Central Extensions of Supersymmetry in Four and Three Dimensions

TL;DR

The paper addresses how tensor central charges extend the supertranslation algebra in $d=4$ and $d=3$, linking string and domain-wall charges to nontrivial moduli configurations. It develops a universal construction where domain-wall charges are governed by the gravitino-mass matrix $M^{AB}$ and string charges arise from the composite $SU(N)$ connection $V^A_{B\,\mu}$, with tensor charges $Z^{(AB)}_{\mu\nu}$ and $Z^{A}_{\mu B}$. For $N=2$ theories from Calabi–Yau compactifications, explicit moduli-dependent realizations are given via wrapped branes, and the discussion extends to 3D where $O(N)$ R-symmetry and $E_{8(8)}$ U-duality organize the scalar charges. The results illuminate how moduli-space geometry controls extended object charges, consider quantum corrections, and suggest fixed-point scenarios for dynamical moduli in the presence of strings and membranes.

Abstract

We consider the maximal central extension of the supertranslation algebra in d=4 and 3, which includes tensor central charges associated to topological defects such as domain walls (membranes) and strings. We show that for all N-extended superalgebras these charges are related to nontrivial configurations on the scalar moduli space. For N=2 theories obtained from compactification on Calabi-Yau threefolds, we give an explicit realization of the moduli-dependent charges in terms of wrapped branes.