arX

iv:2

110.

0477

4v2

[m

ath.

QA

] 9

Nov

202

1

Sewing and Propagation of Conformal Blocks

BIN GUI

Contents

1 Introduction 2

2 The geometric setting 7

3 Sheaves of VOA 10

4 Conformal blocks 13

5 Sewing conformal blocks 15

6 An equivalence of sheaves 18

7 Propagation of conformal blocks 21

8 Multi-propagation 24

9 Sewing and multi-propagation 27

10 A geometric construction of permutation-twisted Vbk-modules 30

A Strong residue theorem for analytic families of curves 35

Index 39

References 40

1

Abstract

Propagation is a standard way of producing new conformal blocks from oldones that corresponds to the geometric procedure of adding new distinct points toa pointed compact Riemann surface. On the other hand, sewing conformal blockscorresponds to sewing compact Riemann surfaces.

In this article, we clarify the relations between these two procedures. Mostimportantly, we show that ”sewing and propagation are commuting procedures”.More precisely: let φ be a conformal block associated to a vertex operator algebraV and a compact Riemann surface to be sewn, and let ≀nφ be its n-times propaga-

tion. If the sewing rSφ converges, then rS ≀nφ (the sewing of ≀nφ) converges to ≀n rSφ(the n-times propagation of the sewing rSφ).

As an application, we can prove the convergence of sewing conformal blocks incertain important cases without assuming V to be CFT-type, C2-cofinite, or ratio-nal. We also provide a new method of explicitly constructing the permutation-twisted modules associated to the tensor product VOA V

bk, originally due to[BDM02]. Our results are crucial for relating the (genus-0) permutation-twistedV

bk-conformal blocks and the untwisted V-conformal blocks (of possibly highergenera) [Gui21].

1 Introduction

Propagating conformal blocks

Let V be a vertex operator algebra (VOA) with vacuum vector 1. Let X “pC; x1, . . . , xN ; η1, . . . , ηNq be an N-pointed compact Riemann surface with local coor-dinates, namely, each connected component of the compact Riemann surface C con-tains at least one of the distinct marked points x1, . . . , xN , and each ηj is an injectiveholomorphic function on a neighborhood of xj sending xj to 0 (i.e., an (analytic) localcoordinate at xj). Associate to each xj a V-module Wj . Then a conformal block asso-ciated to X and W‚ “ W1 b ¨ ¨ ¨ b WN is a linear functional φ : W‚ Ñ C “invariant”under the actions of V (Cf. [Zhu94, FB04, DGT19a]). When C is the Riemann sphereP1, the simplest examples of conformal blocks are as follows. (We let ζ be the standard

coordinate of C.)

1. X “ pP1; 0; ζq, W is associated to the marked point 0. Then each T P HomVpW,V1q(where V1 is the contragredient module of the vacuum V) provides a conformalblock

w P W ÞÑ xTw, 1y

Here x¨, ¨y refers to the standard pairing of V and V1. Of particular interest is the

case that an isomorphism of V-modules T : V»ÝÑ V1 exists and is fixed. Then

there is a canonical conformal block associated to X and V.

2. X “ pP1; 0,8; ζ, ζ´1q, and W,W1 are associated to 0,8. Then we have a conformalblock

τW : W b W1 Ñ C, w b w1 ÞÑ xw,w1y. (1.1)

2

3. X “ pP1; 0, z,8; ζ, ζ´z, ζ´1q, and W,V,W1 are associated to 0, z,8. The the vertexoperation Y for W defines a conformal block

w b v b w1 P W b V b W1 ÞÑ xY pv, zqw,w1y. (1.2)

Now, we add a new point y P Cztx1, . . . , xNu (together with a local coordinate µ)to X and call this new data ≀Xy, and associate the vacuum module V to y. Then eachconformal block φ : W‚ Ñ C associated to X and W‚ canonically gives rise to one≀φy : V b W Ñ C associated to ≀Xy and V b W, called the propagation of φ at y. Thepropagation is uniquely determined by the fact that

≀φp1 b w‚qy “ φpw‚q. (1.3)

For example, it follows easily from such uniqueness that the third example above isthe propagation of the second one at z, i.e.

≀τWpv b w b wqz “ xY pv, zqw,w1y.

More generally, when y P C is close to xi and the local coordinate µ at y is ηi ´ ηipyq,

≀φpv b w‚qy “ φpw1 b ¨ ¨ ¨ b Y pv, ηjpyqqwi b ¨ ¨ ¨ b wNq (1.4)

where the right hand side converges absolutely as a formal Laurent series of ηjpyq. (Cf.[Zhu94, Thm. 6.2], [FB04, Chapter 10], or Thm. 7.1 of this article.) The uniqueness of≀φ satisfying (1.3) is not hard to show; what is more difficult is to prove the existenceof propagation (cf. [TUY89, Zhu94, Zhu96, FB04, Cod19, DGT19a]).

Sewing conformal blocks

It is worth noting that the right hand side of (1.4) is the sewing of φ and ≀τW(“the conformal block defined in (1.2)) corresponding the geometric sewing of Cand P1 along the points xi,8 with respect to their local coordinates ηi, ζ

´1. In gen-

eral, given an pN ` 2q-pointed compact Riemann surface with local coordinates rX “p rC; x1, . . . , xN , x1, x2; η1, . . . , ηN , ξ, q where each connected component of rC intersectstx1, . . . , xNu, if ξ (resp. ) is defined on a neighborhood W 1 of x1 (resp. W 2 of x2) suchthat ξpW 1q is the open disc Dr with radius r (resp. pW 2q “ Dρ), and that W 1 (resp.W 2) contains only one point among x1, . . . , xN , x

1, x2. Then for each 0 ă |q| ă rρ, weremove

F 1 “ ty P W 1 : |ξpyq| ď |q|ρu, F 2 “ ty P W 2 : |pyq| ď |q|ru,

from rC, and glue the remaining part by identifying all y1 P W 1 with y2 P W 2 ifξpy1qpy2q “ q. As a result, we obtain a new compact Riemann surface Cq with markedpoints x1, . . . , xN and local coordinates η1, . . . , ηN . We denote this data by Xq. Cor-responding to this geometric sewing, we associated V-modules W1, . . . ,WN ,M,M1 tothe marked points x1, . . . , xN , x

1, x2 where M1 is contragredient to M, and assume that

the modules are N-gradable (i.e., admissible) with grading operator rL0 such that each

3

graded subspace is finite-dimensional. qrL0 P EndpMqrrqss can be regarded as an ele-

ment of M b M1rrqss, which we denote by qrL0

§ bđ. If ψ : W‚ b M b M1 Ñ C is a

conformal block associated to rX, we define a linear rSψ : W‚ Ñ Crrqss sending eachw P W‚ to

rSψpwq “ ψpw b qrL0

§ bđq. (1.5)

It was shown in [DGT19b, Thm. 8.5.1] that the above linear map defines a “formalconformal block” (i.e., a “conformal block” when q is infinitesimal). If this series con-verges absolutely on |q| ă rρ, then it defines an actual conformal block associated toXq [Gui20, Thm. 11.2], called the sewing of ψ.

In the above process, if rC is connected, then Cq is the self-sewing of rC. For instance,if we sew the X in the above example 3 along 0 and 8 to get a torus, we accordinglysew the conformal block (1.2) to obtain the (normalized) character of W-module v ÞÑTrpY pv, zqqrL0q, which plays an important role in the early development of VOA theory.

If rC has two connected component rC1, rC2, and if we sew rC along x1 P rC1, x2 P rC2, we

obtain a connected sum of rC1 and rC2. If we choose rC “ C\P1 and sew rC along xi P C

and 8 P P1, then at q “ 1, the corresponding sewing of the conformal blocks φ and(1.2) is just (1.4), and the new Riemann surface we get is naturally equivalent to C.

Sewing and propagation

Now, (1.4) indicates that propagation and sewing are related: when the insertedpoint y is close to a marked point xi, the propagation is defined by sewing. Wheny is far from the marked points, the propagation is defined by analytic continuation(provided that it exists).

This observation actually gives us a new proof of the existence of propagation:when y is close to a marked point xi, we simply define the propagation by the series onthe right hand side of (1.4). The convergence of that series follows from [FB04, 10.1.1].Or, if we define conformal blocks to be those vanishing on the actions of certain globalmeromorphic 1-forms of the sheaf of VOA VC on W‚ (which is the definition we takein this article), namely, the “dual definition” in [FB04, 10.1.2], then the convergence ofthe series, as well as the existence of the analytic continuation, follows from the StrongResidue Theorem ([FB04, 9.2.9], see also Thm. A.1). That the right hand side of (1.4)defines a conformal block follows from the previously mentioned fact that the sewingof a conformal block is again a conformal block. See Sec. 7 for details.

In [Gui20], we have proved a very general result on the convergence of sewingconformal blocks, assuming that V is CFT-type and C2-cofinite, and the module M issemi-simple. To prove the convergence, we have to establish a differential equationsatisfied by the formal series (1.5), and the C2-cofiniteness is crucial for finding thatdifferential equation. Here, we see a different pattern for proving the convergence ofsewing: if this sewing process is related to propagation, then its convergence follows fromthe Strong Residue Theorem. In particular, C2-cofinite (or CFT-type) is not needed forproving the convergence. (Note that in [Gui20], the main reason for assuming CFT-type when proving convergence is to use Buhl’s result [Buhl02].)

4

Main result: sewing and propagation are commuting

A main goal of this article (already mentioned in the abstract) is to take the aboveidea (i.e. proving the convergence using the Strong Residue Theorem) to the extreme:

we prove that if (in the setting of (1.5)) rSψ converges absolutely on |q| ă rρ, then

for distinct y1, . . . , yn P rC away from tx1, . . . , xNu and from W 1,W 2 (therefor they are

also points on Cq), rSp≀nψy1,...,ynq, the sewing of the n-propagation of ψ at y1, . . . , yn,

converges absolutely to ≀n rSψy1,...,yn. Moreover, the convergence should be uniform

when y1, . . . , yn vary on compact sets. More generally, we prove such result for the

simultaneous sewing of rC along several pairs of points. See Thm. 9.1 for details. Evenmore generally, we prove such result for families of compact Riemann surfaces (cf.Rem. 9.2).

The above main result of this article can be summarized by

≀n rSψ “ rS ≀

n ψ, (1.6)

i.e. “the propagation of sewing equals the sewing of propagation”. Note that the non-

trivial part of this result is that the convergence of rSψ guarantees the convergence

of rS ≀n ψ. For if we assume or already know the convergence, then the equality (1.6)

follows directly from the fact that ≀n rSψ is the unique conformal block sending 1b¨ ¨ ¨b1 b w P Vbn b W‚ to rSψpwq (cf. (1.3)).

Applications

We give an application of this result. Let Y “ pC; x1, . . . , xN ; η1, . . . , ηNq, associateWj to xj for each j, and choose a conformal block φ : W‚ Ñ C associated to Y. Choose1 ď i ď N . Let P “ pP1; 0,8; ζ, ζ´1q, and associate Wi,W

1i to 0,8. Then ψ :“ φb τWi

:

W‚ b Wi b W1i Ñ C (recall (1.1)) is a conformal block associate to the disjoint union

rX “ Y \ P. If we sew rX along xi P C and 8 P P1 at q, the new pointed Riemannsurface with local coordinates Xq is

Xq “ pC; x1, . . . , xN ; η1, . . . , q´1ηi, . . . , ηNq,

and (setting w‚ “ w1 b ¨ ¨ ¨ b wN as usual)

rSψpw‚q “ φpw1 b ¨ ¨ ¨ b qrL0wi b ¨ ¨ ¨ b wNq, (1.7)

which clearly converges absolutely for all q. Assume ηi is defined on an open discWi Qxi such that ηipWiq “ Dri has radius ri, and that Wi contains only xi among x1, . . . , xN .Choose r ą 0. Then, according to our main result, the sewing of n-propagation

rS ≀n ψpv1 b ¨ ¨ ¨ b vn b w‚qη´1

i pqz1q,...,η´1

i pqznq

“φpw1 b ¨ ¨ ¨ b wi´1 b qrL0

§ bwi`1 b ¨ ¨ ¨ b wNq ¨ ≀nτWipv1 b ¨ ¨ ¨ b vn b wi b đqz1,...,zn

(assuming that the local coordinate at each zj P P1 is ζ ´ zj , and the one at η´1i pqzjq P C

is q´1ηi ´ zj) converges absolutely and uniformly when z1, . . . , zn vary on any compact

5

set of the configuration space ConfnpDˆr q (where Dˆ

r “ tz P C : 0 ă |z| ă ru) and when|q| ă rir.

We are especially interested in the case that q “ 1, which is accessible when r ă ri,

namely, when 0 ă |z1|, . . . , |zn| ă ri. Then ≀n rSψ “ rS ≀n ψ implies (notice (1.7))

≀n φpv1 b ¨ ¨ ¨ b vn b w‚qη´1

i pz1q,...,η´1

i pznq

“φpw1 b ¨ ¨ ¨ b wi´1 b § b wi`1 b ¨ ¨ ¨ b wNq ¨ ≀nτWipv1 b ¨ ¨ ¨ b vn b wi b đqz1,...,zn . (1.8)

In the special case that 0 ă |z1| ă ¨ ¨ ¨ ă |zn| ă ri, the above relation becomes

≀n φpv1 b ¨ ¨ ¨ b vn b w‚qη´1

i pz1q,...,η´1

i pznq

“φ`w1 b ¨ ¨ ¨ b Y pvn, znq ¨ ¨ ¨Y pv1, z1qwi b ¨ ¨ ¨ b wN

˘(1.9)

where the right hand side converges absolutely. Zhu proved relation (1.9) in [Zhu94,Thm. 6.2] when v1, . . . , vn are primary, or when the local coordinates are contained ina projective structure (i.e., an atlas whose transition functions are Mobius transforms).But, as explained below, the general case, especially when 0 ă |z1| “ ¨ ¨ ¨ “ |zn| ă ri, isalso important.

