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Unleash $Q$! Cohomology, Localization, and Interpolation in Parisi-Sourlas Supersymmetry

Bruno Le Floch, Gela Patashuri, Emilio Trevisani

TL;DR

This work presents a unifying framework for dimensional reduction in Parisi–Sourlas supersymmetric theories by organizing the proofs around the cohomology of a carefully chosen supercharge $Q$. It provides three complementary arguments—cohomological decoupling, a modern localization proof, and Cardy-style interpolation—showing that $Q$-exact deformations decouple and that the reduced theory emerges from the $Q$-cohomology. The authors also demonstrate that Cardy’s interpolation extends beyond scalar actions to generic PS-symmetric theories, and they discuss the broader implications for PS uplifts, defects, and potential extensions to curved spaces or gravity. Collectively, the paper clarifies the mathematical structure behind dimensional reduction and offers tools that may apply to a wide class of PS-symmetric models and beyond.

Abstract

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 $Q$. 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 $Q$-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 $Q$-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.

Unleash $Q$! Cohomology, Localization, and Interpolation in Parisi-Sourlas Supersymmetry

TL;DR

This work presents a unifying framework for dimensional reduction in Parisi–Sourlas supersymmetric theories by organizing the proofs around the cohomology of a carefully chosen supercharge . It provides three complementary arguments—cohomological decoupling, a modern localization proof, and Cardy-style interpolation—showing that -exact deformations decouple and that the reduced theory emerges from the -cohomology. The authors also demonstrate that Cardy’s interpolation extends beyond scalar actions to generic PS-symmetric theories, and they discuss the broader implications for PS uplifts, defects, and potential extensions to curved spaces or gravity. Collectively, the paper clarifies the mathematical structure behind dimensional reduction and offers tools that may apply to a wide class of PS-symmetric models and beyond.

Abstract

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 . 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 -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 -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.

Paper Structure

This paper contains 26 sections, 176 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Parisi--Sourlas supersymmetry and preliminaries
  3. Parisi--Sourlas symmetries and superspace
  4. Review of dimensional reduction
  5. Localization: a general review
  6. Unleashing $Q$ in PS models
  7. Cohomological argument
  8. $Q$-exact operators <-> operators built using g(2|2)
  9. Cohomology and consequences of the decoupling
  10. Localization argument
  11. 1-loop determinant and fermionic zero modes
  12. Review of Zaboronsky localization
  13. Interpolation argument
  14. Interpolation method from $Q$-cohomology
  15. Generalizing to other theories
  16. ...and 11 more sections

Figures (4)

  • Figure 1: Diagram showing the relations between the fixed point of a random field theory in $d$ dimensions ($\mathrm{RF}_{d}$ f.p.), a PS $\mathrm{CFT}_{d}$, and a pure $\mathrm{CFT}_{d-2}$.
  • Figure 2: PS uplift conjecture: any $\mathrm{QFT}_{d-2}$ can be embedded into PS $\mathrm{QFT}_{d}$.
  • Figure 3: Dimensional reduction.
  • Figure 4: Diagram for the cohomological argument: among the $OSp(2|2)$-invariant sector of PS QFT$_{d}$, $Q$-exact operators decouple, while the cohomology of $Q$ maps to the operators of the dimensionally reduced theory.