We present a theory unifying the topological responses and anomalies of various gapless fermion systems exhibiting Fermi surfaces, including those with Berry phases, and nodal structures, which applies beyond the noninteracting limit. As our key finding, we obtain a general approach to directly relate gapless fermions and topological insulators in phase space, including first- and higher-order insulators. Using this relation we show that the low-energy properties and response theories for gapless fermionic systems can be directly obtained without resorting to microscopic details. Our results provide a unified framework for describing such systems using well-developed theories from the study of topological phases of matter.

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• Anomalies
• Fermi surface
• Luttinger's theorem
• Quantum anomalous Hall effect
• Topological materials