Localization of N=4 superconformal field theory on S1 X S3 and index

We provide the geometrical meaning of the N = 4 superconformal index. With this interpretation, the N = 4 superconformal index can be realized as the partition function on a Scerk-Schwarz deformed background. We apply the localization method in TQFT to compute the deformed partition function since the deformed action can be written as a delta(epsilon)-exact form. The critical points of the deformed action turn out to be the space of flat connections which are, in fact, zero modes of the gauge field. The one-loop evaluation over the space of flat connections reduces to the matrix integral by which the N = 4 superconformal index is expressed.