Instanton partition functions of 5d \( \mathcal{N}=1 \) gauge theories are Witten indices for the ADHM gauged quantum mechanics with (0, 4) SUSY. We derive the integral contour prescriptions for these indices using the Jeffrey-Kirwan method, for gauge theories with hypermultiplets in various representations. The results can be used to study various 4d/5d/6d QFTs. In this paper, we study 5d SCFTs which are at the UV fixed points of 5d SYM theories. In particular, we focus on the Sp(N ) theories with N_f ≤ 7 fundamental and 1 antisymmetric hypermultiplets, living on the D4-D8-O8 systems. Their superconformal indices calculated from instantons all show \( {E_N}_{{}_f+1} \) symmetry enhancements. We also discuss some aspects of the 6d SCFTs living on the M5-M9 system. It is crucial to understand the UV incompleteness of the 5d SYM, coming from small instantons in our problem. We explain in our examples how to fix them. As an aside, we derive the index for general gauged quantum mechanics with (0, 2) SUSY.

We study a family of 2d \( \mathcal{N}=\left(0,\ 4\right) \) gauge theories which describes at low energy the dynamics of E-strings, the M2-branes suspended between a pair of M5 and M9 branes. The gauge theory is engineered using a duality with type IIA theory, leading to the D2-branes suspended between an NS5-brane and 8 D8-branes on an O8-plane. We compute the elliptic genus of this family of theories, and find agreement with the known results for single and two E-strings. The partition function can in principle be computed for arbitrary number of E-strings, and we compute them explicitly for low numbers. We test our predictions against the partially known results from topological strings, as well as from the instanton calculus of 5d Sp(1) gauge theory. Given the relation to topological strings, our computation provides the all genus partition function of the refined topological strings on the canonical bundle over \( \frac{1}{2}K3 \).

We study the index of 4d $\mathcal{N}=4$ Yang-Mills theory with $U(N)$ gauge group, focusing on the physics of the dual Bogomolny-Prasad-Sommerfield black holes in ${\mathrm{AdS}}_5\times S^5$. Certain aspects of these black holes can be studied from finite $N$ indices with reasonably large $N^2$. We make numerical studies of the index for $N\le 6$, by expanding it up to reasonably high orders in the fugacity. The entropy of the index agrees very well with the Bekenstein-Hawking entropy of the dual black holes, say at $N^2=25$ or 36. Our data clarifies and supports the recent ideas which allowed analytic studies of these black holes from the index, such as the complex saddle points of the Legendre transformation and the oscillating signs in the index. In particular, the complex saddle points naturally explain the $\frac{1}{N}$-subleading oscillating patterns of the index. We also illustrate the universality of our ideas by studying a model given by the inverse of the MacMahon function.