0 } is finite and the number d(M ) =
X
ν∈Zh
4
(µ(ν) − ν)
is finite. The number d(M ) is also written as d(µ) and it is called the degree of the Maya diagram M with characteristic function µ. The ` finite set of Maya diagrams of degree d and change p is denoted Mdp . Clearly Mp = d Mdp . For a Maya diagram M of charge p we define two semi-infinite products |M i = eµ(p−1/2) ∧ eµ(p−3/2) ∧ eµ(p−5/2) ∧ · · · hM | = · · · ∧ eµ(p−5/2) ∧ eµ(p−3/2) ∧ eµ(p−1/2)
Formally, these semi-infinite products is just another notation for the corresponding Maya diagram. This notation is particular convenient for the following discussion. However, by using the basis eν , we clear indicate the relation to the vector spaces W and W † . For any integer p put |pi = ep−1/2 ∧ ep−3/2 ∧ ep−5/2 ∧ · · ·
hp| = · · · ∧ ep−5/2 ∧ ep−3/2 ∧ ep−1/2
Now the fermion Fock space F † (p) of charge p and the dual fermion Fock space F(p) of charge p are defined by M C|M i F(p) = M ∈Mp
Y
F † (p) =
M ∈Mp
We observe that F(p) = where Fd (p) =
M d≥0
ChM |
Fd (p),
M
M ∈Mdp
C|M i.
The dual pairing h·|·i : F † (p) × F(p) → C is given by hM |N i = δM,N ,
M, N ∈ Mp
Put also F F
†
=
M p∈Z
=
M p∈Z
F(p) F † (p)
The vector space F † is called the fermion Fock space and F is called the dual fermion Fock space. These are the semi-infinite exterior products of the vector spaces W † and W respectively, which we shall be interested in. We only define the fermion Fock space by using the basis eν , since we are fixing this basis throughout. 5
The above pairing can be extended to the one on F † × F by assuming that the paring is zero on F † (p) × F(p′ ) if p 6= p′ . Let us introduce the fermion operators ψν and ψ ν for all half integers ν ∈ Zh which act on F from the left and on F † from the right. Left action on F
Right action on
ψν = i(eν ), ν
F†
ψν = ∧e ,
ψ ν = e−ν ∧ −ν
ψ ν = i(e
)
(1.1) (1.2)
where i(·) is the interior product. For example we have ψ−3/2 |0i = i(e−3/2 )e−1/2 ∧ e−3/2 ∧ · · · = −e−1/2 ∧ e−5/2 ∧ e−7/2 ∧ · · · ,
h0|ψ 5/2 = · · · ∧ e−5/2 ∧ e−3/2 ∧ e−1/2 i(e−5/2 ) = · · · ∧ e−7/2 ∧ e−3/2 ∧ e−1/2
Note that ψν maps F(p) to F(p − 1), hence decreases the charge by one, and ψ ν maps F(p) to F(p + 1), hence increase the charge by one. Similarly the right action of ψν maps F † (p) to F † (p + 1) and ψ ν maps F † (p) to F † (p − 1). It is easy to show that for any hu| ∈ F and |vi ∈ F † we have hu|ψν vi = huψν |vi,
hu|ψ ν vi = huψ ν |vi.
Also it is easy to show that ψν |0i = 0
if and only if ν > 0,
ψ ν |0i = 0
if and only if ν > 0.
h0|ψν = 0
if and only if ν < 0,
Similarly we have
h0|ψ ν = 0
if and only if ν < 0.
The fermion operators have the following anti-commutation relations as operators on F and F † . [ψν , ψµ ]+ = 0,
(1.3)
[ψ ν , ψ µ ]+ = 0,
(1.4)
[ψν , ψ µ ]+ = δν+µ,0 ,
(1.5)
where we define [A, B]+ = AB + BA. Note that for each Maya diagram M of charge p we can find non-negative half integers µ1 < µ2 < · · · < µr < 0,
ν1 < ν2 < · · · < νs < 0,
r ≥ 0, s ≥ 0
with r − s = p and µi 6= νj such that Ps
|M i = (−1)
i=1
νi +s/2
ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i. 6
(1.6)
The negative half integers µi ’s and νj ’s are uniquely determined by the Maya diagram M . The normal ordering ·· ·· of the fermion operators are defined as follows. · Aν Bµ · = −Bµ Aν if µ < 0 and ν > 0, · · Aν Bµ otherwise, where A and B is ψ or ψ. By (1.3), (1.4) and (1.5) the normal ordering is non-trivial if ¯−µ . For example we have and only if µ < 0 and Aν = B · ·ψ ψ · 1/2 −1/2 · = −ψ−1/2 ψ 1/2 ,
although as operators on F or F † we have
ψ 1/2 ψ−1/2 = −ψ−1/2 ψ 1/2 + Id . Thus the normal ordering has the effect of subtracting the identity operator, whenever it is non-trivial. The field operators ψ(z) and ψ(z) are defined by X ψ(z) = ψµ z −µ−1/2 , (1.7) µ
ψ(z) =
X
ψ µ z −µ−1/2 .
(1.8)
X
(1.9)
µ
The current operator J(z) is defined by J(z) = ·· ψ(z)ψ(z) ·· =
Jn z −n−1
n∈Z
Note that thanks to the normal ordering, the operator Jn can operate on F and F † even though Jn is an infinite sum of operators. For any integer or half integer j the energy-momentum tensor T (j) (z) is defined by dψ(z) dψ(z) · T (j) (z) = ·· (1 − j) = ψ(z) − jψ(z) dz dz ·
X
−n−2 L(j) . n z
(1.10)
n∈Z
(j)
Again due to the normal ordering, the coefficients Ln operates on F and F † . These operators satisfy the following commutation relations. [Jn , Jm ] = nδn+m,0 , i 1 (j) (j) = (n − m)Ln+m − (6j 2 − 6j + 1)(n3 − n)δm+n.0 , L(j) , L n m 6 i h 1 = −mJn+m − (2j − 1)(n2 + n)δn+m,0 , L(j) , J m n 2 [Jn , ψ(z)] = −z n ψ(z), Jn , ψ(z) = z n ψ(z), h i d (j) n + j(n + 1) ψ(z), Ln , ψ(z) = z z dz h i d n L(j) , + (1 − j)(n + 1) ψ(z) = z ψ(z). z n dz h
7
(1.11) (1.12) (1.13) (1.14) (1.15) (1.16) (1.17)
(j)
Thus the set {Ln }n∈Z forms an infinite-dimensional Lie algebra called the Virasoro algebra with central charge c = −2(6j 2 − 6j + 1). The field operators ψ(z) and ψ(z), the current operator J(z) and the energy-momentum tensor T (j) (z) form the so-called spin j bc-system or ghost system in the physics literature. For j = 1/2 the bc-system is usually called abelian conformal field theory. In the present paper we shall discuss the j = 0 ghost system and we shall also refer to this case as abelian conformal field theory.
2
Universal Grassmann Manifold
In this section we shall briefly recall the theory of the universal Grassmann manifold due to M. Sato ([SA]). Let W † be the vector space introduced in the previous section. Definition 2.1. The universal Grassmann manifold UGMp of charge p, p ∈ Z, is the set of closed subspace U ⊂ W † such that 1. The kernel and the cokernel of the natural linear map f : U → W † /F 0 W † are of finite dimension; 2. dim Ker f − dim Coker f = p. Put UGM =
G
UGMp
p∈Z
Also, we can introduce the induced filtration F m U on U by F mU = U ∩ F mW † . Then we have dim F m U/F m+1 U ≤ 1. Put
M (U ) = m + 1/2 | dim F m U/F m+1 U = 1 .
It is easy to show that M (U ) is a Maya diagram of charge p. Definition 2.2. For U ∈ U GM p a frame Ξ of U is a basis Ξ = {· · · , ζ p−5/2 , ζ p−3/2 , ζ p−1/2 }
of U such that there exists a half integer ν0 such that for any ν ≤ ν0 ζ ν ∈ F ν−1/2 U \ F ν+1/2 U ζ ν ≡ eν
(mod F ν−1/2 U )
For a frame Ξ = {· · · , ζ p−5/2 , ζ p−3/2 , ζ p−1/2 }
of U ∈ UGMp we can define the semi-infinite wedge product · · · ∧ ζ p−5/2 ∧ ζ p−3/2 ∧ ζ p−1/2 8
(2.1)
as an element of the fermion Fock space F † (p) by the following procedure. Put n0 = ν0 − 1/2. For n ≤ n0 the element ζ n+1/2 can be written as ζ
n+1/2
n+1/2
=e
∞ X
+
(n)
ak+1/2 ek+1/2 .
k=n+1
For n ≤ n0 we define the wedge product · · · ∧ en−3/2 ∧ en−1/2 ∧ ζ n+1/2 ∧ · · · ∧ ζ p−1/2 = hn| ∧ ζ n+1/2 ∧ · · · ∧ ζ p−1/2 as the limit of the wedge products n+1/2 p−1/2 hn| ∧ ζm ∧ · · · ζm p n+1 ν is defined by where ζm ν ζm =
X
k 0 we have hΦ|ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i = 0. Moreover, by the second gauge condition if s > 0 and νs < −1/2 we have hΦ|ψνs ψνs−1 · · · ψν1 |0i = 0. Thus we conclude that hΦ| is zero on all the element of the form (3.4) except ψ −1/2 |0i = | − 1i. Therefore, hΦ| is a constant multiple of h−1| and we conclude V † ab ((P1 ; 0; z)) = Ch−1|. Let us assume that C is a non-singular curve of genus g ≥ 1. Let us consider a onepointed curve X = (C; Q; ξ) with a formal coordinate. Choose a basis {ω1 , . . . , ωg } of holomorphic one-forms on C. We let ωi =
∞ X
n=0
an(i) ξ n dξ, 14
i = 1, 2, . . . , g
be the Taylor expansions of ωi ’s at the point Q. For any positive integer j choose a meromorphic one-form ωg+j ∈ H 0 (C, ωC (∗Q)) in such a way that it has the Laurent expansion ∞ X −(j+1) a(g+j) ξ n dξ ωg+j = ξ + n n=−j
at Q. Put
∞ X
e(ωi ) =
n+1/2 a(i) , n e
n=0 −j−1/2
e(ωg+j ) = e
+
∞ X
a(g+j) en+1/2 . n
n=−j
Then, the infinite sums e(ωi ) and e(ωg+j ) are regarded as elements of W † (see section 2). Put hω(X)| = · · · ∧ e(ωg+2 ) ∧ e(ωg+1 ) ∧ e(ωg ) ∧ · · · ∧ e(ω1 ). (3.5) The set {. . . , ωg+2 , ωg+1 , ωg , . . . , ω1 }
is a frame of H 0 (C, ωC (∗Q)) ⊂ C((ξ))dξ. Since H 0 (C, ωC (∗Q)) ∈ UGMg−1 , we see that hω(X)| ∈ F † (g − 1) is non-zero. By a similar arguments as in Example 2.1 we may also regard hω(X)| as an element of the determinant of the canonical sheaf ωC . Note of course that hω(X)| depends on the choice of the basis (ωi ). Lemma 3.1. The semi-infinite wedge product hω(X)| is a non-zero element of F † (g − 1) which satisfies the first and second gauge conditions. Hence hω(X)| ∈ V † ab (X).
P n Proof. For any element ω ∈ H 0 (C, ωC (∗Q)) we let ( ∞ n=−n0 an ξ )dξ be its Laurent expansion. Then we have ∞ X an ψn+1/2 ψ[ω] = n=−n0
whose right action on
F†
is given by ∧
∞ X
n+1/2
an e
n=−n0
!
= ∧e(ω).
Hence we have that hω(X)| ∧ e(ω) = 0. Thus hω(X)| P satisfies the first gauge condition. ∞ m be the Laurent expansion of a meromorphic function f ∈ Now let m=−m0 bm ξ 0 H (C, OC (∗Q)) at Q. Then we have ψ[f ] =
∞ X
m=−m0
15
bm ψm+1/2
and its right action is given by ∞ X
bm i(e−m−1/2 ).
m=−m0
Then we have that e(ω)ψ[f ] = =
X
n+1/2
an e
n=−n0 ∞ X
!
∞ X
−m−1/2
bm i(e
m=−m0
!
)
a−m−1 bm
m=−m0
= Res(f (ξ)ω) = 0. ξ=0
Thus the second gauge condition is also satisfied.
QED
Corollary 3.1. Let 1 = l1 < l2 < · · · < lg ≤ 2g − 1 be the Weierstrass gap values of the curve C at the point Q. Then, ψ[ξ −lg ]ψ[ξ −lg−1 ] · · · ψ[ξ −l1 ]| − 1i = elg −1/2 ∧ elg−1 −1/2 ∧ · · · ∧ el1 −1/2 ∧ e−3/2 ∧ e−5/2 ∧ · · · defines a non-zero element of Vab (X) = F/F(X). Proof.
The Weierstrass gap values have the property (for details see Lemma 6.2): 6 0. he(ωg ) ∧ · · · ∧ e(ω1 )|elg −1/2 ∧ elg−1 −1/2 ∧ · · · ∧ el1 −1/2 i =
On the other hand in the infinite wedge product · · · ∧ e(ωg+n ) ∧ · · · ∧ e(ωg+2 ) ∧ e(ωg+1 ) the term h−1| only appears when we choose the term e−j−1/2 of e(ωg+j ) for each j ≥ 1. Other terms do not have a term e−m−1/2 for a certain positive integer m. Since he(ωg ) ∧ · · · ∧ e(ω1 )| does not contain a term e−m−1/2 for any m ≥ 1 we have that hω(X)|ψ[ξ −lg ]ψ[ξ −lg−1 ] · · · ψ[ξ −l1 ]|−1i = he(ωg )∧· · ·∧e(ω1 )|elg −1/2 ∧elg−1 −1/2 ∧· · ·∧el1 −1/2 i = 6 0. QED Theorem 3.3. The space of ghost vacua V † ab (X) is isomorphic to the determinant of the canonical bundle ωC . Notice that this theorem follows directly from Lemma 3.1, Theorem 3.2 and the discussion in section 2. The isomorphism of course depends on the choice of a basis for H 0 (C, ωC (∗Q)). Let X = (C; Q1 , . . . , QN ; ξ1 , . . . , ξN ) be an N -pointed curve with formal coordinates. Let QN +1 be a non-singular point and choose a formal coordinate ξN +1 of C with center QN +1 . Put e = (C; Q1 , . . . , QN , QN +1 ; ξ1 , . . . , ξN , ξN +1 ). X 16
Then the canonical linear mapping ι : FN
→ FN +1
|vi 7→ |vi ⊗ |0i
induces the canonical mapping
† † ι∗ : FN +1 → FN .
Theorem 3.4. The canonical mapping ι∗ induces an isomorphism e ∼ V † ab (X) = V † ab (X).
This isomorphism is denoted the “Propagation of vacua” isomorphism. Proof. Since we have injective maps H 0 (C, ωC (∗
N X
Qj )) ֒→ H 0 (C, ωC (∗
H 0 (C, OC (∗
j=1
Qj )),
j=1
j=1
N X
N +1 X
Qj )) ֒→ H 0 (C, OC (∗
N +1 X
Qj )),
j=1
and for f (ξ) ∈ C[[ξ]] we have ψ[f (ξ)|0i = 0,
ψ[f (ξ)dξ]|0i = 0,
ˆ is contained in V † ab (X). the image ι∗ (V † ab (X)) Therefore, it is enough to show that any element hφ| ∈ V † ab (X) uniquely determines ˆ such that ι∗ (hΦ|) = hφ|. Note that any element of F is a finite an element hΦ| ∈ V † ab (X) linear combination of elements of the form ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i, µ1 < µ2 < · · · < µr < 0,
ν1 < ν2 < · · · < νs < 0,
r ≥ 0, s ≥ 0.
By a double induction on r and s we shall show that hφ| uniquely determines the value hΦ|ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i in such a way that hΦ| satisfies the first and second gauge conditions. For (r, s) = (0, 0) put hΦ|u ⊗ 0i = hφ|ui for any |ui ∈ FN . Choose |uj i ∈ F(pj ) and put PN
|ui = |u1 i ⊗ · · · ⊗ |uN i.
Choose ω ∈ H 0 (C, ωC (∗ j=1 Qj +(n+1)QN +1 )) in such a way that its Laurent expansion at the point QN +1 has the form ! ∞ X −n−1 n (3.6) an ξN +1 dξ ωN +1 = ξN +1 + n=0
17
Then we have ψ[ωN +1 ]|0i = ψ−n−1/2 |0i = i(e−n−1/2 )|0i. This is no-zero if and only if n is a non-negative integer. For a non-negative integer n define N X p1 +···pN +1 hΦ|ρj (ψ[ωj ])u ⊗ 0i. (3.7) hΦ|u ⊗ ψ−n−1/2 |0i = (−1) j=1
P This is independent of the choice of ω satisfying (3.6), since if ω ′ ∈ H 0 (C, ωC (∗ N j=1 Qj + (n + 1)QN +1 )) satisfies (3.6) ω − ω ′ is holomorphic at QN +1 . Thus hΦ|u ⊗ ψ−n−1/2 |0i is P +1 well-defined for any integer n. Then for any meromorphic form τ ∈ H 0 (C, ω(∗ N n=1 Qj )) we have N +1 X hΦ|ρj (ψ[τj ])u ⊗ 0i = 0. (3.8) j=1
This establishes the first gauge condition on this subspace. Next let us define hΦ|u ⊗ ψ−n2 −1/2 ψ−n1 −1/2 |0i with 0 ≤ n2 < n1 . Choose a meromorphic form ω e ∈ H 0 (C, ωC (∗ which has the Laurent expansion ! ∞ X −n2 −1 n bn ξN +1 dξ. ω eN +1 = ξN +1 +
PN
j=1 Qj +(n2 +1)QN +1 ))
(3.9)
n=0
Define
hΦ|u ⊗ ψ−n2 −1/2 ψ−n1 −1/2 |0i = (−1)p1 +···pN +1
N X j=1
hΦ|ρj (ψ[e ωj ])u ⊗ ψ−n1 −1/2 0i.
