Non-abelian Gerbes and Enhanced Leibniz Algebras

We present the most general gauge-invariant action functional for coupled 1- and 2-form gauge fields with kinetic terms in generic dimensions, i.e. dropping eventual contributions that can be added in particular space-time dimensions only such as higher Chern-Simons terms. After appropriate field redefinitions it coincides with a truncation of the Samtleben-Szegin-Wimmer action. In the process one sees explicitly how the existence of a gauge invariant functional enforces that the most general semi-strict Lie 2-algebra describing the bundle of a non-abelian gerbe gets reduced to a very particular structure, which, after the field redefinition, can be identified with the one of an enhanced Leibniz algebra. This is the first step towards a systematic construction of such functionals for higher gauge theories, with kinetic terms for a tower of gauge fields up to some highest form degree p, solved here for p = 2.Comment: Accepted for Publication in Rapid Communications PRD, submitted originally on April 8, final revised version on June 3

Dirac Sigma Models from Gauging

The G/G WZW model results from the WZW-model by a standard procedure of gauging. G/G WZW models are members of Dirac sigma models, which also contain twisted Poisson sigma models as other examples. We show how the general class of Dirac sigma models can be obtained from a gauging procedure adapted to Lie algebroids in the form of an equivariantly closed extension. The rigid gauge groups are generically infinite dimensional and a standard gauging procedure would give a likewise infinite number of 1-form gauge fields; the proposed construction yields the requested finite number of them. Although physics terminology is used, the presentation is kept accessible also for a mathematical audience.Comment: 20 pages, 3 figure

Poisson Structure Induced (Topological) Field Theories

A class of two dimensional field theories, based on (generically degenerate) Poisson structures and generalizing gravity-Yang-Mills systems, is presented. Locally, the solutions of the classical equations of motion are given. A general scheme for the quantization of the models in a Hamiltonian formulation is found.Comment: 6 pages, LaTeX, TUW940

First Class Constrained Systems and Twisting of Courant Algebroids by a Closed 4-form

We show that in analogy to the introduction of Poisson structures twisted by a closed 3-form by Park and Klimcik-Strobl, the study of three dimensional sigma models with Wess-Zumino term leads in a likewise way to twisting of Courant algebroid structures by closed 4-forms H. The presentation is kept pedagogical and accessible to physicists as well as to mathematicians, explaining in detail in particular the interplay of field transformations in a sigma model with the type of geometrical structures induced on a target. In fact, as we also show, even if one does not know the mathematical concept of a Courant algebroid, the study of a rather general class of 3-dimensional sigma models leads one to that notion by itself. Courant algebroids became of relevance for mathematical physics lately from several perspectives - like for example by means of using generalized complex structures in String Theory. One may expect that their twisting by the curvature H of some 3-form Ramond-Ramond gauge field will become of relevance as well.Comment: 25 pages, invited contribution to the Wolfgang Kummer memorial volum

Lie algebroids, gauge theories, and compatible geometrical structures

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the (possibly singular) foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove furthermore that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized metric structure.Comment: 25 pages. This is the first part of the original preprint that was split into two parts for publication, with a new title, abstract, and introduction. The second, somewhat extended part, entitled 'Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries' is published at Journal of Geometry and Physics 135 (2019) 1-6 and can be found under arXiv:1904.0580

Curving Yang-Mills-Higgs Gauge Theories

Established fundamental physics can be described by fields, which are maps. The source of such a map is space-time, which can be curved due to gravity. The map itself needs to be curved in its gauge field part so as to describe interaction forces like those mediated by photons and gluons. In the present article, we permit non-zero curvature also on the internal space, the target of the field map. The action functional and the symmetries are constructed in such a way that they reduce to those of standard Yang-Mills-Higgs (YMH) gauge theories precisely when the curvature on the target of the fields is turned off. For curved targets one obtains a new theory, a curved YMH gauge theory. It realizes in a mathematically consistent manner an old wish in the community: replacing structures constants by functions depending on the scalars of the theory. In addition, we provide a simple 4d toy model, where the gauge symmetry is abelian, but turning off the gauge fields, no rigid symmetry remains---another possible manifestation of target curvature. It now remains to be seen, if internal curvature in the above sense is realized in nature. Curvature of space-time is proven, but still negligible in particle physics, except for the very early universe where quantum gravity must have played an essential role. An important question therefore is, if glimpses of target curvature can be visible in accelerator physics. We know that at contemporary energy scales, the usual (flat) standard model describes nature to a very high accuracy. Could it be that the alleged deviations in the B to D-star-tau-nu decay reported by BaBar in 2012 and recently also by LHCb are already a manifestation of target curvature? What kind of effects does target curvature have on a YMH theory in general, for what kind of effects do we need to look out for so as to detect it?Comment: 5 pages. Presented by T.S. on several occasions during the first half of 2015. Preprint finished and submitted to Phys. Rev. in July 201

All Symmetries of Non-Einsteinian Gravity in d=2d =2

The covariant form of the field equations for two--dimensional $R^2$--gravity with torsion as well as its Hamiltonian formulation are shown to suggest the choice of the light--cone gauge. Further a one--to--one correspondence between the Hamiltonian gauge symmetries and the diffeomorphisms and local Lorentz transformations is established, thus proving that there are no hidden local symmetries responsible for the complete integrability of the model. Finally the constraint algebra is shown to have no quantum anomalies so that Dirac's quantization should be applicable.Comment: LaTex, 16 pages, TUW9207, (Some smaller corrections, cross-references updated