Thomas Strobl

Characteristic classes associated to Q-bundles

A Q-manifold is a graded manifold endowed with a vector field of degree 1 squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of gauge fields (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. As any principal bundle yields canonically a Q-bundle, this construction generalizes Chern–Weil classes. Novel examples include cohomology classes that are locally de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate the framework to equivariant cohomology and Lecomtes characteristic classes of exact sequences of Lie algebras.

General Yang-Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical framework or From Bianchi identities to twisted Courant algebroids

Starting with minimal requirements from the physical experience with higher gauge theories, i.e. gauge theories for a tower of differential forms of different form degrees, we discover that all the structural identities governing such theories can be concisely recombined into a so-called Q-structure or, equivalently, a Lie infinity algebroid. This has many technical and conceptual advantages: Complicated higher bundles become just bundles in the category of Q-manifolds in this approach (the many structural identities being encoded in the one operator Q squaring to zero), gauge transformations are generated by internal vertical automorphisms in these bundles and even for a relatively intricate field content the gauge algebra can be determined in some lines only and is given by the so-called derived bracket construction. nThis article aims equally at mathematicians and theoretical physicists; each more physical section is followed by a purely mathematical one. While the considerations are valid for arbitrary highest form-degree p, we pay particular attention to p=2, i.e. 1- and 2-form gauge fields coupled non-linearly to scalar fields (0-form fields). The structural identities of the coupled system correspond to a Lie 2-algebroid in this case and we provide different axiomatic descriptions of those, inspired by the application, including e.g. one as a particular kind of a vector-bundle twisted Courant algebroid.

Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions

Abstract We disclose the mathematical structure underlying the gauge field sector of the recently constructed non-abelian superconformal models in six space–time dimensions. This is a coupled system of 1-form, 2-form, and 3-form gauge fields. We show that the algebraic consistency constraints governing this system permit to define a Lie 3-algebra, generalizing the structural Lie algebra of a standard Yang–Mills theory to the setting of a higher bundle. Reformulating the Lie 3-algebra in terms of a nilpotent degree 1 BRST-type operator Q , this higher bundle can be compactly described by means of a Q -bundle; its fiber is the shifted tangent of the Q -manifold corresponding to the Lie 3-algebra and its base the odd tangent bundle of space–time equipped with the de Rham differential. The generalized Bianchi identities can then be retrieved concisely from Q 2 = 0 , which encode all the essence of the structural identities. Gauge transformations are identified as vertical inner automorphisms of such a bundle, their algebra being determined from a Q -derived bracket.

Gauging without initial symmetry

Abstract The gauge principle is at the heart of a good part of fundamental physics: Starting with a group G of so-called rigid symmetries of a functional defined over space–time Σ , the original functional is extended appropriately by additional Lie ( G ) -valued 1-form gauge fields so as to lift the symmetry to Maps ( Σ , G ) . Physically relevant quantities are then to be obtained as the quotient of the solutions to the Euler–Lagrange equations by these gauge symmetries. In this article we show that one can construct a gauge theory for a standard sigma model in arbitrary space–time dimensions where the target metric is not invariant with respect to any rigid symmetry group, but satisfies a much weaker condition: It is sufficient to find a collection of vector fields v a on the target M satisfying the extended Killing equation v a ( i ; j ) = 0 for some connection acting on the index a . For regular foliations this is equivalent to requiring the conormal bundle to the leaves with its induced metric to be invariant under leaf-preserving diffeomorphisms of M , which in turn generalizes Riemannian submersions to which the notion reduces for smooth leaf spaces M / ∼ . The resulting gauge theory has the usual quotient effect with respect to the original ungauged theory: in this way, much more general orbits can be factored out than usually considered. In some cases these are orbits that do not correspond to an initial symmetry, but still can be generated by a finite-dimensional Lie group G . Then the presented gauging procedure leads to an ordinary gauge theory with Lie algebra valued 1-form gauge fields, but showing an unconventional transformation law. In general, however, one finds that the notion of an ordinary structural Lie group is too restrictive and should be replaced by the much more general notion of a structural Lie groupoid.

States in non-associative quantum mechanics: uncertainty relations and semiclassical evolution

A bstractA non-associative algebra of observables cannot be represented as operators on a Hilbert space, but it may appear in certain physical situations. This article employs algebraic methods in order to derive uncertainty relations and semiclassical equations, based on general properties of quantum moments.

Curving Yang-Mills-Higgs gauge theories

We present a Yang-Mills-Higgs (YMH) gauge theory in which structure constants of the gauge group may depend on Higgs fields. The data of the theory are encoded in the bundle

Dirac sigma models from gauging

Eensuremath{rightarrow}M

2d gauge theories and generalized geometry

, where the base

Monopole star products are non-alternative

M

Beyond the standard gauging: gauge symmetries of Dirac sigma models

is the target space of Higgs fields and fibers carry information on the gauge group.