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Starting from an N=1 supersymmetric electric gauge theory with the multiple product gauge group and the bifundamentals, we apply Seiberg dual to each gauge group, obtain the N=1 supersymmetric dual magnetic gauge theories with dual matters…
We study three-dimensional gauge dynamics by using type IIB superstring brane configurations, which can be obtained from the M-theory configuration of M2-branes stretched between two M5-branes with relative angles. Our construction of brane…
We study the homotopy aspects of the twisted Chern classes of torsion bundle gerbe modules. Using Sullivan's rational homotopy theory, we realize the twisted Chern classes at the level of classifying spaces. The construction suggests a…
Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the…
This paper presents the equivariant systems theory and observer design for second order kinematic systems on matrix Lie groups. The state of a second order kinematic system on a matrix Lie group is naturally posed on the tangent bundle of…
We discuss the geometry of string and M-theory gauge fields in Deligne cohomology. In particular, we show how requiring string structure (or loop space Spin-C structure) on the five-brane leads to topological conditions on the flux in the…
Studying the topological aspects of M-branes in M-theory leads to various structures related to Wu classes. First we interpret Wu classes themselves as twisted classes and then define twisted notions of Wu structures. These generalize many…
We propose a unifying mathematical framework describing the higher categorical structures formed by topological defects in quantum field theory equipped with tangential structures, such as orientations, framings, or…
We study the connected components of the space of higher spin bundles on hyperbolic Klein surfaces. A Klein surface is a generalisation of a Riemann surface to the case of non-orientable surfaces or surfaces with boundary. The category of…
We explore the properties of five-dimensional supersymmetric gauge theories living on $5$-brane webs in orientifold $5$-plane backgrounds. This allows constructing quiver gauge theories with alternating $USp(2N)$ and $SO(N)$ gauge groups…
Structure monoids and groups are algebraic invariants of equational varieties. We show how to construct presentations of these objects from coherent categorifications of equational varieties, generalising several results of Dehornoy. We…
The notion of a higher bundle gerbe is introduced to give a geometric realization of the higher degree integral cohomology of certain manifolds. We consider examples using the infinite dimensional spaces arising in gauge theories.
Starting with configurations of fourbranes, fivebranes, sixbranes and orientifolds in Type IIA string theory we derive via M-theory the curves solving N=2 supersymmetric gauge theories with gauge groups SO(N) and Sp(2N). We also obtain new…
We study higher-loop orders of spin bit models underlying the non-planar dynamics of $\mathcal{N}=4$ SYM gauge theory. In particular, we derive a ''tower'' of non-planar identities involving products of site permutation operators. Such…
We study smooth higher symmetry groups and moduli $\infty$-stacks of generic higher geometric structures on manifolds. Symmetries are automorphisms which cover non-trivial diffeomorphisms of the base manifold. We construct the smooth higher…
A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is…
We demonstrate the construction of Poisson structures via Lie algebroids on moduli spaces of twisted stable Higgs bundles over stacky curves. The construction provides new examples of Poisson structures on such moduli spaces. Special…
Decoupling limits of physical interest occur in regions of space--time where the string coupling diverges. This is illustrated in the celebrated example of five-branes. There are several ways to overcome this strong-coupling problem. We…
Tensegrity structures have been extensively studied over the last years due to their potential applications in modern engineering like metamaterials, deployable structures, planetary lander modules, etc. Many of the form-finding methods…
It was proposed the Lie group such that symplectic structure of orbits of co-adjoint representation of the group is revealed symplectic structure of a rigid body dynamics in quaternion variables. It is shown that Poisson brackets of…