The Operadic Modeling of Gauge Systems of the Yang-Mills Type

Abstract

The basis of operadic variational formalism is presented which is necessary when modeling the operadic systems. A general gauge theoretic approach to the abstract operads, based on the physical measurements concepts, is justified and considered. It is explained how the matrix and Poisson algebra relations can be extended to operadic realm. The tangent cohomology spaces of the binary associative flows with their Gerstenhaber algebra structure can be seen as equally natural objects for operadic modeling, just as the matrix and Poisson algebras in conventional modeling. In particular, the relation of the tangent Gerstenhaber algebras to operadic Stokes law for operadic observables is revealed and discussed. Based on this, the rational (cohomological) variational principle and operadic Heisenberg equation for quantum operadic flows are stated. As a modeling selection rule, the operadic gauge equations of the Yang–Mills type are considered and justified from the point of view of the physical measurements and the algebraic deformation theory. It is also shown how the binary weakly non-associative operations are related to approximate operadic (anti-)self-dual models.