Supersymmetry and Homotopy


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\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 Z2\Bbb{Z}_2-grading like the index, here we can detect new N\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

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