The Witten Index and $\mathcal{N}=(2,2)$ Worldsheet Effective Theory of Non-Abelian Vortices

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 $\mathcal{M}_{vortex} = \mathbb{C} \times \mathbb{C} P^{N-1}$. 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.