Theorem 2

Let M be a compact oriented 3 manifoldwithout boundary Suppose that M is

obtained as Dehn surgery on a famedlink L with m components Lj Kj Em

in S3 Then

Zn M Soo Cd Son SonnyKia am

is the Chern Simons partition fund of M

Let us first compute Zi 5 corresponding

to the case with ne link

Lemma 2 2 K S3 Soo

ProofiThe first step is to compute 2 2 5

Viewing S as the time direction weget

EEEtf

2KCExs Trudidim He

In the case of 2 52 we getdim Hss I 2 52 5 1

In case there are Wilson lines passingthrough 52 and along S we get

S 2 52 54427 dim Hs p

I E Cbs Rn Sa o

7 52 5 Ra Rm SirSt Ze Shs Ra Rm Rr Namu

Now 53 can be obtained from 52 5 bythe following surgery

T

inSh S S

We see that 52 5 is obtained by gluingsolid Tori T and T by identifying

2T 2T

similarly S3 is obtained by identifying2T 52T

IS Trf

At the level of conformal blocks this givesF 52 51 R LY l

Zu S3 Ro 5415 Xo

eels xn Eso alla

So 2 52 5 Rn Soo

Sn o 11

Proof of Theorem 2

Let us first deal with the case neo

We know by Lemma 2 that 2 r 5 SoWe thus have to show

Soo CE Seo JCO.ru e2TFdSoomEPCk increase of framing

g

is

no surgery III eeYkuotaloithk.fr

Now we know J in SET

Lemma l for 2 2 0 gives

Ip Son e Soo E CS

nor Inalization FITS Rn2 v S3

The factor C corrects for factors due to

framing ambiguity of 3 manifold M

I Ei Ea