Localization of N=4 SCFT on S^1 x S^3 and Index

We provide the geometric meaning of the N=4 superconformal index. With this consideration, the N=4 index can be interpreted as the partition function on a Scherk-Schwarz deformed background. By standard localization methods, we add a $Q$-exact term to the Scherk-Schwarz deformed action. The critical points of the $Q$-exact term turn out to be the space of flat connections which are, in fact, zero modes of the gauge field. Then one-loop evaluation reduces the matrix integral Minwalla et al obtained in 2005. In addition, if time permits, I will describe the prospect on the relation between the N=1 index and $a$-maximization which would be an analogue in four dimensions of the $Z$-extremization Jafferis has recently argued in three dimensions.