We need to show that this is well-defined. Choose another ω e ′ which has the Laurent ′ expansion of the same type (3.9). Then τ = ω e−ω e is holomorphic at QN +1 , hence ψ[τN +1 ]ψ−n1 −1/2 |0i = 0.
Note that if j > k we have that ρj (ψ[ωj ])ρk (ψ[τk ])|ui = (−1)p1 +···+pk−1 ρj (ψ[ωj ])|u1 ⊗ · · · ⊗ ψ[τk ]uk ⊗ · · · uN i
= (−1)(p1 +···+pk−1 )+(p1 +···+pk−1 +···+pj−1 |u1 ⊗ · · · ⊗ ψ[τk ]uk ⊗ · · · ⊗ ψ[ωj ]uj ⊗ · · · uN i = −ρk (ψ[τk ])ρj (ψ[ωj ])|ui.
18
We have the same result in the case j ≤ k. Hence by (3.7) we have that N X hΦ|ρj (ψ[τj ])u ⊗ ψ−n1 −1/2 0i j=1
= (−1)p1 +···pN +1
N X N X k=1 j=1
= (−1)p1 +···pN
= (−1)p1 +···pN
N N X X hΦ|ρj (ψ[τj ])ρk (ψ[ωk ])u ⊗ 0i j=1 k=1 +1 N N X X hΦ|ρj (ψ[τj ])ρk (ψ[ωk ])u ⊗ 0i k=1
= 0,
hΦ|ρk (ψ[ωk ])ρj (ψ[τj ])u ⊗ 0i
j=1
where the last equality is a consequenceP of (3.8). Now by the same argument as above we +1 can show that for any τ ∈ H 0 (C, ωC (∗ N j=1 )) and any non-negative integer n the first gauge condition holds on the subspace spanned by |u ⊗ ψ−n−1/2 0i: N +1 X j=1
hΦ|ρj (ψ[τj ])u ⊗ ψ−n−1/2 0i = 0.
(3.10)
In this way, by induction on r we can show that for any non-positive integers n1 > n2 > · · · > nr ≥ 0 hΦ|u ⊗ ψ−n1 −1/2 · · · ψ−nr −1/2 0i is well-defined and we have the first gauge condition on the subspace spanned by |u ⊗ ψ−n1 −1/2 · · · ψ−nr −1/2 0i: N +1 X j=1
hΦ|ρj (ψ[τj ])u ⊗ ψ−n1 −1/2 · · · ψ−nr−1 −1/2 0i = 0.
(3.11)
Next let us define hΦ|u ⊗ ψ −n−1/2 ψ−n1 −1/2 · · · ψ−nr −1/2 0i. Choose f ∈ H 0 (C, OC (∗ expansion
PN
j=1 Qj
+ (n + 1)QN +1 )) in such a way that it has the Laurent
−n−1 fN +1 = ξN +1 +
∞ X
n an ξN +1
n=0
at the point QN +1 . Then we have
ψ[fN +1 ]ψ−n1 −1/2 · · · ψ−nr−1 −1/2 |0i = ψ −n−1/2 ψ−n1 −1/2 · · · ψ−nr−1 −1/2 |0i
= en+1/2 ∧ |ψ−n1 −1/2 · · · ψ−nr−1 −1/2 |0i. 19
(3.12)
This is no-zero if n is a non-negative integer. For non-negative integer n define hΦ|u ⊗ ψ −n−1/2 ψ−n1 −1/2 · · · ψ−nr−1 −1/2 0i p1 +···pN +1
= (−1)
N X j=1
hΦ|ρj (ψ[fj ])u ⊗ ψ−n1 −1/2 · · · ψ−nr−1 −1/2 |0i.
This is independent of the choice of f satisfying (3.12), since if f ′ satisfies (3.12) f − f ′ is holomorphic at QN +1 . Thus hΦ|u ⊗ ψ −n−1/2 ψ−n1 −1/2 · · · ψ−nr−1 −1/2 |0i
P +1 is well-defined for any integer n. Then for any meromorphic function h ∈ H 0 (C, O(∗ N n=1 Qj )) we have that N +1 X (3.13) hΦ|ρj (ψ[hj ])u ⊗ ψ−n1 −1/2 · · · ψ−nr−1 −1/2 |0i = 0, j=1
establishing the second gauge condition on this subspace. Now by induction on s we can show that
hΦ|u ⊗ ψ −ns −1/2 · · · ψ −n1 −1/2 ψ−n1 −1/2 · · · ψ−nr−1 −1/2 |0i are well-defined and they satisfy the second gauge condition: N +1 X j=1
hΦ|ρj (ψ[fj ])u ⊗ ψ −ns−1 −1/2 · · · ψ −n1 −1/2 ψ−n1 −1/2 · · · ψ−nr−1 −1/2 |0i = 0.
(3.14)
e has been uniquely constructed from hφ| ∈ V † ab (X). By our construcThus hΦ| ∈ V † ab (X) tion we have that ι∗ (hΦ|) = hφ|.
QED
e be the curve obtained by resolving Next let us consider a curve C with a node P . Let C e the singularity at P and let π : C → C be the natural holomorphic mapping. Then π −1 (P ) consists of two points P+ and P− . Let X = (C; Q1 , . . . , QN ; ξ1 , . . . , ξN )
be an N -pointed curve with formal coordinates and we let e = (C; e P+ , P− , Q1 , . . . , QN ; z, w, ξ1 , . . . , ξN ) X
be the associated N +2-pointed curve with formal coordinates. Define an element |0+,− i ∈ F ⊗ F by |0+,− i = |0i ⊗ | − 1i − | − 1i ⊗ |0i. (3.15)
The natural inclusion
FN
֒→ FN +2
|ui 7→ |0+,− i ⊗ |ui
defines a natural linear mapping † † ι∗+,− : FN +2 → FN .
20
Theorem 3.5. The natural mapping ι∗+,− induces a natural isomorphism
Proof.
First let us show that
e ∼ V † ab (X) = V † ab (X).
e ⊂ V † ab (X). ι∗+,− (V † ab (X))
e put hΦ| = ι∗ (hΦ|). e For |ui ∈ FN we have that e ∈ V † ab (X) For hΦ| +,−
e +,− ⊗ ui. hΦ|ui = hΦ|0 P For a meromorphic one-form ω ∈ H 0 (C, ωC (∗ N e = π ∗ (ω). Then ω e has a j=1 Qj )) put ω Laurent expansions at P+ and P− of the following form: a−1 ω e+ = + a0 + a1 z + a2 z 2 + · · · dz z a−1 + b0 + b1 w + b2 w2 + · · · dw ω e− = − w Then we have that ρ+ (ψ[e ω+ ])|0+,− i = ψ[e ω+ ]|0i ⊗ | − 1i − ψ[e ω+ ]| − 1i ⊗ |0i = a−1 | − 1i ⊗ | − 1i
Similarly we have that ρ− (ψ[e ω− ])|0+,− i = |0i ⊗ ψ[e ω− ]| − 1i + | − 1i ⊗ ψ[e ω− ]|0i = −a−1 | − 1i ⊗ | − 1i.
Since ω and ω e have the same Laurent expansions at Qj , by the above results we have that N X hΦ|ρj (ψ[e ωj ])ui j=1
N X e +,− ⊗ ρj (ψ[e hΦ|0 ωj ])ui = j=1
e + (ψ[e e − (ψ[e = hΦ|ρ ω+ ])0+,− ⊗ ui + hΦ|ρ ω+ ])0+,− ⊗ ui +
=0
N X e +,− ⊗ ρj (ψ[e hΦ|0 ωj ])ui j=1
e Thus hΦ| satisfies the first gauge condition. Similarly by the first gauge condition for hΦ|. we can show that hΦ| satisfies the second gauge condition. Hence hΦ| ∈ V † ab (X). Now let us show that ι∗+,− is bijective. For that purpose it is enough to show that e Choose f ∈ H 0 (C, e ∈ V † ab (X). e O e (∗ PN Qj )) hΦ| ∈ V † ab (X) determines uniquely hΦ| j=1 C such that the Taylor expansions at P± have the forms f+ = −1 + f− =
∞ X
∞ X
21
(3.16)
n=1
bn wn .
n=1
an z n ,
(3.17)
Then we have that ψ[f+ ] = −ψ 1/2 + ψ[f− ] =
∞ X
∞ X
an ψ n+1/2 ,
n=1
bn ψ n+1/2 .
n=1
Hence we have that ρ+ (ψ[f+ ])|0+,− i = ψ[f+ ]|0i ⊗ | − 1i − ψ[f+ ]| − 1i ⊗ |0i = |0i ⊗ |0i, ρ− (ψ[f− ])|0+,− i = |0i ⊗ ψ[f− ]| − 1i + | − 1i ⊗ ψ[f− ]|0i = 0.
Now define e ⊗ 0 ⊗ ui = hΦ|0
N N X X e +,− ⊗ ρj (ψ[fj ])ui = hΦ|0 hΦ|ρj (ψ[fj ])ui. j=1
(3.18)
j=1
PN
e O e (∗ This is well-defined, since if f ′ ∈ H 0 (C, j=1 Qj )) has the same type of Taylor C P ′ expansions (3.16) and (3.17), then h = f − f is an element of H 0 (C, OC (∗ N j=1 Qj )). Hence N X hΦ|ρj (ψ[hj ])ui = 0. j=1
e Now let us show any element PN h ∈ PN that hΦ| just defined satisfies the first gauge condition0 for 0 e e H (C, OCe (∗ j=1 Qj )) and the second gauge condition for any ω ∈ H (C, ωCe (∗ j=1 Qj )) applied to the vectors |0 ⊗ 0 ⊗ ui. Let ω+ = a0 + a1 z + · · · dz ω− = b0 + b1 w + · · · dw be the Taylor expansions of ω at P± . Then we have that ψ[ω+ ]|0i ⊗ |0i = 0,
ψ[ω− ]|0i ⊗ |0i = 0. P Since we may regard ω as an element of H 0 (C, ωC (∗ N j=1 Qj )) having a zero at P , we get by using (3.18) that e + (ψ[ω+ ])0 ⊗ 0 ⊗ ui + hΦ|ρ e − (ψ[ω− ])0 ⊗ 0 ⊗ ui + hΦ|ρ =−
N nX N X k=1
j=1
o hΦ|ρj (ψ[ωj ])ρk [fk ]ui = 0,
by the first gauge condition for hΦ|. Next let h+ = a0 + a1 z + a2 z 2 + · · ·
h− = b0 + b1 w + b2 w2 + · · · 22
N X e j (ψ[ωj ])0 ⊗ 0 ⊗ ui hΦ|ρ j=1
e O e (∗ be the Taylor expansion of h ∈ H 0 (C, C
PN
j=1 Qj ))
at P± . Then we have that
ρ+ (ψ[h+ ])|0i ⊗ |0i = 0, ρ− (ψ[h− ])|0i ⊗ |0i = 0. P If a0 = b0 then h ∈ H 0 (C, OC (∗ N j=1 Qj )). Then by similar arguments as above, we can show that e − (ψ[h− ])0 ⊗ 0 ⊗ ui + e + (ψ[h+ ])0 ⊗ 0 ⊗ ui + hΦ|ρ hΦ|ρ
N X e j (ψ[hj ])0 ⊗ 0 ⊗ ui = 0. hΦ|ρ j=1
If a0 6= b0 , then by subtracting from f the constant b0 and multiplying by a constant, we may assume that h has Taylor expansions of type (3.16) and (3.17). Then by the above argument we have that e ⊗ 0 ⊗ ui = hΦ|0
Then we have that
N X k=1
hΦ|ρk (ψ[hk ])ui.
N N X N X X e ⊗ 0 ⊗ ρj (ψ[hj ])ui = hΦ|ρj (ψ[hj ])ρk (ψ[hk ])ui hΦ|0 j=1 k=1
j=1
=−
N N X X k=1 j=1
hΦ|ρj (ψ[hj ])ρk (ψ[hk ])ui = −
since we have that
N X j=1
e ⊗ 0 ⊗ ρj (ψ[hj ])ui, hΦ|0
ρj (ψ[hj ])ρk (ψ[hk ])|ui = −ρk (ψ[hk ])ρj (ψ[hj ]) |ui. Hence
N X e ⊗ 0 ⊗ ρj (ψ[hj ])ui = 0. hΦ|0 j=1
Thus we conclude that
e − (ψ[h− ])0 ⊗ 0 ⊗ ui + e + (ψ[h+ ])0 ⊗ 0 ⊗ ui + hΦ|ρ hΦ|ρ
N X e j (ψ[hj ])0 ⊗ 0 ⊗ ui = 0. hΦ|ρ j=1
Now we can apply Theorem 3.4 twice and obtained the desired result.
QED
Remark 3.1. The choice of |0+,− i is not unique. Any non-zero multiple of |0+,− i gives a natural isomorphism of the above theorem. But the above normalization of |0+,− i will be seen to be compatible with the preferred elements of the ghost vacua on a nodal curve and on its normalization. For details see Theorem 6.5 below.
23
4
Sheaf of Ghost Vacua
The arguments of this section are almost identical to those in [U2], section 4, but for the readers’ convenience we shall repeat them here in this abelian case. Let F = (π : C → B; s1 , . . . , sN ; ξ1 , . . . , ξN ) be a family of N -pointed semi-stable curves with formal coordinates. That is C and B are complex manifolds, π is a proper holomorphic mapping, and for each point b ∈ B, F(b) = (Cb = π −1 (b); s1 (b), . . . , sN (b); ξ1 , . . . , ξN ) is an N -pointed semi-stable curve with formal coordinates. We let Σ be the locus of double points of the fibers of F and let D be π(Σ). Note that Σ is a non-singular submanifold of codimension two in C, and D is a divisor in B whose irreducible components Di , i = 1, 2, . . . , m′ are non-singular. In this section we use the following notation freely. Sj = sj (B),
S=
N X
Sj .
j=1
Put
† † FN (B) = OB ⊗C FN .
FN (B) = FN ⊗C OB ,
† Definition 4.1. We define the subsheaf V † ab (F) of FN (B) by the gauge conditions: N X j=1
N X j=1
hΦ|ψ[ωj ] = 0,
for all ω ∈ π∗ (ωC/B (∗S)),
hΦ|ψ[fj ] = 0,
for all f ∈ π∗ OC (∗S).
where ωj and fj are the Laurent expansion of ω and f along Sj with respect to the coordinate ξj . The sheaf V † ab (F) is called the sheaf of (j = 0) ghost vacua or the sheaf of abelian vacua of the family F. Similarly the sheaf Vab (F) of (j = 0) dual ghost vacua of the family is defined by Vab (F) = FN (B)/Fab (F).
0 (F) + F 1 (F), where where Fab (F) is the OB -submodule of FN (B) given by Fab (F) = Fab ab 0 (F) is the span of ψ[f ]F (B) for all f ∈ π O (∗S) and F 1 (F) is the span of ψ[ω]F (B) Fab N N ∗ C ab for all ω ∈ π∗ ωC/B (∗S).
Note that we have V † ab (F) = HomOB (Vab (F), OB ).
(4.1)
Moreover, by the right exactness of the tensor product we have that Vab (F) ⊗OB OB,b /mb ∼ = Vab (F(b)). Proposition 4.1. The sheaves Vab (F) and V † ab (F) are coherent OB -modules. 24
(4.2)
Proof First we need to introduce a filtration on F. Note that any element of F is a finite linear combination of elements of the form |ui = ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i with µ1 < µ2 < · · · < µr < 0, ν1 < ν2 < · · · < νs < 0. Let us define an alternative degree d(|ui) by r s X X 1 1 (−µi + ) + (−νj − ). d(|ui) = 2 2 i=1
j=1
Note that this alternative degree d(|ui) is different form the degree of the Maya diagram associated to |ui. We shall only need this alternative degree in this proof and here we shall just refer to it as the degree. Let Fd be the subspace of F spanned by the elements of degree d. The filtration F• on F is defined by M Fp F = Fd . d≤p
Note that Fp F = 0 for p < 0 and F0 F = C|0i + C| − 1i. By definition we have that ψ µ Fp F ⊂ Fp−µ+1/2 F.
ψν Fp F ⊂ Fp−ν−1/2 F, Moreover we have Gr•F F =
M
(4.3)
Fd .
Also let us introduce filtrations on C((ξ)) and C(ξ))dξ by Fp C((ξ)) = C[[ξ]]ξ −p ,
Fp C(ξ))dξ = C[[ξ]]ξ −p dξ.
Then we have that Gr•F C((ξ)) = C[ξ, ξ −1 ],
Gr•F C((ξ))dξ = C[ξ, ξ −1 ]dξ.
Put ψ[Fp C((ξ))dξ] = { ψ[ω(ξ)] | ω(ξ) ∈ Fp C((ξ))dξ } ψ[Fp C((ξ))] = { ψ[f (ξ)] | f (ξ) ∈ Fp C((ξ)) }
By (4.3) it is easy to show that following facts ψ[Fp C((ξ))dξ]Fq F ⊂ Fp+q F,
ψ[Fp C((ξ))]Fq F ⊂ Fp+q F.
let X = (C, Q, ξ) be a one-pointed curve of genus g with coordinate. The filtrations F• on C((ξ)) and C((ξ))dξ induce the ones on H 0 (C, OC (∗Q)) and H 0 (C, ωC (∗Q)) and we have that Fp H 0 (C, OC (∗Q)) = H 0 (C, OC (pQ)),
Fp H 0 (C, ωC (∗Q)) = H 0 (C, ωC (pQ)).
