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We investigate a two-dimensional system of Active Brownian Particles atfinite density. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, webuilt up the generating functional for correlation functions. We study indetail the hydrodynamic regime with a constant density stationary state. Ourfindings reveal that, within a small density fluctuations regime, an emergent$U(1)$ gauge symmetry arises, originated from the conservation of fluidvorticity. Consequently, the interaction between the orientational orderparameter and density fluctuations can be cast into a gauge theory, where theconcept of ``electric charge density" aligns with the local vorticity of theoriginal fluid. We study in detail the case of a microscopic local two-bodyinteraction. We show that, upon integrating out the gauge fields, thestationary states of the rotational degrees of freedom satisfy a non-localFrank free energy. We give explicit expressions for the splay and bend elasticconstants as a function of the P\'eclet number (${\rm Pe}$) and the diffusioninteraction constant ($k_d$).

Emergent Gauge Symmetry in Active Brownian Matter