Global anomalies of Green's function zeros

Anomaly analysis has been an important and powerful tool in studying nonperturbative physics for decades. The anomaly inflow mechanism provides an intuitive interpretation of the bulk-boundary correspondence in topological systems. In this work, we study global anomalies in systems with symmetry-preserving Luttinger surfaces, i.e. the manifolds of fermionic Green's function zeros in the momentum space at zero energy, described by nonlocal effective theories. We view the nonlocal effective theories as a result of integrating out some low energy states. Assuming that the states integrated out do not make extra contributions to the anomalies, we discuss the simplest Lagrangian describing a gapless Dirac zero and a two-pole variant, their global anomalies, and the bulk-boundary correspondence. We then consider the constraints on possible phases with Green's function zeros of Dirac type, such as non-Fermi liquids and emergent gapless quasiparticles on Luttinger surfaces. We also provide some perspectives on why the nonlocal fermionic effective theory discussed by Golterman and Shamir (arXiv: 2311.12790) is not a suitable starting point for a symmetrically gapped phase.