Table Of Contentφ -COORDINATED MODULES FOR VERTEX ALGEBRAS
ǫ
∗
CHENGMING BAI, HAISHENG LI, AND YUFENG PEI
4
1
0
Abstract. We study φǫ-coordinated modules for vertex algebras, where φǫ with ǫ an
2
integerparameteris afamily ofassociatesofthe one-dimensionaladditiveformalgroup.
n
a As the main results, we obtain a Jacobi type identity and a commutator formula for
J
φǫ-coordinatedmodules. We then use these results to study φǫ-coordinated modules for
6
vertex algebras associated to Novikov algebras by Primc.
1
]
A
Q 1. Introduction
.
h
In a programto associate quantum vertex algebras to quantum affine algebras, a theory
t
a
m of φ-coordinated (quasi) modules for quantum vertex algebras was developed in [25],
[ where φ is what was called an associate of the one-dimensional additive formal group
1 (law) F (x,y) = x+y.
a
v
Since the very beginning, it had been recognized that the theory of vertex algebras and
1
8
their modules was governed by the formal group F . This can be seen from the definition
9 a
3 of a vertex algebra V and more generally the definition of a module (W,Y ) for a given
W
.
1 vertex algebra V, where the weak associativity axiom for a V-module (W,Y ) states that
W
0
4 for any u,v ∈ V, w ∈ W, there exists a nonnegative integer l such that
1
: (x+y)lY (u,x+y)Y (v,y)w = (x+y)lY (Y(u,x)v,y)w. (1.1)
v W W W
i
X
The notion of associate of the formal group F was designed in [25] to be an analog of
a
r
a that of G-set of a group G. By definition, an associate of F is a formal series φ(x,z) ∈
a
C((x))[[z]] such that
φ(x,0) = x and φ(x,φ(y,z)) = φ(x,y +z).
Interestingly, it was proved therein that for any p(x) ∈ C((x)), φ (x,z) := ezp(x)d/dxx is
p(x)
an associate of F and every associate of F is of this form. When p(x) = 1, we get the
a a
formal group itself, whereas when p(x) = x, we get φ (x,z) = xez.
p(x)
Let φ(x,z) be a general associate of F . The notion of φ-coordinated (quasi) V-module
a
for a vertex algebra (more generally for a weak quantum vertex algebra in the sense of
[24]) V was defined by replacing the ordinary weak associativity axiom with the property
that for any u,v ∈ V, there is a nonnegative integer k such that
(x −x )kY (u,x )Y (v,x ) ∈ Hom(W,W((x ,x )))
1 2 W 1 W 2 1 2
∗
Corresponding author.
1
2 CHENGMING BAI,HAISHENGLI, ANDYUFENGPEI∗
and
(x −x )kY (u,x )Y (v,x ) | = (φ(x ,z)−x )kY (Y(u,z)v,x ). (1.2)
1 2 W 1 W 2 x1=φ(x2,z) 2 2 W 2
Whe(cid:0)n φ(x,z) = F (x,z) = x+ z, (cid:1)this gives an equivalent definition of the ordinary no-
a
tion of module (cf. [30]). In [25], the main focus is on φ-coordinated (quasi) modules
d
with φ(x,z) = ezxdxx = xez, by which weak quantum vertex algebras were canonically
associated to quantum affine algebras. Among the main results, a Jacobi-type iden-
tity and a commutator formula for φ-coordinated modules were obtained. Later in [28],
φ-coordinated quasi modules were studied furthermore, where a commutator formula,
similar to that for twisted modules (see [15]), was obtained. Just as commutator formulas
for modules and twisted modules are very important and useful in the vertex algebra
theory, such commutator formulas for φ-coordinated (quasi) modules were proved to be
very useful.
In this current paper, we study φ -coordinated modules for vertex algebras, where
ǫ
φǫ(x,z) = ezxǫddxx with ǫ an arbitrary integer. Part of our motivation is the fact that xnddx
with n ∈ Z form a basis of the Witt algebra, which plays a vital role in vertex operator
algebratheoryandinphysics conformalfieldtheory. Conceivably, φ -coordinatedmodules
ǫ
for vertex algebras will be of fundamental importance. Among the main results, we show
that if U is a local subset of Hom(W,W((x))) with W a general vector space, the nonlocal
vertex algebrahUi thatwas obtainedin[25]isavertex algebra. WealsoobtainaJacobi-
φǫ
type identity for φ -coordinated modules for vertex algebras and furthermore we derive a
ǫ
commutator formula.
