Given a finite \(\mathbb {Z}_2\)-graded group \(\hat{\mathsf {G}}\) with ungraded subgroup \(\mathsf {G}\) and a twisted cocycle \(\hat{\lambda } \in Z^n(B \hat{\mathsf {G}}; \mathsf {U}(1)_{\pi })\) which restricts to \(\lambda \in Z^n(B \mathsf {G}; \mathsf {U}(1))\), we construct a lift of \(\lambda \)-twisted \(\mathsf {G}\)-Dijkgraaf–Witten theory to an unoriented...

We introduce a family of loop soup models on the hypercubic lattice. The models involve links on the edges, and random pairings of the link endpoints on the sites. We conjecture that loop correlations of distant points are given by Poisson–Dirichlet correlations in dimensions three and higher. We prove that, in a specific random wire model that is related to the classical XY spin...

In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly...

The Lee–Yang property of certain moment generating functions having only pure imaginary zeros is valid for Ising type models with one-component spins and XY models with two-component spins. Villain models and complex Gaussian multiplicative chaos are two-component systems analogous to XY models and related to Gaussian free fields. Although the Lee–Yang property is known to be...

Global attraction to solitary waves is proved for a model \(\mathbf {U}(1)\)-invariant nonlinear 1D Dirac equation coupled to a nonlinear oscillator: each finite energy solution converges as \(t\rightarrow \pm \infty \) to a set of all “nonlinear eigenfunctions” of the form \(\psi _1(x)e^{-i\omega _1 t}+\psi _2(x)e^{-i\omega _2 t}\). The global attraction is caused by nonlinear...

In this paper we consider a nonlocal energy Iα whose kernel is obtained by adding to the Coulomb potential an anisotropic term weighted by a parameter \({\alpha \in \mathbb{R}}\). The case α = 0 corresponds to purely logarithmic interactions, minimised by the circle law; α = 1 corresponds to the energy of interacting dislocations, minimised by the semi-circle law. We show that...

Stochastic growth processes in dimension (2 + 1) were conjectured by D. Wolf, on the basis of renormalization-group arguments, to fall into two distinct universality classes, according to whether the Hessian \({H_\rho}\) of the speed of growth \({v(\rho)}\) as a function of the average slope \({\rho}\) satisfies \({{\rm det} H_\rho > 0}\) (“isotropic KPZ class”) or \({{\rm det...

We give a description of the delocalized twisted cohomology of an orbifold and the Chern character of a twisted vector bundle in terms of supersymmetric Euclidean field theories. This includes the construction of a twist functor for \({1\vert1}\)-dimensional EFTs from the data of a gerbe with connection.

In this paper, we show that loop groups and the universal cover of \({{\rm Diff}_+(S^1)}\) can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals of S1. Analogous results hold for based loop groups and for the based diffeomorphism group of S1. These results continue to hold for the corresponding centrally extended groups. We use the above...

Umbral moonshine connects the symmetry groups of the 23 Niemeier lattices with 23 sets of distinguished mock modular forms. The 23 cases of umbral moonshine have a uniform relation to symmetries of K3 string theories. Moreover, a supersymmetric vertex operator algebra with Conway sporadic symmetry also enjoys a close relation to the K3 elliptic genus. Inspired by the above two...

We study the adjacency matrices of random d-regular graphs with large but fixed degree d. In the bulk of the spectrum \({[-2\sqrt{d-1}+\varepsilon, 2\sqrt{d-1}-\varepsilon]}\) down to the optimal spectral scale, we prove that the Green’s functions can be approximated by those of certain infinite tree-like (few cycles) graphs that depend only on the local structure of the original...

Let (M, g) be a complete Riemannian 3-manifold asymptotic to Schwarzschild-anti-deSitter and with scalar curvature \({R \geq - 6}\). Building on work of A. Neves and G. Tian and of the first-named author, we show that the leaves of the canonical foliation of (M, g) are the unique solutions of the isoperimetric problem for their area. The assumption \({R \geq -6}\) is necessary...

We study the topology of smectic defects in two and three dimensions. We give a topological classification of smectic point defects and disclination lines in three dimensions. In addition we describe the combination rules for smectic point defects in two and three dimensions, showing how the broken translational symmetry of the smectic confers a path dependence on the result of...

In this paper, we study solutions to the Klein–Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution u to the Klein–Gordon equation, there are smooth functions ui, i = 0,1, on the Lie group under consideration, such that \({u_...

We study the 2-dimensional Ising model at critical temperature on a simply connected subset \({\Omega_{\delta}}\) of the square grid \({\delta\mathbb{Z}^{2}}\). The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and...

The paper studies the locus in the moduli space of rank 2 Higgs bundles over a curve of genus g corresponding to points which are critical for d of the Poisson commuting functions defining the integrable system. These correspond to the Higgs field vanishing on a divisor D of degree d. The degree d critical locus is shown to have an induced integrable system related to K(−D...

We consider the Ginzburg–Landau functional, defined on a two-dimensional simply connected domain with smooth boundary, in the situation when the applied magnetic field is piecewise constant with a jump discontinuity along a smooth curve. In the regime of large Ginzburg–Landau parameter and strong magnetic field, we study the concentration of the minimizing configurations along...

We consider the family of operators \({H^{(\varepsilon)}:=-\frac{d^2}{dx^2}+\varepsilon V}\) in \({\mathbb{R}}\) with almost-periodic potential V. We study the behaviour of the integrated density of states (IDS) \({N(H^{(\varepsilon)};\lambda)}\) when \({\varepsilon\to 0}\) and \({\lambda}\) is a fixed energy. When V is quasi-periodic (i.e. is a finite sum of complex exponentials...

We consider the Einstein flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinian field equations coupled to a wave map type and a Maxwell...

We establish a new connection between moments of \({n \times n}\) random matrices Xn and hypergeometric orthogonal polynomials. Specifically, we consider moments \({\mathbb{E}{\rm Tr} X_n^{-s}}\) as a function of the complex variable \({s \in \mathbb{C}}\) , whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry...

We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann zeta function. It manifests...

Observable properties of a classical physical system can be modelled deterministically as functions from the space of pure states to outcome values; dually, states can be modelled as functions from the algebra of observables to outcome values. The probabilistic predictions of quantum physics are contextual in that they preclude this classical assumption of reality: noncommuting...

We prove that the K-theoretic Nekrasov instanton partition functions have a positive radius of convergence in the instanton counting parameter and are holomorphic functions of the Coulomb parameters in a suitable domain. We discuss the implications for the AGT correspondence and the analyticity of the norm of Gaiotto states for the deformed Virasoro algebra. The proof is based on...

We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of external and interaction potential forces with respect to the 2-Wasserstein distance. We also discuss the overdamped limit to a nonlocal equation used in the modelling of granular...