Take an automorphism g of Vbk to be the permutation associated to the cyclep12 ¨ ¨ ¨kq. Starting from a V-module W, Barron-Dong-Mason constructed in [BDM02]a (canonical) g-twisted V

bk-module structure on the same vector space W. In partic-ular, they explicitly described the twisted vertex operator Y g for vectors in Vbk of theform 1 b ¨ ¨ ¨ b v b ¨ ¨ ¨ b 1. For an arbitrary vector of Vbk, the Y g can then be describedusing normal ordering. Their proof that Y g satisfies the axioms of a g-twisted moduleis algebraic, and in particular relies on a previous algebraic result of [Li96]. Recently,another algebraic proof was given by Dong-Xu-Yu in [DXY21] using Zhu’s algebras.

Now, our observation in this article is that Y gpv1 b ¨ ¨ ¨ b vk, zq can be described by≀kτWi

at pz1, . . . , zkq, where z1, . . . , zk are the distinct k-th roots of unity of z. Thus, us-ing the consequences of our main result such as relation (1.8), we can give a geometricproof that Y g satisfies the axioms of a g-twisted module. See (10) for details. More-over, our point of view will be generalized in [Gui21] to construct permutation twistedconformal blocks from untwisted ones, and vice versa.

Outline

This article is organized as follows. In Section 2, we fix the geometric notationsused in later sections, and define the (multi) propagations for an (analytic) family ofcompact Riemann surfaces. In the case of a single compact Riemann surface C withmarked points S “ tx1, . . . , xNu, its n-propagation is easy to describe: If we let severaldistinct points y1, . . . , yn move on CzS, we obtain a family of compact Riemann sur-faces (all isomorphic to C) with N fixed marked points and n varying points over thebase manifold ConfnpCzSq.

We recall the definitions and basic properties of sheaves of VOAs (i.e. VOA bun-dles) and conformal blocks in Sections 3 and 4. In Section 5, we recall some importantfacts about the sewing of conformal blocks associated to the sewing of a family of com-pact Riemann surfaces. In Section 6, we relate sheaves of VOAs and the W -sheaveswhich were naturally introduced to define (sheaves of) conformal blocks.

6

In Section 7, we give a new proof of conformal block propagation for (analytic)families of compact Riemann surfaces. In particular, we prove that propagation iscompatible (in the complex analytic sense) with the deformation of pointed compactRiemann surfaces. Roughly speaking, this means that if the original conformal blocksare parametrized by τ P B where B is the base manifold of the family, and if thepropagation on each fiber is parametrized by z, then the propagation is a multivariableanalytic function of pz, τq. The precise statement is formulated using the language ofsheaves; see Thm. 7.1. These results were proved in [Cod19, Thm. 3.6] for CFT typeVOAs using a Lie-theoretic method, which relies on the fact that such VOAs havePBW bases. As explained earlier, our proof is based on the idea of sewing, and relieson the Strong Residue Theorem and the fact that the sewing of conformal blocks areconformal blocks [Gui20, Thm. 11.2], whose formal version was proved in [DGT19b].

Note that here we should use the Strong Residue Theorem for analytic families ofcompact Riemann surfaces. This result is well-known, although we are not able to finda proof in the literature. So we include a proof in the Appendix Section A.

We discuss elementary properties of multi-propagation in Section 8. Most of theseimportant properties were more or less known before (cf. [FB04]) but not explicitlywritten down. We collect these results under Thm. 8.2 so that they can be directlycited or used in future works on VOA. These results follow rather directly from thosein the previous sections.

The main theorem of this article, summarized by the slogan “sewing commuteswith propagation”, is proved in Section 9. To give an application of this result, weconstruct in Section 10 permutation-twisted modules for tensor product VOAs.

Acknowledgment

I would like to thank Nicola Tarasca for helpful discussions.

2 The geometric setting

We set N “ t0, 1, 2, . . . u and Z` “ t1, 2, 3, . . . u. Let Cˆ “ Czt0u. For each r ą 0, welet Dr “ tz P C : |z| ă ru and Dˆ

r “ Drzt0u. For any topological space X , we define theconfiguration space ConfnpXq “ tpx1, . . . , xNq P Xn : xi ‰ xj @1 ď i ă j ď nu.

For each complex manifold X , OX is the sheaf of holomorphic functions of X . Foreach x P X and any OX-module E , Ex is the stalk of E at x. mX,x (or simply mx

when no confusion arises) is by definition tf P OX,x : fpxq “ 0u. E |x :“ ExmxEx »E bOX

OX,xmx is the fiber of E at x. More generally, if Y is a closed complex sub-manifold of X with IY being the ideal sheaf (the sheaf of all sections of OX vanishingat Y ), then the restriction E |Y is defined to be E bOX

OXIY (restricted to the set Y ).We suppress the subscript OX under b when taking tensor products of OX-modules.If s is a section of E , then s|Y is the corresponding value s b 1 in E |Y .

(For the readers not familiar with the language of sheaf of modules: we only con-sider the case that E is locally free (with finite or infinite rank), i.e., a holomorphicvector bundle. Then E |Y resp. s|Y is the usual restriction of the vector bundle resp.vector field to the submanifold Y .)

7

If E is locally free, E _ denotes its dual vector bundle.For a Riemann surface C, its cotangent line bundle is denoted by ωC .A family of compact Riemann surfaces X is by definition a holomorphic proper

map of complex manifolds

X “ pπ : C Ñ Bq

that is a submersion and satisfies that each fiber Cb :“ π´1pbq (where b P B) is a (non-necessarily connected) compact Riemann surface.

A family of N-pointed compact Riemann surfaces is by definition

X “ pπ : C Ñ B; ς1, . . . , ςNq (2.1)

where π : C Ñ B is a family of compact Riemann surfaces, each section ςj : B Ñ C isholomorphic and satisfies π ˝ ςj “ 1B, and any two ςipBq, ςjpBq (where 1 ď i ă j ď N)are disjoint. Unless otherwise stated, we also assume that every connected componentof each fiber

Cb “ π´1pbq

(where b P B) contains at least one of ς1pbq, . . . , ςNpbq. We set

Xb “ pCb; ς1pbq, . . . , ςNpbqq,

which is an N-pointed compact Riemann surface. We define closed submanifold

SX “Nď

j“1

ςjpBq,

considered also as a divisor of C. For any sheaf of OC-module E , and for any n P Z, weset

E pnSXq :“ E b OXpnSXq,E p‹SXq “ limÝÑ

nPNE pnSXq.

When E is a vector bundle, E pnSXq is the sheaf of sections of E which possibly haspoles at each ςjpBq with order at most n.

For each 1 ď j ď N , a local coordinate of X at ςj is defined to be a holomorphicfunction ηj P OpWiq (where Wi is a neighborhood of ςipBq) which is injective on eachfiber Wi X π´1pbq and has value 0 on ςipBq. It follows that pπ, ηjq is a biholomorphismfrom Wi to a neighborhood of B ˆ t0u in B ˆC. ηj |Cb is a local coordinate of the fiber Cbat the point ςjpbq, which identifies a neighborhood of ςjpbq (say Wj X Cb) with an opensubset of C such that ςjpbq is identified with the origin. If X is equipped with localcoordinates η1, . . . , ηN at ς1pBq, . . . , ςNpBq respectively, we set

Xb “ pCb; ς1pbq, . . . , ςNpbq; η1|Cb , . . . , ηN |Cbq.

In particular, SXb“ ř

j ςjpbq is a divisor of Cb.

8

Now, we let X “ (2.1) be N-pointed but not necessarily equipped with local co-ordinates. Define the propagated family ≀X as follows. Consider the commutativediagram

C ˆB pCzSXq C

CzSX B

≀π π

π

where C ˆB pCzSXq is the closed submanifold of C ˆ pCzSXq consisting of all px, yq satis-fying πpxq “ πpyq, the first horizontal arrow is the projection onto the first component,and ≀π is the projection onto the second component. We set

≀B “ CzSX, ≀C “ C ˆB pCzSXq.

The holomorphic section σ : CzSX Ñ C ˆB pCzSXq is set to be the diagonal map, i.e.,

σ : x ÞÑ px, xq.

Define sections

≀ςj : CzSX Ñ C ˆB pCzSXq, x ÞÑ pςj ˝ πpxq, xq.

Then we obtain an pN ` 1q-pointed family ≀X of compact Riemann surfaces to be

≀X “ p≀π : ≀C Ñ ≀B; σ, ≀ς1, . . . , ≀ςNq. (2.2)

Intuitively, ≀X is the result of adding one extra marked point to each fiber Cb disjointfrom SXb

, letting this marked point vary on CbzSXbover all b P B, and fixing the other

marked points.One can define multi-propagation inductively by ≀nX “ ≀ ≀n´1X, which corresponds

to varying n extra distinct points of CbzSXb. Write

≀nX “ p≀nπ : ≀nC Ñ ≀

nB; σ1, . . . , σn, ≀

nς1, . . . , ≀nςNq.

Then ≀nX can be described in a more explicit way. Let

nź

B

CzSX “ pCzSXq ˆB ¨ ¨ ¨ ˆB pCzSXqloooooooooooooomoooooooooooooonn

which is the set of all px1, . . . , xnq P śnCzSX satisfying πpx1q “ ¨ ¨ ¨ “ πpxnq. Define the

relative configuration space

ConfnBpCzSXq “!

px1, . . . , xN q Pźn

BCzSX : xi ‰ xj for any 1 ď i ă j ď n

)

which clearly admits a submersion ConfnBpCzSXq Ñ B (sending each px1, . . . , xnq toπpx1q). Take

≀nπ : C ˆB ConfnBpCzSXq Ñ ConfnBpCzSXq.

9

to be the pullback of π : C Ñ B along ConfnBpCzSXq Ñ B. So we have a commutativediagram

C ˆB ConfnBpCzSXq C

ConfnBpCzSXq B

≀nπ π

Then ≀nX is equivalent to

≀nX »

´≀n π : C ˆB ConfnBpCzSXq Ñ ConfnBpCzSXq; σ1, . . . , σn, ≀nς1, . . . , ≀nςN

¯

where

σipx1, . . . , xnq “ pxi, x1, . . . , xnq,≀nςjpx1, . . . , xnq “ pςj ˝ πpx1q, x1, . . . , xnq

for each 1 ď i ď n, 1 ď j ď N , px1, . . . , xnq P ConfnBpCzSXq.

3 Sheaves of VOA

For any (C-)vector space W , we define four spaces of formal series

W rrzss “" ÿ

nPNwnz

n : each wn P W*,

W rrz˘1ss “" ÿ

nPZwnz

n : each wn P W*,

W ppzqq “!fpzq : zkfpzq P W rrzss for some k P Z

),

W tzu “" ÿ

nPCwnz

n : each wn P W*.

Throughout this article, V is a positive-energy vertex operator algebra (VOA) withvacuum 1 and conformal vector c. We write Y pv, zq “ ř

zPZ Y pvqnz´n´1. Then tLn “Y pcqn`1u are Virasoro algebras, andL0 gives grading V “ À

nPN Vpnq, where each Wpnqis finite-dimensional.

In this article, a V-module W means a finitely-admissible V-module. This meansthat W is a weak V-module in the sense of [DLM97] with vertex operators YWpv, zq “ř

nPZ YWpvqnz´n´1, that W is equipped with a diagonalizable operator rL0 satisfying

rrL0, YWpvqns “ YWpL0vqn ´ pn` 1qYWpvqn, (3.1)

that the eigenvalues of rL0 are in N, and that each eigenspace Wpnq is finite-dimensional. Let

W “ànPN

Wpnq

10

be the grading given by rL0. Each

Wďn “

à0ďkďn

Wpkq

is finite-dimensional. We choose the rL0 operator on V to be L0.We can define the contragredient V-module W

1 of W as in [FHL93]. We chooserL0-grading to be

W1 “

ànPN

W1pnq, W

1pnq “ Wpnq˚.

Therefore, if we let x¨, ¨y be the pairing between W and W1, then xrL0w,w1y “ xw, rL0w

1yfor each w P W, w1 P W1.

The vertex operator YW for W (abbreviated as Y in the following) gives a linearmap Y : V b W Ñ Wppzqq sending v b w to Y pv, zqw. We will write YW as Y when thecontext is clear. By identifying V with V b 1 in V b Cppzqq and similarly W with W b 1

in W b Cppzqq, Y can be extended Cppzqq-bilinearly to

Y :´V b Cppzqq

¯b

´W b Cppzqq

¯Ñ W b Cppzqq,

Y pu b f, zqw b g “ fpzqgpzqY pu, zqw(3.2)

(for each u P V, w P W, f, g P Cppzqq). It can furthermore be extended to

Y :´V b Cppzqqdz

¯b

´W b Cppzqq

¯Ñ W b Cppzqqdz (3.3)

in an obvious way. Thus, for each v P V b Cppzqqdz, we can define the residue

Resz“0 Y pv, zqw, (3.4)

which, in case v “ u b fdz, w “ m b g where u P V, m P W, and f, g P Cppzqq, is theW-coefficient of fpzqgpzqY pv, zqmdz before z´1dz.

We define a group G “ tf P OC,0 : fp0q “ 0, f 1p0q ‰ 0u where the stalk OC,0 is theset of holomorphic functions defined on a neighborhood of 0. The multiplication ruleof G is the composition ρ1 ˝ ρ2 of any two elements ρ1, ρ2 P G. By [Hua97], for eachV-module W, there is a homomorphism U : G Ñ W defined in the following way: Ifwe choose the unique c0, c1, c2 ¨ ¨ ¨ P C satisfying

ρpzq “ c0 ¨ exp´ ÿ

ną0

cnzn`1Bz

¯z

then we necessarily have c0 “ ρ1p0q, and we set

Upρq “ ρ1p0qrL0 ¨ exp´ ÿ

ną0

cnLn

¯.

If X is a complex manifold, a (holomorphic) family of transformations ρ : X Ñ G isby definition an analytic function ρ “ ρpx, zq “ ρxpzq on a neighborhood of X ˆ t0u ĂX ˆ C. Then Upρq (on each W) is defined pointwisely, which is an EndpWq-valued

11

function on X whose value at each x P X is Upρxq. Upρq can be regarded as an OX-module automorphism of W bC OX .

