On the Mini-Superambitwistor Space and N=8 Super Yang-Mills Theory

We construct a new supertwistor space suited for establishing a Penrose-Ward transform between certain bundles over this space and solutions to the N=8 super Yang-Mills equations in three dimensions. This mini-superambitwistor space is obtained by dimensional reduction of the superambitwistor space, the standard superextension of the ambitwistor space. We discuss in detail the construction of this space and its geometry before presenting the Penrose-Ward transform. We also comment on a further such transform for purely bosonic Yang-Mills-Higgs theory in three dimensions by considering third order formal "sub-neighborhoods" of a mini-ambitwistor space.Comment: 24 pages, revised and published versio

A Multitrace Matrix Model from Fuzzy Scalar Field Theory

We present the analytical approach to scalar field theory on the fuzzy sphere which has been developed in arXiv:0706.2493 [hep-th]. This approach is based on considering a perturbative expansion of the kinetic term in the partition function. After truncating this expansion at second order, one arrives at a multitrace matrix model, which allows for an application of the saddle-point method. The results are in agreement with the numerical findings in the literature.Comment: 8 pages, talk given by CS at the International Workshop "Supersymmetries and Quantum Symmetries" (SQS'07), Dubna, July 30 - August 4 2007; to appear in the proceeding

Towards an M5-Brane Model I: A 6d Superconformal Field Theory

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.Comment: 1+39 pages, v3: minor improvements, published versio

Self-dual String and Higher Instanton Solutions

We present and discuss explicit solutions to the non-abelian self-dual string equation as well as to the non-abelian self-duality equation in six dimensions. These solutions are generalizations of the 't Hooft-Polyakov monopole and the BPST instanton to higher gauge theory. We expect that these solutions are relevant to the effective description of M2- and M5-branes.Comment: 23 pages, 2 figures, v2 presentation improved, minor errors corrected, new subsection with comment

L∞L_\infty-Algebra Models and Higher Chern-Simons Theories

We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In a first part, we review in detail how higher Chern-Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of $L_\infty$-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In a second part, we demonstrate that Nambu-Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie $p$-algebra extensions of $\mathfrak{so}(p+2)$. Finally, we study a number of $L_\infty$-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.Comment: 44 pages, minor corrections, published versio

M-brane Models from Non-Abelian Gerbes

We make the observation that M-brane models defined in terms of 3-algebras can be interpreted as higher gauge theories involving Lie 2-groups. Such gauge theories arise in particular in the description of non-abelian gerbes. This observation allows us to put M2- and M5-brane models on equal footing, at least as far as the gauge structure is concerned. Furthermore, it provides a useful framework for various generalizations; in particular, it leads to a fully supersymmetric generalization of a previously proposed set of tensor multiplet equations.Comment: 17 pages, v2: new SUSY models, version published in JHE

Branes, Quantization and Fuzzy Spheres

We propose generalized quantization axioms for Nambu-Poisson manifolds, which allow for a geometric interpretation of n-Lie algebras and their enveloping algebras. We illustrate these axioms by describing extensions of Berezin-Toeplitz quantization to produce various examples of quantum spaces of relevance to the dynamics of M-branes, such as fuzzy spheres in diverse dimensions. We briefly describe preliminary steps towards making the notion of quantized 2-plectic manifolds rigorous by extending the groupoid approach to quantization of symplectic manifolds.Comment: 18 pages; Based on Review Talk at the Workshop on "Noncommutative Field Theory and Gravity", Corfu Summer Institute on Elementary Particles and Physics, September 8-12, 2010, Corfu, Greece; to be published in Proceedings of Scienc

Supersymmetric Yang-Mills Theory as Higher Chern-Simons Theory

We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.Comment: v2: 25 pages, conventions improved, typos fixed, published versio

The Mini-Superambitwistor Space

We present the construction of the mini-superambitwistor space, which is suited for establishing a Penrose-Ward transform between certain bundles over this space and solutions to the N=6 super Yang-Mills equations in three dimensions.Comment: 8 pages, talk given at the International Workshop "Supersymmetries and Quantum Symmetries" (SQS'05), Dubna, July 27-31 2005; to appear in the proceeding

Fuzzy Toric Geometries

We describe a construction of fuzzy spaces which approximate projective toric varieties. The construction uses the canonical embedding of such varieties into a complex projective space: The algebra of fuzzy functions on a toric variety is obtained by a restriction of the fuzzy algebra of functions on the complex projective space appearing in the embedding. We give several explicit examples for this construction; in particular, we present fuzzy weighted projective spaces as well as fuzzy Hirzebruch and del Pezzo surfaces. As our construction is actually suited for arbitrary subvarieties of complex projective spaces, one can easily obtain large classes of fuzzy Calabi-Yau manifolds and we comment on fuzzy K3 surfaces and fuzzy quintic three-folds. Besides enlarging the number of available fuzzy spaces significantly, we show that the fuzzification of a projective toric variety amounts to a quantization of its toric base.Comment: 1+25 pages, extended version, to appear in JHE