On Supersymmetry, Boundary Actions and Brane Charges

TL;DR

This paper analyzes how boundary actions can preserve different subalgebras of supersymmetry in 4d theories with boundaries, classifying boundary conditions into A-type and B-type, related to domain walls and strings.

Contribution

It identifies and classifies boundary actions that preserve supersymmetry subalgebras independently of boundary conditions, linking them to physical objects like domain walls and strings.

Findings

Two classes of boundary actions (A-type and B-type) are identified.

A-type preserves a 3d $ ext{N}=1$ subalgebra, B-type preserves a 2d $ ext{N}=(0,2)$ subalgebra.

Boundary terms in the energy-momentum tensor relate to domain wall and string currents.

Abstract

Supersymmetry transformations change the Lagrangian $\mathscr{L}$ into a total derivative $\delta \mathscr{L} = \partial_\mu \mathcal{V}^{\mu}$. On manifolds with boundaries the total derivative term is an obstruction to preserving supersymmetry. Such total derivative terms can be canceled by a boundary action without specifying boundary conditions, but only for a subalgebra of supersymmetry. We study compensating boundary actions for $\mathcal{N}=1$ supersymmetry in 4d, and show that they are determined independently of the details of the theory and of the boundary conditions. Two distinct classes of boundary actions exist, which correspond to preserving either a linear combination of supercharges of opposite chirality (called A-type) or supercharges of opposite chirality independently (B-type). The first option preserves a subalgebra isomorphic to $\mathcal{N} = 1$ in 3d, while the second preserves only a 2d subgroup of the Lorentz symmetry and a subalgebra isomorphic to $\mathcal{N} = (0,2)$ in 2d. These subalgebras are in one to one correspondence with half-BPS objects: the A-type corresponds to domain walls while the B-type corresponds to strings. We show that integrating the full current algebra and taking into account boundary contributions leads to an energy-momentum tensor which contains the boundary terms. The boundary terms come from the domain wall and string currents in the two respective cases.