Let X “ pπ : C Ñ Bq be a family of compact Riemann surfaces. Associated to X

one can define a sheaf of OX-modules VX as follows. (Cf. [FB04, Chapter 6, 17]; ourpresentation follows [Gui20, Sec. 5].) First, suppose U, V Ă C are open subsets, andwe have two holomorphic functions η P OpUq, µ P OpV q locally injective (i.e., etale)on each fiber Ub :“ U X π´1pbq, Vb “ V X π´1pbq (b P B) of U and V respectively. Wecan define a family of transformations pη|µq : U X V Ñ G as follows: for each p P C,both η ´ ηppq and µ ´ µppq restricts to an injective holomorphic function on the fiberpU X V qπppq “ U X V X π´1pπppqq vanishing at p. Then pη|µqp P G is determined by

η ´ ηppqˇpUXV qπppq

“ pη|µqp`µ´ µppq

ˇpUXV qπppq

˘(3.5)

on a neighborhood of 0 P C. Then Uppη|µqq is an OUXV -module automorphism ofV bC OUXV which restricts to an automorphism of Vďn bC OUXV for each n P N. Thecocycle condition pη|µqpµ|νq “ pη|νq holds for any holomorphic function ν on aneighborhood of C which is injective on each fiber.

Thus, we can define V ďnX to be the holomorphic vector bundle on C which asso-

ciates to each open U Ă C and each η P OpUq locally injective on fibers a trivialization(i.e., an isomorphism of OU -modules)

Upηq : V ďnX |U »ÝÑ V

ďn bC OU (3.6)

such that for another similar V Ă C, µ P OpV q, we have the transition function

UpηqUpµq´1 “ Uppη|µqq : Vďn bC OUXV»ÝÑ V

ďn bC OUXV . (3.7)

If n1 ą n, we have clearly an OC-module monomorphism V ďnX Ñ V ďn1

X which, foreach open U Ă C and η as above, is transported under the isomorphisms (3.6) tothe canonical monomorphism Vďn bC OU Ñ Vďn1 bC OU defined by the inclusionVďn

ãÑ Vďn1. Thus we are allowed to define

VX “ limÝÑnPN

V ďnX .

Alternatively, one can directly define VX to be the OC-module which is locally free (ofinfinite rank) and isomorphic to V bC OU via a morphism Upηq, and whose transitionfunction is given by Uppη|µqq. We call VX the sheaf of VOA associated to X and V. IfX is a single compact Riemann surface C, we write VX as VC .

For each fiber Cb (where b P B), we have a canonical equivalence

VX|Cb » VCb” VXb

(3.8)

such that if these two OCb-modules are identified by this isomorphism, then the restric-

tion of the trivialization (3.6) to Ub “ U X π´1pbq equals

Upη|Cbq : VCb|Ub

»ÝÑ V bC OUb.

Definition 3.1. Since the vacuum vector 1 is killed by all Ln (where n ě 0), it is fixed byany change of coordinate Upρq. It follows that we can define a section 1 P VXpCq whichunder any trivialization Upηq is the constant section 1, called the vacuum section.

12

4 Conformal blocks

Let X be a family of N-pointed compact Riemann surfaces as in (2.1). We chooseV-modules W1, . . . ,WN . Set

W‚ “ W1 b ¨ ¨ ¨ b WN .

w P W‚ means a vector in W‚, and w‚ P W‚ means a vector of the form w1 b ¨ ¨ ¨ b wN

where each wi P Wi.The sheaf of conformal blocks is an OB-submodule of an infinite-rank locally free

OB-module WXpW‚q, where the latter is defined as follows. For each open subset V Ă B

such that the restricted family

XV :“ pπ : CV Ñ V ; ς1|V , . . . , ςN |V q

(where CV “ π´1pV q) admits local coordinates η1, . . . , ηN at ς1pV q, . . . , ςNpV q respec-tively, we have a trivialization (i.e., an isomorphism of OV -modules)

Upη‚q ” Upη1q b ¨ ¨ ¨ b UpηN q : WXpW‚q|V »ÝÑ W‚ bC OV .

If V is small enough such that we have another set of local coordinates µ1, . . . , µN atς1pV q, . . . , ςNpV q respectively, for each 1 ď j ď N we choose a family of transforma-tions pηj |µjq : V Ñ G defined by

pηj|µjqb ˝ µj|Cb “ ηj |Cb (4.1)

for each b P V . Then each Upηj |µjq is a holomorphic family of invertible endomor-phisms of Wj associated to pηj|µjq (as defined in Sec. 3). The tensor product of them,as a family of invertible transformations of W‚ (more precisely, an automorphism ofthe OV -module W‚ bC OV ), is the transition function:

Upη‚qUpµ‚q´1 :“ Upη1|µ1q b ¨ ¨ ¨ b UpηN |µNq. (4.2)

This gives the definition of WXpW‚q.In particular, WXb

pW‚q is a vector space equivalent to W‚ through Upη‚|Cbq. It is easyto see that for each b P B, the restriction WXpW‚q|b (i.e., the fiber of the vector bundle atb) is naturally equivalent to WXb

pW‚q:

WXpW‚q|b » WXbpW‚q. (4.3)

This equivalence is uniquely determined by the fact that if we identify the two spaces,then the restriction of Upη‚q to the map WXb

pW‚q Ñ W‚ equals Upη‚|Cbq.To define conformal blocks, we first consider the case that B is a single point. Then

C :“ C is a compact Riemann surface. We can define a linear action of H0pC,VC bωCp‹SXqq on WXpW‚q as follows. Choose any local coordinate ηi of C at the point xj :“ςjpBq, defined on a neighboorhood Wj of xj (so, in particular, ηjpxjq “ 0). Note SX “tx1, . . . , xNu. We assume

Wj X SX “ txju.

13

Note that we have a trivialization

Upηjq : VC |Wi

»ÝÑ V bC OWi» V bC OηipWiq

which, tensored by pη´1j q˚ : ωWj

»ÝÑ ωηjpWjq, gives a trivialization

Vpηjq : VC |Wib ωWi

p‹SXq »ÝÑ V bC ωηjpWiqp‹0q

Then for each v P H0pC,VC bωCp‹SXqq, we have a section Vpηjqv, which is a V-valued(more precisely, Vďn-valued for some n P N) holomorphic 1-form on ηjpWjq but pos-sibly has poles at ηjpxjq “ 0. By taking Laurent series expansions, Vpηjqv can beregarded as an element of V b Cppzqqdz. We then define, (notice that we have an iso-

morphism Upη‚q : WXpW‚q »ÝÑ W‚) an action of v on WXpW‚q by

Upη‚q ¨ v ¨ Upη‚q´1w‚ “Nÿ

j“1

w1 b ¨ ¨ ¨ b Resz“0 Y`Vpηjqv, z

˘wj b ¨ ¨ ¨ b wN

for each w‚ P W‚, where the residue is defined as in (3.4). That this definition is inde-pendent of the choice of local coordinates η‚ follows from [FB04, Thm. 6.5.4] (see also[Gui20, Thm. 3.2]), which relies on a crucial change of variable formula (cf. [Gui20,Thm. 3.3]) proved by Huang [Hua97].

Now that we have a linear action of H0pC,VC b ωCp‹SXqq on WXpW‚q, we say thata linear functional φ : WXpW‚q Ñ C is a conformal block (associated to X and W‚)exactly when φ vanishes on the vector space

J :“ H0pC,VC b ωCp‹SXqq ¨ WXpW‚q

where SpanC is suppressed on the right hand side of the equality.Now we come back to the general setting that X is a family of N-pointed compact

Riemann surfaces. Let φ : WXpW‚q Ñ OB be an OB-module morphism, which can beunderstood in the following way: If locally we identify WXpW‚q|V (where V is an opensubset of B) with W‚ bC OV , then φ associates to each vector w P W‚ (considered asthe constant section w b 1 P W‚ b OpV q) a holomorphic function φpwq on U .

Definition 4.1. Let φ : WXpW‚q Ñ OB be an OB-module morphism. For each b P B,regard φ|b as the restriction of φ to the fiber map WXpW‚q|b » WXb

pW‚q Ñ C. Then,we say φ is a conformal block (over B associated to X and W‚) if for each b P B, φ|bis a conformal block associated to Xb (i.e., φpbq vanishes on H0pCb,VCb

b ωCbp‹SX|bqq ¨

WXbpW‚q).

The following proposition is [Gui20, Prop. 6.4].

Proposition 4.2. Let φ : WXpW‚q Ñ OB be an OB-module morphism. Suppose that eachconnected component of B contains a non-empty open subset V such that the restriction of φto WXV

pW‚q Ñ OV is a conformal block, then the original φ is a conformal block associated toX and W‚.

14

5 Sewing conformal blocks

Let N,M P Z`. Let

rX “ prπ : rC Ñ rB; ς1, . . . , ςN ; ς 11, . . . , ς

1M ; ς2

1 , . . . , ς2M ; q

be a family of pN ` 2Mq-pointed compact Riemann surfaces, assuming that every

connected component of each fiber intersects one of ς1p rBq, . . . , ςNp rBq. For each 1 ď j ďM , we assume rX has local coordinates ξj at ς 1

jp rBq defined on a neighborhood W 1j Ă rC of

ς 1jp rBq and similarly j at ς2

j p rBq defined on a neighborhood W 2j . We assume all W 1

j ,W2j

(1 ď j ď M) are mutually disjoint and are also disjoint from ς1p rBq, . . . , ςNp rBq, so that

ς1p rBq, . . . , ςNp rBq remain after sewing. We also assume that for each 1 ď j ď M , we canchoose rj, ρj ą 0 such that

pξj , rπq :W 1j

»ÝÑ Drj ˆ rB resp. pj, rπq :W 2j

»ÝÑ Dρj ˆ rB (5.1)

is a biholomorphic map. (Recall that Dr is the open disc at 0 P C with radius r.)

We do not assume rX has local coordinates at ς1p rBq, . . . , ςNp rBq.

Sewing families of compact Riemann surfaces

We can sew rX along all pairs ς 1jp rBq, ς2

j p rBq to obtain a new family

X “ pπ : C Ñ B; ς1, . . . , ςNq (5.2)

of compact Riemann surfaces. Here,

B “ Dˆr‚ρ‚

ˆ rB, Dˆr‚ρ‚

“ Dˆr1ρ1

ˆ ¨ ¨ ¨ ˆ DˆrMρM

.

X is described as follows.For each q‚ P Dˆ

r‚ρ‚and b P rB, the fiber Cpq‚,bq is obtained by removing the closed

discs

F 1j,b “ ty P W 1

j X rCb : |ξjpyq| ď |qj |ρju, F 2j,b “ ty P W 2

j X rCb : |jpyq| ď |qj |rju

(for all j) from rCb, and gluing the remaining part of the Riemann surface rCb by iden-

tifying (for all j) y1 P W 1j X rCb with y2 P W 2

j X rCb if ξjpy1qjpy2q “ qj . This procedure

can be performed in a consistent way over all b P rB, which gives π : C Ñ B. See forinstance [Gui20, Sec. 4] for details.1

Since Ω “ rCz ŤjpW 1

j Y W 2j q is not affected by gluing, Dˆ

r‚ρ‚ˆ Ω can be viewed as a

subset of X, and the restriction of π to this set is Dˆr‚ρ‚

ˆΩ1brπÝÝÑ Dˆ

r‚ρ‚ˆ rB “ B. Thus, for

each 1 ď i ď N the section ςi for rX defines the corresponding section 1ˆςi : Dˆr‚ρ‚

ˆ rB ÑDˆ

r‚ρ‚ˆΩ, also denoted by ςi. A local coordinate ηi of rX at ςip rBq extends constantly over

Dˆr‚ρ‚

to a local coordinate of X at ςipBq, also denoted by ηi.

1Indeed, one can extend X to a slightly larger flat family of complex curves (with at worst nodal

singularities) with base manifold Dr‚ρ‚ˆ rB (cf. for instance [Gui20, Sec. 4]).

15

Sewing conformal blocks

We now define sewing conformal blocks associated to rX. Associate to ς1, . . . , ςN

V-modules W1, . . . ,WN . Then we have WrXpW‚q defined by prπ : rC Ñ rB; ς1, . . . , ςNq. For

each connected open rV Ă rB, WrXpW‚qprV q can be identified canonically with a subspace

of WXpW‚qpDˆr‚ρ‚

ˆ rV q consisting of sections of the latter which are constant with re-spect to sewing. More precisely, this identification is compatible with restrictions to

open subsets of rV ; moreover, if rV is small enough such that rX|rV has local coordinates

η1, . . . , ηN at ς1prV q, . . . , ςNprV q which give rise to η1, . . . , ηN of η1, . . . , ηN of X|D

ˆr‚ρ‚ ˆ rV at

ς1pDˆr‚ρ‚

ˆ rV q, . . . , ςNpDˆr‚ρ‚

ˆ rV q (which are constant over Dˆr‚ρ‚

), then the followingdiagram commutes:

WrXpW‚qprV q WXpW‚qpDˆr‚ρ‚

ˆ rV q

W‚ bC OprV q W‚ bC OpDˆr‚ρ‚

ˆ rV q

» Upη‚q »Upη‚q (5.3)

where the bottom horizontal line is defined by pulling pack the projection Dˆr‚ρ‚

ˆ rV ÑrV .

Associate to ς 11, . . . , ς

1M V-modules M1, . . . ,MM , whose contragredient modules

M11, . . . ,M

1M are associated to ς2

1 , . . . , ς2M . We understand W‚ b M‚ b M1

‚ as

W1 b ¨ ¨ ¨ b WN b M1 b M11 b ¨ ¨ ¨ b MM b M

1M ,

where the order has be changed so that each M1j is next to Mj . We can then identify

WrXpW‚ b M‚ b M1‚q “ WrXpW‚q bC M‚ b M

1‚ (5.4)

such that whenever rV Ă rB is open such that rX|rV has local coordinates η1, . . . , ηN at

ς1prV q, . . . , ςNprV q as before, the following diagram commutes:

WrXpW‚ b M‚ b M1‚q

ˇrV WrXpW‚q

ˇrV bC M‚ b M1

‚

W‚ b M‚ b M1‚ bC OrV

“

»Upη‚,ξ‚,‚q

»Upη‚qb1

(5.5)We define

qrL0

j § bjđ “ÿ

nPNqnj

ÿ

aPAj,n

mpn, aq b qmpn, aq P pMj b M1jqrrqjss

where for each n P N, s P C, tmpn, aq : a P Aj,nu is a basis of Wpnq with dual basist qmpn, aq : a P Aj,nu in W1pnq.

