(Co)bordism groups in quantum super PDE's. I: Quantum supermanifolds

Abstract

Following our previous works on noncommutative manifolds and noncommutative PDE's [A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing, Singapore, 1996, 760pp; (Co)bordism groups in quantum PDE's, Acta Appl. Math. 64(2/3) (2000) 111–217; Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. 47/4 (2001) 2609–2620; Quantized Partial Differential Equations, World Scientific Publishing, Singapore, 2004, 500pp; Quantum super Yang–Mills equations: global existence and mass-gap, Dyn. Syst. Appl. 4 (2004) 227–234], we consider in these series of three papers, some further results on quantum supermanifolds and quantum super PDE's [A. Prástaro, (Co)bordism groups in quantum super PDE's. II: quantum super PDE's, Nonlinear Anal.: Real World Appl., to appear; (Co)bordism groups in quantum super PDE's. III: quantum super Yang–Mills equations, Nonlinear Anal.: Real World Appl., to appear]. In particular, in this first part we focus our attention on quantum supermanifolds. These structures globalize the notion of quantum superalgebras, obtaining noncommutative manifolds that are useful to give a fully covariant description of noncommutative geometric structures, hence of quantum physics. We study the geometry of quantum supermanifolds and characterize their (co)homological and (co)bordism properties. Covariant quantizations of super PDE's are given, obtaining examples of quantum supermanifolds justifying their definitions.