Supersymmetry and Homotopy
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 $\Bbb{Z}_2$(evenodd) and integergradings. 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 1connected space out of its de RhamFock 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 1connectedness. 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 noncommutative cases in use of state germs (HaagOjima) corresponding to a Taylor expansion.
 Publication:

arXiv eprints
 Pub Date:
 May 2000
 DOI:
 10.48550/arXiv.mathph/0005027
 arXiv:
 arXiv:mathph/0005027
 Bibcode:
 2000math.ph...5027M
 Keywords:

 Mathematical Physics;
 Algebraic Topology