Gluing II: boundary localization and gluing formulas

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