Generalised instanton counting in two, three and four complex dimensions

In this thesis we study some topics related to instanton counting in (complex) dimension two, three and four. In dimension two, we introduce and study a class of surface defects in $4d$ $mathcal N=4$ Vafa-Witten theory on a product $mathcal{C}_{g,k} imes T^2$ of a $k$-punctured Riemann surface by the two-torus, supporting BPS solutions which we call nested instantons. Exploiting supersymmetric localisation, we compute explicitly the partition function of the Vafa-Witten theory in presence of a general defect in this class and conjecture a specific polynomial behaviour for some nesting profiles. We also study the geometry of the moduli space of representations of the nested instantons quiver, and we prove that it's a quasi-projective variety isomorphic to the moduli space of flags of framed torsion-free sheaves on $mathbb{P}^2$. In dimension three, we extend to higher rank the vertex formalism developed by Maulik-Nekrasov-Okounkov-Pandharipande for Donaldson-Thomas theory. We use the critical structure of the quot scheme of points of $mathbb{A}^3$ and the induced equivariant perfect obstruction theory to compute K-theoretic and cohomological higher rank DT invariants. We also prove two conjectures proposed by Awata-Kanno and Szabo about the plethystic expression of their generating function. Moreover, we define a chiral version of the virtual elliptic genus, and the corresponding elliptic DT invariants. This definition compares to results from string theory found in Benini-Bonelli-Poggi-Tanzini, and we argue that a conjecture about the partition function of elliptic DT invariants can be interpreted in terms of its behaviour in the modular parameter. Finally, in dimension four we study solutions to an analogue to the self-duality equations in (real) four dimensions. We construct a Topological Field Theory describing the dynamics of D(-1)/D7 brane systems on Riemannian manifolds with $Spin(7)$-holonomy. The BPS-bound states counting reproduces the (cohomological) DT theory on four-folds. We also study the local model for eight-dimensional instantons and provide an ADHM-like construction. Finally we generalise the construction to orbifolds of $mathbb{C}^4$ admitting a crepant resolution, and compute the corresponding partition function, which conjecturally encodes the corresponding orbifold DT invariants.