Graded infinite order jet manifolds

The relevant material on differential calculus on graded infinite order jet manifolds and its cohomology is summarized. This mathematics provides the adequate formulation of Lagrangian theories of even and odd variables on smooth manifolds in terms of the Grassmann-graded variational bicomplex.Comment: 30 page

Axiomatic classical (prequantum) field theory. Jet formalism

In contrast with QFT, classical field theory can be formulated in a strict mathematical way if one defines even classical fields as sections of smooth fiber bundles. Formalism of jet manifolds provides the conventional language of dynamic systems (nonlinear differential equations and operators) on fiber bundles. Lagrangian theory on fiber bundles is algebraically formulated in terms of the variational bicomplex of exterior forms on jet manifolds where the Euler--Lagrange operator is present as a coboundary operator. This formulation is generalized to Lagrangian theory of even and odd fields on graded manifolds. Cohomology of the variational bicomplex provides a solution of the global inverse problem of the calculus of variations, states the first variational formula and Noether's first theorem in a very general setting of supersymmetries depending on higher-order derivatives of fields. A theorem on the Koszul--Tate complex of reducible Noether identities and Noether's inverse second theorem extend an original field theory to prequantum field-antifield BRST theory. Particular field models, jet techniques and some quantum outcomes are discussed.Comment: 50 page

Relative non-relativistic mechanics

Dynamic equations of non-relativistic mechanics are written in covariant-coordinate form in terms of relative velocities and accelerations with respect to an arbitrary reference frame. The notions of the non-relativistic reference frame, inertial force, free motion equation, and inertial frame are discussed.Comment: 11 page

Theory of Classical Higgs Fields. III. Metric-affine gauge theory

We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $P\to X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/H\to X$ are treated as classical Higgs fields. Its most comprehensive example is metric-affine gauge theory on the category of natural bundles where gauge fields are general linear connections on a manifold $X$, classical Higgs fields are arbitrary pseudo-Riemannian metrics on $X$, and matter fields are spinor fields. In particular, this is the case of gauge gravitation theory.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1110.117

Differential calculus over N-graded commutative rings

The Chevalley-Eilenberg differential calculus and differential operators over N-graded commutative rings are constructed. This is a straightforward generalization of the differential calculus over commutative rings, and it is the most general case of the differential calculus over rings that is not the non-commutative geometry. Since any N-graded ring possesses the associated Z_2-graded structure, this also is the case of the graded differential calculus over Grassmann algebras and the supergeometry and field theory on graded manifolds.Comment: 71 pages. arXiv admin note: substantial text overlap with arXiv:0910.1515, arXiv:0908.188

Lecture on Gauge Gravitation Theory. Gravity as a Higgs Field

Gravitation theory is formulated as gauge theory on natural bundles with spontaneous symmetry breaking where gauge symmetries are general covariant transformations, gauge fields are general linear connections, and Higgs fields are pseudo-Riemannian metrics.Comment: 46 pages, Invited lecture at the 20th International Summer School on Global Analysis and its Applications. General Relativity: 100 years after Hilbert (Stara Lesna, Slovakia, 2015

Interior-Boundary Conditions for Schrodinger Operators on Codimension-1 Boundaries

Interior-boundary conditions (IBCs) are boundary conditions on wave functions for Schr\"odinger equations that allow that probability can flow into (and thus be lost at) a boundary of configuration space while getting added in another part of configuration space. IBCs are of particular interest because they allow defining Hamiltonians involving particle creation and annihilation (as used in quantum field theories) without the need for renormalization or ultraviolet cut-off. For those Hamiltonians, the relevant boundary has codimension 3. In this paper, we develop (what we conjecture is) the general form of IBCs for the Laplacian operator (or Schr\"odinger operators), but we focus on the simpler case of boundaries with codimension 1.Comment: 13 pages LaTeX, no figures. A previous version of this paper was included as section 4 in arXiv:1505.04847v1, but it will not be contained in subsequent, revised versions of arXiv:1505.0484

Non-perturbative N=1 strings from geometric singularities

The study of curved D-brane geometries in type II strings implies a general relation between local singularities \cx W of Calabi-Yau manifolds and gravity free supersymmetric QFT's. The minimal supersymmetric case is described by F-theory compactifications on \cx W and can be used as a starting point to define minimal supersymmetric heterotic string compactifications on compact Calabi-Yau manifolds with holomorphic, stable gauge backgrounds. The geometric construction generalizes to non-perturbative vacua with five-branes and provides a framework to study non-perturbative dynamics of the heterotic theory.Comment: LaTex, 11 p

Normal frames for general connections on differentiable fibre bundles

The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The existence of bundle coordinates normal at a given point and/or along injective horizontal path is proved. A necessary and sufficient condition of existence of bundle coordinates normal along injective horizontal mappings is derived.Comment: 24 LaTeX pages. The packages AMS-LaTeX and amsfonts are required. In version 2 some results are generalized and proved under weaker conditions. For other papers on the same topic view the "publication" pages at http://theo.inrne.bas.bg/~bozho

Reduction of principal superbundles, Higgs superfields, and supermetric

By virtue of the well-known theorem, a structure Lie group G of a principal bundle P is reducible to its closed subgroup H iff there exists a global section of the quotient bundle P/H. In gauge theory, such sections are treated as classical Higgs fields, and are exemplified by Riemannian and pseudo-Riemannian metrics. This theorem is extended to a certain class of principal superbundles, including a graded frame superbundle with a structure general linear supergroup. Each reduction of this structure supergroup to an orthgonal-symplectic supersubgroup is associated to a supermetric on a base supermanifold.Comment: 21 page