Thus the filtrations are compatible with the actions of ψ[ω] and ψ[f ] to F for ω ∈ H 0 (C, ωC (∗Q)) and f ∈ H 0 (C, OC (∗Q)). Hence the inclusion F(X) ⊂ F is compatible with the filtrations and the quotient space V(X) has the induced filtration F• . Thus we conclude that Gr•F V(X) = Gr•F F/Gr•F F(X). 25
Let 1 = n1 < n2 < · · · < ng = 2g − 1 be the Weierstrass gap values of the curve C at Q. Then Gr•F H 0 (C, OC (∗Q)) ⊂ C[ξ −1 ] does not contain ξ −ni , i = 1, . . . , g but contain ξ −n , n 6= ni , i = 1, . . . , g. Also Gr•F H 0 (C, ωC (∗Q)) contains all ξ −n dξ, n ≤ 0. Thus |ui = ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i may not be in Gr•F V(X) if s = 0 and {µ1 , . . . , µr } ⊂ {−n2 + 1/2, . . . , −ng + 1/2}. Thus Gr•F V(X) is of finite dimensional. Moreover if V is the subspace spanned by ψ −m1 +1/2 ψ −m2 +1/2 · · · ψ −mr +1/2 |0i with 2g − 1 ≥ m1 > m2 > · · · > mr ≥ 0 where 1 ≤ r ≤ 2g, then V is a finite dimensional vector space and the natural linear map V → Gr•F V(X) is surjective. Now for a family F = (π : C → B; s; ξ) of one-pointed curves of genus g with coordinate, we can introduce filtrations on F(B), F(F) , π∗ OC (∗S) and π∗ ωC/B (∗S) of OB -submodules in the same way as above so that we can introduce a filtration on V(F). Moreover, there is an OB -module homomorphism V ⊗C OB → Gr•F V(F) which is surjective by virtue of (4.2) and the above argument. Thus Gr•F V(F) is a coherent OB -module. Therefore, V(F) is also a coherent OB -module. This proves the proposition for the one-pointed case. The general case can be proved similarly or we can use Theorem 3.4 to reduce the general case to the one-pointed case. QED Let us now show the local freeness of the sheaves Vab (F) and V † ab (F). For that purpose we first introduce a certain OB -submodule L(F) of N M j=1
OB ((ξj−1 ))
d dξj
and an action of L(F) on the sheaves Vab (F) and V † ab (F) as first order twisted differential operators. This action will also be used to define a connection on the sheaves Vab (F) and V † ab (F). First recall that we have an exact sequence 0 → ΘC/B (−S) → ΘC/B (mS) →
N M m M j=1 k=0
OB ξj−k
d → 0, dξj
for any positive integer m. From this exact sequence we obtain the exact sequence b
m 0 → π∗ (ΘC/B (mS)) −→
m N M M j=1 k=0
OB ξj−k
d ϑm 1 −→ R π∗ ΘC/B (−S) → 0. dξj
(4.4)
Hence, we have the following exact sequence of OB -modules b
0 → π∗ (ΘC/B (∗S)) −→
N M j=1
OB [ξj−1 ] 26
d ϑ −→ R1 π∗ ΘC/B (−S) → 0. dξj
(4.5)
Note that the mappings b and bm correspond to the Laurent expansions with respect to ξj up to zero’th order. To define the first order differential operators acting on the sheaves Vab (F) and V † ab (F), we need to rewrite the exact sequence (4.5) in the following way. There is an exact sequence dπ
0 → ΘC/B → ΘC −→ π ∗ ΘB → 0, where ΘC/B is the sheaf of vector fields tangent to the fibers of π. Put Θ′C,π = dπ −1 (π −1 ΘB (− log D)). Hence, Θ′C,π is the sheaf of vector field on C tangent to Σ whose horizontal components are constant along the fibers of π. That is, Θ′C,π consists of germs of holomorphic vector fields of the form m′ n X X ∂ ∂ ∂ bi (u)ui + + , a(z, u) bi (u) ∂z ∂ui ∂ui ′ i=1
i=m +1
where (z, u1 , . . . , un ) is a system of local coordinates such that the mapping π is expressed as the projection π(z, u1 , . . . , un ) = (u1 , . . . , un ) and π(Σ) = D is given by the equation u1 · u2 · · · um′ = 0. More generally, we can define the sheaf Θ′C (mS)π as the one consisting of germs of meromorphic vector fields of the form ′
m n X X ∂ ∂ ∂ A(z, u) Bi (u)ui + + , Bi (u) ∂z ∂ui ∂ui ′ i=1
i=m +1
where A(z, u) has a poles of order at most m along S. Now we have an exact sequence dπ
0 → ΘC/B (mS) → Θ′C (mS)π −→ π −1 ΘB (− log D) → 0.
(4.6)
Note that Θ′C (mS)π has the structure of a sheaf of Lie algebras by the usual bracket operation of vector fields and the above exact sequence is one of sheaves of Lie algebras. For m > N1 (2g − 2), by (4.6) we have an exact sequence of OB -modules. dπ
0 → π∗ ΘC/B (mS) −→ π∗ Θ′C (mS)π −→ ΘB (− log D) → 0,
(4.7)
which is also an exact sequence of sheaves of Lie algebras. Taking m → ∞ we obtain the exact sequence dπ
0 → π∗ ΘC/B (∗S) −→ π∗ Θ′C (∗S)π −→ ΘB (− log D) → 0.
27
(4.8)
The exact sequences (4.5) and (4.8) are related by the following commutative diagram.
|| 0 → π∗ ΘC/B (∗S) →
dπ
π∗ Θ′C (∗S)π
0 → π∗ ΘC/B (∗S) →
−→
ΘB (− log D)
→
0
↓ p
↓ ρ
−1 d j=1 OB [ξj ] dξj
−→ R1 π∗ ΘC/B (−S) → 0,
LN
ϑ
where ρ is the Kodaira-Spencer mapping of the family F and p is given by taking the non-positive degree part of the dξdj part of the Laurent expansions of the vector fields in π∗ Θ′C (∗S)π at sj (B). In the following, for simplicity we assume that the Kodaira-Spencer mapping ρ is an injective homomorphism of OB -modules. Therefore, p is an injection. Let pe :
be defined by taking the pe is injective. Put
π∗ Θ′C (∗S)π
−→
N M j=1
OB ((ξj ))
d dξj
d part of the Laurent expansions at sj (B). Since p is injective, dξj L(F) := pe(π∗ Θ′C (∗S)π ).
(4.9)
Then, we have the exact sequence
θ
0 → π∗ ΘC/B (∗S) −→ L(F) −→ ΘB (− log D) → 0
(4.10)
of OB -modules. This is the exact sequence corresponding to (4.5). The Lie bracket [ , ]d on L(F) is obtained from the bracket on π∗ Θ′C (∗S)π by the mapping pe. Thus, for ~ℓ, m ~ ∈ L(F) we have that [~ℓ, m] ~ d = [~ℓ, m] ~ 0 + θ(~ℓ)(m) ~ − θ(m)( ~ ~ℓ)
(4.11)
where [·, ·]0 is the usual bracket of formal vector fields and the action of θ(~ℓ) on d d m ~ = m1 , . . . , mN dξ1 dξN is defined by
d d ~ ~ , . . . , θ(ℓ)(mN ) . θ(ℓ)(m1 ) dξ1 dξN
Then, the exact sequence (4.10) is also an exact sequence of sheaves of Lie algebras. Let us define an action of L(F) on FN (B). Definition 4.2. For ~ ℓ = (ℓ1 , . . . , ℓN ) ∈ L(F), the action of D(~ℓ) on FN (B) is defined by D(~ℓ)(F ⊗ |ui) = θ(~ℓ)(F ) ⊗ |ui − F · 28
N X j=1
ρj (T [ℓj ]) |ui,
(4.12)
where F ∈ OB ,
|ui ∈ FN ,
and T [ℓ] = Res(T (z)ℓ(z)dz). z=0
The following proposition can be proved in the same manner as Proposition 4.2.2 of [U2]. Proposition 4.2. The action D(~ℓ) of ~ℓ ∈ L(F) on FN (B) defined above has the following properties. 1. For any F ∈ OB we have
D(F ~ℓ) = F D(~ℓ).
2. For ~ℓ, m ~ ∈ L(F) we have N
1X Res [D(~ℓ), D(m)] ~ = D([~ℓ, m] ~ d) − ξj =0 6 j=1
d3 ℓj mj dξj dξj3
!
· id.
3. For F ∈ OB and |φi ∈ FN (B) we have D(~ℓ)(F |φi) = (θ(~ℓ)(F ))|φi + F D(~ℓ)|φi. Namely, D(~ℓ) is a first order differential operator, if θ(~ℓ) 6= 0. † We define the dual action of L(F) on FN (B) by
D(~ℓ)(F ⊗ hΦ|) = θ(~ℓ)(F ) ⊗ hΦ| + where F ∈ OB ,
N X j=1
F · hΦ|ρj (T [ℓj ]).
(4.13)
† hΦ| ∈ FN (B).
† Then, for any |ui ∈ FN (B) and hΦ| ∈ FN (B), we have
e + hΦ|{D(~ℓ)|Φi} e = θ(~ℓ)hΦ|Φi. e {D(~ℓ)hΦ|}|Φi
(4.14)
This agrees with the usual definition of the dual connection. Now we shall show that the operator D(~ℓ) acts on Vab (F). Proposition 4.3. For any ~ ℓ ∈ L(F) we have D(~ℓ)(Fab (F)) ⊂ Fab (F).
Hence, D(~ℓ) operates on Vab (F). Moreover, it is a first order differential operator, if θ(~ℓ) 6= 0.
29
Proof.
An element of Fab (F) is an OB -linear combination of elements of the form F⊗
N X j=1
ρj (ψ[ωj ])|ui ,
F′ ⊗
N X j=1
ρj (ψ[fj ])|ui
where F, F ′ ∈ OB ,
ω ∈ π∗ (ωC/B (∗S)),
f ∈ π∗ OC (∗S),
|ui ∈ FN
and fj is the Laurent expansion of f along Sj = sj (B) with respect to the coordinate ξj and likewise for ω. First we shall show the following equality as operators on FN (B) h
D(~ℓ),
N X j=1
N i X ρj (ψ[fj ]) = ρj ψ θ(~ℓ)(fj ) + ℓj (fj ) ,
(4.15)
j=1
where θ(~ℓ) operates on the coefficients of fj . By Proposition 4.2, (3) it is enough to show the equality (4.15) as operators on FN . For |ui ∈ FN , by (4.12) we have N N X X ρj (ψ[fj ])(D(~ℓ)|ui) D(~ℓ) ρj (ψ[fj ])|ui − j=1
j=1
=
N N X X {ρj (ψ[θ(~ℓ)(fj )]) − ρj (T [ℓj ])ρj (ψ[fj ])}|ui + ρj (ψ[fj ])ρj (T [ℓj ])|ui j=1
=
j=1
N X {ρj ψ θ(~ℓ)(fj ) + ℓj (fj ))} |ui. j=1
This implies equation (4.15). Now θ(~ℓ)(fj ) + ℓj (fj ) is nothing but the Laurent expansion at sj (B) of the meromorphic function τ (h) where τ = pe−1 (~ℓ) ∈ π∗ (Θ′C (∗S)π ). Hence, we 0 (F) ⊂ F 0 (F). have the result that D(~ℓ)F(0,1 (0,1 Next let us consider ω ∈ π∗ (ωC/B (∗S)). For τ = pe−1 (~ℓ) ∈ π∗ (Θ′ (∗S)π ) put C
ω e=
d exp{tτ }∗ (ω) . dt
Then the Laurent expansion of ω e along Sj is written as where ωj = Aj (ξj )dξj and
ω ej = ℓj (Aj (ξj )) + ω (ℓj ) , ω (ℓj ) = Aj (ξj )
dℓj (ξj ) . dξj
Now we are ready to prove the following equality as operators on FN (B) h
D(~ℓ),
N X j=1
N i X ρj ψ ω ej . ρj (ψ[ωj ]) = − j=1
30
(4.16)
Note that ω e ∈ π∗ (ωC/B (∗S)). Hence, the equality (4.16) implies the desired result. To prove (4.16) it is enough to show that for ℓ =
∞ X
ln z n
n=−n0
d dz
ω = A(z)dz = (
∞ X
am z m )dz
m=−m0
we have that
h
i T [ℓ], ψ[ω] = −ψ[{ℓ(z)A′ (z) + A(z)ℓ′ (z)}dz].
This can be proved as follows. First note that by (3.2) we have that i h Ln , ψk+1/2 = −(n + k + 1)ψn+k+1/2 . Hence
h
i i h X ak ln+1 Ln , ψk+1/2 T [ℓ], ψ[ω] = n,k
= −
X (n + k + 1)ak ln+1 ψn+k+1/2 n,k
= −ψ[{ℓ(z)A′ (z) + A(z)ℓ′ (z)}dz]. 1 (F) ⊂ F 1 (F). From this we conclude that D(~ℓ)Fab ab
QED
Proposition 4.4. For each element ~ℓ ∈ L(F), D(~ℓ) acts on V † ab (F). Moreover, if θ(~ℓ) 6= 0, then D(~ℓ) acts on V † ab (F) as a twisted first order differential operator. Proof.
Choose
e ∈ V † ab (F). hΨ|
For any element f ∈ π∗ (OC (∗S)) and |ui ∈ FN (B), by Proposition 4.3 and (4.14) we have that ~ℓ)(ψ[f ]|ui)} = 0. e e e {D(~ℓ)hΨ|}(ψ[f ]|ui = θ(~ℓ)(hΨ|ψ[f ]|ui) − hΨ|{D(
Thus we conclude that
e ∈ V † ab (F). D(~ℓ)hΨ|
The remaining statement is an easy consequence of definition (4.13).
QED
Corollary 4.1. The OB -module V † ab (F) is locally free on B \ D. For a proof see [U2], Proposition 4.2.4. For a coherent OB -module G, the locus M consisting of points at which G is not locally free, is a closed analytic subset of B of codimension at least 2. Therefore, we have the following corollary.
31
Corollary 4.2. Let W be the maximal subset of B over which Vab (F) is not locally free. Then, W is an analytic subset of B and W $ D. Since we defined V † ab (F) = HomOB (Vab (F), OB ), we have the following corollary. Corollary 4.3. V † ab (F)|B\D is a locally free OB -module and for any subvariety Y of B \D we have an OY -module isomorphism OY ⊗OB V † ab (F) ≃ V † ab (F|Y ). These two corollaries play a crucial role in proving locally freeness in general. The above corollaries imply the following theorem. Theorem 4.1. If F is a family of N -pointed smooth curves with formal coordinates, then Vab (F) and V † ab (F) are locally free OB -modules and they are dual to each other. Note that for the bilinear pairing h·|·i : Vab (F) × V † ab (F) → OB , we have the equality {D(~ℓ)hΨ|}|ui + hΨ|{D(~ℓ)|ui} = θ(~ℓ)(hΨ|ui).
(4.17)
Now let us define a connection of V † ab (F). First note that by direct calculations we have that dψ(ζ ) j hΦ|ρj ψ(ξj ) |ui dζj ∞ ∞ X X (m + 1)hΦ|ρj (ψm+1/2 ψ n−1/2 )|uiζj−m−2 ξj−n =− m=−∞ n=1 ∞ X
+
∞ X
(m + 1)hΦ|ρj (ψ −n−1/2 ψm+1/2 )|uiζj−m−2 ξjn −
m=−∞ n=0
hΦ|ui . (ζj − ξj )2
This implies hΦ|ρj (T (ξj ))|ui(dξj )2 dψ(ζ ) hΦ|ui j dζ dξ = lim hΦ|ρj ψ(ξj ) |uidζj dξj + j j . ζj →ξj dζj (ζj − ξj )2 Note that we can as in [U2, 3.4] define the correlation functions hΦ|
dψ(w) ψ(z)|uidwdz, dw 32
hΦ|T (z)|uidz 2
(4.18)
such that 2
hΦ|T (z)|uidz = lim
w→z
dψ(w) hΦ|ui hΦ| dwdz . ψ(z)|uidwdz + dw (w − z)2
∗ ∗ Note that hΦ| dψ(w) dw ψ(z)|uidwdz is a meromorphic section of p1 ωC/B ⊗OC×B p2 ωC/B over B whoseLaurent expansion in a neighbourhood of Sj × Sj is equal to the above expansion dψ(ζ )
of hΦ|ρj dζjj ψ(ξj ) |ui, where pi : C ×B C → C is the projection to the i-th factor. Now choose a meromorphic bidifferential ω ∈ H 0 (C ×B C, ωC×B C (2∆)) which has the form ω(w, z) =
dwdz + holomorphic (w − z)2
in a neighbourhood of the diagonal ∆ of C ×B C. Define Te(z) by dψ(w) 2 e hΦ|ρj (T (z))|ui(dz) = lim hΦ| ψ(z)|uidwdz + hΦ|uiω(w, z) . w→z dw
(4.19)
Then hΦ|ρj (Te(z))|ui(dz)2 is a global meromorphic form, that is a meromorphic section of ωC/B over B. The projective connection Sω (z)(dz)2 associated to the bidifferential ω is defined by n dwdz o Sω (z)(dz)2 = 6 lim ω(w, z) − . (4.20) w→z (w − z)2
Then, by (4.19) we have that
1 hΦ|ρj (T (z))|ui(dz)2 = hΦ|Te(z))|ui(dz)2 − hΦ|uiSω (z)2 (dz)2 . 6
(4.21)
Then for any element |ui ∈ FN by (4.12) hΦ|{D(~ℓ)|ui} = −
N X j=1
Res ℓj (ξj )hΦ|ρj (Te(ξj ))|uidξj
ξj =0
X 1 Res ℓj (ξj )Sω (ξj )dξj . + hΦ|ui ξj =0 6 N
j=1
If θ(~ℓ) = 0, that is ~ ℓ is the image of a global vector field τ ∈ π∗ (ΘC/B (S)), then ℓj (ξj )hΦ|ρj (Te(ξj ))|uidξj
is the Laurent expansion of a global meromorphic relative one-form which has poles only at Sj . Hence the first term of the right hand side of the above equality is zero. Therefore, in this case we have that X 1 Res ℓj (ξj )Sω (ξj )dξj . hΦ|{D(~ℓ)|ui} = hΦ|ui ξj =0 6 N
j=1
33
Now put bω (~ℓ) =
N X j=1
Res ℓj (ξj )Sω (ξj )dξj .