In this paper, we also study φ -coordinated modules for some special family of vertex
ǫ
algebras. Note that for any Lie algebra g equipped with a symmetric invariant bilinear
form h·,·i, one has an (untwisted) affine Lie algebra gˆ = g⊗C[t,t−1]+Cc. Furthermore,
for every complex number ℓ, one has a vertex algebra V (ℓ,0). This gives a very important
ˆg
family of vertex algebras. There is another family of vertex algebras which are associated
to Novikov algebras. Recall that a (left) Novikov algebra is a non-associative algebra A
satisfying the condition that
(ab)c−a(bc) = (ba)c−b(ac), (ab)c = (ac)b for a,b,c ∈ A.
A result of Primc (see [36]) is that for any given (left) Novikov algebra A equipped with
a symmetric bilinear form h·,·i such that
hab,ci = ha,bci, hab,ci = hba,ci for a,b,c ∈ A,
one has a Lie algebra L(A) = A ⊗ C[t,t−1] + Cc, and furthermore, for every complex
number ℓ one has a vertex algebra Ve (ℓ,0).
L(A)
e
In this current paper, as an application of our general results we study φ -coordinated
ǫ
modules for vertex algebras Ve (ℓ,0) associated to Novikov algebras A. To determine
L(A)
3
φ -coordinated modules we introduce a Lie algebra Lǫ(A), which has the same underlying
ǫ
space as that of L(A). We show that a φ -coordinated module structure on a vector space
ǫ
e
for vertex algebra Ve (ℓ,0) exactly amounts to a “restricted” module structure for the
L(A)
e
Lie algebra Lǫ(A) of level ℓ.
Throughout this paper, Z and N denote the set of nonnegative integers and the set of
integers, respeectively, and all vector spaces are assumed to be over the field C of complex
numbers. On the other hand, we shall use the standard formal variable notations and
conventions (see [15], [14], and [21]).
This paper is organized as follows: In Section 2, we recall the basic results on associates
and we derive an explicit formula for φ . In Section 3, we prove a Jacobi-type identity and
ǫ
a commutator formula for φ -coordinated modules. In Section 4, we study φ -coordinated
ǫ ǫ
modules for the vertex algebras associated to Novikov algebras.
2. One-dimensional additive formal group and φ -coordinated modules
ǫ
for vertex algebras
2.1. Weherebrieflyrecall thenotionofformalgroup(cf. [17])andthenotionofassociate
of a formal group (see [25]).
Definition 2.1. Aone-dimensionalformal groupover Cisaformalpower series F(x,y) ∈
C[[x,y]] such that
F(x,0) = x, F(0,y) = y, F(x,F(y,z)) = F(F(x,y),z).
The simplest example is the one-dimensional additive formal group
F (x,y) = x+y. (2.2)
a
The following notion of associate of a formal group was introduced in [25]:
Definition 2.3. Let F(x,y) be a one-dimensional formal group over C. An associate of
F(x,y) is a formal series φ(x,z) ∈ C((x))[[z]], satisfying the condition that
φ(x,0) = x, φ(φ(x,x ),x ) = φ(x,F(x ,x )).
0 2 0 2
The following is an explicit classification of associates for F (x,y) obtained in [25]:
a
Proposition 2.4. Let p(x) ∈ C((x)). Set
zn d n
φ(x,z) = ez(p(x)d/dx)x = p(x) x ∈ C((x))[[z]].
n! dx
n≥0 (cid:18) (cid:19)
X
Then φ(x,z) is an associate of F (x,y). Furthermore, every associate of F (x,y) is of
a a
this form with p(x) uniquely determined.
