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Equivalence functors in graded supergeometry

Katarzyna Grabowska, Janusz Grabowski, Mikołaj Rotkiewicz

TL;DR

The paper develops a constructive, geometrical framework that situates $N$-manifolds within the broader realm of graded supergeometry by linking them to purely even $n$-vector bundles via a sequence of canonical equivalences. Central to the approach are the polarization functor and parity reversions, combined with iterated tangent constructions, which yield explicit desuperization functors that connect symmetric and skew-symmetric $n$-vector bundles. A key result is the canonical equivalence $\Xi$ between symmetric $n$-vector bundles and skew-symmetric ones, extended to a reverse polarization that maps NM[n] to purely even skew-symmetric $n$-vector bundles, recovering the desuperization in this setting. The diag construction ties higher tangent structures to holonomic elements, linking $N$-manifolds to the realm of $n$-vector bundles and providing a geometric bridge to results like those of Heuer and Jotz. Overall, the work supplies explicit, canonical tools to navigate graded supergeometry and its desuperization, broadening the operational toolkit for graded structures on supermanifolds.

Abstract

It has been recently proved that the category of N-manifolds of degree $n$, i.e., $\mathbb N$-graded supermanifolds of degree $n$ in which the parity agrees with the gradation, is equivalent to the category of purely even $n$-tuple vector superbundles with a certain action of the symmetry group $S_n$ permuting the vector bundle structures; this can be viewed as a desuperization of N-manifolds. We put this result into a much wider context of graded structures on supermanifolds and describe explicitly several canonical equivalences of the corresponding categories in a purely geometrical and constructive way; the desuperization equivalence functor is a composition of some of them. Our constructions are completely canonical, we use such tools of supergeometry as the iterated tangent functor, the parity reversion in vector superbundles, and the modern view of $n$-tuple vector bundles as a sequence of commuting Euler vector fields of the vector bundle structures in question. All this opens new horizons in the land of graded supergeometry.

Equivalence functors in graded supergeometry

TL;DR

The paper develops a constructive, geometrical framework that situates -manifolds within the broader realm of graded supergeometry by linking them to purely even -vector bundles via a sequence of canonical equivalences. Central to the approach are the polarization functor and parity reversions, combined with iterated tangent constructions, which yield explicit desuperization functors that connect symmetric and skew-symmetric -vector bundles. A key result is the canonical equivalence between symmetric -vector bundles and skew-symmetric ones, extended to a reverse polarization that maps NM[n] to purely even skew-symmetric -vector bundles, recovering the desuperization in this setting. The diag construction ties higher tangent structures to holonomic elements, linking -manifolds to the realm of -vector bundles and providing a geometric bridge to results like those of Heuer and Jotz. Overall, the work supplies explicit, canonical tools to navigate graded supergeometry and its desuperization, broadening the operational toolkit for graded structures on supermanifolds.

Abstract

It has been recently proved that the category of N-manifolds of degree , i.e., -graded supermanifolds of degree in which the parity agrees with the gradation, is equivalent to the category of purely even -tuple vector superbundles with a certain action of the symmetry group permuting the vector bundle structures; this can be viewed as a desuperization of N-manifolds. We put this result into a much wider context of graded structures on supermanifolds and describe explicitly several canonical equivalences of the corresponding categories in a purely geometrical and constructive way; the desuperization equivalence functor is a composition of some of them. Our constructions are completely canonical, we use such tools of supergeometry as the iterated tangent functor, the parity reversion in vector superbundles, and the modern view of -tuple vector bundles as a sequence of commuting Euler vector fields of the vector bundle structures in question. All this opens new horizons in the land of graded supergeometry.
Paper Structure (7 sections, 11 theorems, 105 equations)

This paper contains 7 sections, 11 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Weight vector fields, homogeneity, and ${\mathbf N}$-manifolds
  3. Multiple vector bundles
  4. Parity reversion in $n$-vector bundles
  5. Iterated tangent bundles
  6. Parity reversion for symmetric $n$-vector bundles
  7. Polarizations and desuperization of ${\mathbf N}$-manifolds

Key Result

Theorem 3.1

A homogeneity structure $h$ on a manifold $F$ is regular if and only if it is a multiplication by reals for a vector bundle structure on $h_0:F\to M=h_0(E)$.

Theorems & Definitions (29)

  • Definition 2.1
  • Remark 2.2
  • Theorem 3.1: Grabowski:2009
  • Example 3.2
  • Proposition 3.3
  • Definition 3.4
  • Example 3.5
  • Proposition 3.6
  • Remark 3.7
  • Definition 3.8
  • ...and 19 more