Surface Defect Indices and 2d-4d BPS States

TL;DR

This work constructs an infrared framework that expresses the Schur index of 4d ${\cal N}=2$ theories with (2,2) surface defects in terms of the 2d–4d BPS spectrum on the Coulomb branch, via a refined wall-crossing invariant. The central proposal, $\mathcal{I}_{\mathbb{S}}(q) = (q)_{\infty}^{2r}\,\text{Tr}[\mathcal{S}^{2d-4d}_{\vartheta,\vartheta+\pi}(X_{\gamma})\mathcal{S}^{2d-4d}_{\vartheta+\pi,\vartheta+2\pi}(X_{\gamma})]$, unifies Cecotti–Vafa’s 2d elliptic-genus limit with earlier 4d Schur-index reconstructions and extends to framed (line) defects. The refined, chamber-independent $2d$–$4d$ wall-crossing formalism encodes how 2d solitons, 2d/4d particles, and defect vacua transform across walls, enabling exact defect-indices computations and revealing a universal line-surface relation: a surface defect index can be decomposed into a sum of line-defect indices. The paper confirms the framework in concrete examples, notably SU(2) SYM coupled to a CP^1 sigma model, where localization and IR data agree, and provides a broader program for analyzing defect operators via infrared BPS data and their chiral-algebraic interpretations.

Abstract

We conjecture a formula for the Schur index of four-dimensional $\mathcal{N}=2$ theories coupled to $(2,2)$ surface defects in terms of the $2d$-$4d$ BPS spectrum in the Coulomb phase of the theory. The key ingredient in our conjecture is a refined $2d$-$4d$ wall-crossing invariant, which we also formulate. Our result intertwines recent conjectures expressing the four-dimensional Schur index in terms of infrared BPS particles, with the Cecotti-Vafa formula for limits of the elliptic genus in terms of two-dimensional BPS solitons. We extend our discussion to framed $2d$-$4d$ BPS states, and use this to demonstrate a general relationship between surface defect indices and line defect indices. We illustrate our results in the example of $SU(2)$ super Yang-Mills coupled to the $\mathbb{CP}^1$ sigma model defect.

Figures (1)

• Figure 1: The unwrapping process of a surface defect to a sum of lines. Starting in the upper left we have a surface defect $\mathbb{S}$ (shown in blue) wrapping an equator in $S^{3}$ as well as the $S^{1}$ which is not shown. The defect is cut open using a resolution of the identity with left and right boundary conditions $B^{\alpha}$ and $B_{\alpha}$. The boundary conditions are then parallel transported around the equator using the operator $\mathcal{R}_{2\pi}$. When they collide the surface defect is gone and a sum of line defects $L_{i}$ remains.