4 CHENGMING BAI,HAISHENGLI, ANDYUFENGPEI∗
For any integer ǫ, set
φ (x,z) = ez(xǫd/dx)x, (2.5)
ǫ
an associate of F (x,y). For the rest of this paper, we shall be only concerned about the
a
1-dimensional additive formal group F (x,y) and its associates φ (x,z).
a ǫ
As special cases, we have (see [25])
x
φ (x,z) = x+z, φ (x,z) = xez, φ (x,z) = . (2.6)
0 1 2
1−zx
For the general case, from definition we have
zk k−1
φ (x,z) = x+ xkǫ+1 (1+jǫ). (2.7)
ǫ+1
k!
k≥1 j=0
X Y
Assume ǫ 6= 0. For k ≥ 1, we have
1 k−1 1 k−1 1 −1
(1+jǫ) = (−ǫ)k (− −j) = (−ǫ)k ǫ .
k! k! ǫ k
j=0 j=0 (cid:18) (cid:19)
Y Y
Then
−1
φ (x,z) = x+x (−ǫ)k ǫ (zxǫ)k = x(1−ǫzxǫ)−1ǫ. (2.8)
ǫ+1
k
k≥1 (cid:18) (cid:19)
X
Notice that
limφ (x,z) = xez = φ (x,z). (2.9)
ǫ+1 1
ǫ→0
The following are some simple facts we shall use:
Lemma 2.10. We have
φ (x,z)−x = zh(x,z), (2.11)
ǫ
φ (x,x )−φ (x,x ) = (x −x )g(x,x ,x ), (2.12)
ǫ 1 ǫ 2 1 2 1 2
where h(x,z) is a unit in C((x))[[z]] and g(x,x ,x ) is a unit in C((x))[[x ,x ]].
1 2 1 2
Proof. By definition we have
j
1 d
φ (x,z)−x = z zj−1 xǫ x.
ǫ
j! dx
j≥1 (cid:18) (cid:19)
X
Set h(x,z) = 1zj−1 xǫ d jx ∈ C((x))[[z]]. As h(x,0) = 1, h(x,z) is a unit in
j≥1 j! dx
C((x))[[z]]. On the other hand, by definition we have
P (cid:0) (cid:1)
φ (x,x )−φ (x,x ) = ex1xǫddx −ex2xǫddx x = e(x1−x2)xǫddx −1 ex2xǫddx ·x
ǫ 1 ǫ 2
(cid:16) j (cid:17) (cid:16) (cid:17)
1 d
= (x −x ) (x −x )j−1 xǫ φ (x,x ).
1 2 1 2 ǫ 2
j! dx
j≥1 (cid:18) (cid:19)
X
5
Set
j
1 d
g(x,x ,x ) = (x −x )j−1 xǫ φ (x,x ).
1 2 1 2 ǫ 2
j! dx
j≥1 (cid:18) (cid:19)
X
We have g(x,x ,x ) ∈ C((x))[[x ,x ]] and g(x,x ,x ) = φ (x,x ). Note that φ (x,x ) is
1 2 1 2 2 2 ǫ 2 ǫ 2
a unit in C((x))[[x ]] as φ (x,0) = x (nonzero in C((x))). It then follows that g(x,x ,x )
2 ǫ 1 2
is a unit in (C((x))[[x ]])[[x ]] = C((x))[[x ,x ]]. (cid:3)
2 1 1 2
2.2. Next we recall the definition of a (nonlocal) vertex algebra and the definitions of a
module and a φ -coordinated module for a (nonlocal) vertex algebra. The notion of non-
ǫ
local vertex algebra was studied in [23] (under the name “axiomatic G -vertex algebra”)
1
and in [24], and it was also independently studied in [4] (under the name “field algebra”).
The theory of φ-coordinated modules for quantum vertex algebras was developed in [26],
whereas φ -coordinated modules were the main focus therein.
1
Definition 2.13. A nonlocal vertex algebra is a vector space V equipped with a linear
map
Y(·,x) : V → Hom(V,V((x))) ⊂ (EndV)[[x,x−1]]
v 7→ Y(v,x) = v x−n−1 (where v ∈ EndV)
n n
Z
X
and with a distinguished vector 1 ∈ V, satisfying the conditions that
Y(1,x) = 1, Y(v,x)1 ∈ V[[x]] and limY(v,x)1 = v for v ∈ V
x→0
and that for any u,v,w ∈ V, there exists a nonnegative integer l such that
(x +x )lY(u,x +x )Y(v,x )w = (x +x )lY(Y(u,x )v,x )w. (2.14)
0 2 0 2 2 0 2 0 2
Furthermore, a vertex algebra is a nonlocal vertex algebra V satisfying the condition that
for any u,v ∈ V, there exists a nonnegative integer k such that
(x −x )kY(u,x )Y(v,x ) = (x −x )kY(v,x )Y(u,x ). (2.15)
1 2 1 2 1 2 2 1
For each nonlocal vertex algebra V, there is a canonical operator D on V, which is
defined by
d
D(v) = v 1 = Y(v,x)1 | for v ∈ V.
