(Co)bordism groups in quantum super PDEs. II: Quantum super PDEs

Abstract

Following our previous works on the integral (co)bordism groups of quantum PDEs [A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing Co., Singapore, 1996, 760pp; A. Prástaro, (Co)bordisms in PDE's and quantum PDE's, Rep. Math. Phys. 38(3) (1996) 443–455; A. Prástaro, Quantum and integral (co)bordism groups in partial differential equations, Acta Appl. Math. 51(3) (1998) 243–302; A. Prástaro, (Co)bordism groups in PDE's, Acta Appl. Math. 59(2) (1999) 111–202; A. Prástaro, (Co)bordism groups in quantum PDE's, Acta Appl. Math. 64(2/3) (2000) 111–217; A. Prástaro, Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. 47/4 (2001) 2609–2620; A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing Co., Singapore, 2004, 500pp; A. Prástaro, Quantum super Yang-Mills equations: global existence and mass-gap, Dynamic Syst. Appl. 4 (2004) 227–234; A. Prástaro, Th.M. Rassias, A geometric approach to a noncommutative generalized d’Alembert equation, C. R. Acad. Sci. Paris 330(I-7) (2000) 545–550; A. Prástaro, Th.M. Rassias, Results on the J.D’Alembert equation, Ann Acad. Paed. Crac. Stud. Math. 1 (2001) 117–128.], we specialize, now, on quantum super partial differential equations, i.e., partial differential equations built in the category of quantum supermanifolds. These are manifolds modeled on locally convex topological vector spaces built starting from quantum algebras endowed also with a Z 2 -gradiation, and a Z 2 -graded Lie algebra structure, ( quantum superalgebra). Then, we extend to these manifolds, with such richer structure, our previous results, and build a geometric theory of quantum super PDEs, that allows us to obtain theorems of existence of (smooth) local and global solutions in the category of quantum supermanifolds. Some quantum (super) PDEs, arising from the Dirac quantization of some classical (super) PDEs, are considered in some details.