We study bosonic quantum field theory on the double covering \(\widetilde{dS}_{2}\) of the two-dimensional de Sitter universe, identified to a coset space of the group \(\mathrm{SL}(2, {{\mathbb {R}}})\). The latter acts effectively on \(\widetilde{dS}_{2}\) and can be interpreted as it relativity group. The manifold is locally identical to the standard the Sitter spacetime \({dS}_2\); it is globally

We study monodromy reduction of Fuchsian connections from a sheave theoretic viewpoint, focusing on the case when a singularity of a special connection with four singularities has been resolved. The main tool of study is based on a bundle modification technique due to Drinfeld and Oblezin. This approach via invariant spaces and eigenvalue problems allows us not only to explain Erdélyi’s classical infinite

We provide an overview of the tools and techniques of resurgence theory used in the Borel-Écalle resummation method, which we then apply to the massless Wess–Zumino model. Starting from already known results on the anomalous dimension of the Wess–Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and

We construct a one-parameter family of static and spherically symmetric solutions to the Einstein-Vlasov system bifurcating from the Schwarzschild spacetime. The constructed solutions have the property that the spatial support of the matter is a finite, spherically symmetric shell located away from the black hole. Our proof is mostly based on the analysis of the set of trapped timelike geodesics and

We study the semigroup generated by the hypoelliptic Laplacian on the circle and the maximal bounded holomorphic extension of this semigroup. Using an orthogonal decomposition into harmonic oscillators with complex shifts, we describe the domain of this extension and we show that boundedness in a half plane corresponds to absolute convergence of the expansion of the semigroup in eigenfunctions. This

A Correction to this paper has been published: 10.1007/s00023-020-01014-z

Lattices of polynomial KP and BKP \(\tau \)-functions labelled by partitions, with the flow variables equated to finite power sums, as well as associated multipair KP and multipoint BKP correlation functions, are expressed via generalizations of Jacobi’s bialternant formula for Schur functions and Nimmo’s Pfaffian ratio formula for Schur Q-functions. These are obtained by applying Wick’s theorem to

This paper revisits the proof of Anderson localization for multi-particle systems. We introduce a multi-particle version of the eigensystem multi-scale analysis by Elgart and Klein, which had previously been used for single-particle systems.

Random noncommutative geometry can be seen as a Euclidean path-integral quantization approach to the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of investigating phase transitions in random NCG of arbitrary dimension, we study the nonperturbative Functional Renormalization Group for multimatrix models whose action consists of noncommutative polynomials in Hermitian

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result

Wigner and Yanase conjectured the more general problem of the concavity of the function \(\rho \mapsto \text {Trace}\ \rho ^pK^*\rho ^{1-p}K\). Lieb gave an actual connection with something then known as the strong subadditivity of the quantum entropy conjecture, which was due to Ruelle and Robinson. Lieb proved the concavity of that function, and this deep result is known as the Lieb’s concavity theorem

It is emphasized that for interactions with derivative couplings, the Ward Identity (WI) securing the preservation of a global symmetry should be modified. Scalar QED is taken as an explicit example. More precisely, it is rigorously shown in scalar QED that the naive WI and the improved Ward Identity (‘Master Ward Identity’, MWI) are related to each other by a finite renormalization of the time-ordered

We consider a system of M particles in contact with a heat reservoir of \(N\gg M\) particles. The evolution in the system and the reservoir, together with their interaction, is modeled via the Kac’s master equation. We chose the initial distribution with total energy \(N+M\) and show that if the reservoir is initially in equilibrium, that is, if the initial distribution depends only on the energy of

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates that are consistent with central limit theorems that have been established in the last years.

In Roesch (Proof of a null penrose conjecture using a new quasi-local mass, arXiv:1609.02875 [gr-qc], 2016), the notion of Double Convexity for a foliation of a conical null hypersurface was introduced to give a proof, if satisfied, of the Null Penrose Inequality. Double Convexity constrains the geometry of a Marginally Outer Trapped Surface (MOTS), called a quasi-round MOTS. In the first part of this

For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different

We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these

We consider interaction energies \(E_f[L]\) between a point \(O\in {\mathbb {R}}^d\), \(d\ge 2\), and a lattice L containing O, where the interaction potential f is assumed to be radially symmetric and decaying sufficiently fast at infinity. We investigate the conservation of optimality results for \(E_f\) when integer sublattices kL are removed (periodic arrays of vacancies) or substituted (periodic

The space of vacua of many four-dimensional, \({\mathcal {N}}=2\) supersymmetric gauge theories can famously be identified with a family of complex curves. For gauge group SU(2), this gives a fully explicit description of the low-energy effective theory in terms of an elliptic curve and associated modular fundamental domain. The two-dimensional space of vacua for gauge group SU(3) parametrizes an intricate

In Mizera (J High Energy Phys 8:097, 2017) , Sebastian Mizera discovered a tree expansion formula of a homology intersection number on the configuration space \(\mathcal {M}_{0,n}\). The formula originates in a study of Kawai–Lewellen–Tye relation in string theory. In this paper, we give an elementary proof of the formula. The basic ingredients are the combinatorics of the real moduli space \(\overline{\mathcal

We introduce a global thermostat on Kac’s 1D model for the velocities of particles in a space-homogeneous gas subjected to binary collisions, also interacting with a (local) Maxwellian thermostat. The global thermostat rescales the velocities of all the particles, thus restoring the total energy of the system, which leads to an additional drift term in the corresponding nonlinear kinetic equation.

