In this thesis , we focus on the partition function of two dimensional conformal field theory . From invariance of partition function under T transformation , we conclude that every state have integer spin and the difference between the left and the right central charges are integer multiple of 24 . By using the T-invariance of partition function in theories with real partition function , we show that the theory is parity even . In prior research , using the S - invariance of partition function in the canonical ensemble , an upper bound on the lowest primary field is obtained . In order to improve the upper bound , we study medium temperature expansion in grand canonical ensemble . For an arbitrary order of derivatives , we show that by increasing the order of derivatives , the better upper bound can be obtained; moreover , the order of derivatives can not be an arbitrary number . We obtain the optimal values of the order of derivatives which leads to the best upper bound . We also , study the theories with chiral algebra . Using the medium temperature expansion in different manner , we show that corresponding to $S$ invariance non-chiral partition function , there exists a chiral S-invariance function . Then , we focus on special Finally , we study the partition function of CFT corresponding to the pure gravity with negative cosmological constant . Using the pure gravity in Chern-Simons approach , it is shown that the dual CFT is extremal and the partition function is holomorphicly factorizable . Focusing on the holomoorphicaly factorizable partition function , we show that the weight of primary fields are integer and the left and the right central charges are integer multiple of 8 . Using the relation between the central charges and the Chern-Simons coupling , we show that the gauge group of Chern-Simons theory is SO(2,1)× SO(2,1) and its three-fold cover . In order to obtain the partition function dual to the pure gravity , we introduce the Heck operator . Using the expansion of partition function in terms of generalized Heck operator , we show that micro canonical entropy is equal to Bekenstein-Hawking entropy plus its logarithmic corrections .
