[Paper Review] The Geometry of Supersymmetry / A concise introduction

Norbert Poncin, Sarah Schouten|arXiv (Cornell University)|Jul 26, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper presents a comprehensive, insight-driven introduction to supergeometry and higher supergeometry, focusing on $bZ_2^n$-manifolds and their integration theory. It develops a novel $bZ_2^n$-Berezinian integration framework that generalizes classical supergeometry by incorporating Laurent series and $bZ_2^n$-graded differential forms, enabling coordinate-independent integration over manifolds of dimension $1|(1,1,1)$.

ABSTRACT

This text is a short but comprehensive introduction to the basics of supergeometry and includes some of the recent advances in colored supergeometry. We do not aim for a standard text that states results and proves them more or less rigorously, but all too often offers little insight to the uninformed reader. Instead we opted for a smooth exposition of the successive themes, choosing an order and an approach which are close to the way these pieces of mathematics could have been or were discovered, thereby highlighting the reasons for the various choices and facilitating deeper understanding. We hope that the text will be useful for PhD students and researchers who wish to acquire knowledge in the geometry of supersymmetry.

Motivation & Objective

• To provide a pedagogically motivated, conceptually grounded introduction to supergeometry and higher supergeometry for PhD students and researchers.
• To address the lack of intuitive understanding in standard treatments by reconstructing the development of supergeometry as a natural progression of ideas.
• To extend classical supergeometry to $bZ_2^n$-manifolds, particularly in the context of differential calculus and integration theory.
• To develop a consistent integration theory for $bZ_2^n$-manifolds using generalized $bZ_2^n$-Berezinian sections and Laurent series expansions.
• To establish coordinate independence of the integral via the coherence of pullbacks and the use of $bZ_2^n$-local cohomology

Proposed method

• Introduces $bZ_2^n$-manifolds via a graded function sheaf with $bZ_2^n$-grading, where coordinates commute according to the standard scalar product on $bZ_2^n$.
• Develops differential calculus on $bZ_2^n$-manifolds using sheaf-theoretic methods, including the super tangent bundle and super differential forms.
• Defines the $bZ_2^n$-Berezinian as a generalization of the classical Berezinian, crucial for integration and transformation laws.
• Constructs integration via Laurent series expansions of generalized $bZ_2^n$-Berezinian sections, with integration over even and odd coordinates treated via residue-like operations.
• Establishes coordinate independence by proving the coherence of pullbacks: $( ho ho')^{* ilde{}} = ho^{* ilde{}} ho'^{* ilde{}}$ for composable morphisms.
• Generalizes the integration theory to higher $bZ_2^n$-manifolds using $bZ_2^n$-local cohomology and generalized fractions, linking to Grothendieck duality

Experimental results

Research questions

• RQ1 How can supergeometry be taught in a way that reveals the underlying intuition and motivation behind its constructions?
• RQ2 What are the structural and technical differences between classical $bZ_2$-supergeometry and $bZ_2^n$-supergeometry, especially in differential calculus?
• RQ3 How can integration theory be consistently extended to $bZ_2^n$-manifolds, particularly when even-degree parameters are involved?
• RQ4 What role do Laurent series and generalized fractions play in defining $bZ_2^n$-Berezinian sections and their integrals?
• RQ5 How can the pullback of generalized sections be made coherent across coordinate transitions in $bZ_2^n$-geometry?

Key findings

• The paper constructs a coordinate-independent integration theory for $bZ_2^2$-manifolds of dimension $1|(1,1,1)$ using Laurent series expansions of $bZ_2^2$-Berezinian sections.
• The integral of a compactly supported generalized $bZ_2^2$-Berezinian section $rak s$ is defined as $ extstyleigint@ extstyleigint@rak s = igint@ dx hinspace f_{-111}(x)$, where $f_{-111}$ is the residue coefficient of $y^{-1}$ in the Laurent expansion.
• The pullback of generalized sections satisfies the coherence condition $( ho ho')^{* ilde{}} = ho^{* ilde{}} ho'^{* ilde{}}$, ensuring consistency under composition.
• The $bZ_2^n$-Berezinian sheaf is defined via the determinant of the Jacobian matrix with $bZ_2^n$-grading, generalizing the classical Berezinian for $bZ_2$-manifolds.
• The integration theory is extended to higher $bZ_2^n$-manifolds using the $q_0$-th $bZ_2^n$-local cohomology module $rak H^{q_0}_{rak J}(U,rak O)$, which describes generalized fractions.
• The framework is compatible with Grothendieck duality and requires a carefully chosen group of admissible coordinate transformations to avoid problematic monomials in the Laurent series