Post on 11-Apr-2022
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