Surface defects and instanton partition functions

TL;DR

This work develops a Higgsing (vortex) prescription to compute protected quantities, namely the 4d superconformal index and the 5d index/partition function, in the presence of co-dimension two defects. By specializing UV partition functions to Higgs-branch poles, the authors derive residue formulas that yield the IR defect data, including contributions from 3d chiral sectors and vortex modes, and they test these against dualities and brane constructions. The paper extends the framework to 3d/5d coupled systems, introduces a JK-residue formalism for computing the 5d instanton sum with defect degrees of freedom, and demonstrates that Wilson loops and difference equations quantify the Seiberg-Witten data in these enriched theories. They also establish S-transformation behavior of defects via hemisphere indices and connect the results to integrable systems in the NS limit, offering a powerful computation scheme for a broad class of co-dimension-two defects with potential links to q-deformed algebras and topological-string/AGT structures.

Abstract

We study the superconformal index of five-dimensional SCFTs and the sphere partition function of four-dimensional gauge theories with eight supercharges in the presence of co-dimension two half-BPS defects. We derive a prescription which is valid for defects which can be given a "vortex construction", i.e. can be defined by RG flow from vortex configurations in a larger theory. We test the prescription against known results and expected dualities. We employ our prescription to develop a general computational strategy for defects defined by coupling the bulk degrees of freedom to a Gauged Linear Sigma Model living in co-dimension two.

Figures (8)

• Figure 1: The NS5-D4 brane description of a baryonic Higgs branch of an $SU(2) \times SU(2)$ quiver gauge theory which flow to a pure $SU(2)$ gauge theory in the IR. On the right, we include a D2 brane segment corresponding to a position-dependent Higgs branch
• Figure 2: The NS5-D4 brane description of a baryonic Higgs branch of an $SU(3)$ theory with two flavors which flow to a pure $SU(2)$ gauge theory in the IR. On the right, we include a D2 brane segment corresponding to a position-dependent Higgs branch
• Figure 3: The NS5-D4 brane description of a particular Higgs branch of a $SU(4)^3$ linear quiver gauge theory. On the left: the un-Higgsed theory. In the middle: a Higgs branch deformation which flows to the product of two $SU(2)$ SQCD theories, each with four flavors. On the right: a position dependent Higgs branch vev flows to a surface defect associated to D2 branes stretched between the two sub-systems.
• Figure 4: Left: A generic fivebrane web, engineering an SCFT with an $SU(2) \times U(1)$ flavor symmetry. Center: the Newton polygon for the web and a generic Coulomb branch deformation. Right: A Higgs branch deformation of the 5d SCFT. At low energy, we find a simpler SCFT, associated to the web with edges $(1,1)$, $(1,-2)$, $(-2,1)$.
• Figure 5: A position dependent Higgs branch vev in the theory of Figure \ref{['fig:two']} produces a $D^{(1)}_{(1,1)}$ defect in the simpler SCFT. A position dependent Higgs branch vev in a different theory produces a $D^{(1)}_{(0,1)}$ defect in the simpler SCFT. The two defects should be related by a 3d $SL(2,{\mathbb Z})$ transformation.