# Surface defects and instanton-vortex interaction

## TL;DR

This paper introduces a simple formula for the 4d-2d partition function of half-BPS surface defects in 4d N=2 gauge theories, unifying various defect calculations and highlighting the role of negative vortices.

## Contribution

It provides a universal formula for the 4d-2d partition function applicable to multiple types of surface defects, connecting different computational approaches.

## Key findings

- Reproduces recent Higgsing defect calculations
- Matches orbifold defect calculations by Kanno and Tachikawa
- Highlights the significance of negative vortices as negative D0 branes

## Abstract

We propose a simple formula for the 4d-2d partition function of half-BPS surface defects in $d=4,\ \mathcal{N}=2$ gauge theories: $Z^{\text{4d-2d}}=\langle Z^{\text{2d}} \rangle_{\text{4d}}$. Our results are applicable for any surface defect obtained by gauging a 2d flavour symmetry using a 4d gauge group. For defects obtained via the Higgsing procedure, our formula reproduces the recent calculation by Pan and Peelaers. For Gukov-Witten defects our results reproduce the orbifold calculation by Kanno and Tachikawa. We emphasize the role of "negative vortices" which are realized as negative D0 branes.