One of the great powers of global symmetry is its ability to constrain the possible phases of many-body quantum systems. In this talk, we will present a symmetry that enforces every symmetric model to be in a phase with a Fermi surface. This constraint is entirely non-perturbative and a strong form of symmetry-enforced gaplessness. We construct this symmetry in fermionic quantum lattice models on a $d$-dimensional Bravais lattice, and it is generated by a U(1) fermion-number symmetry and Majorana translation symmetry. The resulting symmetry group is an infinite-dimensional non-abelian Lie group closely related to the Onsager algebra. We will comment on the topology of these symmetry-enforced Fermi surfaces and the UV symmetry's relation to the IR LU(1) symmetry of ersatz Fermi liquids. (This talk is based on ongoing work with Shu-Heng Shao and Luke Kim.)
