Heisenberg Spin Chain And Supersymmetric Gauge Theory

We show how a Heisenberg spin chain emerges from the two-dimensional ${\mathcal N}$=(2,2) gauge theory at an intermediate scale, which relies on the renormalization group flow guided by the global symmetries and the dynamics of domain walls. The discussion of higher-dimensional gauge theories with four supercharges is similar by compactifying them into two dimensions. Instead of utilizing the bosonic fields solely in the literature, adopting the fermionic degrees of freedom of gauge theory is a crucial step in our construction. From the perspective of the spin chain, we observe that the Seiberg(-like) duality between two gauge theories is manifest since it is a finite symmetry in the closed spin chain. Based on this, we further conjecture gauge theories with the same global symmetries could be unified into a single system: the Heisenberg spin chain. At least, we prove this conjecture at the intermediate scale. Finally, we also comment on other formulations of an integrable system centered on the Heisenberg spin chain.