ξj =0
(4.22)
Then this defines an OB -module homomorphism and if θ(~ℓ) = 0 then we have that
bω : L(F) → OB ,
1 hΦ|{D(~ℓ)|ui} = bω (~ℓ)hΦ|ui. 6 For a vector field X ∈ Θ(− log D) on B tangent to D choose ~ℓ ∈ L(F) such that θ(~ℓ) = X. Then the connection on Vab (F) is defined by 1 (ω) ∇X ([|Ψi]) = D(~ℓ)([|Ψi]) − bω (~ℓ)([|Ψi]), 6
(4.23)
for |Ψi ∈ Vab (F). Dually the connection on V † ab (F) is defined by 1 (ω) ∇X (hΦ|) = D(~ℓ)(hΦ|) + bω (~ℓ)(hΦ|), 6
(4.24)
for hΦ| ∈ V † ab (F). These are well-defined, that is the right hand sides of (4.23) and (4.24) are independent of the choice of ~ ℓ ∈ L(F) with θ(~ℓ) = X. Just like for the non-abelian conformal field theory (see for example [U2], section 5) we can prove the following theorem. Theorem 4.2. The operator ∇(ω) defines a projectively flat holomorphic connection of the sheaves Vab (F) and V † ab (F). Moreover, the connection has a regular singularity along the locus D ⊂ B which is the locus of the singular curves. The connection ∇(ω) depends on the choice of bidifferential ω and if we choose another bidifferential ω ′ then there exists a holomorphic one-form φω,ω′ on B such that (ω)
(ω ′ )
∇X − ∇X
1 = hφω,ω′ , Xi. 6
(4.25)
Moreover, the curvature form R is given by X o d3 ℓj 1n Res bω (~n) − X(bω (m)) ~ + Y (bω (~ℓ)) − mj dξj , ξj =0 dξj 6 N
R(X, Y ) =
(4.26)
j=1
where X,Y ∈ ΘC/B (∗S)), ~ℓ, m ~ ∈ L(F) with X = θ(~ℓ), Y = θ(m), ~ and ~n ∈ L(F) is defined ~ by ~n = [ℓ, m] ~ d (see (4.11)). Remark 4.1. For the connection on the sheaf of non-abelian vacua V †~λ (F) of level l over B with gauge symmetry g, a complex simple Lie algebra, the curvature form Rg is given by cv Rg(X, Y ) = R(X, Y ) · id 2 where l · dim g , g∗ is the dual Coxeter number of g. cv = l + g∗ For the details see [U2], section 5. 34
5
Degeneration of Curves and Sewing
From a curve with one ordinary double point, we construct via the usual sewing a oneparameter family of curves over a disk, whose fibers over the punctured disk are smooth curves. We shall show that the sheaf of j = 0 ghost vacua gives an invertible sheaf over the whole disk. This implies that the dimension of the space of ghost vacua is one for any pointed curve. The arguments of this section are almost the same, as those from [U2] section 5, we just need to modify a few arguments in our setting. First we recall certain basic facts about the energy-momentum tensor and the fermion operators. In the j = 0 ghost system, the energy-momentum tensor T (z) is given by dψ(z) T (0) (z) = ·· ψ(z) ·· . dz Note that we have the formula Ln =
∞ X
kψ n−k+1/2 ψk−1/2 +
∞ X
kψ−k−1/2 ψ n+k+1/2 .
(5.1)
k=1
k=0
If n 6= 0 then by the anti-commutation relation (1.5) we have that ψ n−k+1/2 ψk−1/2 = −ψk−1/2 ψ n−k+1/2 and ψ−k−1/2 ψ n+k+1/2 = −ψ n+k+1/2 ψ−k−1/2 . Thus the normal ordering (0)
is non-trivial only for L0 and we have that X X (0) L0 = (ν + 1/2)ψ −ν ψν + (ν − 1/2)ψ−ν ψ ν . ν>0
(5.2)
ν>0
(0)
In the following, we often use the notation T (z), Ln instead of T (0) (z) and Ln . By (3.2) and (3.3) we have that [Ln , ψν ] = −(n + ν + 1/2)ψn+ν ,
[Ln , ψ ν ] = −(n + ν + 1/2)ψ n+ν + (n + 1)ψ n+ν .
(5.3) (5.4)
In particular we have that [L0 , ψν ] = −(ν + 1/2)ψν ,
[L0 , ψ ν ] = −(ν − 1/2)ψ ν .
(5.5)
Note that the degree of the element |ui = ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i with µ1 < µ2 < · · · < µr < 0, ν1 < ν2 < · · · < νs < 0 is given by d(|ui) = −
r X i=1
µi −
s X j=1
νj − p2 /2
where p = r − s is the charge of the element. By (5.2) and (5.5) we thus see that L0 |ui = (d + p(p + 1)/2)|ui. Hence we have the following lemma. 35
(5.6)
Lemma 5.1. On Fd (p) the operator L0 acts by multiplication by the scalar d + p(p + 1)/2. Lemma 5.2. For any element |ui ∈ Fd (p) and any half integer ν ∈ Zh we have that deg(ψν |ui) = d + p − (ν + 1/2),
deg(ψ ν |ui) = d − p − (ν + 1/2).
Proof. Note that the charge of ψν |ui is p − 1 and the charge of ψ ν |ui is p + 1. By (5.5) we have L0 (ψν |ui) = ψν (L0 |ui) − (ν + 1/2)ψν |ui = {d + p(p + 1)/2 − (ν + 1/2)}ψν |ui. Hence, we get that deg(ψν |ui) = d + p(p + 1)/2 − (ν + 1/2) − p(p − 1)/2 = d + p − (ν + 1/2). By the same way we can show that L0 (ψ ν |ui) = {d + p(p + 1)/2 − (ν − 1/2)}ψν |ui. Hence deg(ψ ν |ui) = d − p − (ν + 1/2). QED Let |ui = ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i µ1 < µ2 < · · · < µr < 0,
ν1 < ν2 < · · · < νs < 0,
and M ∈ M(p) be the Maya diagram corresponding to |ui. Then Ps
|M i = (−1)
i=1
νi +s/2
ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i.
Let us introduce a bilinear pairing ( | ) : F(p) × F(p) → C by putting (|M i||N i) = hM |N i = δM,N for Maya diagrams M, N ∈ M(p). The pairing is perfect on Fd (p) × Fd (p) and zero on Fd (p) × Fd′ (p) for d 6= d′ . Note that the bilinear pairing ( | ) is symmetric. Lemma 5.3. (ψν |ui | |vi) = (|ui | ψ −ν |vi),
for |ui ∈ F(p + 1), |vi ∈ F(p),
(ψ ν |ui | |vi) = (|ui | ψ−ν |vi),
for |ui ∈ F(p − 1), |vi ∈ F(p).
36
Proof. Assume
It is enough to consider the case in which |ui and |vi are Maya diagrams. Ps
|M i = (−1)
i=1
νi +s/2
ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i,
with µ1 < µ2 < · · · < µr < 0, ν1 < ν2 < · · · < νs < 0. Then we have that Ps
ψν |M i = (−1)
i=1
νi +s/2+r
ψ µ1 ψ µ2 · · · ψ µr ψν ψνs ψνs−1 · · · ψν1 |0i.
For simplicity assume that ν > νs . Then (ψν |M i | |N i) 6= 0 if and only if the Maya diagram N has the form Ps
|N i = (−1)
νi +ν+s/2+1/2
i=1
ψ µ1 ψ µ2 · · · ψ µr ψν ψνs ψνs−1 · · · ψν1 |0i.
In this case (ψν |M i | |N i) = (−1)ν+1/2+r . On the other hand ψ −ν |N i = (−1)ν+1/2+r |M i. Hence (ψν |M i | |N i) = (|M i | ψ −ν |N i). Other cases can be treated similarly.
QED
Let r : F(p) → F(−p) be a mapping defined by taking the mirror image of the Maya diagram with respect to 0 and interchange white and black. Namely, if Ps
|M i = (−1)
i=1
νi +s/2
ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i,
with µ1 < µ2 < · · · < µr < 0, ν1 < ν2 < · · · < νs < 0, then r(M ) is defined by Pr
|r(M )i = (−1)
j=1
µj +r/2
ψ ν1 ψ ν2 · · · ψ νs ψµr ψµr−1 · · · ψµ1 |0i.
Note that we have charge(r(M )) = − charge(M ) and deg(r(M )) = deg(M ). Hence, we have a linear map r : Fd (p) → Fd (−p). Now for any integer p let us introduce a bilinear pairing { | } : Fd (p) × Fd (−p) → C. For that purpose put α(n) =
1, n ≡ 1, 2 −1, n ≡ 0, 3
(mod 4) (mod 4)
Then it is easy to show that we for any n ∈ Z get that α(n + 1) = (−1)n+1 α(n).
37
(5.7)
The bilinear pairing { | } is defined by {|ui | |vi} = α(charge(|ui))(|ui | r(|vi)).
(5.8)
In particular for Maya diagrams M ∈ M(p) and N ∈ M(−p) we have that {|M i||N i} = α(p)hM |r(N )i = α(p)δM,r(N ) . Note that the pairing { | } : Fd (p) × Fd (−p) → C is perfect and that it can be extended to F(p) × F(−p) → C by requiring that the pairing is zero on Fd (p) × Fd′ (−p) for d 6= d′ . Now from the above Lemma 5.3 we obtain the following results. Lemma 5.4. {ψν |ui | |vi} = (−1)−ν−1/2 {|ui | ψ−ν |vi},
{ψ ν |ui | |vi} = (−1)−ν−1/2 {|ui | ψ −ν |vi}. Proof.
By the definition of the mapping r we have that ψν r(|vi) = (−1)−ν+1/2+charge(|vi) r(ψ −ν |vi),
ψ −ν r(|vi) = (−1)−ν+1/2+charge(|vi) r(ψ−ν |vi).
Hence by Lemma 5.3 and (5.7) for |ui ∈ F(p + 1) and |vi ∈ F(−p) we see that {ψν |ui | |vi} = α(p)(ψν |ui | r(|vi)) = α(p)(|ui | ψ −ν r(|vi)) = (−1)−ν+1/2+p α(p)(|ui | r(ψ−ν |vi))
= (−1)−ν−1/2 α(p + 1)(|ui | r(ψ−ν |vi))
= (−1)−ν−1/2 {|ui | ψ−ν |vi}. By a similar argument we can show the second equality.
QED
Let us introduce the shift operator s : F(p) → F(p + 1) by defining s(µ)(ν) = µ(ν − 1) + 1, for a Maya diagram M with characteristic function µ. E.g. s(ψ µ1 ψ µ2 · · · ψ µr ψνs ψνs−1 · · · ψν1 |0i)
= ψ µ1 −1 ψ µ2 −1 · · · ψ µr −1 ψνs +1 ψνs−1 +1 · · · ψν1 +1 |1i
= (−1)s ψ µ1 −1 ψ µ2 −1 · · · ψ µr −1 ψ −1/2 ψνs +1 ψνs−1 +1 · · · ψν1 +1 |0i +δνs ,− 1 ψ µ1 −1 ψ µ2 −1 · · · ψ µr −1 ψνs−1 +1 · · · ψν1 +1 |0i 2
with µ1 < µ2 < · · · < µr < 0, ν1 < ν2 < · · · < νs < 0. Now the bilinear pairings { | }+ : F(p) × F(−p − 1) → C is defined by {|ui | |vi}+ = {|ui | s(|vi)} = α(charge(|ui))(|ui, | r(s(|vi)).
(5.9)
Note that the pairing { | }+ is perfect on Fd (p) × Fd (−p − 1) and zero on Fd (p) × Fd′ (−p − 1) for d 6= d′ . Finally we obtain the following result which play an important role in the sewing procedure. 38
Lemma 5.5. We have the following equalities {ψν |ui | |vi}+ = (−1)−ν−1/2 {|ui | ψ−ν−1 |vi}+ ,
{ψ ν |ui | |vi}+ = (−1)−ν−1/2 {|ui | ψ −ν+1 |vi}+ . Proof.
These results follow from Lemma 5.4 and (5.9).
QED
Let us now consider a one-parameter family of curves which is a family of smooth curves degenerating to a single nodal curve. Let C0 be a complete curve with only one ordinary double point P such that C0 \{P } is e0 → C0 non-singular. Let Q1 , Q2 , . . . , QN be distinct non-singular points on C0 . Let ν : C −1 be the normalization of the singular curve. Put {P+ , P− } = ν (P ). e0 , ΘC (∗ PN Qj )) and a Lemma 5.6. There exist a meromorphic vector field ˜l ∈ H 0 (C 0 j=1 coordinate neighbourhood U+ (resp. U− ) of the point P+ (resp. P− ) with local coordinate z with center P+ (resp. w with center P− ) such that we have U+ = { z | |z| < 1 }, ˜l|U = 1 z d , + 2 dz
U− = { w | |w| < 1 } ˜l|U = 1 w d . − 2 dw
This lemma is a special case of lemma 5.3.1 in [U2]. For the convenience of the read we give the proof in this special case here. Proof. Choose coordinate neighbourhood U+ (resp. U− ) and a local coordinate x with center P+P(y with center P− ). Then we can find a meromorphic vector field ˜l ∈ H 0 (C e0 , ΘC (∗ N Qj )) such that 0 j=1 ˜l|U = ( 1 x + a2 x2 + · · · ) d , + 2 dx
˜l|U = ( 1 y + b2 y 2 + · · · ) d . − 2 dy
Then, by solving the following differential equations ˜l(z) = ˜ l(w) =
1 dz z , z(0) = 0, 2 dx 1 dw w , w(0) = 0, 2 dy
we obtain holomorphic functions z and w in neighbourhoods of P+ and P− , respectively. It is easy to show that z and w give local coordinates with center P+ and P− , respectively. Moreover, choosing U± smaller if necessary, by the differential equations we have 1 d ˜ l|U+ = z , 2 dz
˜l|U = 1 w d . − 2 dw
Replacing z by cz with c 6= 0 and w by c′ w with c′ 6= 0 we may assume that U+ = { z | |z| < 1 },
U− = { w | |w| < 1 } QED
39
Let us now construct a flat one-parameter deformation of C0 . We do this just like in section 5.3 in [U2]. Let D = { q ∈ C | |q| < 1 }
S = { (x, y, q) ∈ C3 | xy = q, |x| < 1, |y| < 1, |q| < 1 } e0 × D | P ∈ C e0 \ (U+ ∪ U− ), or P ∈ U+ , |z(P )| > |q| } Z = { (P, q) ∈ C e0 × D | Q ∈ C e0 \ (U+ ∪ U− ), or Q ∈ U− , |w(Q)| > |q| } W = { (Q, q) ∈ C
Now introduce an equivalence relation ∼ on Z ⊔ S ⊔ W as follows. 1) 2) 2)
Z ∩ (U+ × D) ∋ (P, q) ∼ (x, y, q ′ ) ∈ S ⇐⇒ (x, y, q ′ ) = (z(P ), q/z(P ), q)
W ∩ (U− × D) ∋ (Q, q) ∼ (x, y, q ′ ) ∈ S ⇐⇒ (x, y, q ′ ) = (q/w(P ), w(P ), q) Z ∋ (P, q) ∼ (Q, q ′ ) ∈ W ⇐⇒ (P, q) = (Q, q ′ )
Let C be the two-dimensional complex manifold obtained as the quotient of Z ⊔ S ⊔ W by this equivalence relation. There is a proper holomorphic map π : C → D such that the fiber over the origin is C0 and for q 6= 0 the fiber Cq = π −1 (q) is a non-singular curve of e0 is connected or not, where the genus of C e0 is g. The family genus g +1 or g according as C π : C → D is a flat deformation (or smoothing) of C0 . Let us assume that Qj ∈ / U+ ∪ U− . Then, the point Qj ∈ C0 \ {P } defines a holomorphic section σi : D → C. Let us choose a local coordinate ξj with center Qi . Put F = (π : C → D; σ1 , . . . , σN ; ξ1 , . . . , ξN ). The meromorphic vector field ˜l given in Lemma 5.6 defines local vector fields ~l = (l1 (ξ1 ) d , l2 (ξ2 ) d , . . . , lN (ξN ) d ) dξ1 dξ2 dξN where lj (ξj ) dξdj is the Laurent expansion of ˜l with respect to the coordinate ξj with center Qj . Lemma 5.7. The local vector fields ~l is an element of L(F) and we have d θ(~l) = q . dq Proof.
The meromorphic vector field ˜l − q q dq
P is a holomorphic section of π∗ Θ′C (∗ N j=1 σj (D))π over D. This show the first part of the proposition. A proof of the second part can be found in [U2, Corollary 5.3.3 ]. QED Theorem 5.1. Put
and
e = (C e0 ; P+ , P− , Q1 , . . . , QN , z, w, ξ1 , . . . , ξN ) X X = (C0 ; Q1 , . . . , QN , ξ1 , . . . , ξN ). 40
e there exists a unique global flat section hΦ(q)| e Then for any element hΦ| ∈ V † ab (X) of † † e the sheaf of ghost vacua V ab (F) over the disk such that hΦ(0)| is the element of V ab (X) corresponding to the element hΦ|, i.e. e hΦ(0)|ui = hΦ|0+,− ⊗ ui
for any |ui ∈ FN .
Proof. Let {vi (d, p)}i=1,...,md be a basis of Fd (p) for any p ∈ Z and {v i (d, p)}i=1,...,md be the dual basis of Fd (−p − 1) with respect to the pairing { | }+ . Note that F0 (p) is spanned by |pi over C and we choose v1 (0, p) = |pi for all p ∈ Z. Then v 1 (0, −p − 1) = α(p)| − p − 1i. e Define hΦ(q)| by e hΦ(q)|ui =
md ∞ X o XnX (−1)p+d hΦ|vi (d, p) ⊗ v i (d, −p − 1) ⊗ ui q d+p(p+1)/2 .
(5.10)
p∈Z d=0 i=1
e Then by Theorem 3.5 hΦ(0)| ∈ V † ab (X) is the element corresponding to hΦ|. e e Let us show that hΦ(q)| satisfies the gauge conditions. Since hΦ(q)| is defined as a formal power series, we will first prove the formal gauge conditions. Later we shall show e that hΦ(q)| satisfies a differential equation of Fuchsian type so that it converges. Then, the formal gauge conditions implies the usual gauge conditions. Choose τ ∈ π∗ (ωC/B (∗S)). In a neighbourhood of the double point P in C, τ can be expressed in the form X an,m z n wm dz ∧ dw/dq τ=
n,m≥0
where dz ∧ dw/dq is a local basis of ωC/B such that
Now put
dz ν ∗ ( dz ∧ dw/dq )|U+ = − , z τ+ =
X
∞
an,m z
n
n,m≥0
τ− =
X
dw ν ∗ ( dz ∧ dw/dq )|U− = . w
an,m
n,m≥0
∞
X X q m dz am,n z m−n−1 dz q n (− ) = − z z
q n m dw = w w w
n=0 m=0 ∞ ∞ X X
n=0 m=0
an,m wm−n−1 dw) q n ,
and (n)
τ+
(n)
τ−
= − =
∞ X
m=0
∞ X
m=0
Then τ+ =
∞ X
am,n z m−n−1 dz,
an,m wm−n−1 dw.
