Elliptic Quantum Curves of 6d SO(N) theories

TL;DR

This paper studies supersymmetric defects in 6d SO(N) theories, revealing that their partition functions satisfy elliptic difference equations that quantize the Seiberg-Witten curve, with implications for twisted compactifications.

Contribution

It introduces a novel connection between defect partition functions and elliptic difference equations quantizing Seiberg-Witten curves in 6d SO(N) theories.

Findings

Partition function with codimension 2 defect satisfies elliptic difference equation.

Expectation value of codimension 4 defect forms an elliptic curve section.

RG flows of defects yield quantum Seiberg-Witten curves for twisted compactifications.

Abstract

We discuss supersymmetric defects in 6d $\mathcal{N}=(1,0)$ SCFTs with $\mathrm{SO}(N_c)$ gauge group and $N_c-8$ fundamental flavors. The codimension 2 and 4 defects are engineered by coupling the 6d gauge fields to charged free fields in four and two dimensions, respectively. We find that the partition function in the presence of the codimension 2 defect on $\mathbb{R}^4\times \mathbb{T}^2$ in the Nekrasov-Shatashvili limit satisfies an elliptic difference equation which quantizes the Seiberg-Witten curve of the 6d theory. The expectation value of the codimension 4 defect appearing in the difference equation is an even (under reflection) degree $N_c$ section over the elliptic curve when $N_c$ is even, and an odd section when $N_c$ is odd. We also find that RG-flows of the defects and the associated difference equations in the 6d $\mathrm{SO}(2N+1)$ gauge theories triggered by Higgs VEVs of KK-momentum states provide quantum Seiberg-Witten curves for $\mathbb{Z}_2$ twisted compactifications of the 6d $\mathrm{SO}(2N)$ gauge theories.