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We calculate the superconformal indices of the $\mathcal N=2$ superconformalfield theories realized on $N$ coincident D3-branes in 7-brane backgrounds withconstant axiodilaton via the AdS/CFT correspondence. We include the finite-$N$corrections as the contribution of D3-branes wrapped around 3-cycles in theinternal space. We take only single-wrapping contributions into account forsimplicity. We also determine the orders of the next-to-leading correctionswhich we do not calculate. The orders are relatively high, and we obtain manytrustable terms. We give the results for $N=1,2,3$ explicitly, and find niceagreement with known results.

Holographic index calculation for Argyres-Douglas and Minahan-Nemeschansky theories