Principal 2-bundles and their gauge 2-groups

In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the last section we provide some explicit examples.Comment: 40 pages; v3: completely revised and extended, classification corrected, name changed, to appear in Forum Mat

Central extensions of groups of sections

If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context we provide sufficient conditions for integrability in terms of data related only to the group K.Comment: 54 pages, revised version, to appear in Ann. Glob. Anal. Geo

A supergeometric approach to Poisson reduction

This work introduces a unified approach to the reduction of Poisson manifolds using their description by graded symplectic manifolds. This yields a generalization of the classical Poisson reduction by distributions (Marsden-Ratiu reduction). Further it allows one to construct actions of strict Lie 2-groups and to describe the corresponding reductions.Comment: 40 pages. Final version accepted for publicatio

Shared decision making in breast cancer treatment guidelines: Development of a quality assessment tool and a systematic review

Background: It is not clear whether clinical practice guidelines (CPGs) and consensus statements (CSs) are adequately promoting shared decision making (SDM). Objective: To evaluate the recommendations about SDM in CPGs and CSs concerning breast cancer (BC) treatment. Search strategy: Following protocol registration (Prospero no.: CRD42018106643), CPGs and CSs on BC treatment were identified, without language restrictions, through systematic search of bibliographic databases (MEDLINE, EMBASE, Web of Science, Scopus, CDSR) and online sources (12 guideline databases and 51 professional society websites) from January 2010 to December 2019. Inclusion criteria: CPGs and CSs on BC treatment were selected whether published in a journal or in an online document. Data extraction and synthesis: A 31-item SDM quality assessment tool was developed and used to extract data in duplicate. Main results: There were 167 relevant CPGs (139) and CSs (28); SDM was reported in only 40% of the studies. SDM was reported more often in recent publications after 2015 (42/101 (41.6 %) vs 46/66 (69.7 %), P = .0003) but less often in medical journal publications (44/101 (43.5 %) vs 17/66 (25.7 %), P = .009). In CPGs and CSs with SDM, only 8/66 (12%) met one-fifth (6 of 31) of the quality items; only 14/66 (8%) provided clear and precise SDM recommendations. Discussion and conclusions: SDM descriptions and recommendations in CPGs and CSs concerning BC treatment need improvement. SDM was more frequently reported in CPGs and CSs in recent years, but surprisingly it was less often covered in medical journals, a feature that needs attention

Properties of field functionals and characterization of local functionals

Functionals (i.e. functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the proper space of test functions (smooth functions) and of the relevant concept of differential (Bastiani differential) are discussed. The relation between the multiple derivatives of a functional and the corresponding distributions is described in detail. It is proved that, in a neighborhood of every test function, the support of a smooth functional is uniformly compactly supported and the order of the corresponding distribution is uniformly bounded. Relying on a recent work by Yoann Dabrowski, several spaces of functionals are furnished with a complete and nuclear topology. In view of physical applications, it is shown that most formal manipulations can be given a rigorous meaning. A new concept of local functionals is proposed and two characterizations of them are given: the first one uses the additivity (or Hammerstein) property, the second one is a variant of Peetre's theorem. Finally, the first step of a cohomological approach to quantum field theory is carried out by proving a global Poincar\'e lemma and defining multi-vector fields and graded functionals within our framework.Comment: 32 pages, no figur

Z_2 Invariants of topological insulators as geometric obstructions

We consider a gapped periodic quantum system with time-reversal symmetry of fermionic (or odd) type, i.e. the time-reversal operator squares to −1. We investigate the existence of periodic and time-reversal invariant Bloch frames in dimensions 2 and 3. In 2d, the obstruction to the existence of such a frame is shown to be encoded in a Z2-valued topological invariant, which can be computed by a simple algorithm. We prove that the latter agrees with the Fu-Kane index. In 3d, instead, four Z2 invariants emerge from the construction, again related to the Fu-Kane-Mele indices. When no topological obstruction is present, we provide a constructive algorithm yielding explicitly a periodic and time-reversal invariant Bloch frame. The result is formulated in an abstract setting, so that it applies both to discrete models and to continuous ones

Quasi-periodic paths and a string 2-group model from the free loop group

We address the question of the existence of a model for the string 2-group as a strict Lie-2-group using the free loop group L-Spin (or more generally LG for compact simple simply-connected Lie groups G). Baez-Crans-Stevenson-Schreiber constructed a model for the string 2-group using a based loop group. This has the deficiency that it does not admit an action of the circle group S^1, which is of crucial importance, for instance in the construction of a (hypothetical) S1-equivariant index of (higher) differential operators. The present paper shows that there are in fact obstructions for constructing a strict model for the string 2-group using LG. We show that a certain infinite-dimensional manifold of smooth paths admits no Lie group structure, and that there are no nontrivial Lie crossed modules analogous to the BCSS model using the universal central extension of the free loop group. Afterwards, we construct the next best thing, namely a coherent model for the string 2-group using the free loop group, with explicit formulas for all structure. This is in particular important for the expected representation theory of the string group that we discuss briefly in the end.Michael Murray, David Michael Roberts, Christoph Wocke