−2 x=0
dx
(cid:18) (cid:19)
This operator D satisfies the following property:
d
[D,Y(v,x)] = Y(Dv,x) = Y(v,x). (2.16)
dx
6 CHENGMING BAI,HAISHENGLI, ANDYUFENGPEI∗
Definition 2.17. Let V be a vertex algebra. A V-module is a vector space W equipped
with a linear map
Y (·,x) : V → Hom(W,W((x))) ⊂ (EndW)[[x,x−1]],
W
v 7→ Y (v,x),
W
satisfying the conditions that Y (1,x) = 1 (the identity operator on W) and that for
W W
u,v ∈ V, w ∈ W, there exists l ∈ N such that
(x +x )lY (u,x +x )Y (v,x ) = (x +x )lY (Y(u,x )v,x )w. (2.18)
0 2 W 0 2 W 2 0 2 W 0 2
Remark 2.19. It was shown in [30] (Lemma 2.9) that the weak associativity axiom in
the definition of a V-module can be equivalently replaced by the condition that for any
u,v ∈ V, there exists k ∈ N such that
(x −x )kY (u,x )Y (v,x ) ∈ Hom(W,W((x ,x ))), (2.20)
1 2 W 1 W 2 1 2
xkY (Y(u,x )v,x ) = (x −x )kY (u,x )Y (v,x ) | . (2.21)
0 W 0 2 1 2 W 1 W 2 x1=x2+x0
Definition 2.22. Let V be a nonloc(cid:0)al vertex algebra and let φ b(cid:1)e an associate of the
one-dimensional additive formal group F . A φ-coordinated V-module is a vector space W
a
equipped with a linear map Y (·,x) as in Definition 2.17, satisfying the conditions that
W
Y (1,x) = 1 and that for u,v ∈ V, there exists k ∈ N such that
W W
(x −x )kY (u,x )Y (v,x ) ∈ Hom(W,W((x ,x ))), (2.23)
1 2 W 1 W 2 1 2
(φ(x ,x )−x )kY (Y(u,x )v,x )
2 0 2 W 0 2
= ((x −x )kY (u,x )Y (v,x ))| . (2.24)
1 2 W 1 W 2 x1=φ(x2,x0)
Itisclearthatthenotionofφ -coordinatedV-moduleisequivalent tothatofV-module.
0
3. Jacobi-type identity for φ -coordinated modules
ǫ
In this section, we shall present some axiomatic results on φ -coordinated modules for
ǫ
vertex algebras. In particular, we establish a Jacobi-type identity and a commutator
formula.
Let W be a vector space. Set
E(W) = Hom(W,W((x))) ⊂ (EndW)[[x,x−1]]. (3.1)
A subset U of E(W) is said to be local if for any a(x),b(x) ∈ U there exists a nonnegative
integer k such that
(x −x )ka(x )b(x ) = (x −x )kb(x )a(x ). (3.2)
1 2 1 2 1 2 2 1
A pair (a(x),b(x)) in E(W) is said to be compatible if there exists k ∈ N such that
(x −x )ka(x )b(x ) ∈ Hom(W,W((x ,x ))). (3.3)
1 2 1 2 1 2
7
Note that (3.2) implies (3.3). Thus each pair in a local subset is always compatible.
Fix an integer ǫ throughout this section. Let (a(x),b(x)) be any compatible pair in
E(W) with k ∈ N such that (3.3) holds. We define a(x)ǫb(x) ∈ E(W) for n ∈ Z in terms
n
of generating function
Yǫ(a(x),z)b(x) = a(x)ǫb(x)z−n−1
E n
n∈Z
X
by
Yǫ(a(x),z)b(x) = (φ (x,z)−x)−k (x −x)ka(x )b(x) | , (3.4)
E ǫ 1 1 x1=φǫ(x,z)
where (φ (x,z)−x)−k is viewed as an eleme(cid:0)nt of C((x))((z)) (r(cid:1)ecalling Lemma 2.10).