16

Now, for any conformal block ψ : WrXpW‚ bM‚ bM1‚q Ñ O rB associated to the family

rX and W‚ b M‚ b M1‚, we define an O rB-module morphism

rSψ : WrXpW‚q Ñ O rBrrq1, . . . , qM ss

by sending each section w over an open rV Ă rB to

rSψpwq “ ψ

´w b pqrL0

1 § b1đq b ¨ ¨ ¨ b pqrL0

M § bMđq¯

P OprV qrrq1, . . . , qM ss. (5.6)

The identification (5.4) is used in this definition. rSψ is called the (normalized) sewingof ψ.

Definition 5.1. Let X be a complex manifold. Consider an element

f “ÿ

n1,...,nMPCfn1,...,nM

qn1

1 ¨ ¨ ¨ qnM

M P OpXqtq1, . . . , qMu

where each fn1,...,nMP OpXq. Let R1, . . . , RM P r0,`8s and D

ˆR‚

“ DˆR1

ˆ ¨ ¨ ¨ ˆ DˆRM

.For any locally compact subset Ω of Dˆ

R‚ˆ X , we say that formal series f converges

absolutely and locally uniformly (a.l.u.) on Ω, if for any compact subsets K Ă Ω, wehave

suppq‚,xqPK

ÿ

n1,...,nMPC

ˇfn1,...,nM

pxqqn1

1 ¨ ¨ ¨ qnM

M

ˇă `8.

In the case that f P OpXqrrq˘11 , . . . , q˘1

M ss, it is clear from complex analysis that f con-verges a.l.u. on D

ˆR‚

ˆ X if and only if f is the Laurent series expansion of an element(also denoted by f ) of OpDˆ

R‚ˆ Xq.

Definition 5.2. We say that rSψ converges a.l.u. (on B “ Dˆr‚ρ‚

ˆ rB), if for any open

subset rV Ă rB and any section w of WrXpW‚qprV q, rSψpwq converges a.l.u. on Dˆr‚ρ‚

ˆ rV .

Theorem 5.3 ([Gui20], Thm. 11.2). If rSψ converges a.l.u. on B “ Dˆr‚ρ‚

ˆ rB, then rSψ (resp.Sψ), when extended OB-linearly to an OB-module homomorphism WXpW‚q Ñ OB using theinclusion WrXpW‚q Ă WXpW‚q defined by (5.3), is a conformal block associated to X and W‚.

Example 5.4. Let Y “ pC; x1, . . . , xNq be an N-pointed compact Riemann surface withlocal coordinates η1, . . . , ηN at x1, . . . , xN , defined on neighborhoods W1, . . . ,WN satis-fying Wj X tx1, . . . , xNu “ xj for each 1 ď j ď N . Assume η1pW1q “ Dr for some r ą 0.

Let ζ be the standard coordinate of C. Let rX be the disjoint union of Y and pP1; 0, 1,8q,namely, we have an pN ` 3q-pointed compact Riemann surface

rX “ pC \ P1; x1, . . . , xN , 0, 1,8q.

We equip rX with local coordinates η1, . . . , ηN , ζ, pζ ´ 1q, ζ´1. The local coordinate ζ at 0should be defined at |z| ă 1 so that no marked points other than 0 is inside this region.

We sew rX along x1 P C and 8 P P1 using the chosen local coordinates η1 and 1ζ toobtain a family X. Then

X “ pπ : C ˆ Dˆr Ñ D

ˆr ; x1, x2, . . . , xN , ςq

17

where π is the projection onto the Dˆr -component, the sections x1, . . . , xN are (rigor-

ously speaking) sections sending q to px1, qq, . . . , pxN , qq. The section ς is defined byςpqq “ pη´1

1 pqq, qq, where η´11 sends Dr biholomorphically to W1. Moreover, the local

coordinates of X defined naturally by those of rX are described as follows: For each|q| ă r, their restrictions to

Xq “ pC; x1, x2, . . . , xN , η´11 pqqq (5.7)

are q´1η1, η2, . . . , ηN , q´1pη1 ´ qq.

Attach V-modules W1, . . . ,WN ,W1,V,W11 with simple L0-grading to the marked

points x1, . . . , xN , 0, 1,8 respectively of rX. Fix the trivializations of W -sheaves usingthe chosen local coordinates. Let φ : W1 b ¨ ¨ ¨ b WN Ñ C be a conformal block associ-ated to pC; x1, . . . , xNq and W1, . . . ,WN . Let

ω : W1 b V b W11 Ñ C,

w b ub w1 ÞÑ xY pu, 1qw,w1y “ÿ

nPZxY puqnw,w1y,

which is a conformal block associated to pP1; 0, 1,8q and W1,V,W11. Thenψ :“ φbω is

a conformal block for rX. Note that when u, w1 are rL0-homogeneous (i.e. eigenvectors

of rL0) with eigenvalues (weights) Ăwtpuq,Ăwtpw1q P N respectively, by (3.1), Y puqnw1 isrL0-homogeneous with weight Ăwtpuq ` Ăwtp1q ´ n´ 1. Then

rSψ : W1 b ¨ ¨ ¨WN b V Ñ CrrqssrSψpw1 b ¨ ¨ ¨ b wN b uq “ ř

nPZq

Ăwtpuq`Ăwtpw1q´n´1 ¨ψ`Y puqnw1 b w2 b ¨ ¨ ¨ b wN

˘ (5.8)

when u, w1 are rL0-homogeneous.From [FB04, Sec. 10.1], this series converges a.l.u. on Dˆ

r (i.e. when 0 ă |q| ă r).(See the proof of Thm. 7.1 for the detailed explanation.) Then, by Theorem 5.3, foreach 0 ă |q| ă r, (5.8) converges to a conformal block associated to Xq and the localcoordinates mentioned after (5.7). If we change the coordinates at x1 and η´1

1 pqq to η1and η1 ´ q respectively, then in the formula (5.8), u and w1 should be multiplied both

by q´rL0 . Under the trivialization given by the new coordinates, rSψpw1 b ¨ ¨ ¨ bwN b uqequals

ψ`Y pu, qqw1 b w2 b ¨ ¨ ¨ b wN

˘:“

ÿ

nPZq´n´1 ¨ψ

`Y puqnw1 b w2 b ¨ ¨ ¨ b wN

˘. (5.9)

We conclude that (once the a.l.u. convergence is established) for all 0 ă |q| ă r, (5.9) isa conformal block associated to Xq, local coordinates η1, η2 . . . , ηN , η1 ´ q, and modulesW1, . . . ,WN ,V.

6 An equivalence of sheaves

Recall ≀X “ p≀π : ≀C Ñ ≀B; σ, ≀ς1, . . . , ≀ςNq in (2.2). In particular, ≀C “ C ˆB pCzSXq,≀B “ CzSX. The goal of this section is to establish a canonical isomorphism

W≀XpV b W‚q » VX b π˚WXpW‚q|CzSX,

18

which relates the sheaves of VOAs and the W -sheaves.Note that π˚WXpW‚q is the pullback sheaf WXpW‚q bOB

OC . This is the sheaf forthe presheaf associating to each open U Ă C the OpUq-module WXpW‚q

`πpUq

˘bOpπpUqq

OpUq. (Note that π is an open map.) Assume the restriction XπpUq has local coordinatesη1, . . . , ηN at ς1pπpUqq, . . . , ςN pπpUqq. We write

π˚w :“ w b 1 P WXpW‚q bB OC “ π˚WXpW‚q

for any section w P WXpW‚q. Sheafifying the tensor product Upη‚q b 1 on the presheafprovides an isomorphism of OC-modules

π˚Upη‚q ” Upη‚q b 1 : WXpW‚q

ˇU

bOπpUqOU

»ÝÑ´W‚ bC OπpUq

¯bOπpUq

OU (6.1)

or simply a trivialization

π˚Upη‚q : π˚WXpW‚q

ˇU

»ÝÑ W‚ bC OU . (6.2)

Choose µ P OpUq injective on each fiber of U . Then we have a trivialization

Upµq b π˚Upη‚q : VX b π˚WXpW‚q

ˇU

»ÝÑ V b W‚ bC OU (6.3)

Now assume U Ă CzSX “ ≀B. Then we can equip the family ≀XU with lo-cal coordinates as follows. For the local coordinate at each submanifold ≀ςjpUq of≀CU “ ≀C X ≀π´1pUq, we choose ≀ηj defined by

≀ηjpx, yq “ ηjpxq (6.4)

whenever px, yq P C ˆB CzSX makes the above definable. The local coordinate at σpUqis µ given by

µpx, yq “ µpxq ´ µpyq (6.5)

when px, yq P U ˆB U . (Recall that σ is the diagonal map.) We can then use µ, ≀η‚ “p≀η1, . . . , ≀ηNq to obtain a trivialization

Upµ, ≀η‚q : W≀XpV b W‚q|U »ÝÑ V b W‚ bC OU (6.6)

We shall relate the two trivializations. First, we need a lemma. Recall U Ă CzSX.Recall (3.5) and (4.1).

Lemma 6.1. Suppose η11, . . . , η

1N are local coordinates of XπpUq at ς1pπpUqq, . . . , ςNpπpUqq re-

spectively, and µ1 P OpUq is injective on each fiber of U . Then, for each x P U , we have

p≀ηj| ≀ η1jqx “ pηj|η1

jqπpxq, pµ|µ1qx “ pµ|µ1qx.

Note that p≀ηj| ≀ η1jq is a family of transformations over U Ă ≀B “ CzSX, and the

transformation over the point x is p≀ηj| ≀η1jqx. pµ|µ1qx is understood in a similar way.

19

Proof. We identify ≀Cx with Cπpxq by identifying py, xq P C ˆB CzSX with y P Cπpxq. Then,from the definition of ≀ηj, ≀η

1j , we clearly have ≀ηj|≀Cx “ ηj|Cπpxq

and ≀η1j|≀Cx “ η1

j|Cπpxq. By

(4.1), we have

p≀ηj | ≀ η1jqx ˝ ≀η1

j|≀Cx “ ≀ηj|≀Cx ,pηj|η1

jqπpxq ˝ η1j|Cπpxq

“ ηj|Cπpxq.

This proves p≀ηj| ≀ η1jqx “ pηj |η1

jqπpxq.Similarly,

pµ|µ1qx ˝ µ1|≀Cx “ µ|≀Cx.By (6.5), we have µ|≀Cx “ pµ ´ µpxqq|Cπpxq

and µ1|≀Cx “ pµ1 ´ µ1pxqq|Cπpxq. These imply

pµ|µ1qx ˝ pµ1 ´ µ1pxqq|Cπpxq“ pµ´ µpxqq|Cπpxq

.

Comparing this relation with (3.5) shows that pµ|µ1qx “ pµ|µ1qx.

Proposition 6.2. We have a unique isomorphism

ΨX : W≀XpV b W‚q »ÝÑ VX b π˚WXpW‚qˇCzSX

(6.7)

such that for any open U Ă CzSX and µ,µ, ≀η‚ as above, the restriction of this isomorphismto U makes the following diagram commutes.

W≀XpV b W‚qˇU

VX b π˚WXpW‚qˇU

V b W‚ bC OU

»ΨX

»Upµ,≀η‚q

»Upµqbπ˚Upη‚q (6.8)

Proof. One can define an isomorphism ΨX such that the above diagram commutes.Such isomorphism is clearly unique. Thus, it remains to check that ΨX is well-defined.We will do so by checking that the transition functions of the two sheaves agree.

Assume U is small enough such that we can have another set of µ1, η1‚ similar to

µ, η‚. Then by (4.2) and Lemma 6.1, for each x P U , we have equalities

Upµ, ≀η‚qx ¨ Upµ1, ≀η1‚q´1

x “ Upµ|µ1qx b Up≀η1| ≀ η11qx b ¨ ¨ ¨ b Up≀ηN | ≀ η1

Nqx“Uppµ|µ1qxq b Upη1|η1

1qπpxq b ¨ ¨ ¨ b UpηN |η1Nqπpxq (6.9)

for transformations on V b W‚ bC OU |x » V b W‚.By (4.2) and (6.1), we have

`π˚

Upη‚q˘x

¨`π˚

Upη1‚q

˘´1

x“ Upη1|η1

1qπpxq b ¨ ¨ ¨ b UpηN |η1Nqπpxq (6.10)

for automorphisms of W‚ bC OU |x » W‚. Thus, by (4.2) and (3.7),`Upµq b π˚

Upη‚q˘x

¨`Upµ1q b π˚

Upη1‚q

˘´1

x

“Uppµ|µ1qqx b Upη1|η11qπpxq b ¨ ¨ ¨ b UpηN |η1

Nqπpxq,

which equals (6.9).

20

7 Propagation of conformal blocks

Letφ : WXpW‚q Ñ OB be a conformal block associated to W‚ “ W1b¨ ¨ ¨bWN and afamily X of N-pointed compact Riemann surfaces. Recall ≀C “ C ˆB pCzSXq, ≀B “ CzSX.The goal of this section is to prove the following theorem. (Cf. [Zhu94, Sec. 6], [FB04,Thm. 10.3.1], [Cod19, Thm. 3.6].)

Theorem 7.1. There is a unique OCzSX-module morphism ≀φ : W≀XpV b W‚q Ñ OCzSX

satis-fying the following property:

”Choose any open subset V Ă B such that the restricted family XV has local coordinatesη1, . . . , ηN at ςjpV q. For each j, we choose a neighborhood Wj Ă CV of ςjpV q on which ηj isdefined, such that Wj intersects only ςjpV q among ς1pV q, . . . , ςNpV q. Identify

Wj “ pηj, πqpWjq via pηj , πqso that Wj is a neighborhood of t0u ˆ V in C ˆ V . Let

Uj :“ WjzSX “ Wjzpt0u ˆ V qwhich is inside Cˆ ˆ V . Let z be the standard coordinate of C. Identify

WXpW‚qˇV

“ W‚ bC OV via Upη‚q.Identify

W≀XpV b W‚qˇUj

“ V b W‚ bC OUjvia Upηj, ≀η‚q (7.1)

(cf. (6.6)). For each u P V, w‚ P W‚, consider each vector of W‚ as a constant section ofW‚ b OpUjq and u b w‚ as a constant section of V b W‚ bC OpUjq. Then the followingequation holds on the level of OpV qrrz˘1ss:

φ`w1 b ¨ ¨ ¨ b Y pu, zqwj b ¨ ¨ ¨ b wN

˘“ ≀φpu b w‚q (7.2)

where Y pu, zqw :“ řnPZ Y puqnw ¨ z´n´1 is an element of Wjppzqq, and ≀φpu b w‚q P OpUjq

is regarded as an element of OpV qrrz˘1ss by taking Laurent series expansion.”Moreover, ≀φ is a conformal block associated to ≀X and V b W‚.

