Download the pdf here, preprint, SSRN accepted on 15th August, 2026. Abstract:-The Witten Index for the worldsheet theory of BPS cosmic strings in $\mathcal{N} = 2$ supersymmetry gauge theories have been investigated. Starting from $\mathcal{N} = 2$ supersymmetry alegbra with central charges, BPS bounds and representation theory of short multiplets has been reviewed. Cosmic strings of codimension $2$ is used as the foundation for constructing the moduli space of the vortex solutions for both abelian and non-abelian $U (N)$ gauge theories.

Major focus has been devoted to calculating the zero-modes of the vortex moduli. For the non-abelian case, the vortex moduli space acquires an internal orientational structure described by the compact $\mathbb{C}P^{N-1}$ , arising from color-flavor locking. The low-energy dynamics of the worldsheet is governed by an $\mathcal{N} = (2, 2)$ sigma model with the target space $\mathcal{M}_{vortex} = \mathbb{C} × \mathbb{C}P^{N-1}$. Using supersymmetry localization, it is shown that the Witten index of this worldsheet theory, receives the entire non-trivial contribution from $\mathbb{C}P^{N-1}$. The fermionic/odd contributions of the ground state vanish by virtue of the Betti Numbers. The main result demonstrates that the index is a topological invariant, protected by the BPS condition. This provides a non-perturbative link between the bulk topological defect and the worldsheet supersymmetric quantum mechanics.

Published on 10th August 2026,

Citation

@misc{nandi_2026_21875864,
  author       = {Nandi, Soumyadip},
  title        = {The Witten Index and \$\mathcal{N}=(2,2)\$
                   Worldsheet Effective Theory of Non-Abelian
                   Vortices
                  },
  month        = aug,
  year         = 2026,
  publisher    = {Zenodo},
  doi          = {10.5281/zenodo.21875864},
  url          = {https://doi.org/10.5281/zenodo.21875864},
}