(n)
τ+ q n ,
τ− =
∞ X
n=0
n=0
41
(n)
τ− q n .
e0 , ω e (∗(P+ + P− + It is easy to show that there is a unique τ (n) ∈ H 0 (C C0
PN
j=1 Qj )) such (n) (n) (n) {τ+ , τ− , τ (ξ)} define a
(n) {τ±
is the Taylor expansion of τ (n) at P± . the data that meromorphic one-form . Hence by the first gauge condition for hΦ| we see that (n)
hΦ|ψ[τ+ ]vi (d, p) ⊗ v i (d, −p − 1) ⊗ ui (n)
+ (−1)p hΦ|vi (d, p) ⊗ ψ[τ− ]v i (d, −p − 1) ⊗ ui
−
N X (n) hΦ|vi (d, p) ⊗ v i (d, −p − 1) ⊗ ρj (ψ[τj ])ui = 0. j=1
Hence N X ∞ ∞ N X X X (n) (n) n n e e hΦ(q)|ρ τj q ])ui = hΦ(q)|ρj (ψ[ j (ψ[τj ])uiq j=1 n=0
n=0 m ∞ ∞ d XXXX
j=1
=
p∈Z n=0 d=0 i=1
n (n) (−1)p+d hΦ|ψ[τ+ ]vi (d, p) ⊗ v i (d, −p − 1) ⊗ ui
o (n) +(−1)p hΦ|vi (d, p) ⊗ ψ[τ− ]v i (d, −p − 1) ⊗ ui q d+n+p(p+1)/2 .
Note that
(n) ψ[τ+ ] (n)
= −
ψ[τ− ] =
∞ X
(5.11)
am,n ψm−n−1/2 ,
m=0 ∞ X
an,m ψm−n−1/2 .
m=0
Then by Lemma 5.5 md X (n) hΦ|ψ[τ+ ]vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+n+p(p+1)/2 i=1
=− =−
md X ∞ X
am,n hΦ|ψm−n−1/2 vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+n+p(p+1)/2
i=1 m=0 md md+p−m+n ∞ X X X
m=0 i=1
j=1
am,n {ψm−n−1/2 vi (d, p) | v j (d + p − m + n, −p)}+
· hΦ|vj (d + p − m + n, p − 1) ⊗ v i (d, −p − 1) ⊗ uiq d+n+p(p+1)/2 =−
md md+p−m+n ∞ X X X
m=0 i=1
j=1
(−1)n−m am,n {vi (d, p) | ψn−m−1/2 v j (d + p − m + n, −p)}+
· hΦ|vj (d + p − m + n, p − 1) ⊗ v i (d, −p − 1) ⊗ uiq d+n+p(p+1)/2
=
∞ md+p−m+n X X
m=0
j=1
(−1)n−m+1 am,n hΦ|vj (d + p − m + n, p − 1) ⊗ ψn−m−1/2 v j (d + p − m + n, −p) ⊗ uiq d+n+p(p+1)/2 . 42
Hence md ∞ X X (n) (−1)d+p hΦ|ψ[τ+ ]vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+n+p(p+1)/2 n=0 i=1 ∞ md+p−n+m ∞ X X X
=
n=0 m=0
j=1
(−1)d−n+m+p+1 an,m hΦ|vj (d + p − n + m, p − 1)
⊗ ψm−n−1/2 v j (d + p − n + m, −p) ⊗ uiq d+m+p(p+1)/2 .
Put
de = d + p − n + m.
Then
de + n + p(p − 1)/2 = d + m + p(p + 1)/2.
Hence the right hand side of the last equality can be rewritten in the form m
∞ X ∞ X de X e e e p − 1) ⊗ ψm−n−1/2 v j (d, e −p) ⊗ uiq d+n+p(p−1)/2 (−1)d+1 an,m hΦ|vj (d,
n=0 m=0 j=1 ∞ mde
=
XX e e e −p) ⊗ uiq d+n+p(p−1)/2 e p − 1) ⊗ ψ[τ (n) ]v j (d, . (−1)d+1 hΦ|vj (d, −
n=0 j=1
Hence
md ∞ X ∞ X XX
(n)
(−1)p+d hΦ|ψ[τ+ ]vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+n+p(p+1)/2
p∈Z n=0 d=0 i=1 md ∞ X ∞ X XX
=− =−
(n)
(−1)d hΦ|vj (d, p − 1) ⊗ ψ[τ− ]v j (d, −p) ⊗ uiq d+n+p(p−1)/2
p∈Z n=0 d=0 i=1 md ∞ X ∞ X XX
p∈Z n=0 d=0 i=1
(n)
(−1)d hΦ|vj (d, p) ⊗ ψ[τ− ]v j (d, −p − 1) ⊗ uiq d+n+p(p−1)/2
Therefore, by (5.11) we conclude that ∞ N X X (n) e τj q n ])ui = 0. hΦ(q)|ρ j (ψ[ n=0
j=1
This proves the first formal gauge condition. Similarly, we can show that the second formal gauge condition also holds. Next we shall show that the formal power series (5.10) is a formal solution of a Fuchsian differential equation. Hence the power series (5.10) indeed converges. Therefore the formal gauge conditions are nothing but the usual gauge conditions. Note that on the punctured disk D∗ we have the connection introduced in section 4. For the local vector field ~l defined by the meromorphic vector field e l in Lemma 5.6 the corresponding connection has the form N
q
X d e 1 ~ e e hΦ(q)|ρ (hΦ(q)|ui) − j (T [lj ])ui − bω (l)hΦ(q)|ui = 0 dq 6 j=1
43
(5.12)
where ω is an element of ⊠2 H 0 (C ×D C, ωC/D (2∆))
with ω=
dvdu + holomorphic at ∆, (v − u)2
where ∆ is the diagonal of C ×D C and if pj : C ×D C → C is the projection onto the j’th factor, then ⊠2 ωC/D = p∗1 ωC/D ⊗ p∗2 ωC/D . 2 e e The formal correlation function PN hΦ(q)|T (z)|uidz which will be later proved to be an 0 ⊗2 element of H ((C, ωC/D ) (∗ j=1 sj (D)) is defined in (4.19):
1 e dψ(w) e ψ(z)|uidwdz − ω(w, z)hΦ(q)|uidwdz . hΦ(q)| w→z 2 dw
e hΦ(q)| Te(z)|uidz 2 = lim
Then, by (4.21) we have that
1 e 2 e e hΦ(q)| Te(z)|uidz 2 = hΦ(q)|T (z)|uidz 2 + hΦ(q)|uiS ω (z)dz . 6
(5.13)
d Let ˜l = l(z) be the meromorphic vector field defined in Lemma 5.6. For q 6= 0 dz ˜ e e l · hΦ(q)| Te(z)|uidz 2 = l(z)hΦ(q)| Te(z)|uidz
is a meromorphic one-form on
Cq′ = Cq \ { (x, y, q) ∈ S | |x| ≤ ǫ, |y| ≤ ǫ } where ǫ < 1 is a sufficiently small positive number.
∧
∨
γ+
γ−
∧
∨
44
The boundary of Cq′ consists of two disjoint curves γ± . We choose the orientation of γ± in such a way that Cq′ lies to the left of γ± . Then, we have that Z Z 1 1 e e e √ l(w)hΦ(q)| Te(w)|uidw l(z)hΦ(q)|T (z)|uidz + √ 2π −1 γ+ 2π −1 γ− =
N X j=1
=
N X j=1
=
N X j=1
On the curve γ+
e ResQj {l(z)hΦ(q)| Te(z)|uidz}
e hΦ(q)| Res (l(ξj )T (ξj )dξj )ui + ξj =0
N
1X e Res {l(ξj )Sω (ξj )dξj hΦ(q)|ui} ξj =0 6 j=1
˜ e e hΦ(q)|ρ j (T [l(ξj )])ui + bω (l)hΦ(q)|ui. ˜l = 1 z d . 2 dz
Hence 1 √
Z
e l(z)hΦ(q)| Te(z)|uidz Z 1 1 e e dz. = √ z hΦ(q)|T (z)|ui + Sω (z)hΦ(q)|ui 6 4π −1 γ+
2π −1
γ+
Since Sω (z)dz 2 is holomorphic at z = 0 Z Z 1 1 e e √ l(z)hΦ(q)| Te(z)|uidz = √ zhΦ(q)|T (z)|uidz 2π −1 γ+ 4π −1 γ+
On the other hand Z 1 e √ zhΦ(q)|T (z)|uidz 4π −1 γ+ Z ∞ md (−1)p+d 1 XXX √ zhΦ|T (z)|vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+p(p+1)/2 dz = 2 2π −1 γ+ p∈Z d=0 i=1 ∞ m
d 1 XXX (−1)p+d hΦ|L0 (vi (d, p)) ⊗ v i (d, −p − 1) ⊗ uiq d+p(p+1)/2 = 2
=
1 2
p∈Z d=0 i=1 md ∞ X XX p∈Z d=0 i=1
(−1)p+d (d +
p(p + 1) )hΦ|vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+p(p+1)/2 . 2
Similarly 1 √
2π −1
Z
γ−
e l(w)hΦ(q)| Te(w)|uidw ∞ m
d p(p + 1) 1 XXX (−1)p+d (d + )hΦ|vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+p(p+1)/2 . = 2 2
p∈Z d=0 i=1
45
Thus we obtain that N X ˜ e hΦ(q)|ρ j (T [l(ξj )])ui + bω (l)hΦ|ui j=1
=
md ∞ X XX p(p + 1) (−1)p+d (d + )hΦ|vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+p(p+1)/2 . 2
p∈Z d=0 i=1
On the other hand ∞ m
q
d XXX p(p + 1) d e (−1)p+d (d + hΦ(q)|ui = )hΦ|vi (d, p) ⊗ v i (d, −p − 1) ⊗ uiq d+p(p+1)/2 . dq 2
p∈Z d=0 i=1
e Hence hΦ(q)|ui is a formal solution of the differential equation N
X 1 ˜ d e e hΦ(q)|ρ q hΦ(q)|ui − j (T [l(ξj )])ui − bω (l)hΦ|ui = 0 dq 6 j=1
e which is of Fuchsian type. Hence hΦ(q)|ui converges for all u ∈ FN .
QED
Now let us give a proof of Theorem 3.2. By (4.2) and Theorem 4.1 dim V † ab (X) depends only on the genus of the curve C and the number N of the points, if the curve C is non-singular. Moreover, by Theorem 3.4 dim V † ab (X) depends only on the genus of the curve C provided C is non-singular. Put d(g) = dim V † ab (X)
for a non-singular curve C of genus g. Now assume that the curve C has one double point P . Let F = (π : C → D; s+ , s− , ξ1 , . . . , ξN ) be the family constructed above so that F(q), q 6= 0 is a non-singular curve of genus g. Then, by (4.2) and Proposition 4.1 d(g) = dim V † ab (X) = dim Vab (F(0)) ≥ dim Vab (F(q)) = dim V † ab (F(q)). Moreover, by Theorem 3.5 we have that e = d(g − 1). dim V † ab (X) = dim V † ab (X)
On the other hand by Theorem 5.1
dim V † ab (F(0)) ≤ dim V † ab (F(q)). Hence d(g − 1) = dim V † ab (F(0)) ≤ dim V † ab (F(q)) = d(g). If the curve C has m double points, applying Theorem 3.5 m times we conclude that dim V † ab (X) is independent of X. In Example 3.1, we showed that dim V † ab ((P1 ; 0; z)) = 1. Hence, by Theorem 3.4 for any X we have that dim V † ab (X) = 1. QED 46
Corollary 5.1. The connection (5.12) extends holomorphically over q = 0. e Proof. Since hΦ(q)| is a single valued holomorphic function and the differential equation (5.12) is of Fuchsian type, N X 1 ~ e e hΦ(q)|ρ j (T [lj ])ui + bω (l)hΦ(q)|ui 6 j=1
must be divisible by q. Hence the differential equation has no singularity at q = 0. QED Theorem3.2 implies the following important fact. Theorem 5.2. For a family F = (π : C → B; s1 , . . . , sN ; ξ1 , . . . , ξN ) of N -pointed semi-stable curves with formal neighbourhoods the sheaf V † ab (F) of ghost vacua and the sheaf Vab (F) of dual ghost vacua are invertible OB -modules. Moreover, they are dual to each other. Proof. By Theorem3.2 and (4.2), for any point b ∈ B the vector space Vab (F) ⊗OB OB,b /mb is one-dimensional. As Vab (F) is a coherent OB -module, this implies that Vab (F) is an invertible OB -module. Then, by (4.1) V † ab (F) is also an invertible OB -module. QED Next we shall consider smoothing of a family of nodal curves with sections and formal coordinates. We need to generalize the above construction. Let e = (e F π : Ce → E; s1 , . . . , sN , t+ , t− ; ξ1 , . . . , ξN , z, w)
be a family of (N + 2)-pointed smooth curves with formal coordinates. Assume that N ≥ 1 and E is a small polydisk {(u1 , . . . , um ) ∈ Cm | |ui | < ε, 1 ≤ i ≤ m }, and z, w are holomorphic coordinates such that there exists a meromorphic vector field e l ∈ e Θ e (∗S)) with H 0 (C, C/E
1 d 1 d e l|X = z , e l|Y = w . 2 dz 2 dw There always exist such coordinates by Lemma 5.6. Put S=
N X
sj (E),
S± = t+ (E) + t− (E).
j=1
Then for any positive integer M we have and exact sequence de π
e ∗ ΘE → 0 0 → ΘC/E e (−S± + M S) → ΘCe(−S± + M S) → π
is the sheaf of relative holomorphic vector fields and ΘZ denotes the sheaf of where ΘC/E e holomorphic vector fields on some complex manifold Z. Put π −1 (e π −1 ΘE ). ΘCe(−S± + M S)πe = de Taking the inductive limit on M we can also define ΘCe(−S± + ∗S)πe . 47
Similarly for any positive integer M we have an exact sequence 0 → ΘC/E e (−S± + M S) → e (−S± − S) → ΘC/E
N M M M j=1 k=0
OE ξj−k
d → 0. dξj
From these exact sequences we have a commutative diagram de π
e∗ ΘCe(−S± + ∗S)πe → 0 → π e∗ ΘC/E e (−S± + ∗S) → π k
ΘE
↓p
0 → π e∗ ΘC/E e (−S± + ∗S) →
LN
−1 d j=1 OE [ξj ] dξj
→ 0
↓ρ
→ R1 π e∗ ΘC/E e (−S± − S) → 0
For simplicity assume that the OE -homomorphism ρ is injective. Therefore, p is also injective. Let N M d OE ((ξj )) pe : π e∗ ΘCe(−S± + ∗S)πe → dξj j=1
be the Laurent expansions of the
d dξj
part along sj (E) and put
b = pe(e L(F) π∗ ΘCe(−S± + ∗S)πe ) ⊂
N M j=1
OE ((ξj ))
d . dξj
Then we have an exact sequence
θb
b 0→π e∗ ΘC/E e (−S± + ∗S) → L(F) → ΘE → 0.
Hence we can find local vector fields
(i)
m ~ (i) = m1 b such that which are sections of L(F)
d (i) d , . . . , mN dξ1 dξN
d bm θ( ~ (i) ) = . dui
Note that by (4.9) we can define N M d d d e ⊂ ⊕ OE ((z)) ⊕ OE ((w)) OE ((ξj )) L(F) dξj dz dw j=1
and we have the exact sequence
b θe 0→π e∗ ΘC/E e (∗S) → L(F) → ΘE → 0. 48
(5.14)
Then the above argument shows that if we put ~e (i) = m(i) d , . . . , m(i) d , 0, 0 , m 1 N dξ1 dξN
then
~e (i) ) = d . em θ( dui
(5.15)
b = (b Let us identify t+ (u) with t− (u) for each u ∈ E and obtain a family F π : Cb → e b E; s1 , . . . , sN ; ξ1 , . . . , ξN ) of N -pointed nodal curves. Let ν : C C be this identification map. b can be constructed as follows. For further details see A flat deformation of this family F section 5.3 in [U2]. For 0 < ε ≤ 1 put Xε = { P ∈ Ce | |z(P )| < ε } Yε = { P ∈ Ce | |w(P )| < ε }
and also put X = X1 , Y = Y1 . Choose 0 < ε1 < ε2 < 1 and choose an open covering {Uα }4≤α≤s of Ce \ (Xε2 ∪ Yε2 ) such that Uα ∩ Xε1 = ∅,
Uα ∩ Yε1 = ∅.
Put D = { q ∈ C | |q| < 1 }
S0 = { (x, y, q) ∈ C3 | xy = q, |x| < 1, |y| < 1, |q| < 1 } S = S0 × E
Z = { (P, q) ∈ Ce × D | P ∈ Ce \ (X ∪ Y ) , or P ∈ X & |z(P )| > |q| } W = { (P, q) ∈ Ce × D | P ∈ Ce \ (X ∪ Y ) , or P ∈ Y & |w(P )| > |q| }.
On Z ∪ S ∪ W let us introduce an equivalence relation ∼ as follows:.
1. A point (P, q) ∈ Z ∩ (X × D) and a point (x, y, q ′ , u) ∈ S are equivalent if and only if q , q, π e(P ) . (x, y, q ′ , u) = z(P ), z(P )
2. A point (P, q) ∈ W ∩ (Y × D) and a point (x, y, q ′ , u) ∈ S are equivalent if and only if q (x, y, q ′ , u) = , w(P ), q, π e(P ) . w(P ) 3. A point (P, q) ∈ Z and a point (Q, q ′ ) ∈ W are equivalent if and only if (P, q) = (Q, q ′ ).