Note that the left hand side of (7.2) is understood asÿ

nPZφ

`w1 b ¨ ¨ ¨ b Y puqnwj b ¨ ¨ ¨ b wN

˘z´n´1,

which is in OpUjqppzqq.

Proof of the uniqueness of ≀φ. It suffices to restrict to the propagation of each fiber Xb,i.e., restrict ≀φ to a morphism φ|≀pXbq : W≀pXbqpV b W‚q Ñ OCbzSXb

. (Note that ≀pXbq isCb ˆ pCbzSXb

q Ñ CbzSXbwith marked points.) By (7.2), we know ≀φ|≀pXbq is uniquely

determined on pW1 Y ¨ ¨ ¨ Y WNq X Cb. For two possible propagations ≀1φ, ≀2φ, let Ω bethe set of all x P CbzSXb

on a neighborhood of which ≀1φ|≀pXbq agrees with ≀2φ|≀pXbq. ThenΩ is open and intersect any connected component of Cb. By complex analysis, it is clearthat if U is a connected open subset of CbzSXb

intersecting Ω such that the restrictionW≀pXbqpVbW‚q|U is equivalent to V bW‚ bC OU , then U Ă Ω. So Ω is closed, and hencemust be CzSXb

. This proves the uniqueness.

21

Proof of the existence of ≀φ. We identify W≀XpVbW‚q with VXbπ˚WXpW‚qˇCzSX

as in Prop.

6.2, and construct an OCSX

-module morphism ≀φ : VX b π˚WXpW‚qˇCzSX

Ñ OCzSXsatis-

fying (7.2). By the uniqueness proved above, we can safely restrict the base manifoldB to V . So we assume in the following that B “ V and hence X has local coordinatesη‚ at marked points. So we identify WXpW‚q with W‚ bC OB through Upη‚q, which yields

VX b π˚WXpW‚q “ VX bC W‚ (7.3)

For each k P N, we let

E “ pV ďkX q_

be the dual bundle of V ďkX . Then the identifications Wj “ pηj , πqpWjq and

V ďkX |Wj

“ Vďk bC OWj

via Upηjq (7.4)

are compatible with the identifications in Sec. A if we set the Ei in that section tobe pVďkq_. Choose any w‚ P W‚. Let sj “ ř

nPZ ej,n ¨ zn as in Sec. A where eachej,n P pVďkq_ bC OpBq is defined by

u P Vďk ÞÑ φ

`w1 b ¨ ¨ ¨ b Y puq´n´1wj b ¨ ¨ ¨ b wN

˘P OpBq.

For each b P B, sinceφ|b is a conformal block, it vanishes onH0pCb,V ďkCb

bωCbp‹SXb

qq¨w‚.This means that s1, . . . , sN satisfy condition (c) of Theorem A.1. Hence, by that the-orem, s1, . . . , sN are series expansions of a unique element s P H0pC, pV ďk

X q_p‹SXqq,which restricts to s P H0pCzSX, pV ďk

X q_q and hence defines an OCzSX-module morphism

V ďkX |CzSX

bC w‚ Ñ OCzSX. These morphisms are compatible for different k, and is ex-

tended OCzSX-linearly to a morphism ≀φ : VX b π˚WXpW‚q

ˇCzSX

Ñ OCzSX(recall (7.3)).

By Prop. 6.2, we can regard ≀φ as a morphism ≀φ : W≀XpV bW‚q Ñ OCzSX. Note that

the identifications (7.3) and (7.4) are compatible with (7.1), thanks to the commutativediagram (6.8). Thus, ≀φ satisfies (7.2) under the required identifications.

Proof that ≀φ is a conformal block. Since being a conformal block is a fiberwise condition,we may prove ≀φ is a conformal block by restricting it to each fiber Xb and its propaga-tion ≀pXbq. Therefore, we may assume that B is a single point. So C :“ C is a compactRiemann surface. We trim each Wj so that ηjpWjq “ Drj for some rj ą 0.

From the previous proof, we have a morphism ≀φ : W≀XpV b W‚q Ñ OCzSXwhich,

given the trivializations in the statement of Theorem 7.1, is equal to (7.2) when re-stricted to WjzSX “ Wjztςju. This shows that the series (7.2) converges a.l.u. on0 ă |z| ă rj . Therefore, as explained in Example 5.4, we can use Thm. 5.3 to con-clude that ≀φ is a conformal block when restricted to each Wj . By Prop. 4.2, ≀φ isglobally a conformal block.

The proof of Thm. 7.1 is completed.We now give an application of this theorem. Suppose E is a set of vectors in

a V-module W. We say E generates W if W is spanned by vectors of the formY pu1qn1

¨ ¨ ¨Y pukqnkw where k P Z`, u1, . . . , uk P V, n1, . . . , nk P Z, w P E.

22

Proposition 7.2. Let X “ pC; x1, . . . , xNq be an N-pointed connected compact Riemannsurface, where N ě 2. Choose local coordinate ηj at xj . Associate V-modules W1, . . . ,WN tox1, . . . , xN . Identify WXpW‚q “ W1 b ¨ ¨ ¨ b WN via Upη‚q. Suppose that for each 2 ď i ď N ,Ei is a generating subset of Wi. Then any conformal block φ : W1 b W2 b ¨ ¨ ¨WN Ñ C isdetermined by its values on W1 b E2 b ¨ ¨ ¨ b EN .

Proof. Assume φ vanishes on W1 b E2 b ¨ ¨ ¨ b EN . We shall show that φ vanishes onW1 b Y puqnE2 b ¨ ¨ ¨ b EN for each u P V, n P Z. Then, by successively applying thisresult, we see that φ vanishes on W1 b W2 b E3 b ¨ ¨ ¨ b EN , and hence (by repeatingagain this procedure several times) vanishes on W1 b W2 b ¨ ¨ ¨ b WN .

Identify W≀XpV b W‚q “ VX

ˇCzSX

bC W‚ using (6.7). Then we can consider ≀φ as a

morphism ≀φ : VX

ˇCzSX

bC W‚ Ñ OCzSX. Let Ω be the open set of all x P CzSX such that

x has a neighborhood U Ă CzSX such that the restriction

≀φ|U : VX|U bC W1 b E2 b ¨ ¨ ¨ b EN Ñ OU

vanishes. We note that if U is connected, and if we can find an injective η P OpUq (sothat VX|U is trivialized to VbC OU ), then by complex analysis, ≀φ|U vanishes whenever≀φ|V vanishes for some non-empty open V Ă U . We conclude that if such U intersectsΩ, then U must be inside Ω. So Ω is closed. It is clear that for each w1 P W1, w2 PE2, . . . , wN P EN , the following formal series of z

φpY pu, zqw1 b w2 b ¨ ¨ ¨ b wNq

vanishes. Thus, by Thm. 7.1, Ω contains W0ztx0u for some neighborhood W0 of x0.Therefore Ω “ CzSX. By Thm. 7.1 again, we see

φpw1 b Y pu, zqw2 b ¨ ¨ ¨ b wNq

also vanishes. This finishes the proof.

Remark 7.3. Since 1 generates V, we see that if V,W2, . . . ,WN (where N ě 2) are asso-ciated to a connected X “ pC; x1, . . . , xN q, then any conformal block φ : VbW2 b ¨ ¨ ¨ bWN Ñ C is determined by its values on 1 b W2 b ¨ ¨ ¨ b WN . This proves the followingtwo well-known results. In fact, in the literature, the propagation of conformal blocksis best known in the form of the following two corollaries.

Corollary 7.4. Let X “ pC; x1, . . . , xNq be an N-pointed compact Riemann surface associatedwith V-module W1, . . . ,WN . Identify W≀XpV b W‚q “ VX

ˇCzSX

bC WXpW‚q via (6.7). Then

for each x P CzSX, ≀φ|x is the unique linear map VX

ˇx

bC WXpW‚q Ñ C which is a conformalblock and satisfies

≀φ|xp1 b wq “ φpwq

for each vector w P WXpW‚q.

Proof. The uniqueness follows from the previous remark. We shall show that ≀φp1 bwq, which is an element of OpCzSXq, equals the constant function φpwq. By complexanalysis, it suffices to prove ≀φp1 b wq “ φpwq when restricted to each Wjztxju (whereWj is a small disc containing xj on which a local coordinate is defined). This is true by(7.2).

23

Corollary 7.5. Let X “ pC; x1, . . . , xNq be an N-pointed connected compact Riemann surfaceassociated with V-module W1, . . . ,WN . Choose x P Cztx1, . . . , xNu. Then the space of con-formal blocks associated to X and W‚ is isomorphic to the space of conformal blocks associatedto p≀Xqx “ pC; x, x1, . . . , xNq and V,W1, . . . ,WN .

Proof. We assume the identifications in Cor. 7.4. The linear map F from the first spaceto the second one is defined by φ ÞÑ ≀φ|x. The linear map G from the second one to thefirst one is defined by ψ ÞÑ ψp1 b ¨q. By Cor. 7.4, we have G ˝ F “ 1. By Remark 7.3, Gis injective. So G is bijective.

8 Multi-propagation

Let X “ pC; x1, . . . , xN q be an N-pointed compact Riemann surface. Recall SX “tx1, . . . , xNu. We choose local coordinates η1 P OpW1q, . . . , ηN P OpWNq of X atx1, . . . , xN , where each Wj is a neighborhood of xj satisfying Wj X SX “ txju.

Let n P Z`. By Section 2, ≀nX is

≀nX “ p≀nπ : C ˆ ConfnpCzSXq Ñ ConfnpCzSXq; σ1, . . . , σn, ≀nx1, . . . , ≀nxNq

where ≀nπ is the projection onto the second component, and the sections are given by

≀nxjpy1, . . . , ynq “ pxj , y1, . . . , ynq,σipy1, . . . , ynq “ pyi, y1, . . . , ynq.

We define local coordinate

≀nηjpx, y1, . . . , ynq “ ηjpxq (8.1)

of ≀nX at xj ˆConfnpCzSXq, defined onWj ˆConfnpCzSXq. Suppose U is an open subsetof CzSX which admits an injective µ P OpUq. Then a local coordinate iµ of p≀nXqU atσipUq is defined by

iµpx, y1, . . . , ynq “ µpxq ´ µpyiq (8.2)

whenever this expression is definable.We shall relate the W -sheaves with the exterior product V bn

C , which is an OCn-module defined by

V bnC :“ pr˚

1VC b pr˚2VC b ¨ ¨ ¨ b pr˚

nVC . (8.3)

Here, each pri : Cn “ C ˆ ¨ ¨ ¨ ˆ Cloooooomoooooon

n

Ñ C is the projection onto the i-th component. The

tensor products are over OCn as usual. Similar to the description in Section 6, the OCn-module pr˚

i VC is the pullback of the (infinite-rank) vector bundle VC along pri to Cn,i.e., VC bOC

OCn where the action of f P OC on OCn is defined by the multiplication off ˝ pri. If U Ă C is open and µ P OpUq is injective, we then have a trivilization

pr˚i Upµq : pr˚

i VC

ˇpr

´1

i pUq»ÝÑ V bC Opr

´1

i pUq.

24

Proposition 8.1. We have a unique isomorphism

W≀nXpVbn b W‚q »ÝÑ V bnC

ˇConfnpCzSXq bC WXpW‚q (8.4)

such that for any n mutually disjoint open subsets U1, . . . , Un Ă CzSX and any injectiveµ1 P OpU1q, . . . , µn P OpUnq, the restriction of this isomorphism to U makes the followingdiagram commutes.

W≀nXpVbn b W‚qˇU1ˆ¨¨¨ˆUn

V bnC

ˇU1ˆ¨¨¨ˆUn

bC WXpW‚q

V b W‚ bC OU1ˆ¨¨¨ˆUn

»

»Up‚µ‚,≀

nη‚q»

pr˚1Upµ1qb¨¨¨bpr˚

nUpµnqbUpη‚q

(8.5)Here,

p‚µ‚, ≀nη‚q :“ p1µ1, . . . ,nµn, ≀

nη1, . . . , ≀nηnq.

Proof. Suppose we have another injective µ1i P OpUiq. Similar to the proof of Lemma

6.1, we see that for each yi P Ui,

piµi|iµ1iqpy1,...,ynq “ pµi|µ1

iqyi .(See (3.5) and (4.1) for the meaning of notations.) Using this relation, one shows, as inthe proof of Prop. 6.2, that the transition functions for the two trivializations in (8.5)are equal. This finishes the proof.

Choose a conformal block φ : W‚ Ñ C associated to X and WXpW‚q. By Theorem7.1, we have n-propagation ≀nφ, which is a conformal block associated to ≀nX and VbnbW‚. By Prop. 8.1, we can regard ≀nφ as a morphism

≀nφ : V bn

C

ˇConfnpCzSXq bC WXpW‚q Ñ OConfnpCzSXq.

Important facts about ≀nφ

Choose (non-necessarily disjoint) open U1, . . . , Un Ă C and write

ConfpU‚zSXq “ pU1 ˆ ¨ ¨ ¨ ˆ Unq X ConfnpCzSXq.For any sections vi P VCpUiq and any w P WXpW‚q, we write

≀nφpv1, . . . , vn, wq :“ ≀

nφ

´pr˚

1v1 b ¨ ¨ ¨ b pr˚nvn b w

ˇConfpU‚zSXq

¯

P O`ConfpU‚zSXq

˘. (8.6)

We now summarize some important properties of ≀nφ in this setting.As an elementary fact, the map pv1, . . . , vnq ÞÑ ≀nφpv1, . . . , vn, wq intertwines the

action of each OpUiq on the i-th component. (Here, each f P OpUiq acts onOpConfpU‚zSXqq by the multiplication of pf ˝priq|ConfpU‚zSXq). Moreover, it is compatiblewith restricting to open subsets of Ui.

We set ≀0φ “ φ.

25

Theorem 8.2. Identify

WXpW‚q “ W‚ via Upη‚q.

Choose any w‚ P W‚. For each 1 ď i ď n, choose an open subset Ui of C equipped with aninjective µi P OpUiq. Identify

VC

ˇUi

“ V bC OUivia Upµiq.

