Unleash $Q$! cohomology, localization, and interpolation in Parisi-Sourlas supersymmetry

Parisi—Sourlas supersymmetric models are known to undergo dimensional reduction; that is, their physics is captured by models in two fewer dimensions. In this work, we revisit dimensional reduction, providing new arguments and reformulating existing proofs in terms of the cohomology of a supercharge Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> . We obtain three main results. First, we show that the recently developed picture of dimensional reduction via decoupling of operators admits a natural explanation in terms of Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -exactness. Second, we provide a new proof of dimensional reduction using the supersymmetric localization argument. Third, we revisit Cardy’s ``interpolation’’ proof—which is reminiscent of localization but does not rely on saddle-point methods—and show that it can be understood as a consequence of deforming the action by a Q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Q</mml:mi> </mml:math> -exact term. Finally, we show that, while existing nonperturbative proofs of dimensional reduction apply only to scalar Lagrangians, our formulation of Cardy’s argument extends to any theory with Parisi—Sourlas supersymmetry.