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The first steps towards linearisation of partial orders and equivalence relations are described. The definitions of partial orders and equivalence relations (on sets) are formulated in a way that is standard in category theory and that makes the linearisation (almost) automatic. The linearisation is then achieved by replacing sets by coalgebras and the Cartesian product by the tensor product of vector spaces. As a result, definitions of orders and equivalence relations on coalgebras are proposed. These are illustrated by explicit examples that include relations on coalgebras spanned by grouplike elements (or linearised sets), the diagonal relation, and an order on a three-dimensional non-cocommutative coalgebra. Although relations on coalgebras are defined for vector spaces, all the definitions are formulated in a way that is immediately applicable to other braided monoidal categories.

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We discuss two examples of infinite random graphs obtained as limits of finite statistical mechanical systems: a model of two-dimensional discretized quantum gravity defined in terms of causal triangulated surfaces, and the Ising model on generic random trees. For the former model we describe a relation to the so-called uniform infinite tree and results on the Hausdorff and spectral dimension of two-dimensional space-time obtained in B. Durhuus, T. Jonsson, J.F. Wheater, J. Stat. Phys. 139, 859 (2010) are briefly outlined. For the latter we discuss results on the absence of spontaneous magnetization and argue that, in the generic case, the values of the Hausdorff and spectral dimension of the underlying infinite trees are not influenced by the coupling to an Ising model in a constant magnetic field (B. Durhuus, G.M. Napolitano, in preparation).

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The aim of the paper is to answer the following question: does \(\kappa \)-deformation fit into the framework of noncommutative geometry in the sense of spectral triples? Using a compactification of time, we get a discrete version of \(\kappa \)-Minkowski deformation via \(C^*\)-algebras of groups. The dynamical system of the underlying groups (including some Baumslag–Solitar groups) is used in order to construct finitely summable spectral triples. This allows to bypass an obstruction to finite-summability appearing when using the common regular representation.

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We explain how the Calabi problem on a smooth projective complex manifold can be discussed from the point of view of quantum formalism. We derive from this approach a natural flow on the space of Kähler potentials that has an interpretation in terms of moments maps. Finally, we discuss briefly how such techniques could be adapted to the study of the J-flow.

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More than 50 years ago it was realized that General Relativity could be expressed in Hamiltonian form. Unfortunately, just like electromagnetism and Yang–Mills theory, the Einstein equations split into evolution equations and constraints which complicates matters. The 4 constraints are expressions of the gauge freedom of the theory, general covariance. One can cleanly pose initial data for the gravitational field, but this data has to satisfy the constraints. To find the independent degrees of freedom, one needs to factor the initial data by the constraints. There are many ways of doing this. I can do so in such a way as to implement the model suggested by Poincaré for a well-posed dynamical system: Pick a configuration space and give the free initial data as a point of the configuration space and a tangent vector at the same point. Now, the evolution equations should give a unique curve in the same configuration space. This gives a natural definition of what I call the true gravitational degrees of freedom.

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The concept of moduli is illustrated in several problems from geometry and physics. These problems range from complex geometry to supersymmetric gauge theories, integrable models, and string theory. Some of them are quite classical, but others have emerged only relatively recently, for example in the interplay between complex geometry and two-dimensional supergravity.

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These notes summarise a talk surveying the combinatorial or Hamiltonian quantisation of three dimensional gravity in the Chern–Simons formulation, with an emphasis on the role of quantum groups and on the way the various physical constants (\(c,G,{\mit \Lambda },\hbar )\) enter as deformation parameters. The classical situation is summarised, where solutions can be characterised in terms of model space-times (which depend on \(c\) and \({\mit \Lambda }\)) together with global identifications via elements of the corresponding isometry groups. The quantum theory may be viewed as a deformation of this picture, with quantum groups replacing the local isometry groups, and non-commutative space-times replacing the classical model space-times. This point of view is explained, and open issues are sketched.

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We overview the color–flavor transformation which has various applications to problems as diverse as lattice gauge theory, random network models, and dynamical systems with disorder. We present this transformation in the context of the fermionic sector of lattice QCD and induced lattice gluodynamics. Application to low energy QCD on a lattice leads to a theory where the inverse number of colors appears as expansion parameter. We use a saddle point approximation to estimate the partition function both in the pure mesonic sector and in the case of a single baryon on a mesonic background. The effective chiral Lagrangian of QCD is recovered up to the terms of order O(\(p^4\)). We also consider the color–flavor transformation applied to the pure lattice gluodynamics, in which the gauge theory is induced by a heavy chiral scalar field. The effective, color–flavor transformed theory is expressed in terms of gauge singlet matrix fields carried by the lattice links.

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The theory of gerbes provides us with powerful cohomological and geometric tools that have been successfully employed in the construction and classification of consistent two-dimensional non-linear bosonic \(\sigma \)-models, in both the classical and the quantum régime. The theory does, in particular, naturally accommodate the concept of a duality map between two such models and affords a rigorous formulation of the gauge principle. In the present note, I review recent progress in understanding the geometry of the \(\sigma \)-model from the gerbe-theoretic perspective.

abstract

We describe the statistical mechanics of a melting crystal in three dimensions and its relation to a diverse range of models arising in combinatorics, algebraic geometry, integrable systems, low-dimensional gauge theories, topological string theory and quantum gravity. Its partition function can be computed by enumerating the contributions from noncommutative instantons to a six-dimensional cohomological gauge theory, which yields a dynamical realization of the crystal as a discretization of spacetime at the Planck scale. We describe analogous relations between a melting crystal model in two dimensions and \(\mathcal {N}=4\) supersymmetric Yang–Mills theory in four dimensions. We elaborate on some mathematical details of the construction of the quantum geometry which combines methods from toric geometry, isospectral deformation theory and noncommutative geometry in braided monoidal categories. In particular, we relate the construction of noncommutative instantons to deformed ADHM data, torsion-free modules and a noncommutative twistor correspondence.

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We survey our recent work on degenerations of Ricci-flat Kähler metrics on compact Calabi–Yau manifolds with Kähler classes approaching the boundary of the Kähler cone.

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A sketch of an approach towards Lorentzian spectral geometry (based on joint work with Mario Paschke) is described, together with a general way to define abstractly the quantized Dirac field on such Lorentzian spectral geometries. Moyal–Minkowski spacetime serves as an example. The scattering of the quantized Dirac field by a non-commutative (Moyal-deformed) action of an external scalar potential is investigated. It is shown that differentiating the \(S\)-matrix with respect to the strength of the scattering potential gives rise to quantum field operators depending on elements of the non-commutative algebra entering the spectral geometry description of Moyal–Minkowski spacetime, in the spirit of “Bogoliubov’s formula”, in analogy to the situation found in external potential scattering by a usual scalar potential.