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We provide the geometrical meaning of the ${\cal N}=4$ superconformal index.With this interpretation, the ${\cal N}=4$ superconformal index can be realizedas the partition function on a Scherk-Schwarz deformed background. We apply thelocalization method in TQFT to compute the deformed partition function sincethe deformed action can be written as a $\delta_\epsilon$-exact form. Thecritical points of the deformed action turn out to be the space of flatconnections which are, in fact, zero modes of the gauge field. The one-loopevaluation over the space of flat connections reduces to the matrix integral bywhich the ${\cal N}=4$ superconformal index is expressed.

Localization of N=4 Superconformal Field Theory on S^1 x S^3 and Index