ǫ
Remark 3.5. A notion of compatible subset of E(W) was introduced in [23] and it
was proved therein that each compatible subset generates a nonlocal vertex algebra in a
certain canonical way. It was proved in [25] (Theorem 4.10) that any compatible subset U
of E(W) generates a nonlocal vertex algebra hUi with W as a canonical φ -coordinated
φǫ ǫ
module. On the other hand, it was proved in [23] that every local subset of E(W) is
compatible. Thus each local subset U of E(W) generates a nonlocal vertex algebra hUi .
φǫ
It was proved in [25] that hUi is a vertex algebra, whereas it was proved in [22] that
φ1
hUi is a vertex algebra.
φ0
In the following, we shall prove that for every integer ǫ, hUi is a vertex algebra,
φǫ
generalizing the corresponding results of [22, 25].
Proposition 3.6. Let W be a vector space and let V be a local subspace of E(W), which
is Yǫ-closed in the sense that
E
u(x)ǫv(x) ∈ V for u(x),v(x) ∈ V, n ∈ Z.
n
Let a(x),b(x) ∈ V. Suppose
(x −x )ka(x )b(x ) = (x −x )kb(x )a(x ) (3.7)
1 2 1 2 1 2 2 1
for some nonnegative integer k. Then
(x −x )kYǫ(a(x),x )Yǫ(b(x),x ) = (x −x )kYǫ(b(x),x )Yǫ(a(x),x ). (3.8)
1 2 E 1 E 2 1 2 E 2 E 1
Proof. Let c(x) ∈ V be arbitrarily fixed. There exists l ∈ N with l ≥ k such that
(z −x)la(z)c(x) = (z −x)lc(x)a(z), (z −x)lb(z)c(x) = (z −x)lc(x)b(z).
Using this and (3.7) we get
(y −z)l(y −x)l(z −x)la(y)b(z)c(x) ∈ Hom(W,W((x,y,z))).
8 CHENGMING BAI,HAISHENGLI, ANDYUFENGPEI∗
By Lemma 4.7 in [25], we have
(φ (x,x )−φ (x,x ))l(φ (x,x )−x)l(φ (x,x )−x)lYǫ(a(x),x )Yǫ(b(x),x )c(x)
ǫ 1 ǫ 2 ǫ 1 ǫ 2 E 1 E 2
= (y −z)l(y −x)l(z −x)la(y)b(z)c(x)| .
y=φǫ(x,x1),z=φǫ(x,x2)
Set
f(x,x ,x ) = (φ (x,x )−φ (x,x ))l(φ (x,x )−x)l(φ (x,x )−x)l,
1 2 ǫ 1 ǫ 2 ǫ 1 ǫ 2
which lies in C((x))((x ,x )). Then
1 2
f(x,x ,x )(φ (x,x )−φ (x,x ))kYǫ(a(x),x )Yǫ(b(x),x )c(x)
1 2 ǫ 1 ǫ 2 E 1 E 2
= (y −z)l(y −x)l(z −x)l(y −z)ka(y)b(z)c(x)|
y=φǫ(x,x1),z=φǫ(x,x2)
= (y −z)l(y −x)l(z −x)l(y −z)kb(z)a(y)c(x)|
z=φǫ(x,x2),y=φǫ(x,x1)
= f(x,x ,x )(φ (x,x )−φ (x,x ))kYǫ(b(x),x )Yǫ(a(x),x )c(x).
