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