Now put C = Z ∪ S ∪ W/ ∼. Then, C is a complex manifold and there is a natural e holomorphic mapping π : C → E × D. Moreover C \ (X ∪ Y ) × D is contained in C as an open subset. Hence we can define holomorphic sections sj by sj : E × D → C
(u, q) → (sj (u), q) ∈ Z. 49
Then it is easy to show that F = (π : C → B = E × D; s1 , . . . , sN ; ξ1 , . . . , ξN ) is a family of N -pointed curves with formal coordinates. It is also easy to show that the family π : C → B = E × D is a versal family of N -pointed curves. Note that each fiber over q = 0 is a nodal curve. We identify E with E × {0}. Then we have an exact sequence. θ
0 → π∗ ΘC/B (∗S) → L(F) → ΘB (− log E) → 0. Lemma 5.8. We can choose a local vector field m ~ (i) in (5.14) in such a way that it can be regarded as a section of L(F) over B with θ(m ~ (i) ) =
d dui
Proof. Since dud i ∈ H 0 (B, ΘB (− log E)), we can choose a τ (i) ∈ π∗ Θ′C,π (∗S) such that θ(e p(τ (i) )) = dud i , provided we choose B small enough. e \ Xε ∪ Yε2 chosen above. Put Let {Uα }4≤α≤s be the open covering of C U0 = {(x, y, u1 , . . . , un , q) ∈ S | x 6= 0 },
U2 = X × D,
U3 = Y × D,
U1 = {(x, y, u1 , . . . , un , q) ∈ S | y 6= 0 }
Uα = Uα × D,
4 ≤ α ≤ s.
Then, {Uα }1≤α≤s is an open covering on C. Let (wα , u1 , . . . , um , q) be local coordinates of Uα , where w0 = y, w1 = x, w2 = x, w3 = y. In these coordinates our vector field τ (i) is expressed as τ (i) |Uα = Aα (wα , q, u)
d d d + + B(u, q)q . dwα dui dq
ˇ On the other hand the Kodaira-Spencer class ρ( dud i ) is expressed by a Cech cocycle θαβ =
∂fαβ d ∂ui dwα
where on Uα ∩ Uβ 6= ∅ we write wα = fαβ (wβ , u1 , . . . , um , q). Hence, if we put θα = −Aα (wα , q, u)
d d − B(u, q)q dwα dq
then we have θαβ = θβ − θα .
Note that for α, β ≤ 4, fαβ is independent of q. Therefore on Ce \ Xε2 ∪ Yε2 × D Aα (wα , 0, u)
d d d + + B(u, 0)q , dwα dui dq 50
α≥4
′
define a global meromorphic vector filed τe(i) . On the other hand for α, β ≤ 3, fαβ is independent of u1 , . . . , um . On U0 ∩ U2 we have w2 = q/w0 .
Hence we have d d d w2 d d d A0 (w0 , u, q) + +B(u, q)q = (− A0 (q/w2 , u, q)+w2 B(u, q)) + +B(u, q)q . dw0 dui dq w0 dw2 dui dq Therefore
w2 A0 (q/w2 , u, q) + w2 B(u, q). w0 Since A0 (w0 , u, q) and A2 (w2 , u, q) are holomorphic, we conclude that A0 (w1 , u, q) has the form A0 (w0 , u, q) = a0 (u, q)w0 , A2 (w2 , u, q) = −
and on U0 ∩ U2
A2 (w2 , u, q) = (−a0 (u, q) + B(u, q))w2 ,
where a0 (u, q) is holomorphic in u and q. In this way we can show that on Uα , α ≤ 3, Aα has the form Aα (wα , u, q) = aα (u, q)wα , where aα (u, q) is holomorphic in u and q. Also the above calculations show that aα (u, 0)wα
d d d + + B(u, 0)q , dwα dui dq
α≤3 ′
define a holomorphic vector field on ∪3α=0 Uα . This vector field coincides with τe(i) on the intersection of Ce \ Xε ∪ Yε2 × D and ∪3α=0 Uα . Thus we have a global meromorphic (i) vector field τe on C. Then, by our construction if we put m ~ (i) = pe(e τ (i) ), it is independent of q and we have θ(m ~ (i) ) = dud i . Now restrict τe(i) to q = 0 and denote this restriction τe(i) (0). Then the above calculation shows that τe(i) (0)I ⊂ I where I is the ideal defining the double point. Hence, ν ∗ (e τ (i) (0)) is d ∗ (i) ∗ (i) ˜ p(ν τe (0))) = e and θ(˜ ˜(ν τe (0)) = m ~ (i) . a meromorphic vector field on C dui . Note that p QED † † e we can construct hΨ(q)| ∈ V ab (F) using formula (5.10): Now from hΦ| ∈ V ab (F) e hΦ(q)|ui =
md ∞ X o XnX (−1)p+d hΦ|vi (d, p) ⊗ v i (d, −p − 1) ⊗ ui q d+p(p+1)/2 .
(5.16)
p∈Z d=0 i=1
Then by (4.12), (5.14) and (5.15) e hΦ(q)|D( m ~ (i) )|ui = − =−
N X (i) e hΦ(q)|ρ j (T [m ])|ui j
j=1
md ∞ X N XnX X
o ′ (i) (−1)p+d hΦ|ρj (T [mj ])vi′ (d, p) ⊗ v i (d, −p − 1) ⊗ ui q d+p(p+1)/2
j=1 p∈Z d=0 i′ =1 md ∞ X X nX
=−
o ~e (i) )vi′ (d, p) ⊗ v i′ (d, −p − 1) ⊗ ui q d+p(p+1)/2 . (−1)p+d hΦ|D(m
p∈Z d=0 i′ =1
51
Let ω(q) = ω(x, y, u.q)dxdy ∈ H 0 (C⊗E×D C, ω ⊠2 (2∆)) be a bidifferential with Res2 (ω) = 1. e Note that ω(0) is a bidifferential on Ce ×E C. (ω(q)) Then, by (4.24) the connection ∇ d is defined by dui
∇
(ω(q)) d dui
1 e e e hΦ(q)| = D(m ~ (i) )hΦ(q)| + bω(q) (m ~ (i) )hΦ(q)|. 6
Therefore, if we start from hΨ| = ∇ the sewing procedure, we have that
d dui
e and construct hΨ(q)| e hΦ| ∈ V † ab (F) ∈ V † ab (F) by
1 e e e ~ (i) )hΦ(q)|. hΨ(q)| = D(m ~ (i) )hΦ(q)| + bω(0) (m 6
Therefore, we have ∇
(ω(0))
(ω(q)) d dui
1 e e e hΦ(q)| − hΨ(q)| = bω(q) (m ~ (i) ) − bω(0) (m ~ (i) ) hΦ(q)|. 6
(5.17)
Thus we obtain the following theorem.
Theorem 5.3. Let F = (π : C → B = E × D; s1 , . . . , sN ; ξ1 , . . . , ξN ) be a family of N pointed curves with formal coordinates such that the restriction of the family to E is the family of nodal curves F = (π : Cb → E; s1 , . . . , sN ; ξ1 , . . . , ξN ). Let e = (e F π : Ce → E; s1 , . . . , sN , t+ , t− ; ξ1 , . . . , ξN , z, w)
b be the family of (N + 2)-pointed curves obtained by normalization of each fiber of F. † † 0 e e For hΦ| ∈ V ab (F) let hΦ(q)| ∈ V ab (F) be obtained by sewing. Let ω(q) ∈ H (C ⊗E×D e C, ω ⊠2 (2∆)) be a bidifferential with Res2 (ω) = 1. Let hΨ(q)| ∈ V † ab (F) be constructed from (ω(0)) e by the sewing. Then we have that hΨ| = ∇ d hΦ| ∈ V † ab (F) dui
∇
(ω(q)) d dui
1 e e e hΦ(q)| − hΨ(q)| = bω(q) (m ~ (i) ) − bω(0) (m ~ (i) ) hΦ(q)|. 6
Remark 5.1. For the non-abelian case the same result holds, provided one replaces the coefficient 1/6 by cv /12.
6
Formal Coordinates and Preferred Sections
In this section we shall study the behavior of the ghost vacua under coordinate changes. We shall also define a preferred section which is a local holomorphic section of the sheaf of ghost vacua V † ab (F) of a family of one-pointed smooth curves with formal coordinates. First we recall some basic facts about the relationship between coordinates change and local vector fields. We let D be the automorphism group Aut C((ξ)) of the field C((ξ)) of formal Laurent series as a C-algebra. The group D may be regarded as the automorphism group of the ring C[[ξ]] of formal power series. There is a natural isomorphism D
≃
{
∞ X
n=0
an ξ n+1 | a0 6= 0 }
h 7→ h(ξ)
52
where the composition h ◦ g of h, gP ∈ D corresponds to a formal power series h(g(ξ)). In n+1 | a 6= 0 }. the following we identify D with { ∞ 0 n=0 an ξ Put D p = { h ∈ D | h(ξ) = ξ + ap ξ p+1 + · · · } for a positive integer p. Then we have a filtration D = D0 ⊃ D1 ⊃ D2 ⊃ . . . Put also d = C[[ξ]]ξ
d dξ
dp = C[[ξ]]ξ p+1
d dξ
p = 0, 1, 2, . . .
Then, we have a filtration d = d0 ⊃ d1 ⊃ d2 ⊃ · · · For any l ∈ d and f (ξ) ∈ C[[ξ]] define exp(l)(f (ξ)) by ∞ X 1 k exp(l)(f (ξ)) = (l f (ξ)). k! k=0
Also put 0 = { h ∈ D | h(ξ) = aξ + a1 ξ 2 + · · · , a > 0 } D+ d d0+ = { l(ξ) | l(ξ) = αξ + α1 ξ 2 + · · · , α ∈ R } dξ
Then, we have the following result. Lemma 6.1. The exponential map exp : d → D
l 7→ exp(l)
is surjective. Moreover, the exponential map induces an isomorphism 0 exp : d0+ ≃ D+ .
Since, for any integer n, we have √ d exp(2πn −1ξ ) = id, dξ the exponential mapping is not injective on d. 0 we define exp(T [l]) by For the energy-momentum tensor T (z) and any element l ∈ D+ exp(T [l]) =
∞ X 1 T [l]k . k! k=0
53
Then, exp(T [l]) operates on F from the left and on F † from the right. 0 , there exist a unique l ∈ d0 with By Lemma 6.1, for any automorphism h ∈ D+ + 0 exp(l) = h. Now for h ∈ D+ define the operator G[h] by G[h] = exp(−T [l]) where exp(l) = h. Then, G[h] operates on F from the left and on F † from the right. Then we have the following important results. 0 , f (ξ)dξ ∈ C((ξ))dξ, g(ξ) ∈ C((ξ)) and l = l(ξ) d ∈ Theorem 6.1. For any h ∈ D+ dξ d † C((ξ)) dξ , we have the following equalities as operators on F and F .
G[h](ψ[f (ξ)dξ])G[h]−1
= ψ[h∗ (f (ξ)dξ)] = ψ[f (h(ξ))h′ (ξ)dξ]
G[h](ψ[g(ξ)])G[h]−1 = ψ[h∗ (g(ξ))] = ψ[g(h(ξ))] G[h1 ◦ h2 ] = G[h1 ]G[h2 ] G[h]T [l]G[h]−1 = T [ad(h)(l)] + where {f (ξ); ξ} is the Schwarzian derivative.
1 Res {h(ξ); ξ}l(ξ)dξ . 6 ξ=0
A proof is easily given by applying (5.3) and (5.4). From this theorem we infer easily the following proposition. 0 , j = 1, 2, . . . , N and N -pointed curve Proposition 6.1. For any hj ∈ D+
X = (C; Q1 , Q2 , . . . , Qn ; ξ1 , ξ2 , . . . , ξN ) with formal coordinates, put X(h) = (C; Q1 , Q2 , . . . , QN ; h1 (ξ1 ), h2 (ξ2 ), . . . , hN (ξN )). b · · · ⊗G[h b Then, the isomorphism G[h1 ]⊗ N] † FN
† → FN
b · · · ⊗φ b N | 7→ hφ1 G[h1 ]⊗ b · · · ⊗φ b N G[hN ]| hφ1 ⊗
induces the canonical isomorphism
† † b · · · ⊗G[h b G[h1 ]⊗ N ] : V ab (X) → V ab (X(h) )
Let X = (C; Q; ξ) be a one-pointed smooth curve of genus g with a formal coordinate. We shall show that if we fix a symplectic basis {α1 , . . . , αg , β1 , . . . , βg } of H1 (C, Z), then there is a canonical preferred non-zero vector hω(X, {α, β})| ∈ V † ab (X) which is a refinement of the construction given in Lemma 3.1. Let us choose a normalized basis {ω1 , . . . , ωg } of holomorphic one-forms on C which is characterized by Z ωj = δij , 1 ≤ i, j ≤ g. (6.1) βi
54
The matrix τ = (τij ),
τij =
Z
ωj
αi
is then called the period matrix of the curve C. Now the numbers Ini , n = 1, 2, . . ., i = 1, . . . g are defined by ∞ X Ini ξ n−1 )dξ. ωi = ( n=1
Ini
depend on the symplectic basis {α1 , . . . , αg , β1 , . . . , βg } and the Note that the numbers coordinate ξ. (n) For a positive integer n ≥ 1 let ωQ be a meromorphic one-form on C which has a pole of order n + 1 at Q and holomorphic elsewhere such that √ Z Z 2π −1Ini (n) (n) ωQ = − (6.2) , ωQ = 0, 1 ≤ i ≤ g n αi βi ∞ X 1 (n) qn,m ξ m−1 dξ. (6.3) + ωQ = n+1 ξ m=1 (n)
These conditions uniquely determine ωQ . Note that the second equality of (6.2) and (6.3) imply the first equality of (6.2). The preferred element hω(X, {α, β})| ∈ V † ab (X) is defined by hω(X, {α, β})| = · · · e(ωg+2 ) ∧ e(ωg+1 ) ∧ e(ωg ) ∧ · · · ∧ e(ω1 ),
where
(n)
ωg+n = ωQ . For details see Lemma 3.1. We call {ωn }, n = 1, 2, . . . a normalized basis for X. Note that the normalized basis depends on the choice of a symplectic basis of H1 (C, Z) and the coordinate ξ. 0 put X = {C; Q; η = h(ξ)}. Then Theorem 6.2. For h(ξ) ∈ D+ h
hω(X, {α, β})|G[h] = hω(Xh , {α, β})|, where G[h] : V † (Xh ) → V † (X) is the canonical isomorphism given in Proposition 6.1 Proof. First consider the case in which h(ξ) = aξ for a positive number a. Put (n) η = aξ. Let ωi , i = 1, . . . , g and ωg+n = ωQ be chosen for X as above. Then ωi = (n) ωQ
=
∞ X
n=1 an
a−n−1 Ini η n−1 dη,
η n+1
Put ω e i = ωi ,
(n)
ω eQ
+
∞ X
m=1
a−m−1 qn,m η m−1 dη.
1≤i≤g (n)
= a−n ωQ =
1
η n+1 55
+
∞ X
m=1
m−1 η dη. a−n−m−1 qn,m
Then, the element hXh , {α, β}| ∈ V † ab (Xh ) is given by he ω (X, {α, β})| = · · · e(e ωg+2 ) ∧ e(e ωg+1 ) ∧ e(e ωg ) ∧ · · · e(e ω1 ), where
(n)
ω eg+n = ω eQ .
Note that hω(X, {α, β})|, he ω (X, {α, β})| ∈ F(g − 1). Let α be the positive number such that exp(α) = a. Put d . dξ
l = αξ Then we have T [l] = αL0 ,
G[h] = exp(−αL0 ).
On Fd† (g − 1), G[h] operates by the multiplication by a−(d+g(g−1)/2) . The coefficient of · · · e−l1 −5/2 ∧e−l1 −3/2 ∧em1 +1/2 ∧e−l1 +1/2 ∧· · ·∧emk +1/2 ∧· · ·∧e−3/2 ∧eng +1/2 ∧· · ·∧en1 +1/2 (6.4) in hω(X, {α, β})| is A = ql1 ,m1 +1 · · · qlk ,mk +1 Ingg +1 · · · In11 +1 . The degree d of this term is given by formula (5.6) d=
g X
ni +
l X g(g − 1) . (mj + lj + 1) − 2 j=1
i=1
Thus we have that −d − g(g − 1)/2 = −
g X i=1
ni −
l X (mj + lj + 1). j=1
On the other hand the coefficient of hω(Xh , {α, β})| is a−
Pg
i=1
ni −
Pl
j=1 (mj +lj +1)
A = a−(d+g(g−1)/2) A.
This implies hω(X, {α, β})|G[h] = hω(Xh , {α, β})|.
Now let us consider the case in which h(ξ) is an element of D 1 ⊂ C((ξ)). Let {e ωn }, n = 1, 2, . . . be a normalized basis of Xh . Put ωn = h∗ (e ωn ). Since h(ξ) = ξ + a1 ξ 2 + · · · , the normalized basis of hX| is given by {ωn }, n = 1, 2, . . .. For any positive integer m all terms in hφm | = · · · e−m−5/2 ∧ e−m−3/2 ∧ e(ωm+g ) ∧ · · · ∧ e(ω1 ) 56
are also present in hω(X, {α, β})| by its definition. Moreover, we can express hφm | = h−m − 1|ψ[ωm+g ] · · · ψ[ω1 ]
= h−m − 1|ψ[h∗ (e ωm+g )] · · · ψ[h∗ (e ω1 )].
Then by Theorem 6.1 hφm | = h−m − 1|G[h]ψ[e ωm+g ] · · · ψ[e ω1 ]G[h]−1 . Note that h−m − 1| ≡ h−m − 1| mod F1 (−m − 1). It is easy to show that each term appearing in hφm |G[h] − h−m − 1|ψ[e ωm+g ] · · · ψ[e ω1 ]G[h]−1 . does not appear in hφm+1 | = h−m − 2|G[h]ψ[e ωm+1+g ] · · · ψ[e ω1 ]G[h]−1 . Taking the limit m → ∞ we conclude hω(X, {α, β})| = hω(Xh , {α, β})|G[h]−1 . QED Next let us study the dependence of the section hω(X, {α, β})| on the choice of symplectic basis. α1 , . . . α eg , βe1 , . . . , βeg } be symplectic bases Theorem 6.3. Let {α1 , . . . αg , β1 , . . . , βg } and {e 1 of H (C, Z) of the non-singular curve C. Assume that {β1 , . . . , βg } and {βe1 , . . . , βeg } span the same Lagrangian sublattice in H 1 (C, Z). Then e hω(X, {α, β})| = det U hω(X, {e α , β})|,
where U ∈ GL(g, Z) is defined by Proof.
