M5-branes, orientifolds, and S-duality

We study the instanton partition functions of 5d maximal super Yang-Mills theories with all classical gauge groups. They are computed from the ADHM quantum mechanics of the D0-D4-O4 systems. Our partition functions respect S-dualities of the circle compactified Yang-Mills theories and various orientifold backgrounds. We also compute and study the $S^5$ partition functions that correspond to the 6d (2,0) superconformal indices. Our SO(2N) index takes the form of the vacuum character of $\mathcal{W}_D$ algebra in a special limit, supporting the $\mathcal{W}$ algebra conjecture. We propose new indices for (2,0) theories with outer automorphism twists along the temporal circle, obtained from non-simply-laced SYMs on $S^5$.