We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d N = 4 superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2-and 3-point functions of Higgs branch operators in the full 3d N = 4 theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d N = 2 subalgebra of the N = 4 algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on S 3 that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.

We develop new techniques for computing exact correlation functions of a class of local operators, including certain monopole operators, in three-dimensional N = 4 abelian gauge theories that have superconformal infrared limits. These operators are position-dependent linear combinations of Coulomb branch operators. They form a onedimensional topological sector that encodes a deformation quantization of the Coulomb branch chiral ring, and their correlation functions completely fix the (n ≤ 3)-point functions of all half-BPS Coulomb branch operators. Using these results, we provide new derivations of the conformal dimension of half-BPS monopole operators as well as new and detailed tests of mirror symmetry. Our main approach involves supersymmetric localization on a hemisphere HS 3 with half-BPS boundary conditions, where operator insertions within the hemisphere are represented by certain shift operators acting on the HS 3 wavefunction. By gluing a pair of such wavefunctions, we obtain correlators on S 3 with an arbitrary number of operator insertions. Finally, we show that our results can be recovered by dimensionally reducing the Schur index of 4D N = 2 theories decorated by BPS 't Hooft-Wilson loops.

We develop an approach to the study of Coulomb branch operators in 3D N = 4 gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative and noncommutative star product algebra on the Coulomb branch. For "good" and "ugly" theories (according to the Gaiotto-Witten classification), we also exhibit a trace map on this algebra, which allows for the computation of correlation functions and, in particular, guarantees that the star product satisfies a truncation condition. This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.

We propose a way of computing 4-manifold invariants, old and new, as chiral correlation functions in half-twisted 2d N = (0, 2) theories that arise from compactification of fivebranes. Such formulation gives a new interpretation of some known statements about Seiberg-Witten invariants, such as the basic class condition, and gives a prediction for structural properties of the multi-monopole invariants and their non-abelian generalizations. 5 Non-abelian generalizations 51 5.1 Coulomb-branch index and new 4-manifold invariants 54 A Curvature of the canonical connection on the universal bundle 57 A.1 SU (N f ) invariance and the moment map 59 B Topological twist of SQED 60 1.1 Unorthodox invariants of smooth 4-manifolds Searching for new invariants of smooth 4-manifolds, that potentially could go beyond the Seiberg-Witten and Donaldson invariants, it was proposed in [1] to consider a twodimensional quantum field theory T [M 4 , G] as a rather unusual invariant of smooth structures on a 4-manifold M 4 . Specifically, T [M 4, G] is a 2d N = (0, 2) superconformal theory that, apart from M 4 , also depends on a choice of a root system G and is invariant under the Kirby moves. Luckily, conformal field theories in two dimensions exhibit rich mathematical structure which, on the one hand, is rich enough to (potentially) describe the wild world of smooth 4-manifolds and, on the other hand, is rigorous enough to hope for a precise mathematical definition of the invariant T [M 4 , G]. In fact, for many practical purposes and applications in this paper, a mathematically inclined reader can think of T [M 4 , G] as a functor that assigns a vertex operator algebra (VOA) to a smooth 4-manifold M 4 (and a "gauge" group G).Composing it with other functors that assign various quantities to 2d conformal theories, one can obtain more conventional invariants of smooth 4-manifolds:Here, Z can be any invariant of a 2d conformal theory with N = (0, 2) supersymmetry, e.g., its elliptic genus, chiral ring, moduli space of marginal couplings, or central charge. Since 2d theory T [M 4 ; G] is systematically determined by M 4 and invariant under the Kirby moves, all such invariants lead to various 4-manifold invariants; some are simple and some are quite powerful. In particular, it was conjectured in [1] that chiral ring of the theory T [M 4; G] for G = SU (2) or, equivalently, its Q + -cohomology knows about Donaldson invariants of M 4 . One of the main goals in this paper is to present some evidence to this conjecture and to build a bridge between VOA G [M 4 ] and more traditional 4-manifold invariants. Conjecturally, at least for manifolds with b + 2 > 1, one can trade SU (2) Donaldson invariants for Seiberg-Witten invariants defined in a simpler gauge theory, with gauge group G = U (1), which would be too trivial if not for an extra ingredient, the additional spinor fields. We wish to study how these invariants are realized in 2d theory T [M 4 ; G] with G = SU (2) and G = U (1), respectively. In particular, we shall see that in 2d realizati...