Choose vi P VCpUiq “ V bC OpUiq. Choose py1, . . . , ynq P ConfpU‚zSXq. Then the followingare true.

(1) If U1 “ Wj (where 1 ď j ď N) and contains only y1, xj among all x‚, y‚, if µ1 “ ηj , andif U1 contains the closed disc with center xj and radius |ηjpy1q| (under the coordinateηj), then

≀n φpv1, v2, . . . , vn, w‚q

ˇy1,y2,...,yn

“ ≀n´1 φ

`v2, . . . , vn, w1 b ¨ ¨ ¨ b Y pv1, zqwj b ¨ ¨ ¨ b wN

˘ˇy2,...,yn

ˇz“ηjpy1q (8.7)

where the series of z on the right hand side converges absolutely, and v1 is considered asan element of V b Cppzqq by taking Taylor series expansion with respect to the variableηj at xj .

(2) If U1 “ U2 and contains only y1, y2 among all x‚, y‚, if µ1 “ µ2, and if U2 contains theclosed disc with center y2 and radius |µ2py1q ´ µ2py2q| (under the coordinate µ2), then

≀n φpv1, v2, v3, . . . , vn, w‚q

ˇy1,y2,...,yn

“ ≀n´1 φ

`Y pv1, zqv2, v3, . . . , vn, w‚

˘ˇy2,...,yn

ˇz“µ2py1q´µ2py2q (8.8)

where the series of z on the right hand side converges absolutely, and v1 is considered asan element of V b Cppzqq by taking Taylor series expansion with respect to the variableµ2 ´ µ2py2q at y2.

(3) We have

≀nφp1, v2, v3, . . . , vn, w‚q “ ≀

n´1φpv2, . . . , vn, w‚q. (8.9)

(4) For any permutation π of the set t1, 2, . . . , nu, we have

≀nφpvπp1q, . . . , vπpnq, w‚q

ˇyπp1q,...,yπpnq

“ ≀nφpv1, . . . , vn, w‚q

ˇy1,...,yn

. (8.10)

Proof. When v1, v2 are constant sections (i.e. in V), (1) and (2) follow from Thm. 7.1and especially formula (7.2). The general case follows immediately. (3) follows fromCor. 7.4. By (3), part (4) holds when v1, . . . , vn are all the vacuum section 1. Thus, ithols for all v1, . . . , vn due to Prop. 7.2.

26

9 Sewing and multi-propagation

We assume, in addition to the setting of Section 5, that rB is a single point. Namely,we have an pN ` 2Mq-pointed compact Riemann surface

rX “ p rC; x1, . . . , xN ; x11, . . . , x

1M ; x2

1, . . . , x2Mq,

where each connected component of rC contains one of x1, . . . , xN . For each 1 ď j ď M ,rX has local coordinates ξj at x1

j and j at x2j defined respectively on neighborhoods

W 1j Q x1

j ,W2j Q x2

j . All W 1j ,W

2j (where 1 ď j ď M) are mutually disjoint and do not

contain x1, . . . , xN . ξjpW 1jq “ Drj , and jpW 2

j q “ Dρj . For each marked point xi weassociate a V-module Wi. To x1

j and x2j to we associate respectively a V-module Mj

and its contragredient M1j . We set

SrX “ tx1, . . . , xNu.

Also, for each 1 ď i ď N , choose a local coordinate ηi at xi. Identify

WrXpW‚ b M‚ b M1‚q “ W‚ b M‚ b M

1‚ via Upη‚, ξ‚, ‚q.

We sew rX along each x1j , x

2j to obtain a family

X “ pπ : C Ñ Dˆr‚ρ‚

; x1, . . . , xNq,

where the points x1, . . . , xN on rC and the local coordinates η1, . . . , ηN at these pointsextend constantly (over Dˆ

r‚ρ‚) to sections and local coordinates of X, denoted by the

same symbols. (Cf. Sec. 5.) For each q‚ P Dˆr‚ρ‚

, we identify

WXq‚pW‚q “ W‚ via Upη‚q.

Let φ : W‚ b M‚ b M1‚ Ñ C be a conformal block associated to rX that converges

a.l.u. on Dˆr‚ρ‚

. Let U1, . . . , Un Ă rC be open and disjoint from each W 1j ,W

2j . For each

q‚ P Dˆr‚ρ‚

, since the fiber Cq‚ is obtained by removing a small part of each W 1j ,W

2j Ă rC

and gluing the remaining part of rC, we see that each Ui can be regarded as an opensubset of the fiber Cq‚ . By Thm. 5.3,

rSq‚φ :“ rSφ|q‚

is a conformal block associated to Xq‚. Thus, we can consider its n-propagation ≀n rSq‚φ.In the setting of Thm. 8.2, and setting

ConfpU‚zSrXq “ pU1 ˆ ¨ ¨ ¨ ˆ Unq X Confnp rCzSrXq,

for each vi P V rCpUiq “ VCq‚pUiq and w‚ P W‚,

≀n rSq‚φpv1, . . . , vn, w‚q P OpConfpU‚zSrXqq.

27

This expression relies holomorphically on q‚ due to Thm. 7.1 (applied n times). Thus,by varying q‚, we obtain

≀n rSφpv1, . . . , vn, w‚q P O

`D

ˆr‚ρ‚

ˆ ConfpU‚zSrXq˘. (9.1)

Since ≀nφ is a conformal block associated to ≀nrX, we can talk about the a.l.u. con-vergence of its sewing rS ≀n φ, which is a conformal block by Thm. 5.3 again. In thesetting of Thm. 8.2, this means for each vi P V rCpUiq and w‚ P W‚ the a.l.u. convergenceof

rS ≀n φpv1, . . . , vn, w‚q :“ ≀

nφ

´v1, . . . , vn, w‚ b pqrL0

1 § b1đq b ¨ ¨ ¨ b pqrL0

M § bMđq¯

(9.2)

P OpConfpU‚zSrXqqrrq1, . . . , qM ss

on Dˆr‚ρ‚

ˆ ConfpU‚zSrXq in the sense of Def. 5.1. We may ask whether this convergenceis true, and if it is true, whether the value of this expression at q‚ equals (9.1). Theanswer is Yes.

Theorem 9.1. Assume rSφ converges a.l.u. on Dˆr‚ρ‚

. Then for each open U1, . . . , Un Ă rCdisjoint from W 1

j ,W2j (1 ď j ď N), each vi P V rCpUiq and w‚ P W‚, the relation

rS ≀n φpv1, . . . , vn, w‚q “ ≀

n rSφpv1, . . . , vn, w‚q (9.3)

holds on the level of P OpConfpU‚zSrXqqrrq˘11 , . . . , q˘1

M ss. In particular, the left hand side con-verges a.l.u. on Dˆ

r‚ρ‚ˆ ConfpU‚zSrXq.

We note that the right hand side of (9.3) is considered as a series of q1, . . . , qM bytaking Laurent series expansion.

Proof. We prove this theorem by induction on n. Let us assume the case for n ´ 1 is

proved. For each 1 ď i ď N we choose a neighborhood Wi Ă rC of xi on which ηiis defined. We assume Wi is small enough such that it does not intersect any W 1

j ,W2j

(1 ď j ď N) and contains only x1 of x1, . . . xN .Step 1. Note that we can clearly shrink Dˆ

r‚ρ‚since the formal series in (9.3) are

independent of the size of this punctured polydisc. Therefore, we can also shrink each

W 1j ,W

2j to smaller discs, so that the interior of rCz Ť

1ďjďMpW 1j YW 2

j q (denoted by H) is

homotopic to H0 “ rCztx11, . . . , x

1M , x

21, . . . , x

2Mu. Therefore, since each connected com-

ponent of rC (and hence each one of H0) intersects x1, . . . , xN , each one of H0 containsat least one of W1, . . . ,WN . The same is true for H. So each connected component ofHzSrX contains at lease one Wjztxju.

Fix U2, . . . , Un and v2, . . . , vn, w‚ as in the statement of this theorem. Let Ω be theopen set of all y1 P HzSrX contained in an open U1 Ă HzSrX such that (9.3) holds for allv1 P V rCpU1q. By complex analysis, if V1 Ă HzSrX is open such that V rC |V1

is trivializable(e.g., when there is an injective element of OpV1q), then V1 Ă Ω whenever V1 X Ω ‰ H.So Ω is closed. Thus, if Ω intersects W1ztx1u, . . . ,WNztxNu, then Ω “ HzSrX, whichfinishes the proof.

Step 2. We show Ω intersects W1ztx1u, and hence intersects the other Wiztxiu by asimilar argument. Indeed, we shall show that (9.3) holds whenever U1 “ W1.

28

Note w‚ “ w1 b w2 b ¨ ¨ ¨ b wN by convention. We let w˝ “ w2 b ¨ ¨ ¨ b wN . Iden-tify W1 with η1pW1q via η1 so that η1 is identified with the standard coordinate z. Let

ConfpU˝zSrXq “ pU2 ˆ ¨ ¨ ¨ ˆ Unq X Confn´1p rCzSrXq. Identify V rC |W1with V bC OW1

usingUpη1q. Choose any v1 P V bC OpW1q. Then by Thm. 8.2,

rS ≀n φpv1, v2, . . . , vn, w‚q “ rS ≀

n´1 φpv2, . . . , vn, Y pv1, zqw1 b w˝qon the level of OpConfpU˝zSrXqqrrz˘1, q˘1

1 , . . . , q˘1M ss. By our assumption on the pn ´ 1q-

case, this expression can be regarded as an element of (and hence this equation holdson the level of) OpDˆ

r‚ρ‚ˆ ConfpU˝zSrXqqrrz˘1ss, and we have

rS ≀n φpv1, v2, . . . , vn, w‚q “ ≀

n´1 rSφpv2, . . . , vn, Y pv1, zqw1 b w˝qalso on this level. By Thm. 8.2 again, this expression equals

≀n rSφpv1, v2, . . . , vn, w1 b w˝q

on this level. Since the above is an element of OpDˆr‚ρ‚

ˆ ConfpU‚zSrXqq, by the unique-ness of Laurent series expansion, we see the left hand side of (9.3) is also an elementof this ring, and (9.3) holds on this level.

Remark 9.2. We discuss how to generalize Thm. 9.1 to the case that rX is a family ofcompact Riemann surfaces as in Sec. 5. We assume the setting of that section, together

with one more assumption that rX has local coordinates η1, . . . , ηN at ς1p rBq, . . . , ςNp rBqso that we can identify the W -sheaves with the free ones using the trivialization Upη‚qor Upη‚, ξ‚, ‚q.

We use freely the notations in Sec. 5. Let SrX “ Ť1ďiďM ςip rBq. Let

φ : W‚ b M‚ b M1‚ bC O rB Ñ O rB

be a conformal block associated to rX converging a.l.u. on B “ Dˆr‚ρ‚

ˆ rB. Choose any

open U1, . . . , Un Ă rC disjoint from all W 1j ,W

2j . Choose vi P VrXpUiq and w‚ P W‚. Let

Conf rBpU‚zSrXq be the set of all py1, . . . , ynq P ConfpU‚zSrXq satisfying rπpy1q “ ¨ ¨ ¨ “ rπpynq.For each mj P Mj , m

1j P M1

j , we have

≀nφpv1, . . . , vn, w‚ b m‚ b m1

‚q P OpConf rBpU‚zSrXqq

whose restriction to each rCˆnb (where b P rB is such that rCb intersects U1, . . . , Un) is

≀npφ|bqpv1, . . . , vn, w‚ b m‚ b m1‚q. (Indeed, this expression is a priori only a function

holomorphic when restricted to each rCˆnb ; that it is holomorphic on Conf rBpU‚zSrXq (i.e.,

holomorphic when b also varies) is due to Thm. 7.1.) Thus, we can define

rS ≀n φpv1, . . . , vn, w‚q P OpConf rBpU‚zSrXqqrrq˘1

1 , . . . , q˘1M ss (9.4)

using (9.2). Similarly, with the aid of Thm. 7.1 we can define

≀n rSφpv1, . . . , vn, w‚q P O

`D

ˆr‚ρ‚

ˆ Conf rBpU‚zSrXq˘

(9.5)

whose restriction to each Dˆr‚ρ‚

ˆ rCˆnb is ≀n rSpφ|bqpv1, . . . , vn, w‚q.

Consider (9.5) on the level of OpConf rBpU‚zSrXqqrrq˘11 , . . . , q˘1

M ss. By applying Thm.9.1 to φ|b, we see that the coefficients before q1, . . . , qN of (9.4) and (9.5) agree when

restricted to each rCˆnb . So (9.4) “ (9.5). In particular, (9.4) converges a.l.u. on Dˆ

r‚ρ‚ˆ

Conf rBpU‚zSrXq.

29

10 A geometric construction of permutation-twisted

Vbk-modules

Let U be a (positive energy) VOA, and let g be an automorphism of U fixing thevacuum and the conformal vector of U. In particular, g preserves the L0-grading of U.We assume g has finite order k.

A (finitely-admissible) g-twisted U-module is a vector space W together with a

diagonalizable operator rLg0, and an operation

Y g : U b W Ñ Wrrz˘1kssu b w ÞÑ Y gpu, zqw “

ÿ

nP 1

kZ

Y gpuqnw ¨ z´n´1

satisfying the following conditions:

1. W has rLg0-grading W “ À

nP 1

kNWpnq, each eigenspace Wpnq is finite-

dimensional, and for any u P U we have

rrLg0, Y

gpuqns “ Y gpL0uqn ´ pn` 1qY gpuqn. (10.1)

In particular, for each w P W the lower truncation condition follows: Y gpuqnw “0 when n is sufficiently small.

