Proposition 2 system of fundamentalsolutions of KZ eq around 4 42 0

written as

Io Ui Uz 4 u us uAuzktfentrist

with holomorphic function 4 u us

OI is matrix valued and diagonalizedwith respect to a a

j L

pal sa

tree basis of conformal blocksSimilarly can construct horizontal sections

of E associated to1 Az Ry

J L

pen am

perform coordinate transformation32 J 402 52 02

40 Zz Zz 02 23 Z o 3722

connection matrix w is expressedaround 4 4 0 as

w Iz do I t tr det wa

where Ws is hot I form around 4 4 0

ICM and R2acts dC23 are diagonalize

simultaneously with respect to pigSolutions of KZ eq around 4 02 0 become

I 4 4 4 4 uz uN qts tr t I

where 414 vz is hot around 4 02 0

Note that Us Zz Z 4 2272,7 and

U2 Zz Z O 73 2223 Z

I corresponds to asymptotic regionA Z L Zz G Zz and I to the regionB Z K Zz L 23

Analytic continuation from region A

to region D gives

g

Io OI F where F is calledconnection matrix

12 Az 12 A

a Eunn

R Ay1 14

I 2 3 4 I 237 4

In terms of chiral vertex operators this givesLemma 3

In the region 0 111 1721 we have

II G 42nd's.Ixoidth

Eu.it ICHlidHa.xo4Ia.GisDOI and I can be understood as solutions

of Fuchsian differential equationG x 5 t GG

with regular singularities at 0,1 and as

Ifa zz zj.cz zfr R R Gfz z

Io and Is then correspond to the two solutions

G x H Cx x x o H hal

Glx Hix i923

around 0 and x l We have

G x Glx F

Next let IT be a trivalent tree with ntt

external edges

i i

Take htt points p pm pm on theRiemann sphere with put as and setzcp.kz

space of conformal blocksH pi r Pu put A oh 22

with basis given by labellings of abovetree

Each trivalent tree represents a system ofsolutions of the KZ equation

consider tree of type f 1213 a nti

perform coordinate transformations

µ 2 Z and k Uk 4kt Un l

K I e n Ihave solutions of KZ equations around4 Um O

OI 4 u un ykd ud t D sin

UmDlitt

where 4 u un is matrix valued hot

functionOI is diagonalized with respect to basis

corresponding to f 1213 4 htt

Consider the case a 4

For the tree of type 41271343 5 performcoordinate transformation 31 403 33 32 0203

33 03 and associate

E 4 G Oz os v Kd uke E.ge

where k G Oz Us is holomorphic around0 Uz Uz5 types of I Ei

Fyx 25

I Iµ

Penne 8.9 a a a EAs a as

I 112 43 Ay

42 14X V

i As

One can go from graphI 2 3 4 5

to graph I 2 341 5 by the connectionmatrix of Lemma 3 as follows

I VII 3 Hana 1 idHa

For b id 454th 3DIn general the connection matrix for eachedge of the Pentagon above is represented

by the composition of connectionmatrices

for n 3 call n 3 connection matrix

elementary c in and the linear mapis called elementary fusion operationwe also have without proofProposition 3

For the two edge paths connecting two

distinct trees in the Pentagon diagramthe corresponding compositions of elementaryconnection matrices coincide

Monodromy representation of braid groupTake n distinct points pups pn withcoordinates satisfying ocz.az a Zu

associate level k highest weights a in

to pi por Take po o put as with

no O and the 0

corresponding conformal blocksCpo pi iput i Ao Ai a A n

Hong Kj E

Denote image of above map by Va an

with basis given by in such tha

any triple mm aj nj satisfies quantumClebsch Gordan condition at level K

a generator ri of the braid group Bndefines a linear map

p Ti Va init an Va dit di in

interchange of points Pi and pit

TranePhi only depends on homotopy class ofabove path as KZ connection is flatIn general we have for r c Bn

PG Va An Vittore Arorawhere Ti Bu Su is natural surjectionWe also have plot p G PE o t c BuLet us deal with the ease a 3

For the solutionE Un Us 4 u us qtTR12ytth 2 5213 223

with u Zz Z u 7 we seeZz Z

pcr.lu u PG I P expft e a Iwhere Piz Vain 7 Vain Asdialk is diagonalized for tree basis1273 4 with eigenvalue Ix Aa Aa

On the other handPbs Is _Paz expftFR231k Ea

where 5223 12 is diagonalized with respectto basis Ct 23 4

To combine these local monodromieswe need connection matrix F

I Eat