More on the Tensorial Central Charges in N=1 Supersymmetric Gauge Theories (BPS Wall Junctions and Strings)
A. Gorsky, M. Shifman
TL;DR
The paper analyzes tensorial central charges in ${\cal N}=1$ supersymmetric gauge theories for solitons with axial geometry (strings and wall junctions) and derives a general expression for the central extension $Z_\mu$. It shows that, at weak coupling, BPS strings exist only in SQED with a Fayet–Iliopoulos term, while the $(1/2,1/2)$ central charge receives ambiguous matter contributions that do not affect the string tension; in non-Abelian theories these strings cannot saturate at weak coupling. The discussion extends to ${\cal N}=2$ theories, where strong-coupling BPS strings may arise on Coulomb or Higgs branches, with a brane picture offering geometric interpretations (D2 between NS5s for FI strings and M2 between M5s in M-theory). The work also highlights a precise interplay between ambiguities in the $(1/2,1/2)$ and $(1,0)$ charges that yields unambiguous wall-junction tensions, and it connects these results to two-dimensional sigma-model instantons and brane constructions. Overall, the paper clarifies when BPS axial objects can exist in ${\cal N}=1$ and ${\cal N}=2$ gauge theories and how brane realizations illuminate their central charges and saturations.
Abstract
We study the central extensions of the N=1 superalgebras relevant to the soliton solutions with the axial geometry - strings, wall junctions, etc. A general expression valid in any four-dimensional gauge theory is obtained. We prove that the only gauge theory admitting BPS strings at weak coupling is supersymmetric electrodynamics with the Fayet-Iliopoulos term. The problem of ambiguity of the (1/2,1/2) central charge in the generalized Wess-Zumino models and gauge theories with matter is addressed and solved. A possibility of existence of the BPS strings at strong coupling in N=2 theories is discussed. A representation of different strings within the brane picture is presented.