We analyse the Casimir effect of two nonsingular centers of interaction in three space dimensions, using the framework developed by Herdegen. Our model is mathematically well-defined and all physical quantities are finite. We also consider a scaling limit, in which the problem tends to that with two Dirac \(\delta \)’s. In this limit the global Casimir energy diverges, but we obtain its asymptotic

We propose a new generalization to quantum states of the Wasserstein distance, which is a fundamental distance between probability distributions given by the minimization of a transport cost. Our proposal is the first where the transport plans between quantum states are in natural correspondence with quantum channels, such that the transport can be interpreted as a physical operation on the system

We investigate the spectrum of Schrödinger operators on finite regular metric trees through a relation to orthogonal polynomials that provides a graphical perspective. As the Robin vertex parameter tends to \(-\,\infty \), a narrow cluster of finitely many eigenvalues tends to \(-\,\infty \), while the eigenvalues above this cluster remain bounded from below. Certain “rogue” eigenvalues break away

We analyse the boundary structure of general relativity in the coframe formalism in the case of a lightlike boundary, i.e. when the restriction of the induced Lorentzian metric to the boundary is degenerate. We describe the associated reduced phase space in terms of constraints on the symplectic space of boundary fields. We explicitly compute the Poisson brackets of the constraints and identify the

In this note, we prove an existence result for the Einstein conformal constraint equations for metrics with vanishing Yamabe invariant assuming that the mean curvature satisfies an explicit near-CMC condition and that the TT-tensor is small in \(L^2\).

We consider independent electrons in a periodic crystal in their ground state, and turn on a uniform electric field at some prescribed time. We rigorously define the current per unit volume and study its properties using both linear response and adiabatic theory. Our results provide a unified framework for various phenomena such as the quantization of Hall conductivity of insulators with broken time-reversibility

We consider rapidly mixing dynamical systems and link the decay of the shortest distance between multiple orbits with the generalized fractal dimension. We apply this result to multidimensional expanding maps and extend it to the realm of random dynamical systems. For random sequences, we obtain a relation between the longest common substring between multiple sequences and the generalized Rényi entropy

We prove that the only asymptotically flat spacetimes with a suitably regular event horizon, in a generalised Majumdar–Papapetrou class of solutions to higher-dimensional Einstein–Maxwell theory, are the standard multi-black holes. The proof involves a careful analysis of the near-horizon geometry and an extension of the positive mass theorem to Riemannian manifolds with conical singularities. This

We define a random model for the moments of the new eigenfunctions of a point scatterer on a two-dimensional rectangular flat torus. In the deterministic setting, S̆eba conjectured these moments to be asymptotically Gaussian, in the semi-classical limit. This conjecture was disproved by Kurlberg–Ueberschär on Diophantine tori. In our model, we describe the accumulation points in distribution of the

Equation (5.20), should read as follows.

A class of deep Boltzmann machines is considered in the simplified framework of a quenched system with Gaussian noise and independent entries. The quenched pressure of a K-layers spin glass model is studied allowing interactions only among consecutive layers. A lower bound for the pressure is found in terms of a convex combination of K Sherrington–Kirkpatrick models and used to study the annealed and

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are

We determine the 2-group structure constants for all the six-dimensional little string theories (LSTs) geometrically engineered in F-theory without frozen singularities. We use this result as a consistency check for T-duality: the 2-groups of a pair of T-dual LSTs have to match. When the T-duality involves a discrete symmetry twist, the 2-group used in the matching is modified. We demonstrate the matching

We propose an abstract framework describing energy-renormalized Hamiltonians in terms of local algebras. Within the framework, we examine the positivity improvingness of the semigroup generated by the renormalized Hamiltonian. As examples, we discuss the renormalized Nelson Hamiltonian and the renormalized Nelson Hamiltonian at fixed total momentum. The characteristic features of our approach are as

We introduce a family of ‘spatial’ random cycle Huang–Yang–Luttinger (HYL)-type models in which the counter-term only affects cycles longer than some cut-off that diverges in the thermodynamic limit. Here, spatial refers to the Poisson reference process of random cycle weights. We derive large deviation principles and explicit pressure expressions for these models, and use the zeroes of the rate functions

We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.

