We study the multiplicity of BPS domain walls in $\mathcal{N}=1\mathrm{}$ super Yang-Mills theory, by passing to a weakly coupled Higgs phase through the addition of fundamental matter. The number of domain walls connecting two specified vacuum states is then determined via the Witten index of the induced world volume theory, which is invariant under the deformation to the Higgs phase. The world volume theory is a sigma model with a Grassmanian target space which arises as the coset associated with the global symmetries broken by the wall solution. Imposing a suitable infrared regulator, the result is found to agree with recent work of Acharya and Vafa in which the walls were realized as wrapped $D4\mathrm{}$-branes in type IIA string theory.

• Received 4 June 2002

DOI:https://doi.org/10.1103/PhysRevD.66.065015