2. Y gp1, zq “ 1W .

3. (g-equivariance) For each u P U,

Y gpgu, zq “ Y gpu, e´2iπzq :“ÿ

nP 1

kZ

Y gpuqnw ¨ e2pn`1qiπz´n´1. (10.2)

4. (Jacobi identity-analytic version) Let W 1 “ ÀnP 1

kNWpnq˚. Let Pn be the projec-

tion of W “ šnP 1

kNWpnq˚ (the dual space of W 1) onto Wpnq and similarly U (the

dual space of U1) onto Upnq. Then for each u, v P U, w P W, w1 P W 1, and for eachz ‰ ξ in Cˆ with chosen arg ξ, the following series of n

xY gpu, zqY gpv, ξqw,w1y :“ÿ

nP 1

kN

xY gpu, zqPnYgpv, ξqw,w1y (10.3)

xY gpv, ξqY gpu, zqw,w1y :“ÿ

nP 1

kN

xY gpv, ξqPnYgpu, zqw,w1y (10.4)

xY gpY pu, z ´ ξqv, ξqw,w1y :“ÿ

nPNxY gpPnY pu, z ´ ξqv, ξqw,w1y (10.5)

(where ξ is fixed) converge a.l.u. for z in |z| ą |ξ|, |z| ă |ξ|, |z ´ ξ| ă |ξ| respec-tively. Moreover, for any fixed ξ P Cˆ with chosen argument arg ξ, let Rξ be theray with argument arg ξ from 0 to 8, but with 0, ξ,8 removed. Any point on Rξ

30

is assumed to have argument arg ξ. Then the above three expressions, consid-ered as functions of z defined on Rξ satisfying the three mentioned inequalitiesrespectively, can be analytically continued to the same holomorphic function onthe open set

∆ξ “ Cztξ,´tξ : t ě 0u,

which can furthermore be extended to a multivalued holomorphic function fξpzqon Cˆztξu (i.e., a holomorphic function on the universal cover of Cˆztξu).

In the above Jacobi identity, if we let the seriesř

n hnpzq be any of (10.3), (10.4),(10.5), then by saying that this series converges a.l.u. for z in an open set Ω, we meansupzPK

řn |fnpzq| ă `8 for each compact K Ă Ω; the sup is over all z P K with all

possible arg z.

Remark 10.1. The above analytic version of Jacobi identity is equivalent to the usualalgebraic one. Indeed, assume without loss of generality that gu “ e2ijπku. Then the

g-equivariance condition shows that zjkY gpu, zq is single-valued over z. Thus, z

jk times

(10.3), (10.4), (10.5) are series expansions on |z| ą |ξ|, |z| ă |ξ|, |z ´ ξ| ă |ξ| respectively

(not necessarily restricting to Rξ) of the same single-valued holomorphic function zjk fξ

on Cˆztξu. By Strong Residue Theorem, this is equivalent to that for each m,n P Z,

´ ¿

|z|“2|ξ|

´¿

|z|“|ξ|3

´¿

|z´ξ|“|ξ|3

¯z

jk

`mpz ´ ξqnfξpzqdz “ 0,

where in these integrals, fξpzq is replaced by (10.3), (10.4), (10.5) respectively. Equiva-lently,

ÿ

lPN

ˆj

k` m

l

˙@Y g

`Y puqn`lv, ξ

˘w,w1Dξ j

k`m´l

“ÿ

lPN

ˆn

l

˙p´1ql

@Y gpuq j

k`m`n´lY

gpv, ξqw,w1Dξl

´ÿ

lPN

ˆn

l

˙p´1qn´l

@Y gpv, ξqY gpuq j

k`m`lw,w

1Dξn´l. (10.6)

By comparing the coefficients before ξ´h´1, the above is equivalent to that for eachm,n P Z, h P 1

kZ, (suppressing w,w1)

ÿ

lPN

ˆj

k` m

l

˙Y g

`Y

`u

˘n`l

v˘

jk

`m`h´l

“ÿ

lPN

ˆn

l

˙p´1qlY g

`u

˘jk

`m`n´lY g

`v

˘h`l

´ÿ

lPN

ˆn

l

˙p´1qn´lY g

`v

˘n`h´l

Y g`u

˘jk

`m`l

(10.7)

which is the algebraic Jacobi identity.

31

Construction of twisted representations associated to cyclic permuta-

tion actions of Vbk

We let U “ Vbk with conformal vector c b 1 b ¨ ¨ ¨ b 1 ` ¨ ¨ ¨ ` 1 b ¨ ¨ ¨ b 1 b c, and g

an automorphism defined by

g : pv1, v2, . . . , vkq P Vbk ÞÑ pvk, v1, . . . , vk´1q.

For each V-module with rL0-operator, we define an associated g-twisted U-module W

as follows.As a vector space, W “ W. We define rLg

0 “ 1k

rL0.Let ζ be the standard coordinate of C. Let X “ pP1; 0,8q. We associate to 0,8 local

coordinates local coordinates ζ, ζ´1 and V-modules W,W1. Note

Upζ, ζ´1q : WXpW b W1q »ÝÑ W b W

1

Let x¨, ¨y be the pairing for W and W1. We define a conformal block

τW : WXpW b W1q Ñ C,

Upη0, η8q´1pw b w1q ÞÑ xw,w1y

whenever the local coordinates η0, η8 at 0,8 are such that pP1; 0,8; η0, η8q »pP1; 0,8; ζ, ζ´1q. It is easy to see that this definition is independent of the choice ofsuch η0, η8.

In the setting of Thm. 8.2, we have

≀kτW : VXpCˆq b ¨ ¨ ¨ b VXpCˆqloooooooooooooomoooooooooooooon

k

bWXpW b W1q Ñ OpConfkpCˆqq

where all the b are over C. Let

ωk “ e´2iπk.

Since ζk : z ÞÑ zk is locally injective holomorphic on Cˆ, we have a trivilization

Upζkq : VX|Cˆ»ÝÝÝÝÑ V bC OCˆ.

Then, for each w P W, w1 P W1, and for each v1, . . . , vn P V (considered as a constantsection of V bC OpCˆq) we define, for v‚ “ v1 b ¨ ¨ ¨ b vk P Vbk,

xY gpv‚, zqw,w1y“ ≀kτW

`Upζkq´1v1, . . . ,Upζkq´1vk,Upζ, ζ´1q´1pw b w1q

˘ˇˇω‚´1

kk?z

(10.8)

where, for each z P Cˆ with argument arg z,

ω‚´1k

k?z :“ p k

?z, ωk

k?z, ω2

kk?z, . . . , ωk´1

kk?zq P ConfkpCˆq, (10.9)

and k?z is assumed to have argument 1

karg z.

32

(10.8) is a multi-valued function of z, single-valued of k?z P C

ˆ. So we have Laurentseries expansion

xY gpv‚, zqw,w1y “ÿ

nP 1

kZ

xY gpv‚qnw,w1yz´n´1

which defines Y gpv‚qn as a linear map W b W1 Ñ C.

Lemma 10.2. Each Y gpv‚qn is a linear operator on W. Moreover, (10.1) is satisfied.

Proof. For each q P Cˆ with chosen arg q, by (4.2) we have

Upq 1

k ζ, q´ 1

k ζ´1qUpζ, ζ´1q´1 “ q1

krL0 b q´ 1

krL0 “ q

rLg0 b q´rLg

0 .

Thus

xY gpv‚, zqq´rLg0w, q

rLg0w1y

“ ≀k τW

`Upζkq´1v1, . . . ,Upζkq´1vk,Upq 1

k ζ, q´ 1

k ζ´1q´1pw b w1q˘ˇˇω‚´1

kk?z. (10.10)

We have an equivalence of pointed Riemann spheres with locally injective functionsand local coordinates (at the last two marked points)

pP1;ω‚´1k

k?z, 0,8; ζk, q

1

k ζ, q´ 1

k ζ´1q»pP1;ω‚´1

kk?qz, 0,8; q´1ζk, ζ, ζ´1q

defined by z P P1 ÞÑ k?qz P P1, where k

?q has argument 1

karg q. By (3.7) and (3.5), on V

we have

UpζkqUpq´1ζkq´1 “ Uppζk|q´1ζkqq “ qL0 .

So (10.10) equals

≀k τW

`Upq´1ζkq´1v1, . . . ,Upq´1ζkq´1vk,Upζ, ζ´1q´1pw b w1q

˘ˇˇω‚´1

kk?qz

“ ≀k τW

`Upζkq´1qL0v1, . . . ,Upζkq´1qL0vk,Upζ, ζ´1q´1pw b w1q

˘ˇˇω‚´1

kk?qz

We conclude

xY gpv‚, zqq´rLg0w, q

rLg0w1y “ xY gpqL0v‚, qzqw,w1y.

So, if L0v‚ “ αv‚, rLg0w “ βw, rLg

0w1 “ γw1, then

xY gpv‚, zqw,w1y “ qα`β´γxY gpv‚, qzqw,w1y,

which shows, by looking at the coefficients before z´n´1, that xY gpv‚qnw,w1y equals 0

unless α`β´γ´n´1 “ 0. This proves Y gpv‚qnWpβq Ă Wpα`β´n´1q. In particular,Y gpv‚qn can be regarded as a linear operator on W .

33

Using part (3) and (4) of Thm. 8.2, it is easy to show Y gp1, zq “ 1W and show (10.2).Moreover:

Theorem 10.3. Y g satisfies the Jacobi identity. Therefore, pW, Y gq is a g-twisted Vbk-module.

Proof. Choose the two vectors of U to be u‚ “ u1 b ¨ ¨ ¨ b uk, v‚ “ v1 b ¨ ¨ ¨ b vk P Vbk.Identify WXpW b W1q “ W b W1 via Upζ, ζ´1q. Identify VX|Cˆ “ V bC OCˆ via Upζkq.For each ξ P C

ˆ with chosen arg ξ, we define

fξpzq “ ≀2kτWpu1, . . . , uk, v1, . . . , vk, w b w1q

ˇˇω‚´1

kk?z, ω‚´1

kk?ξ

(10.11)

where ω‚´1k

k?ξ is a k-tuple understood in a similar way as (10.9). Then fξ is a multi-

valued holomorphic function which lifts to a single-valued one on the k-fold coveringspace Cˆzpω‚´1

kk?ξq of Cˆztξu.

Let pmn,αqαPA be a set of basis of Wpnq with dual basis p qmn,αqαPA. Assume 0 ă |z| ă|ξ|. We shall show that the following infinite sum over n

xY gpv‚, ξqY gpu‚, zqw,w1y“

ÿ

nPN

ÿ

αPA≀kτWpu1, . . . , uk, w b qmn,αqω‚´1

kk?z ¨ ≀kτWpv1, . . . , vk, mn,α b w1qω‚´1

kk?ξ (10.12)

converges a.l.u. to fξpzq. Indeed, this expression is the sewing at q “ 1 of the 2k-propagation of the conformal block

φ : W b W1 b W b W

1 Ñ C,

w1 b w11 b w2 b w1

2 ÞÑ xw1, w11y ¨ xw2, w

12y

associated to pP1a \ P1

b ; 0a,8a, 0b,8bq. Here, P1a,P

1b are two identical Riemann spheres.

The sewing is along 8a and 0b using local coordinates ζ, ζ´1, and by choosing suitableopen discs W 1 Q 8a,W

2 Q 0b with radius r, ρ satisfying rρ ą 1 such that W 1,W 2 donot intersect ω‚´1

kk?z and ω‚´1

kk?ξ. (Note that |z| ă |ξ| guarantees the existence of such

W 1,W 2.) Since the sewing of φ clearly converges a.l.u. on Dˆrρ, by Thm. 9.1, the sewing

at q “ 1 of ≀2kφ (which is (10.12)) converges a.l.u. (for varying z) to the 2k-propagationof the sewing, which is just fξpzq. A similar argument shows that when 0 ă |ξ| ă |z|,xY gpu‚, zqY gpv‚, ξqw,w1y converges a.l.u. (for varying z) to fξpzq.

Consider gξ P ConfkpCzω‚´1k

k?ξq defined by

gξpz1, . . . , zkq “ ≀2kτWpu1, . . . , uk, v1, . . . , vk, w b w1q

ˇˇz1,...,zk, ω

‚´1

kk?ξ.

The region Ω “ tz P Cˆ : |zk ´ ξ| ă |ξ|u has k connected components Ω1, . . . ,Ωk, eachone Ωi contains exactly one element ωi´1

kk?ξ of ω‚´1

kk?ξ, and Ωi » ζkpΩiq where ζkpΩiq

is the open disc with center ξ and radius |ξ|. By Thm. 8.2 and the definition (10.8),whenever zi P Ωi for each i, we have (letting x1, . . . , xk be formal variables)

gξpz1, . . . , zkq“ ≀

k τWpY pu1, x1qv1, . . . , Y puk, xkqvk, w b w1qˇˇω‚´1

kk?ξ

ˇˇxk“zk

k´ξ

¨ ¨ ¨ˇˇx1“zk

1´ξ

34

“xY gpY pu1, x1qv1 b ¨ ¨ ¨ b Y puk, xkqvk, ξqw,w1yˇˇxk“zk

k´ξ

¨ ¨ ¨ˇˇx1“zk

1´ξ. (10.13)

where the right hand side converges absolutely and successively for xk, xk´1, . . . , x1.Since the simultaneous Laurent series expansion of the holomorphic functionhpκ1, . . .κkq “ gξp k

?ξ ` κ1, ωk

k?ξ ` κ2, . . . , ω

k´1k

k?ξ ` κkq in the region 0 ă |κi| ă |ξ|

(for all i) clearly converges a.l.u., and since the coefficients of these series agree withthose before the powers of x1, . . . , xk on the right hand side of (10.13) (by taking Lau-rent series expansion through contour integrals), we see that (10.13) converges abso-lutely (as a multi-variable series) to gξpz1, . . . , zkq at the desired points.

Now we assume 0 ă |z ´ ξ| ă |ξ|, assume arg z is such that k?z P Ω1 Q k

?ξ

(which is true when arg z “ arg ξ), and set pz1, . . . , zkq “ ω‚´1k

k?z. Then we see that

xY gpY pu‚, z ´ ξqv‚, ξqw,w1y converges a.l.u. to gξpω‚´sk

k?zq “ fξpzq. This finishes the

verification of the Jacobi identity.

Remark 10.4. Using Thm. 8.2, it is easy to see that

≀kτWp1, ¨ ¨ ¨ ,Upζq´1vi, ¨ ¨ ¨ , 1, w b w1q|z “ xY pv, zqw,w1y.

By (3.7), UpζqUpζkq´1 “ Uppζ |ζkqq. Thus, when v‚ “ v1 b 1 b ¨ ¨ ¨ b 1, (10.8) becomes

xY pUppζ |ζkq k?zqv1, k

?zqw,w1y.

By (3.5), pζ |ζkq k?z sends zk1 ´ z to z1 ´ k

?z when z1 is close to k

?z. Hence this transfor-

mation equals δk,z where

δk,zptq “ pz ` tq 1

k ´ z1

k .

