Supersymmetry and Homotopy

TL;DR

This paper explores how supersymmetric structures encode hidden homotopical information by using deformations of manifolds, extending classical homotopy theory to include new invariants through a novel approach involving Taylor expansions and locality.

Contribution

It introduces a new method linking differential geometry and homotopy via Taylor expansion, allowing detection of N-graded invariants beyond traditional Z2-grading.

Findings

Relates different supersymmetries N=2n and N=(n,n) despite grading differences

Extends homotopy invariants to include N-graded invariants beyond index

Links supersymmetry with homotopy through a local, deformation-based approach

Abstract

The homotopical information hidden in a supersymmetric structure is revealed by considering deformations of a configuration manifold. This is in sharp contrast to the usual standpoints such as Connes' programme where a geometrical structure is rigidly fixed. For instance, we can relate supersymmetries of types N=2n and N=(n, n) in spite of their gap due to distinction between $\Bbb{Z}_2$(even-odd)- and integer-gradings. Our approach goes beyond the theory of real homotopy due to Quillen, Sullivan and Tanr\'e developed, respectively, in the 60's, 70's and 80's, which exhibits real homotopy of a 1-connected space out of its de Rham-Fock complex with supersymmetry. Our main new step is based upon the Taylor (super-)expansion and locality, which links differential geometry with homotopy without the restriction of 1-connectedness. While the homotopy invariants treated so far in relation with supersymmetry are those depending only on $\Bbb{Z}_2$-grading like the index, here we can detect new $\Bbb{N}$-graded homotopy invariants. While our setup adopted here is (graded) commutative, it can be extended also to the non-commutative cases in use of state germs (Haag-Ojima) corresponding to a Taylor expansion.