l = xab(m + n) if (a,&) = -1 (8.5.46) 
r n(m + n) + m&,+n,O if (a,a) = 2 and b = a-', 
recovering Theorem 7.2.1 in case the lattice L is integral. The results in the 
next section include Theorem 7.2.1 in general. 
8.6. Commutators of General Vertex Operators 
We shall now show that the assertions of Corollaries 8.5.1, 8.5.4 and 8.5.6 
remain valid when r(a) and r(b) are replaced by arbitrary elements of V, 
[recall that Y(a, z) = Y(r(a), z)]. In particular, the operators u, or x,(n) span 
