Supersymmetry and Homotopy 1 the Algebraic Data of a Physical System

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 Z2(even-odd)and integer-gradings. Our approach goes beyond the theory of real homotopy due to Quillen, Sullivan and Tanré 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 Z2-grading like the index, here we can detect new 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. 1 The Algebraic Data of a Physical System The purpose of this section is to fix notation and our general viewpoint. For simplicity, let us start with a classical system on a finite-dimensional affine space V with its coordinate space V . In reference to a chosen probability