We initiate the study of intersecting surface operators/defects in 4D quantum field theories (QFTs). We characterize these defects by coupled 4D/2D/0D theories constructed by coupling the degrees of freedom localized at a point and on intersecting surfaces in spacetime to each other and to the 4D QFT. We construct supersymmetric intersecting surface defects preserving just two supercharges in N = 2 gauge theories. These defects are amenable to exact analysis by localization of the partition function of the underlying 4D/2D/0D QFT. We identify the 4D/2D/0D QFTs that describe intersecting surface operators in N = 2 gauge theories realized by intersecting M2 branes ending on N M5 branes wrapping a Riemann surface. We conjecture and provide evidence for an explicit equivalence between the squashed four-sphere partition function of these intersecting defects and correlation functions in Liouville/Toda CFT with the insertion of arbitrary degenerate vertex operators, which are labeled by two representations of SU(N).

We propose a set of novel expansions of Nekrasov's instanton partition functions. Focusing on 5d supersymmetric pure Yang-Mills theory with unitary gauge group on C-q,t(2) 1 x S-1, we show that the instanton partition function admits expansions in terms of partition functions of unitary gauge theories living on the 3d subspaces C-q x S-1, Ct-1 x S-1 and their intersection along S-1. These new expansions are natural from the BPS/CFT viewpoint, as they can be matched with W-q,W-t correlators involving an arbitrary number of screening charges of two kinds. Our constructions generalize and interpolate existing results in the literature.

We consider type IIB SL(2, Z) symmetry to relate the partition functions of different 5d supersymmetric Abelian linear quiver Yang-Mills theories in the Omega-background and squashed S-5 background. By Higgsing S-dual theories, we extract new and old 3d mirror pairs. Generically, the Higgsing procedure yields 3d defects on intersecting spaces, and we derive new hyperbolic integral identities expressing the equivalence of the squashed S-3 partition functions with additional degrees of freedom on the S-1 intersection.

We consider U(N) SQCD on S-5 and propose a Higgs branch-like expression for its partition function. We support the result by arguing that the knowledge of certain BPS codimension 2 and 4 defects arising from Higgsing is enough to reconstruct the bulk partition function, and that the defect partition functions satisfy a set of non-perturbative Schwinger-Dyson equations. We show that the result is consistent with, and naturally come from, the BPS/CRT perspective. In this language, the defect partition functions are identified with free boson correlators of the q-Virasoro modular triple, and the constraint equations with Ward identities satisfied by the corresponding Dotsenko-Fateev q-conformal blocks, providing a natural basis to expand the S-5 partition function.

Inspired by 5d supersymmetric Yang-Mills theories placed on the compact space S5, we propose an intriguing algebraic construction for the q-Virasoro algebra. We show that, when multiple q-Virasoro chiral sectors have to be fused together, a natural SL(3,Z) structure arises. This construction, which we call the modular triple, is consistent with the observed triple factorization properties of supersymmetric partition functions derived from localization arguments. We also give a 2d CFT-like construction of the modular triple, and conjecture for the first time a (non-local) Lagrangian formulation for a q-Virasoro model, resembling ordinary Liouville theory.

Four-dimensional N = 2 superconformal quantum field theories contain a subsector carrying the structure of a chiral algebra. Using localization techniques, we show for the free hypermultiplet that this structure can be accessed directly from the path integral on the four-sphere. We extend the localization computation to include supersymmetric surface defects described by a generic 4d/2d coupled system. The presence of a defect corresponds to considering a module of the chiral algebra: our results provide a calculational window into its structure constants.

We analyze intersecting surface defects inserted in interacting four-dimensional N = 2 supersymmetric quantum field theories. We employ the realization of a class of such systems as the infrared fixed points of renormalization group flows from larger theories, triggered by perturbed Seiberg-Witten monopole-like configurations, to compute their partition functions. These results are cast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provide concrete expressions for the instanton partition function in the presence of intersecting defects and we study the corresponding ADHM model.