1 2 ǫ 1 ǫ 2 E 2 E 1
Noticingthat(φ (x,x )−x)l(φ (x,x )−x)l isinvertibleinC((x))((x ,x )),bycancellation,
ǫ 1 ǫ 2 1 2
we get
(φ (x,x )−φ (x,x ))l+kYǫ(a(x),x )Yǫ(b(x),x )c(x)
ǫ 1 ǫ 2 E 1 E 2
= (φ (x,x )−φ (x,x ))l+kYǫ(b(x),x )Yǫ(a(x),x )c(x). (3.9)
ǫ 1 ǫ 2 E 2 E 1
By Lemma 2.10, we have
(φ (x,x )−φ (x,x ))l+k = (x −x )l+kg(x,x ,x )l+k,
ǫ 1 ǫ 2 1 2 1 2
where g(x,x ,x ) is a unit in C((x))[[x ,x ]]. By cancellation we get
1 2 1 2
(x −x )l+kYǫ(a(x),x )Yǫ(b(x),x ) = (x −x )l+kYǫ(b(x),x )Yǫ(a(x),x ). (3.10)
1 2 E 1 E 2 1 2 E 2 E 1
Combining this with the weak associativity obtained in [25] we get
x −x x −x
x−1δ 1 2 Yǫ(a(x),x )Yǫ(b(x),x )−x−1δ 2 1 Yǫ(b(x),x )Yǫ(a(x),x )
0 x E 1 E 2 0 −x E 2 E 1
(cid:18) 0 (cid:19) (cid:18) 0 (cid:19)
x −x
= x−1δ 1 0 Yǫ(Yǫ(a(x),x )b(x),x ). (3.11)
2 x E E 0 2
(cid:18) 2 (cid:19)
From (3.7) we have
(x −x )ka(x )b(x ) ∈ Hom(W,W((x ,x ))),
1 2 1 2 1 2
so that
(φ (x ,x )−x )kYǫ(a(x),x )b(x) = (x −x)ka(x )b(x)| .
ǫ 2 0 2 E 0 1 1 x1=φǫ(x2,x0)
Multiplying both sides of (3.11) by (φ (x ,x )−x )k and then taking Res we get
ǫ 2 0 2 x0
(φ (x ,x )−x )kYǫ(a(x),x )Yǫ(b(x),x )
ǫ 2 0 2 E 1 E 2
= (φ (x ,x )−x )kYǫ(b(x),x )Yǫ(a(x),x ).
ǫ 2 0 2 E 2 E 1
9
By Lemma 2.10, we have
(φ (x ,x )−x )k = xkh(x ,x )k
ǫ 2 0 2 0 2 0
where h(x ,x ) is a unit in C((x ))[[x ]]. By cancellation we obtain
2 0 2 0
(x −x )kYǫ(a(x),x )Yǫ(b(x),x ) = (x −x )kYǫ(b(x),x )Yǫ(a(x),x ),
1 2 E 1 E 2 1 2 E 2 E 1
(cid:3)
as desired.
Now we have:
Theorem 3.12. Let W be a vector space and let U be any local subset of E(W). Then
hUi is a vertex algebra and W is a canonical φ -coordinated hUi -module.
φǫ ǫ φǫ
Proof. We already knew that hUi is a nonlocal vertex algebra and W is a φ -coordinated
φǫ ǫ
hUi -module with Y (α(x),z) = α(z) for α(x) ∈ hUi . As hUi is the smallest Yǫ-
φǫ W φǫ φǫ E
closed local subspace containing U and 1 , we see that hUi as a nonlocal vertex algebra
W φǫ
is generated by U. Given that U is local, by Proposition 3.6 we see that
{Yǫ(a(x),z) | a(x) ∈ U}
E
is a local subset of E(hUi ). It follows that hUi is a vertex algebra and W is a φ -
φǫ φǫ ǫ
coordinated hUi -module. (cid:3)
φǫ
We also have the following results:
Proposition 3.13. Let V be a vertex algebra and let (W,Y ) be a φ -coordinated V-
W ǫ
module. Suppose that for some fixed u,v ∈ V, k ∈ N,
(x −x )kY (u,x )Y (v,x ) ∈ Hom(W,W((x ,x ))).
1 2 W 1 W 2 1 2
Then
(x −x )kY (u,x )Y (v,x ) = (x −x )kY (v,x )Y (u,x ). (3.14)
1 2 W 1 W 2 1 2 W 2 W 1
Proof. Recall the skew-symmetry of V:
Y(u,x)v = exDY(v,−x)u for u,v ∈ V.
From the definition, there exists l ∈ N such that
(x −x )lY (u,x )Y (v,x ), (x −x )lY (v,x )Y (u,x ) ∈ Hom(W,W((x ,x ))).