βe1 .. . =U βeg
β1 .. . . βg
By the assumption on the β-cycles we have that t −1 α e U B α = , 0 U β βe
where B is a g × g integral matrix. For a normalized basis {ω1 , . . . , ωg } of holomorphic one-forms with respect to the symplectic basis {α, β} we have that Z ωj = U. βei
57
e Hence the normalized basis of holomorphic one-forms {e ω1 , . . . , ω fg } with respect to {e α, β} is given by (e ω1 , . . . , ω fg ) = (ω1 , . . . , ωg )U −1 . Thus
(Ien1 , . . . , Ieng ) = (In1 , . . . , Ing )U −1 ,
(n)
for all n’s. Therefore, for the normalized meromorphic one-form ωQ with respect to the symplectic basis {α, β} we have that R R R (n) (n) (n) ω ω ω Q Q Q αe1 . α1 . β1 . = t U −1 +B .. .. . R R R . (n) (n) (n) α e g ωQ α e g ωQ βg ωQ 1 1 Ien In = t U −1 ... = ... Ing Ieng R R (n) (n) ω β1 ωQ βe1 Q .. .. =0 = U . . R R (n) (n) ω ω βg Q βeg Q (n)
Thus ωQ is also the normalized meromorphic one-form with respect to the symplectic e Hence we have that basis {e α, β}. e hω(X, {α, β})| = det U hω(X, {e α , β})|.
(6.5)
QED
Note that det U = ±1. Next we shall show that hω(X)| is independent of the point on the curve C. For that purpose we need the following lemma. We use the basis {ω1 , . . . , ωg } of holomorphic one-forms on the curve C normalized by (6.1). Lemma 6.2. The numbers 1 = n1 < n2 values if and only if 1 In 1 . det .. 1 In g Proof.
If
det
In11 .. . In1g
< · · · < ng ≤ 2g − 1 are the Weierstrass gap ··· .. . ··· ··· .. . ···
Ingg .. .
Ingg Ingg .. .
Ingg
6= 0
= 0,
then there exists a vector (a1 , . . . , ag ) 6= (0, . . . , 0) such that 1 In1 · · · Ingg . .. .. (a1 , . . . , ag ) .. . . = (0, . . . , 0). In1g
···
58
Ingg
Form this we infer Res (ag ξ −ng + · · · + a1 ξ −n1 )ωi = 0,
i = 1, . . . , g.
ξ=0
This means that there is a meromorphic function f ∈ H 0 (C, OC (∗Q)) whose principal part at the point Q is ag ξ −ng + · · · + a1 ξ −n1 . Thus ng is not a Weierstrass gap value. On the other hand if 1 In · · · Ing1 1 . .. .. det .. . . 6= 0, 1 In · · · Ingg g
for any vector (a1 , . . . , ag ) 6= 0, we have 1 In1 .. (a1 , . . . , ag ) . In1g
This means that
··· .. . ···
Ing1 .. 6 (0, . . . , 0). . = g Ing
Res (ag ξ −ng + · · · + a1 ξ −n1 )ωi ) 6= 0, ξ=0
for a suitable 1 ≤ i ≤ g. Hence there does not exists a meromorphic function f ∈ QED H 0 (C, OC (∗Q)) whose principal part at Q is ag ξ −ng + · · · + a1 ξ −n1 . Let {P, Q} be two smooth points on the curve C with formal coordinates ξ, η, respectively. Put X0 = (C; P, Q; ξ, η), X1 = (C; P ; ξ), X2 = (C; Q; η). Then the natural imbeddings ι1 : F ֒→ F2
|ui 7→ |ui ⊗ |0i
ι2 : F ֒→ F2
|ui 7→ |0i ⊗ |ui
induce canonical isomorphisms V † ab (X0 ) ι∗1
∗
ց ι2
ւ
V † ab (X1 )
V † ab (X2 )
by Theorem 3.4. Theorem 6.4. Under the above notation we have ι∗2 ◦ (ι∗1 )−1 (hω(X1 , {α, β})|) = hω(X2 , {α, β})|.
59
Proof.
e = (ι∗ )−1 (hω(X1 , {α, β})|). Then, by Theorem 3.4 we have Put hΦ| 1 e ⊗ 0i = hω(X1 , {α, β})|ui hΦ|u
for any |ui ∈ F. Since the space of ghost vacua is one-dimensional, we have that e ⊗ ui = chω(X2 , {α, β})|ui hΦ|0
(6.6)
for a constant independent of |ui ∈ F. To determine c it is enough to choose a |ui such that we can calculate both sides of the equality (6.6). Let 1 = n1 < n2 < · · · < ng be the gap values at the point Q and put |ui = eng −1/2 ∧ · · · en1 −1/2 ∧ | − 1i. Let fj ∈ H 0 (C, OC (∗(P + Q)), j = 1, 2, . . . , g be chosen in such a way that fj has the Laurent expansion (j) (j) fj = η −nj + a0 + a1 η + · · · at Q. Also let ωQ,P be a meromorphic one-form having poles of order one at Q with residue 1 and at P with residue −1 and else holomorphic. Then we have that e ⊗ ui = hΦ|0 e ⊗ ρQ (ψ[fg ])ρQ (ψ[fg−1 ]) · · · ρQ (ψ[f1 ])| − 1i hΦ|0 e ⊗ ρQ (ψ[fg ])ρQ (ψ[fg−1 ]) · · · ρQ (ψ[f1 ])ρQ (ψ[ωQ,P ])0i = hΦ|0
e ⊗ ρQ (ψ[ωQ,P ])ρQ (ψ[fg ])ρQ (ψ[fg−1 ]) · · · ρQ (ψ[f1 ])0i = (−1)g hΦ|0 e P (ψ[ωQ,P ])0 ⊗ ρQ (ψ[fg ])ρQ (ψ[fg−1 ]) · · · ρQ (ψ[f1 ])0i = (−1)g+1 hΦ|ρ
e − 1 ⊗ ρQ (ψ[fg ])ρQ (ψ[fg−1 ]) · · · ρQ (ψ[f1 ])0i = (−1)g hΦ| e P (ψ[fg ])| − 1 ⊗ ρQ (ψ[fg−1 ]) · · · ρQ (ψ[f1 ])0i = (−1)g hΦ|ρ
e P (ψ[f1 ]) · · · ρP (ψ[fg−1 ])ρP (ψ[fg ])| − 1 ⊗ 0i = (−1)g+g(g−1)/2 hΦ|ρ e P (ψ[fg ])ρP (ψ[fg−1 ]) · · · ρP (ψ[f1 ])| − 1 ⊗ 0i = (−1)g hΦ|ρ
= (−1)g hω(X1 , {α, β})|ρP (ψ[fg ])ρP (ψ[fg−1 ]) · · · ρP (ψ[f1 ])| − 1i The last term can be expressed in a matrix form ResP (fg ω1 ) ResP (fg ω2 ) ResP (fg−1 ω1 ) ResP (fg−1 ω2 ) .. .. (−1)g . . ResP (f2 ω1 ) ResP (f2 ω2 ) ResP (f1 ω1 ) ResP (f1 ω2 )
··· ··· .. . ··· ···
But fj ωk has poles only at P and Q, thus
ResP (f2 ωg ) ResP (f1 ωg )
ResP (fg ωg ) ResP (fg−1 ωg ) .. .
ResP (fj ωk ) = − ResQ (fj ωk ) = −Inkj (Q) where ωk =
∞ X
Ink (Q)η n−1 dη.
n=1
60
Hence the above determinant is 1 In (Q) · · · Ingg (Q) g .. .. .. = hω(X2 , {α, β})|e ng −1/2 ∧ · · · ∧ en1 −1/2 ∧ | − 1i. . . . I 1 (Q) · · · Ing (Q) 1 n1 Hence we have that
e ⊗ en −1/2 ∧ · · · ∧ en −1/2 ∧ | − 1i = hω(X2 , {α, β})|e n −1/2 ∧ · · · ∧ en −1/2 ∧ | − 1i. hΦ|0 g g 1 1
This shows c = 1.
QED
e be the curve obtained by resolving the Let us consider a curve C with a node P . Let C e → C be the natural holomorphic mapping. Then π −1 (P ) singularity at P and let π : C e is connected. consists of two points P+ and P− . Assume that C Let X = (C; Q; ξ) be a one-pointed curve with formal coordinates and we let e = (C; e P+ , P− , Q; z, w, ξ) X
be the associated 3-pointed curve with formal coordinates. Then Theorem 3.5 says that the natural isomorphism |0+,− i ⊗ F ∼ =F induces the natural isomorphism
where
e ∼ ι∗+,− : V † ab (X) = V † ab (X).
|0+,− i = |0i ⊗ | − 1i − | − 1i ⊗ |0i. We can define hω(X, {α, β})| similar to the non-singular case, by choosing a basis {α, β} = {α1 , . . . , αg−1 , αg , β1 , . . . , βg−1 } of H1 (C, Z), in such a way that α1 , α2 , . . . , αg−1 and e Z) under natural map to β1 , β2 , . . . , βg−1 is the image of a symplectic basis of H1 (C, H1 (C, Z) and αg corresponds to the invariant cycle of a flat deformation of the curve C. Then we can choose a basis {ω1 , . . . , ωg−1 , ωg , ωg+1 , ωg+2 , . . .} of H 0 (C, ω(∗Q)) such e ω e (∗Q)) as in that {π ∗ ω1 , . . . , π ∗ ωg−1 , π ∗ ωg+1 , π ∗ ωg+2 , . . .} is a normalized basis of H 0 (C, C (6.1), (6.2) and (6.3) where we put (n)
π ∗ ωg+n = ωQ ,
n = 1, 2, . . . ,
e which has poles of order one at P+ and P− and π ∗ ωg is a meromorphic one-form on C with residue −1 and 1, respectively, is holomorphic outside P± and Z P− π ∗ ωg = 1. P+
Then put hω(X, {α, β})| = h· · · ∧ e(ωm ) ∧ · · · ∧ e(ω2 ) ∧ e(ω1 )|. 61
The proof of Lemma 3.1 applies also in this case and shows that hω(X, {α, β})| is an element of V † ab (X). Let b = (C, e Q; ξ). X
Then, by applying Theorem 3.4 at the points P± we have a canonical isomorphism e ∼ b ι∗ : V † ab (X) = V † ab (X).
Theorem 6.5. Under the above assumptions and notation we have that b {b b ι∗ ◦ (ι∗+,− )−1 (hω(X, {α, β})|) = (−1)g hω(X, α, β})|
b = {α1 , . . . , αg−1 , β1 , . . . , βg−1 }. where {b α, β} Proof.
Put
e = (ι∗ )−1 (hω(X, {α, β})|). hΦ| +,−
By the definition of ι∗ we have that
e ⊗ 0 ⊗ ui = hι∗ ◦ (ι∗+,− )−1 (hω(X, {α, β})|)|ui. hΦ|0
e O e (∗Q)) such that f (P+ ) = −1 and f (P− ) = Choose a meromorphic function f ∈ H 0 (C, C 0. Then by (3.18) (using the notation hω(X)| = hω(X, {α, β})|), we have that e ⊗ 0 ⊗ ui = hω(X)|ρQ (ψ[f ])ui hΦ|0
for any |ui ∈ F. The proof of lemma 3.1 show that for a meromorphic form ωj on C we have that ∧e(ωj )ψ[f ] = ResQ (f ωj ) = ResQ (f π ∗ ωj ) = ∧e(π ∗ ωj )ψ[f ]. For j 6= g, the meromorphic forms π ∗ ωj have only poles at Q. Hence ∧e(ωj )ψ[f ] = 0,
j 6= g.
On the other hand, since π ∗ ωg has a pole of order one at P+ and P− with residues −1 and 1, respectively, and is holomorphic outside the points P± and Q and f (P+ ) = −1, f (P− ) = 0 we have that ResQ (f π ∗ ωg ) = − ResP+ (f π ∗ ωg ) = −1. Thus we conclude that e ⊗ 0 ⊗ ui = hω(X)|ρQ (ψ[f ])ui = hω(X)ρQ (ψ[f ])|ui hΦ|0
= (−1)g−1 ResQ (f ωg )h· · · ∧ e(ωg+1 ) ∧ e(ωg−1 ) ∧ · · · ∧ e(ω1 )|ui b {b b = (−1)g hω(X, α, β})|ui.
QED
For an N -pointed curve X = (C; Q1 , . . . , QN ; ξ1 , . . . , ξN ) we define the preferred element hω(X, {α, β})| by hω(X, {α, β})| = ι∗ −1 hω(X1 , {α, β})| 62
(6.7)
where X1 = (C; Q1 , ξ1 ) and ι∗ : V † ab (X) → V † ab (X1 ) is defined by applying Theorem 3.4 several times. Note by Theorem 6.4 that if we choose to use the pair (Qj , ξj ) instead of (Q1 , ξ1 ) in defining X1 , we get the same preferred element. For an N -pointed nodal curve X = (C; Q1 , . . . , QN ; ξ1 , . . . , ξN ) with formal coordinates e of C is an irreducible curve, we can define the preferred such that the normalization C element by generalizing the discussion just before Theorem 6.5 in the obvious way. We can however not apply the same method to define the preferred element for onee is disconnected. This is simply pointed nodal curves (C; Q, ξ) whose normalization C because a holomorphic one-form with support on one component will have zero Taylor expansion at all points of the curve not contained in that component. Therefore, we need to find another definition. For simplicity, in the following, we shall consider a two-pointed nodal curve X = (C; Q1 , Q2 ; ξ1 , ξ2 ) with formal coordinates e has two connected components C1 and C2 . Assume such that the normalized curve C e1 = that Q1 , P+ ∈ C1 and Q2 , P− ∈ C2 . Put X1 = (C1 ; Q1 , ξ1 ), X2 = (C2 ; Q2 , ξ2 ) X e e (C1 ; Q1 , P+ , ξ1 , z), X2 = (C2 ; Q2 , P− , ξ2 , w), X = (C1 ∪ C2 ; Q1 , Q2 , P+ , P− , ξ1 , ξ2 , z, w). Then by Theorem 3.4 and Theorem 3.5 we have isomorphisms ej) ι∗j : V † ab (X e ι∗ : V † ab (X) +,−
∼ = V † ab (Xj ), ∼ = V † ab (X).
j = 1, 2
Moreover, by Proposition 3.1 we have
e = V † ab (X e 1 ) ⊗ V † ab (X e 2 ). V † ab (X)
Choose symplectic bases {α(i) , β (i) } of H1 (Ci , Z). Then, {α, β} = {α(1) , α(2) , β (1) , β (2) } is a symplectic basis of H1 (C, Z). Put hΦ1 | = ι∗1 −1 (hω(X1 , {α(1) , β (1) })|),
hΦ2 | = ι∗2 −1 (hω(X2 , {α(2) , β (2) })|), hΦ| = hΦ1 | ⊗ hΦ2 |.
Finally define hω(X, {α, β})| = ι∗+,− (hΦ|).
(6.8)
For a general N -pointed nodal curve X = (C; Q1 , . . . , QN ; ξ1 , . . . , ξN ) we define the preferred element hω(X, {α, β})| in a similar way. The following lemma plays an important role in characterizing the preferred section. Lemma 6.3. Under the same notation as above, let the genera of C1 and C2 be g1 and g2 , respectively. Let 1 = m1 < m2 < · · · < mg1 ≤ 2g1 − 1 and 1 < n1 < n2 < · · · < ng2 ≤ 2g2 − 1 be the Weierstrass gap values of C1 at Q1 and C2 at Q2 , respectively. Put |u1 i = emg1 −1/2 ∧ · · · ∧ em1 −1/2 ∧ | − 1i,
|u2 i = eng2 −1/2 ∧ · · · ∧ en1 −1/2 ∧ |0i.
Then we have that 1 Im (Q1 ) · · · g1 .. ∗ −1 .. h(ι+,− ) ω(X, {α, β})|u1 ⊗u2 i = . . I 1 (Q1 ) · · · m1 63
g1 Im g1 (Q1 ) .. . g1 Im (Q 1) 1
g1 +1 Ing (Q2 ) · · · 2 .. .. · . g +1. In1 (Q2 ) · · · 1
6= 0 g1 +g2 In1 (Q2 )
Ing1g2+g2 (Q2 ) .. .
where {ω1 , . . . , ωg1 } and {ωg1 +1 , . . . , ωg1 +g2 } are the normalized bases of holomorphic oneforms on C1 and C2 , respectively, and ∞ X
m=1
k Im (Q1 )ξ1m−1 dξ1 ,
1 ≤ k ≤ g1
is the Taylor expansion of ωk at Q1 and ∞ X
n=1
Ing1 +i (Q2 )ξ2n−1 dξ2 ,
1 ≤ i ≤ g2 .
is the Taylor expansion of ωg1 +i at Q2 .
