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There is a growing interest in the logical possibility that exceptional mathematical structures (exceptional Lie and superLie algebras, the exceptional Jordan algebra, etc.) could be linked to an ultimate "exceptional" formulation for a…
Symmetry formulated by group theory plays an essential role with respect to the laws of nature, from fundamental particles to condensed matter systems. Here, by combining symmetry analysis and tight-binding model calculations, we elucidate…
Based on a fundamental symmetry between space, time, mass and charge, a series of group structures of physical interest is generated, ranging from C2 to E8. The most significant result of this analysis is a version of the Dirac equation…
These are introductory notes on symmetries in quantum field theory and how they apply to particle physics. The notes cover the fundamentals of group theory, their representations, Lie groups, and Lie algebras, along with an elaborate…
Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs…
There are many Lie groups used in physics, including the Lorentz group of special relativity, the spin groups (relativistic and non-relativistic) and the gauge groups of quantum electrodynamics and the weak and strong nuclear forces.…
Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this…
In part 1, we review the structure theory of $\mathbb{F} S_n$, the group algebra of the symmetric group $S_n$ over a field of characteristic 0. We define the images $\psi(E^\lambda_{ij})$ of the matrix units $E^\lambda_{ij}$ ($1 \le i, j…
We generalize, to any space-time dimension, the unitarity bounds of highest weight UIR's of the conformal groups with Lie algebras $so(2,d)$. We classify gauge theories invariant under $so(2,d)$, both integral and half-integral spins. A…
We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in…
Harmonic functions of the three dimensional Lie groups defined on certain manifolds related to the Lie groups themselves and carrying all their unitary representations are explicitly constructed. The realisations of these Lie groups are…
A new structure, based on joining copies of a group by means of a \emph{twist}, has recently been considered to describe the brackets of the two exceptional real Lie algebras of type $G_2$ in a highly symmetric way. In this work we show…
In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the…
We build nearly topological quantum field theories in various dimensions. We give special attention to the case of 8 dimensions for which we first consider theories depending only on Yang-Mills fields. Two classes of gauge functions exist…
A ladder algebraic structure for $L^2(\mathbb{R}^+)$ which closes the Lie algebra $h(1)\oplus h(1)$, where $h(1)$ is the Heisenberg-Weyl algebra, is presented in terms of a basis of associated Laguerre polynomials. Using the Schwinger…
We study higher-form global symmetries and a higher-group structure of a low-energy limit of $(3+1)$-dimensional axion electrodynamics in a gapped phase described by a topological action. We argue that the higher-form symmetries should have…
We construct a bi-Hamiltonian structure for the holomorphic spin Sutherland hierarchy based on collective spin variables. The construction relies on Poisson reduction of a bi-Hamiltonian structure on the holomorphic cotangent bundle of…
A topological theory for euclidean gravity in eight dimensions is built by enforcing octonionic self-duality conditions on the spin connection. The eight-dimensional manifold must be of a special type, with G_2 or Spin(7) holonomy. The…
A recent series of works by M. Dubois-Violette, I. Todorov and S. Drenska characterised the SM gauge group GSM as the subgroup of SO(9) that, in the octonionic model of the later, preserves the split O=C+C3 of the space of octonions into a…
General features of microscopic and macroscopic chiral structures can be discussed under the standard of orthogonal group theory. Configuration space of systems, not physical space, is taken into account. This change of perspective allows…