1 2 W 1 W 2 1 2 W 2 W 1 1 2
Then, using Lemmas 3.6 and 3.7 in [25] we get
((x −x )lY (u,x )Y (v,x ))|
1 2 W 1 W 2 x1=φǫ(x2,x0)
= (φ (x ,x )−x )lY (Y(u,x )v,x )
ǫ 2 0 2 W 0 2
= (φ (x ,x )−x )lY (ex0DY(v,−x )u,x )
ǫ 2 0 2 W 0 2
= (φ (x ,x )−x )lY (Y(v,−x )u,φ (x ,x )).
ǫ 2 0 2 W 0 ǫ 2 0
10 CHENGMING BAI,HAISHENGLI, ANDYUFENGPEI∗
On the other hand, we have
((x −x )lY (v,x )Y (u,x ))| = (x −φ (x ,−x ))lY (Y(v,−x )u,x ).
1 2 W 2 W 1 x2=φǫ(x1,−x0) 1 ǫ 1 0 W 0 1
Hence
((x −x )lY (u,x )Y (v,x ))|
1 2 W 1 W 2 x1=φǫ(x2,x0)
= (((x −x )lY (v,x )Y (u,x ))| )|
1 2 W 2 W 1 x2=φǫ(x1,−x0) x1=φǫ(x2,x0)
= ((x −x )lY (v,x )Y (u,x ))| .
1 2 W 2 W 1 x1=φǫ(x2,x0)
It follows that
(x −x )lY (u,x )Y (v,x ) = (x −x )lY (v,x )Y (u,x ),
1 2 W 1 W 2 1 2 W 2 W 1
which implies
(x −x )l(x −x )kY (u,x )Y (v,x ) = (x −x )l(x −x )kY (v,x )Y (u,x ).
1 2 1 2 W 1 W 2 1 2 1 2 W 2 W 1
Noticing that (x −x )kY (v,x )Y (u,x ) ∈ Hom(W,W((x ))((x ))) and
1 2 W 2 W 1 2 1
(x −x )kY (u,x )Y (v,x ) ∈ Hom(W,W((x ,x ))) ⊂ Hom(W,W((x ))((x )))
1 2 W 1 W 2 1 2 2 1
by assumption, we can multiply both sides by the inverse of (x − x )l in C((x ))((x ))
1 2 2 1
(cid:3)
to obtain the desired relation (3.14).
Proposition 3.15. Let V be a vertex algebra and let (W,Y ) be a φ -coordinated V-
W ǫ
module. Set V = {Y (v,x)|v ∈ V}. Then V is a local subspace of E(W), (V ,Yǫ,1 )
W W W W E W
is a vertex algebra, and Y is a homomorphism of vertex algebras.
W
Proof. By Definition 2.22 and Proposition 3.13, V is a local subspace of E(W). Let
W
u,v ∈ V. By definition, there exists k ∈ N such that
(x −x)kY (u,x )Y (v,x) ∈ Hom(W,W((x ,x)))
1 W 1 W 1
and
(φ (x ,z)−x )kYǫ(Y(u,x),z)Y (v,x) = (x −x)kY (u,x )Y (v,x)| .
ǫ 2 2 E W 1 W 1 W x1=φǫ(x2,z)
On the other hand, from the definition of Yǫ(·,x) we have
E
(φ (x ,z)−x )kYǫ(Y (u,x),z)Y (v,x) = (x −x)kY (u,x )Y (v,x)| .
ǫ 2 2 E W W 1 W 1 W x1=φǫ(x2,z)
It follows that
(φ (x ,z)−x )kYǫ(Y(u,x),z)Y (v,x) = (φ (x ,z)−x )kYǫ(Y (u,x),z)Y (v,x).
ǫ 2 2 E W ǫ 2 2 E W W
Since both Yǫ(Y(u,x),z)Y (v,x) and Yǫ(Y (u,x),z)Y (v,x) involve only finitely many
E W E W W
negative powers of z, and since (φ (x ,z)−x )k is a unit in C((x ))((z)) (by Lemma 2.10),
ǫ 2 2 2
by cancellation we get
Yǫ(Y(u,x),z)Y (v,x) = Yǫ(Y (u,x),z)Y (v,x).
E W E W W