We conclude

Y gpv1 b 1 b ¨ ¨ ¨ b 1, zq “ Y pUpδk,zqv1, k?zq. (10.14)

This equation uniquely determines the g-twisted module structure of W , since Vbk isg-generated by vectors of the form v1 b 1 b ¨ ¨ ¨ b 1.

It is not hard to check that Upδk,zq agrees with the operator ∆kpzq in [BDM02]. Thus,our g-twisted module pW, Y gq agrees with the one pT k

g pWq, Ygq in [BDM02, Thm. 3.9].

A Strong residue theorem for analytic families of curves

Let X “ pπ : C Ñ B; ς1, . . . , ςNq be a (holomorphic) family of N-pointed compactRiemann surfaces. Recall the definition in Sec. 2. In particular, we assume each con-nected component of each fiber Cb “ π´1pbq contains at least one of ς1pbq, . . . , ςNpbq.We let E be a holomorphic vector bundle on C with finite rank, and let E _ be its dualbundle.

We assume that X is equipped with local coordinates η1, . . . , ηN at ς1pBq, . . . , ςNpBqrespectively. Assume for each j that ηj is defined on a neighborhood Wj Ă C of ςjpBqwhich intersects only ςjpBq among ς1pBq, . . . , ςNpBq, and that there is a trivialization

Ej |Wj» Ej bC OWj

35

with dual trivialization

E _j |Wj

» E_j bC OWj

,

where Ej is a finite-dimensional vector space and E_j is its dual space. We identify

Ej|Wjand E _|Wj

with their trivializations.For each j, we identify

Wj “ pπ, ηjqpWjq via pπ, ηjq.

Then Wj is a neighborhood of B ˆ t0u in B ˆ C. We let z be the standard coordinate ofC. Consider

sj “ÿ

nPZej,n ¨ zn P

`Ej bC OpBq

˘ppzqq, (A.1)

where each ej,n P Ej bC OpBq is 0 when n is sufficiently small. Considering ej,n as anEj-valued holomorphic on OpBq, we let ej,npbq P Ej be its value at b P B. Then sjpbq,the restriction of sj to Cb, is represented by

sjpbq “ÿ

n

ej,npbqzn P Ejppzqq.

Suppose that s is a section of E p‹SXq defined on Wj . Then s|Wj“ s|Wj

pb, zq is an Ej-valued meromorphic function on Wj with poles at z “ 0. We say that s has seriesexpansion sj at ςjpBq if for each b P B, the meromorphic function s|Wj

pb, zq of z hasLaurent series expansion (A.1) at z “ 0.

For each b P B, choose σb P H0pCb, E _|Cb b ωCbp‹SXb

qq. Then in Wj,b “ Wj X π´1pBq,σb can be regarded as an E_

j bdz-valued holomorphic function but with possibly finitepoles at z “ 0. So it has series expansion at z “ 0:

σb|Wj,bpzq “

ÿ

n

φj,nzndz P E_

j ppzqqdz

where φj,n P E_j . We define the residue pairing

Resjxsj , σby “Resz“0xsjpbq, σb|Uj ,bpzqy

“Resz“0

ˆA ÿ

n

ej,npbqzn,ÿ

n

φj,nznEdz

˙. (A.2)

in which the pairing between Ej and E_j is denoted by x¨, ¨y.

We now prove the Strong Residue Theorem for E . Our proof is inspired by that of[Ueno08, Thm. 1.22].

Theorem A.1. For each 1 ď j ď N , choose sj as in (A.1). Then the following statements areequivalent.

(a) There exists s P H0pC, E p‹SXqq whose series expansion at ςjpBq (for each 1 ď j ď N)is sj .

(b) For each b P B, there exists sb P H0pCb, E |Cbp‹SXbqq whose series expansion at ςjpbq (for

each 1 ď j ď N) is sjpbq.

36

(c) For any b P B and any σb P H0`Cb, E

_|Cb b ωCbp‹SXb

q˘,

Nÿ

j“1

Resjxsj, σby “ 0. (A.3)

Moreover, when these statements hold, there is only one s P H0pC, E p‹SXqq satisfying (a).

Proof. (a) trivially implies (b). That (b) implies (c) follows from Residue theorem (i.e.,Stokes theorem): The evaluation between sb and σb is an element of H0pCb, ωCb

p‹SXbqq

whose total residue over all poles is 0.If s satisfies (a), then for each b P B, s|Cb is uniquely determined by its series expan-

sions near ς1pbq, . . . , ςNpbq (since each component of Cb contains some ςjpbq). Thereforethe sections satisfying (a) is unique.

Now assume (c) is true. We shall prove (a). Suppose that for each b P B we can finda neighborhood V Ă B such that an s satisfying (a) exists for the family XV . Then, bythe uniqueness proved above, we can glue all these locally defined s to a global one.Thus, we may shrink B to a small neighborhood of a given b0 P B when necessary.

We first note that, by replacing B with a neighborhood of a given b0 P B, we mayassume π˚E pkSXq “ 0 for sufficiently large k. Indeed, choose any b0 P B. Then by Serreduality,

H0`Cb, E |Cbp´kSXb

q˘

» H1`Cb, E

_|Cb b ωCbpkSXb

q˘, (A.4)

which, by Serre vanishing theorem, equals 0 for some k “ k0 when b “ b0. Since π isopen, X is a flat family ([GPR, Thm. II.2.13] or [Fis76, Sec. 3.20]). Thus, we can applythe upper-semicontinuity theorem ([GPR, Thm. III.4.7] or [BS76, Thm. III.4.12]) to seethat (A.4) vanishes for k “ k0 and (by shrinking B to a neighborhood of b0) any b P B.Since the vector space H0

`Cb, E |Cbp´kSXb

q˘

shrinks as k increases, (A.4) is constantlyzero for all b P B and k ě k0. This implies π˚E p´kSXq “ 0 for all k ě k0 ([GPR, Thm.III.4.7-(d)] or [BS76, Cor. III.3.5]).

Choose p P N such that for each 1 ď j ď N , the ej,n in (A.1) equals 0 when n ă ´p.For any k ě k0, as π˚E p´kSXq “ 0, the short exact sequence

0 Ñ E p´kSXq Ñ E ppSXq Ñ E ppSXqE p´kSXq Ñ 0

induces a long one

0 Ñ π˚E ppSXq Ñ π˚`E ppSXqE p´kSXq

˘ δÝÑ R1π˚E p´kSXq. (A.5)

For each 1 ď j ď N , set sj|k “ řnăk ej,n ¨ zn, which can be regarded as a section in

E ppSXqpWjq. Let W0 “ CzSX. Then U “ tW0,W1, . . . ,WNu is an open cover of C. DefineCech 0-cocycle ψ “ pψjq0ďjďN P Z0pU, E ppSXqE p´kSXqq by setting

ψ0 “ 0, ψj “ sj|k p1 ď j ď Nq.

Then δψ “`pδψqi,j

˘0ďi,jďN

P Z1pU, E p´kSXqq is described as follows: pδψq0,0 “ 0; if

i, j ą 0 then pδψqi,j is not defined since Wi X Wj “ H; if 1 ď j ď N then pδψqj,0 “´pδψq0,j equals sj |k (considered as a section in E p´kSXqpWj X W0q).

37

Consider δψ as a section of R1π˚E p´kSXq. We shall show that δψ “0. By the fact that (A.4) vanishes and the invariance of Euler characteris-tic, dimH1

`Cb, pE |Cbqp´kSXb

q˘

is locally constant over b P B, which shows thatR1π˚pC, E p´kSXqq is locally free and its fiber at b is naturally equivalent toH1

`Cb, pE |Cbqp´kSXb

q˘. (Cf. [GPR, Thm. III.4.7] or [BS76, Thm. III.4.12].) Thus, it

suffices to show that for each fiber Cb, the restriction δψ|Cb P H1pCb, E |Cbp´kSXbqq is

zero.The residue pairing for the Serre duality

H1pCb, E |Cbp´kSXqq » H0`Cb, E

_|Cb b ωCbpkSXb

q˘˚

applied to δψ|Cb and any σb P H0`Cb, E

_|Cb b ωCbpkSXb

q˘, is given by

xδψ|Cb, σby “Nÿ

j“1

Resjxsj|k, σby.

Since for each 1 ď j ď N , xsj ´ sj|k, σby has removable singularity at z “ 0, we haveResjxsj ´ sj |k, σby “ 0. Therefore,

xδψ|Cb, σby “Nÿ

j“1

Resjxsj, σby “ 0.

Thus δψ|Cb “ 0 for any b. This proves that δψ “ 0.By (A.5), for each k ě k0, there is a unique s|k P

`π˚E ppSXq

˘pBq “ H0pC, E ppSXqq

which is sent to ψ P π˚`E ppSXqE p´kSXq

˘pBq. So near ςjpBq, s|k has series expansion

s|k “ sj|k ` ‚zk ` ‚zk`1 ` ¨ ¨ ¨ . (A.6)

By this uniqueness, we must have s|k0 “ s|k0`1 “ s|k0`2 “ ¨ ¨ ¨ . Let s “ s|k0 . Then s hasseries expansion sj at ςjpBq for each j.

We remark that the above proof also applies to locally free sheaves over a properflat family of pointed complex curves (with at worst nodal singularities) such thateach SXb

does not intersect the node of Cb, and that SXbintersects each irreducible

component of Cb. This is because the residue pairing for Serre duality is described inthe same way as in the smooth case.

38

Index

The vacuum section 1, 12

Cˆ, 7Cb,Xb, 8ConfpU‚zSXq, 25ConfnpXq, 7ConfnB, 9

Dr,Dˆr , 7

Dˆr‚ρ‚

, 15

E pnSXq, E p‹SXq, 8

G, 11

rL0, 10

Res, 11

rSψ, 17rSq‚φ, 27SX, 8

Upη‚q, 13Upρq, 11Upηq, 12π˚Upη‚q, 19

Vpηjq, 14V ďnX ,VX,VC , 12

Wpnq,Wpnq, 10W‚, w‚ “ w1 b ¨ ¨ ¨ b wN , 13WXpW‚q, 13

≀X, ≀C, ≀B, ≀π, 9≀nX, ≀nC, ≀nB, ≀nπ, 10XV , CV , 13

YW “ Y , 11

rrzss, rrz˘1ss, ppzqq, tzu, 10pηj |µjq, 13ωC , ωCb

, 8ωk “ e´2iπk, 32ω‚´1k

k?z, 32

µ, iµ, 19, 24pη|µq, 12≀ηj , ≀

nηj , 19, 24

≀ςj, ≀nςj , 9, 10

≀φ, 21≀nφ, 25

39

References

[BDM02] Barron, K., Dong, C. and Mason, G., 2002. Twisted sectors for tensor product vertex oper-ator algebras associated to permutation groups. Communications in mathematical physics,227(2), pp.349-384.

[BS76] Banica, C. and Stanasila, O., 1976. Algebraic methods in the global theory of complex spaces.

[Buhl02] Buhl, G., 2002. A spanning set for VOA modules. Journal of Algebra, 254(1), pp.125-151.

[Cod19] Codogni, G., 2019. Vertex algebras and Teichmuller modular forms. arXiv preprintarXiv:1901.03079.

[DGT19a] Damiolini, C., Gibney, A., and Tarasca, N., 2019. Conformal blocks from vertex algebras andtheir connections on Mg,n. To appear in Geometry & Topology, arXiv:1901.06981.

[DGT19b] Damiolini, C., Gibney, A. and Tarasca, N., 2019. On factorization and vector bundles ofconformal blocks from vertex algebras. arXiv preprint arXiv:1909.04683.

[DLM97] Dong, C., Li, H. and Mason, G., 1997. Regularity of Rational Vertex Operator Algebras. Ad-vances in Mathematics, 132(1), pp.148-166.

[DXY21] Dong, C., Xu, F. and Yu, N., 2021. Permutation orbifolds and associative algebras. ScienceChina Mathematics, pp.1-10.

[FB04] Frenkel, E. and Ben-Zvi, D., 2004. Vertex algebras and algebraic curves (No. 88). AmericanMathematical Soc..

[FHL93] Frenkel, I., Huang, Y. Z., & Lepowsky, J. (1993). On axiomatic approaches to vertex operatoralgebras and modules (Vol. 494). American Mathematical Soc..

[Fis76] Fischer, G., Complex analytic geometry. Lecture Notes in Mathematics, Vol. 538. Springer-Verlag, Berlin-New York, 1976. vii+201 pp.

[GPR] Grauert, H., Peternell, T., Remmert, R. eds., 1994. Several complex variables VII: sheaf-theoretical methods in complex analysis (Vol. 74, p. 369). Springer.

[Gui20] Gui, B., 2020. Convergence of Sewing Conformal Blocks. arXiv preprint arXiv:2011.07450.

[Gui21] Gui, B., 2021. Genus-zero Permutation-twisted Conformal Blocks for Tensor Product VertexOperator Algebras: The Tensor-separable Case. arXiv: 2111.04662.

[Hua97] Huang, Y.Z., 1997. Two-dimensional conformal geometry and vertex operator algebras (Vol.148). Springer Science & Business Media.

[Li96] Li, H.S., 1996. Local Systems of Twisted Vertex Operators. In Moonshine, the Monster, andRelated Topics: Joint Research Conference on Moonshine, the Monster, and Related Topics,June 18-23, 1994, Mount Holyoke College, South Hadley, Massachusetts (Vol. 193, p. 203).American Mathematical Soc..

[TUY89] Tsuchiya, A., Ueno, K. and Yamada, Y., 1989. Conformal field theory on universal family ofstable curves with gauge symmetries. In Integrable Sys Quantum Field Theory (pp. 459-566).Academic Press.

[Ueno08] Ueno, K., 2008. Conformal field theory with gauge symmetry (Vol. 24). American Mathe-matical Soc..

[Zhu94] Zhu, Y., 1994. Global vertex operators on Riemann surfaces. Communications in Mathemat-ical Physics, 165(3), pp.485-531.

[Zhu96] Zhu, Y., 1996. Modular invariance of characters of vertex operator algebras. Journal of theAmerican Mathematical Society, 9(1), pp.237-302.

YAU MATHEMATICAL SCIENCES CENTER, TSINGHUA UNIVERSITY, BEIJING, CHINA.

E-mail: binguimath@gmail.com bingui@tsinghua.edu.cn

40