Non-Abelian Lie algebroids over jet spaces

Abstract:
Lie algebra-valued zero-curvature representations are the pre-requisites for solving Cauchy's problems for nonlinear PDEs via the Inverse Scattering Transform (IST). Geometrically, such representations are flat connection one-forms in principal fibre bundles over the jet spaces where nonlinear PDEs live. If the equation's unknowns depend on two (but not three or more) variables (x,t), the zero-curvature representation must contain a parameter which can not be removed under gauge transformations (for three or more variables, having such parameter is not obligatory). M. Marvan has developed a very convenient cohomological technique which checks whether the parameter in a given family of representations is or is not removable; that theory's differential is constructed explicitly for every ZCR and then it is used in the verification procedure.

In the paper arXiv:1305.4598 [math.DG] (joint with A. Krutov) we relate Marvan's idea to a natural class of variational Lie algebroids in which his (that is, Marvan's not Lie's) operators are the anchors. Extending the fibres in vector bundles for Lie algebra-valued ZCRs by the dual of that algebra and by the opposite-parity neighbours of the two vector spaces, we enlarge the IST geometry and construct another complex. Of course, it stems from the realisation of variational Lie algebroid at hand via homological evolutionary vector field - or by the master-functional S satisfying the classical master-equation [[S,S]]=0 with respect to the variational Schouten bracket. This reveals an interesting approach to quantisation of the setup: because the cohomological realisation is well defined at any number of independent variables, we argue that its quantum extension does not reduce to the standard Batalin-Vilkovisky quantisation of Chern-Simons models over threefolds but that it should be obtained via deformation quantisation of the IST problem for the PDE under study.

This is joint work with A.O. Krutov (Department of Higher Mathematics, Ivanovo State Power University, Russia).