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The Kaehler quotient of a complex reductive Lie group relative to the conjugation action carries a complex algebraic stratified Kaehler structure which reflects the geometry of the group. For the group SL(n,C), we interpret the resulting…

It has been known for some time that generalised geometry provides a particularly elegant rewriting of the action and symmetries of 10-dimensional supergravity theories, up to the lowest nontrivial order in fermions. By exhibiting the full…

We consider the quantum version of Arnold's generalisation of a rigid body in classical mechanics. Thus, we quantise the motion on an arbitrary Lie group manifold of a particle whose classical trajectories correspond to the geodesics of any…

It is well established that the mass parameter breaks the conformal symmetries in the case of geodesic motion. The proper conformal Killing vectors cease to generate conserved charges when non-null geodesics are considered. We examine how…

In pregeometry a metric arises as a composite object at large distances. We investigate if its signature, which distinguishes between time and space, could be a result of the dynamics rather than being built in already in the formulation of…

A common feature of the extended phase space of gauge theory, the crossed product of quantum theory, and quantum reference frames (QRFs) is the adjoining of degrees of freedom followed by a constraining procedure for the resulting total…

This paper develops a unified framework for observables in n-plectic geometry, extending the L_infty-algebra of Hamiltonian (n-1)-forms to Hamiltonian forms of all degrees via a degree-shifting Grassmann variable u that encodes submanifold…

We study the Killing vectors of the quantum ground-state manifold of a parameter-dependent Hamiltonian. We find that the manifold may have symmetries that are not visible at the level of the Hamiltonian and that different quantum phases of…

It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics…

We revisit and construct new examples of supersymmetric 2D topological sigma models whose target space is a Poisson supermanifold. Inspired by the AKSZ construction of topological field theories, we follow a graded-geometric approach and…

We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local…

Let $p$ be a prime, $E$ be a supersingular elliptic curve defined over $\bar{\mathbb{F}}_p$, and $\mathscr{O}$ be its (geometric) endomorphism ring. Earlier results of Chevyrev-Galbraith and Goren-Love have shown that the successive minima…

Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a non-degenerate Poisson…

Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such…

We show that R-matricies of all simple quantum groups have the properties which permit to present quantum group twists as transitions to other coordinate frames on quantum spaces. This implies physical equivalence of field theories…

We classify the allowed structures of the discrete 1-form gauge sector in six-dimensional supergravity theories realized as F-theory compactifications. This provides upper bounds on the 1-form gauge factors $\mathbb{Z}_m$ and in particular…

$q$-analogues of quantities in mathematics involve perturbations of classical quantities using the parameter $q$, and revert to the original quantities when $q$ goes $1$. An important example is the $q$-analogues of binomial coefficients…

We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the…

We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of…

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…