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Gluing II: Boundary Localization and Gluing Formulas

We describe applications of the gluing formalism discussed in the companion paper. When a $d$-dimensional local theory $\text{QFT}_d$ is supersymmetric, and if we can find a supersymmetric polarization for $\text{QFT}_d$ quantized on a $(d-1)$-manifold $W$, gluing along $W$ is described by a non-local $\text{QFT}_{d-1}$ that has an induced supersymmetry. Applying supersymmetric localization to $\text{QFT}_{d-1}$, which we refer to as the boundary localization, allows in some cases to represent gluing by finite-dimensional integrals over appropriate spaces of supersymmetric boundary conditions. We follow this strategy to derive a number of `gluing formulas' in various dimensions, some of which are new and some of which have been previously conjectured. First we show how gluing in supersymmetric quantum mechanics can reduce to a sum over a finite set of boundary conditions. Then we derive two gluing formulas for 3D $\mathcal{N}=4$ theories on spheres: one providing the Coulomb branch representation of gluing, and another providing the Higgs branch representation. This allows to study various properties of their $(2,2)$-preserving boundary conditions in relation to Mirror Symmetry. After that we derive a gluing formula in 4D $\mathcal{N}=2$ theories on spheres, both squashed and round. First we apply it to predict the hemisphere partition function, then we apply it to the study of boundary conditions and domain walls in these theories. Finally, we mention how to glue half-indices of 4D $\mathcal{N}=2$ theories.