We review in detail the Batalin–Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an L∞‐algebra and how quasi‐isomorphisms between L∞‐algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern–Simons theories and give some useful shortcuts in usually rather involved computations.

We present an extended version of Riemannian geometry suitable for the description of current formulations of double field theory (DFT). This framework is based on graded manifolds and it yields extended notions of symmetries, dynamical data and constraints. In special cases, we recover general relativity with and without 1-, 2-and 3-form gauge potentials as well as DFT. We believe that our extended Riemannian geometry helps to clarify the role of various constructions in DFT. For example, it leads to a covariant form of the strong section condition. Furthermore, it should provide a useful step towards global and coordinate invariant descriptions of T-and U-duality invariant field theories.

We establish a Penrose-Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly N = (1,0) and N = (2,0) supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings. © 2014 Springer-Verlag Berlin Heidelberg

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor field theories via a Penrose-Ward transform. The resulting higher or categorified gauge theories significantly generalise those obtained previously from principal 2-bundles in that the so-called Peiffer identity is relaxed in a systematic fashion. This transform also exposes various unexplored structures of higher gauge theories modelled on principal 3-bundles such as the relevant gauge transformations. We thus arrive at the non-Abelian differential cohomology that describes principal 3-bundles with connective structure. © 2014 Springer Science+Business Media Dordrecht

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from theČech groupoid to weak Lie 2-groups. As is demonstrated, some of these Lie 2-groups can be differentiated to semistrict Lie 2-algebras by a method due toŠevera. We further derive the full description of connective structures on semistrict principal 2-bundles including the non-linear gauge transformations. As an application, we use a twistor construction to derive superconformal constraint equations in six dimensions for a nonAbelian N = (2, 0) tensor multiplet taking values in a semistrict Lie 2-algebra.

We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet. All of the ingredients of this action have been available in the literature. We bring these pieces together by choosing the string Lie 2-algebra as a gauge structure, which we motivated in previous work. The kinematical data contains a connection on a categorified principal bundle, which is the appropriate mathematical description of the parallel transport of self-dual strings. Our action can be written down for each of the simply laced Dynkin diagrams, and each case reduces to a four-dimensional supersymmetric Yang-Mills theory with corresponding gauge Lie algebra. Our action also reduces nicely to an M2-brane model which is a deformation of the ABJM model. While this action is certainly not the desired M5-brane model, we regard it as a key stepping stone towards a potential construction of the (2,0)-theory.

We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras, as well as the approach based on harmonic analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms of foliations of Ê n by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.

We discuss chiral zero-rest-mass field equations on six-dimensional space-time from a twistorial point of view. Specifically, we present a detailed cohomological analysis, develop both Penrose and Penrose-Ward transforms, and analyse the corresponding contour integral formulae. We also give twistor space action principles. We then dimensionally reduce the twistor space of six-dimensional space-time to obtain twistor formulations of various theories in lower dimensions. Besides well-known twistor spaces, we also find a novel twistor space amongst these reductions, which turns out to be suitable for a twistorial description of self-dual strings. For these reduced twistor spaces, we explain the Penrose and Penrose-Ward transforms as well as contour integral formulae.1 One may also conformally compactify space-time to obtain È 3 as twistor space. The line È 1 that is deleted from È 3 to obtain È 3 • corresponds on space-time to the point infinity which is used for this conformal compactification.2 These moduli spaces are obtained from the solution spaces by quotienting with respect to the group of gauge transformations. 3 Examples of twistor spaces for higher-dimensional space-times including Penrose and Penrose-Ward transforms can be found e.g. in [20][21][22]. P 6 . In particular, we find that the twistor space P 6 contains naturally the ambitwistor space, which provides a twistor description of the Maxwell and Yang-Mills equations, a twistor space we shall refer to as the hyperplane twistor space and which turns out to be suitable for a twistor description of the self-dual string equation, and the minitwistor space underlying a twistor description of monopoles. To our knowledge, the hyperplane twistor space has not been discussed in the literature before. Therefore, we shall be explicit in constructing both Penrose and Penrose-Ward transforms over this twistor space.This paper is structured as follows. We begin our considerations with a brief review of spinors and free fields in six dimensions. We then present the construction of the twistor 4 There is a non-Abelian extension of the self-dual string equation on loop space [25]. Also there have been some recent proposals for non-Abelian M5-brane models, see e.g. [33]. 5 An alternative proof can be found in [21]; see also [29].3 space for six-dimensional space-time from various perspectives in Section 3. In Section 4, we lay down the cohomological foundation on which all of our later analysis is based.This section also contains a detailed proof of the Penrose transform and explicit integral formulae yielding zero-rest-mass fields. The Penrose-Ward transform is presented in Section 5, where we also comment on the aforementioned action principle. We then continue with discussing various dimensional reductions in Section 6. In particular, we show how the six-dimensional picture reduces to the ambitwistor space describing Maxwell fields in four dimensions, the twistor description of self-dual strings and the twistor description of monopoles. We summarise our conclusions...