We clarify the derivation of high-energy QCD evolution equations from the fundamental gauge symmetry of QCD. The gauge-fixed classical action of the Color Glass Condensate (CGC) is shown to be invariant under a suitable BRST symmetry, that holds after the separation of the gluon modes into their fast classical (background) part, the soft component and the semifast one, over which the one-step quantum evolution is carried out. The resulting Slavnov-Taylor (ST) identity holds to all orders in perturbation theory and strongly constrains the CGC effective field theory (EFT) arising from the integration of the soft modes. We show that the ST identity guarantees gauge-invariance of the EFT. It also allows to control the dependence on the gauge-fixing choice for the semifast modes (usually the lightcone gauge in explicit computations). The formal properties of the evolution equations valid in different regimes (BKFL, JIMWLK, ...) can be all derived in a unified setting within this algebraic approach.
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