Infrared Computations of Defect Schur Indices

Abstract

We conjecture a formula for the Schur index of N=2 four-dimensional theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS excitations. We test our proposal in a variety of examples for SU(2) gauge theories, either conformal or asymptotically free. We use the conjecture to compute these defect-enriched Schur indices for theories which lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate in various examples that line defect indices can be expressed as sums of characters of the associated two-dimensional chiral algebra and that for Argyres-Douglas theories the line defect OPE reduces in the index to the Verlinde algebra.

Figures (14)

• Figure 1: The geometry of line defects. When we conformally map to $S^3\times S^1$, each (half) line defect $L_i$ wraps around the $S^1$ and sits at a point on a great circle (blue line) on the $S^3$ according to their phases $\vartheta_i$. Here $\vartheta_{ij}=\vartheta_i-\vartheta_j$. The worldline of the line defect is shown in red.
• Figure 2: The BPS quiver for the $\mathcal{N}=2$$SU(2)$ gauge theory with $N_f=4$ hypermultiplets in the fundamental representation. The Dirac pairings $\langle \gamma_{i}, \gamma_{j}\rangle$ are given by the arrows between the nodes.
• Figure 3: The conformal map (together with the compactification of $\mathbb{R}$ to $S^1$) of (a) a half line defect and (b) a full line defect from $\mathbb{R}^4$ to $S^3\times S^1$. The worldline of the line defect is shown in red.
• Figure 4: The BPS quiver for the $\mathcal{N}=2$ pure $SU(2)$ gauge theory.
• Figure 5: The framed quiver for a Wilson line in the $\mathbf{2}$ in the $SU(2)$ gauge theory with $N_f=4$ flavors. The core charge of the doublet Wilson line is given by $\gamma_c= -{1\over2} ( \gamma_1 + \gamma_2 + \gamma_3 +\gamma_5)$.