Proof. Let ω1 be a meromorphic one-form on C1 which is holomorphic outside P+ and Q1 such that at P+ and Q1 , ω1 has the Laurent expansions: dz + holomorphic z dξ1 = − + holomorphic. ξ1
ω1,+ = ω1,Q1
Similarly let ω2 be a meromorphic one-form on C2 which is holomorphic outside P− and Q2 such that at P− and Q2 ω2 has the Laurent expansions: dw + holomorphic w dξ2 + holomorphic. ξ2
ω2,− = − ω2,Q2 = Then, we have that
ψ[ω1,+ ]|0i = | − 1i,
−ψ[ω1,− ]|0i = | − 1i.
and therefore, we have that h(ι∗+,− )−1 ω(X, {α, β})|u1 ⊗ u2 i = hΦ|0+,− ⊗ u1 ⊗ u2 i
= hΦ|0 ⊗ −1 ⊗ u1 ⊗ u2 i − hΦ| − 1 ⊗ 0 ⊗ u1 ⊗ u2 i
= hΦ1 |0 ⊗ u1 ihΦ2 | − 1 ⊗ u2 i − hΦ1 | − 1 ⊗ u1 ihΦ2 |0 ⊗ u2 i
= hω(X1 , {α(1) , β (1) })|u1 ihΦ2 |ψ[ω1,+ ]|0i ⊗ u2 i
+hΦ1 |ψ[ω1,− ]|0i ⊗ u1 ihω(X2 , {α(2) , β (2) })|u2 i
= hω(X1 , {α(1) , β (1) })|u1 ihω(X2 , {α(2) , β (2) })|ψ[ω1,Q2 ]u2 i
+hω(X1 , {α(1) , β (1) })|ψ[ω1,Q1 ]u1 ihω(X2 , {α(2) , β (2) })|u2 i
Since ψ[ω1,Q2 ]|u2 i = eng2 −1/2 ∧ · · · ∧ en1 −1/2 ∧ | − 1i + ∗ ∧ |0i
and hω(X2 , {α(2) , β (2) })| does not contains the term e−1/2 , we have that
hω(X2 , {α(2) , β (2) })|ψ[ω1,Q2 ]u2 i = hω(X2 , {α(2) , β (2) })|eng2 −1/2 ∧ · · · en1 −1/2 ∧ | − 1i g1 +1 Ing (Q2 ) · · · Ing1g+g2 (Q2 ) 2 2 .. .. .. = 6= 0 . . . g +1 In1 (Q2 ) · · · Ing1 +g2 (Q2 ) 1 1 64
Since by the same reason hω(X1 , {α(1) , β (1) })|ψ[ω1,Q1 ]u1 ihω(X2 , {α(2) , β (2) })|u2 i = 0 and 1 g1 Im (Q1 ) · · · Im (Q ) 1 g g1 1 .. .. .. hω(X1 , {α(1) , β (1) })|u1 i = 6= 0 . . . g1 I 1 (Q1 ) · · · Im 1 (Q1 ) m1
we obtain the desired result.
QED
Let F = (π : C → B, s1 , . . . , sN , ξ1 , . . . , ξN ) be a family of N -pointed smooth curves. For any point b ∈ B there exists an open neighbourhood Ub such that π −1 (Ub ) is topologically trivial so that we can choose smoothly varying symplectic bases {α1 (t), . . . , αg (t), β1 (t), . . . , βg (t)},
t ∈ B.
Then we can define hω(Xt , {α(t), β(t)})| where Xt = (π −1 (t), s1 (t), ξ1 ). Theorem 6.6. The section hω(Xt , {α(t), β(t)})| is a holomorphic section of V † ab (F) over Ub . Proof. Put F′ = (π : C → B, s1 , ξ1 ). Then, by Theorem 3.4 it is easy to show that we get an isomorphism V † ab (F) ∼ = V † ab (F′ ) by the propagation of vacua construction. Hence it is enough to show that hω(Xt , {α(t), β(t)})| depends holomorphically on t. This follows, (n) since ωs1 (t) varies holomorphically in t. QED Finally, we analyze the preferred section for families of deformations of nodal curves. Let C0 be a complete curve with only one ordinary double point P such that C0 \{P } is e0 → C0 non-singular. Let Q1 , Q2 , . . . , QN be distinct non-singular points on C0 . Let ν : C −1 be the normalization of the singular curve. Put {P+ , P− } = ν (P ). Note that the normalization might or might not be connected but assume that each component contains at least one Qj . Let F = (π : C → D; σ1 , . . . , σN ; ξ1 , . . . , ξN ) be the family constructed from (C0 , Q1 , . . . , QN , ξ1 , . . . ξN ) as described right before Lemma 5.7. Suppose we now have a continuous basis {αi (t), βi (t)} of H1 (π −1 (t), Z), t ∈ (0, 1) ⊂ D, such that we get a well defined limit as t goes to zero, which gives a symplectic basis, say {α1 (0), . . . , αg−1 (0), αg (0), β1 (0), . . . , βg−1 (0)} of H1 (C0 , Z) as described above for nodal curves and βg (0) = 0. Let Xt = (π −1 (t), s1 (t), . . . sN (t), ξ1 , . . . ξN ). Theorem 6.7. We have that hω(X0 , {α(0), β(0)})| = lim hω(Xt , {α(t), β(t)})|. t→0
e0 is connected. It is then enough to prove the theorem Proof. Assume the curve C when N = 1. Note by [F, Proposition 3.7] that if {ω1 , . . . , ωg } is a normalized basis of holomorphic one-forms on C0 , then a normalized bases of holomorphic one-forms for the family is of the form: 1 ωi (x, t) = ωi (x) + t(ωi (P+ ) − ωi (P− ))(ω(x, P+ ) − ω(x, P− )) + O(t2 ), 4 ωg (x, t) = ωP+ −P− (x) + tug (x) + O(t2 ).
65
1≤i≤g−1
e0 , and Here {ω1 (x), . . . , ωg−1 (x)} is a normalized basis of holomorphic one-forms of C ωi (P+ ) is the number fi (0) where in a neighbourhood of P+ , ωi is expressed as fi (z)dz. The number ωi (P− ) is defined similarly. Moreover, ω(x, y) is the normalized bidifferential e0 and in a neighbourhood of (x, P+ ) if we express ω(x, y) = f (x, z)dxdz of the curve C then ω(x, P+ ) is defined as f (x, 0)dx. The one-form ω(x, P− ) defined similarly. The form e0 which has a pole of order one at P+ with ωP+ −P− (x) is a meromorphic one-form of C residue 1, pole of order one at P− with residue −1 and holomorphic outside P+ and P− , e0 which has only poles at P± of order three. and ug is a meromorphic one-form on C e0 \ {P+ , P− } Finally, the expression O(t2 ) means that it is a holomorphic one-form on C O(t2 ) e0 \ {P+ , P− }. and limt→0 t2 is a holomorphic one-form on C Let ∞ X Ini (t)ξ n−1 )dξ ωi (x, t) = ( n=1
for i = 1, . . . g. The meromorphic one-forms ωg+n (t) on Ct , as defined by (6.1) and (6.2) has a pole only at Q1 = s1 (t) of order n + 1 with Laurent expansion at s1 (t) of the form dξ1 + holomorphic ξ1n+1 and satisfies Z
√ Z 2π −1Ini (t) ωg+n (t) = 0, , ωg+n (t) = − n βi (t) αi (t)
1 ≤ i ≤ g,
n ≥ 0.
Then, hω(Xt , {α(t), β(t)})| = h· · · ∧ e(ωm (t)) ∧ · · · ∧ e(ω2 (t)) ∧ e(ω1 (t))|. Thus we conclude that lim hω(Xt , {α(t), β(t)})| = hω(X0 , {α(0), β(0)})|.
t→0
e0 has two connected components C1 and C2 , and that Next, assume that the curve C Q1 , P+ ∈ C1 and Q2 , P− ∈ C2 . Moreover, it is enough to consider the case N = 2. Let {ω1 , . . . , ωg1 } and {ωg1 +1 , . . . , ωg1 +g2 } be normalized bases of holomorphic one-forms of C1 and C2 , respectively. Then we can find a family {ω1 (t), . . . , ωg1 +g2 (t)} of normalized bases of the family π : C ∗ = π −1 (D \ {0}) → D \ {0} such that limt→0 ωk (t) = ωk according to Proposition 3.1 [F]. Here we regard ωk as a holomorphic section of the dualizing sheaf ωC0 of the nodal curve, by extending it by zero to the other component. Over D \ {0} hω(Xt , {α(t), β(t)})| is a holomorphic section of the sheaf of ghost vacua. Let 1 = m1 < m2 < · · · < mg1 ≤ 2g1 − 1 and 1 = n1 < n2 < · · · < ng2 ≤ 2g2 − 1 be the Weierstrass gap values of C1 at Q1 and C2 at Q2 , respectively. Put |u1 i = emg1 −1/2 ∧ · · · ∧ em1 −1/2 ∧ | − 1i,
|u2 i = eng2 −1/2 ∧ · · · ∧ en1 −1/2 ∧ |0i.
Choose sections fj of π∗ OC (nj s2 (D)+∗s1 (D)) over D such that it has a Laurent expansion fj,2 =
1 n + holomorphic ξ2 j 66
along s2 (D). Put also
X′t = (π −1 (t); s1 (t); ξ1 ).
Then we have that hω(Xt , {α(t), β(t)})|u1 ⊗ u2 i = hω(Xt , {α(t), β(t)})|u1 ⊗ ψ[fg2 ,2 ]ψ[fg2 −1,2 ] · · · ψ[f1,2 ]|0i
= (−1)g2 −1 hω(Xt , {α(t), β(t)})|ψ[f1,2 ]ψ[f2,2 ] · · · ψ[fg2 ,2 ]|u1 ⊗ 0i
= (−1)g2 −1 hω(X′t , {α(t), β(t)})|ψ[f1,2 ]ψ[f2,2 ] · · · ψ[fg2 ,2 ]|u1 i ResQ1 (f1 ω1 (t)) · · · ResQ1 (f1 ωg1 +g2 (t)) .. .. .. . . . g2 −1 ResQ1 (fg2 ω1 (t)) · · · ResQ1 (fg2 ωg1 +g2 (t)) = (−1) g +g 1 2 1 Im1 (Q1 , t) ··· Im1 (Q1 , t) .. . . . . . . . g +g 1 2 1 ··· Img (Q1 , t) I (Q1 , t) mg1
1
where the Taylor expansion of ωk (t) along s1 (D) and s2 (D) are written as ωk (t) = ωk (t) =
∞ X
k Im (Q1 , t)ξ1m−1
m=1 ∞ X
Ink (Q2 , t)ξ1n−1
n=1
Now by the residue theorem we have ResQ1 (fj ωk (t)) = − ResQ2 (fj ωk (t)) = −Inkj (Q2 , t). Since for any positive integers m, n we have k lim Im (Q1 , t) t→0 lim Ing1 +i (Q1 , t) t→0 k lim Im (Q2 , t) t→0 lim Ing1 +i (Q2 , t) t→0
k = Im (Q1 ),
1 ≤ k ≤ g1 ,
= 0,
1 ≤ i ≤ g2 ,
= 0,
1 ≤ k ≤ g1 ,
= Ing1 +i (Q2 ), 1 ≤ i ≤ g2 ,
67
we conclude that lim hω(Xt , {α(t), β(t)})|u1 ⊗ u2 i −I 1 (Q2 , t) · · · −Ing1 +g2 (Q2 , t) 1 n1 .. .. .. . . . −I 1 (Q , t) · · · −I g1 +g2 (Q , t) ng2 2 2 ng2 g2 −1 = (−1) lim 1 g1 +g2 t→0 Im (Q , t) · · · I (Q , t) m1 1 1 1 .. .. .. . . . 1 g1 +g2 Img (Q1 , t) · · · Im (Q1 , t) g1 1 g1 +1 1 g1 In1 (Q2 ) · · · Ing11+g2 (Q2 ) Im (Q1 ) · · · Im (Q ) 1 1 1 .. .. .. .. .. .. = − · . . . . . . 1 g1 g +g g +1 1 1 Im (Q1 ) · · · Im Ing2 (Q2 ) · · · Ing2 2 (Q2 ) g1 (Q1 ) g1 t→0
= −hω(X0 , {α(0), β(0)})|u1 ⊗ u2 i = 6 0.
Hence if limt→0 hω(Xt , {α(t), β(t)})| exists, then it is −hω(X0 , {α(0), β(0)})|. Now let us show that limt→0 hω(Xt , {α(t), β(t)})| exists. Note that for t 6= 0, hω(X′t , {α(t), β(t)})| is given by h· · · ∧ e(ωm (t)) ∧ · · · ∧ e(ω2 (t)) ∧ e(ω1 (t))|,
where {ω1 (0), . . . , ωg (0)} is a normalized basis of holomorphic one-forms of C0 and ωg+i (0) is a normalized meromorphic one-form which has only a pole at s1 (0) = Q1 . Now for t 6= 0 {ω1 (0), . . . , ωg (0)} is defined by using Theorem 3.4. This means that for |v1 ⊗ v2 i ∈ F2 , the evaluation hω(Xt , {α(t), β(t)})|v1 ⊗ v2 i is reduced to the evaluation of hω(X′t , {α(t), β(t)})|v1′ i so that we can use similar arguments as above. For example let us calculate hω(Xt , {α(t), β(t)})|v1 ⊗ v2 i for v1 = ek1 +1/2 ∧ |0i,
1 ≤ k1 ,
v2 = elg −1/2 ∧ · · · ∧ elg −1/2 ∧ i(e−m−1/2 )| − 1i, Choose meromorphic functions fi ∈ rent expansion along s2 (D):
H 0 (D, π
fi,2 =
1 ξ2li
0 ≤ l1 < · · · < lg ,
∗ OC (∗(s1 (D)
0 ≤ m.
+ s2 (D))) which have the Lau-
+ holomorphic.
Also choose τ ∈ H 0 (D, π∗ ωC (∗(s1 (D) + s2 (D))) which has the Laurent expansion along s2 (D): 1 τ2 = m + holomorphic dξ2 . ξ2 Then by a similar argument as above hω(Xt , {α(t), β(t)})|v1 ⊗ v2 i = ±hω(X′t , {α(t), β(t)})|ψ[τ1 ]ψ[f1,1 ] · · · ψ[fg,1 ]| − 1i. By a simple calculation we have that g X (−1)i−1 Ress1 (t) (fi τ )ψ[f1,1 ] · · · ψ[fi−1,1 ]|ψ[fi+1,1 ] · · · | − 1i ψ[τ1 ]ψ[f1,1 ] · · · ψ[fg,1 ]| − 1i = i=1
ψ[f1,1 ] · · · ψ[fg,1 ]ψ[τ1 ]| − 1i. 68
Moreover, we have that ψ[τ1 ]| − 1i =
s X
n=1
\ ∧ e−n3 /2 ∧ · · · , an e−3/2 ∧ · · · ∧ e−n−1/2
Then, as above hω(X′t , {α(t), β(t)})|ψ[τ1 ]ψ[f1,1 ] · · · ψ[fg,1 ]| − 1i Ress1 (t) (f1 ω1 (t)) · · · Ress1 (t) (f1 ωg (t)) g X .. .. .. (−1)i−1 Ress1 (t) (fi τ ) =± . . . i=1 Ress (t) (fg ω1 (t)) · · · Ress (t) (fg ωg (t)) 1 1 Ress1 (t) (f1 ωg+n (t)) Ress1 (t) (f1 ω2 (t)) · · · Ress1 (t) (f1 ωg (t)) s X .. .. .. + . . . n=1 Ress1 (t) (fg ωg+n (t)) Ress1 (t) (fg ω2 (t)) · · · Ress1 (t) (fg ωg (t)) Ress1 (t) (f1 ω1 (t)) Ress1 (t) (f1 ωg+n (t)) · · · Ress1 (t) (f1 ωg (t)) .. .. .. + . . . Ress (t) (fg ω1 (t)) Ress (t) (fg ωg+n (t)) · · · Ress (t) (fg ωg (t)) 1 1 1 Ress1 (t) (f1 ω1 (t)) Ress1 (t) (f1 ω2 (t)) · · · Ress1 (t) (f1 ωg+n (t)) .. .. .. + . . . Ress (t) (fg ω1 (t)) Ress (t) (fg ω2 (t)) · · · Ress (t) (fg ωg+n (t)) 1 1 1
which can be rewritten in the following form by using Ress1 (t) (fi ωk ) = − Ress2 (t) (fi ωk ). − Ress2 (t) (f1 ω1 (t)) · · · .. .. . . g X − Ress (t) (fg1 ω1 (t)) · · · i−1 2 (−1) Ress1 (t) (fi τ ) =± Ress1 (t) (fg ωg1 +1 (t)) · · · i=1 .. .. . . Ress (t) (fg ω1 (t)) ··· 1 − Ress2 (t) (f1 ωg+n (t)) − Ress2 (t) (f1 ω2 (t)) .. .. . . s X − Ress (t) (fg1 ωg+n (t)) − Ress (t) (fg1 ω2 (t)) 2 2 + Ress (t) (fg1 +1 ωg+n (t)) Ress (t) (fg1 +1 ω2 (t)) 1 1 n=1 .. .. . . Ress (t) (fg ωg+n (t)) Ress (t) (fg ω2 (t)) 1
1
69
− Ress2 (t) (fg1 ωg (t)) Ress1 (t) (fg ωg1 +1 (t)) .. . Ress1 (t) (fg ωg (t)) − Ress2 (t) (f1 ωg (t)) .. .
··· .. .
··· ··· .. . ···
− Ress2 (t) (f1 ωg (t)) − Ress2 (t) (fg1 ωg (t)) Ress1 (t) (fg1 +1 ωg (t)) Ress (t) (fg ωg (t)) 1
+ +
− Ress2 (t) (f1 ωg (t)) − Ress2 (t) (fg1 ω1 (t)) − Ress2 (t) (fg1 ωg+n (t)) · · · − Ress2 (t) (fg1 ωg (t)) Ress1 (t) (fg1 +1 ω1 (t)) Ress1 (t) (fg1 +1 ωg+n (t)) · · · Ress1 (t) (fg1 +1 ωg (t)) .. .. .. . . . Ress1 (t) (fg ω1 (t)) Ress1 (t) (fg ωg+n (t)) ··· Ress1 (t) (fg ωg (t)) Ress1 (t) (f1 ω1 (t)) Ress1 (t) (f1 ω2 (t)) ··· Ress1 (t) (f1 ωg+n (t)) .. .. .. . . . ··· Ress1 (t) (fg1 ωg+n (t)) Ress1 (t) (fg1 ω2 (t)) Ress1 (t) (fg1 ω1 (t)) . Ress1 (t) (fg1 +1 ω1 (t)) Ress1 (t) (fg1 +1 ω2 (t)) · · · Ress1 (t) (fg1 +1 ωg+n (t)) .. .. .. . . . Ress1 (t) (fg ω1 (t)) Ress1 (t) (fg ω2 (t)) ··· Ress1 (t) (fg ωg+n (t)) − Ress2 (t) (f1 ω1 (t)) .. .
− Ress2 (t) (f1 ωg+n (t)) .. .
··· .. .
Now it is easy to see that the last expression has a meaning when t goes to 0. Hence limt→0 hω(Xt , {α(t), β(t)})|v1 ⊗ v2 i always exist.
QED
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