We consider the Dirac equation for the classical spinor field placed in an external, time-dependent electromagnetic field of the form typical for scattering settings: \(F=F^\mathrm{ret}+F^\mathrm{in}=F^\mathrm{adv}+F^\mathrm{out}\), where the current producing \(F^{\mathrm{ret}/\mathrm{adv}}\) has past and future asymptotes homogeneous of degree \(-3\), and the free fields \(F^{\mathrm{in}/\mathrm{out}}\)

We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation \(\text {i}\partial _t u + \Delta u + |u|^{p-1}u=0\) on \(\mathbb {R}^d\), close to the mass critical case and for any space dimension \(d\ge 1\). These profiles bifurcate from the ground-state solitary wave. The argument relies on the classical matched asymptotics method suggested

We introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties

We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a one-dimensional tiling space \(\Omega \) with finite local complexity and study self-maps F that are homotopic to the identity and whose displacements are strongly pattern equivariant. In place of the familiar rotation number, we define a cohomology class \([\mu ] \in {\check{H}}^1(\Omega , {\mathbb {R}})\). We prove

The point-interaction approximation (or the Foldy–Lax approximation) of the electromagnetic fields generated by a cluster of small-scaled inhomogeneities is derived in the mesoscale (i.e., mesoscopic scale) regime, that is, when the minimum distance \(\varvec{\delta }\) between the particles is proportional to their maximum diameter \({\varvec{a}}\) in the form \(\varvec{\delta }={\varvec{c}}_r \,

For two dimensional Schrödinger operator H with point interactions, we prove that wave operators of scattering for the pair \((H,H_0)\), \(H_0\) being the free Schrödinger operator, are bounded in the Lebesgue space \(L^p({{\mathbb {R}}}^2)\) for \(1

Given a nonnegative self-adjoint operator H acting on a separable Hilbert space and an orthogonal projection P such that \(H_P := (H^{1/2}P)^*(H^{1/2}P)\) is densely defined, we prove that \(\lim _{n\rightarrow \infty } (P\,\mathrm {e}^{-itH/n}P)^n = \mathrm {e}^{-itH_P}P\) holds in the strong operator topology. We also derive modifications of this product formula and its extension to the situation

We consider algebraic varieties canonically associated with any Lie superalgebra, and study them in detail for super-Poincaré algebras of physical interest. They are the locus of nilpotent elements in (the projectivized parity reversal of) the odd part of the algebra. Most of these varieties have appeared in various guises in previous literature, but we study them systematically here, from a new perspective:

We investigate the spectral properties of discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the elements of the Fibonacci subshift with a locally constant function. The fundamental trace map formalism for this model is presented and related to its spectral features via an extension of a multitude of works on the classical model, where the sampling function

In this work, we provide results on the long-time localization in space (dynamical localization) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form \(H=H_0+W\), where \(H_0\) is rotationally symmetric and has dense point spectrum and W is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the

Consider a finite triangulation of a surface M of genus g and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev’s work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation

We give a complete characterization of the reflectionless Schrödinger operators on the line with integrable potentials, solve the inverse scattering problem of reconstructing such potentials from the eigenvalues and norming constants, and derive the corresponding generalized soliton solutions of the Korteweg–de Vries equation.

The probability distribution of a function of a subsystem conditioned on the value of the function of the whole, in the limit when the ratio of their values goes to zero, has a limit law: It equals the unconditioned marginal probability distribution weighted by an exponential factor whose exponent is uniquely determined by the condition. We apply this theorem to explain the canonical equilibrium ensemble

One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries

We consider a family of globally stationary (horizonless), asymptotically flat solutions of five-dimensional supergravity. We prove that massless linear scalar waves in such soliton spacetimes cannot have a uniform decay rate faster than inverse logarithmically in time. This slow decay can be attributed to the stable trapping of null geodesics. Our proof uses the construction of quasimodes which are

We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form H. There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum \(\mathfrak {sl}_N\) invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes

We consider suspension (semi)flows over \({C}^{1+\alpha }\) full-branch Markov piecewise expanding interval maps and piecewise hyperbolic maps and prove exponential decay of correlations with respect to Gibbs measures associated with piecewise Hölder continuous potentials. As a consequence, typical codimension one attractors of \(C^{1+\alpha }\) Axiom A flows have exponential decay of correlations

In general, linear response theory expresses the relation between a driving and a physical system’s response only to first order in perturbation theory. In the context of charge transport, this is the linear relation between current and electromotive force expressed in Ohm’s law. We show here that in the case of the quantum Hall effect, all higher-order corrections vanish. We prove this in a fully

Quantum trajectories are Markov processes that describe the time evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called Stochastic Schrödinger Equations, which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum

We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous, or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators