Gluing II: Boundary Localization and Gluing Formulas

Understand

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.