Showing papers in "Communications in Mathematical Physics in 2021"

Symmetries in Quantum Field Theory and Quantum Gravity

Abstract: In this paper we use the AdS/CFT correspondence to refine and then establish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any “long-range” bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irreducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general gauge groups, reproducing the “convex hull condition” of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the bulk and boundary. Quantum field theory results we meet while assembling the necessary tools include continuous global symmetries without Noether currents, new perspectives on spontaneous symmetry-breaking and ’t Hooft anomalies, a new order parameter for confinement which works in the presence of fundamental quarks, a Hamiltonian lattice formulation of gauge theories with arbitrary discrete gauge groups, an extension of the Coleman–Mandula theorem to discrete symmetries, and an improved explanation of the decay $$\pi ^0\rightarrow \gamma \gamma $$ in the standard model of particle physics. We also describe new black hole solutions of the Einstein equation in $$d+1$$ dimensions with horizon topology $${\mathbb {T}}^p\times {\mathbb {S}}^{d-p-1}$$ .

Long-Time Asymptotics for the Spin-1 Gross–Pitaevskii Equation

Abstract: On the basis of the spectral analysis of the $$4\times 4$$ Lax pair associated with the spin-1 Gross–Pitaevskii equation and the scattering matrix, the solution to the Cauchy problem of the spin-1 Gross–Pitaevskii equation is transformed into the solution to the corresponding Riemann–Hilbert problem The Deift–Zhou nonlinear steepest descent method is extended to the Riemann–Hilbert problem, from which a model Riemann–Hilbert problem is established with the help of distinct factorizations of the jump matrix for the Riemann–Hilbert problem and a decomposition of the matrix-valued spectral function Finally, the long-time asymptotics of the solution to the Cauchy problem of the spin-1 Gross–Pitaevskii equation is obtained

Positive Configuration Space

Abstract: We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel.

Geometric Rényi Divergence and its Applications in Quantum Channel Capacities

Abstract: Having a distance measure between quantum states satisfying the right properties is of fundamental importance in all areas of quantum information. In this work, we present a systematic study of the geometric Renyi divergence (GRD), also known as the maximal Renyi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties, which are not satisfied by other quantum Renyi divergences. For example we prove a chain rule inequality that immediately implies the “amortization collapse” for the geometric Renyi divergence, addressing an open question by Berta et al. [Letters in Mathematical Physics 110:2277–2336, 2020, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Renyi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter and efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.

Log-Modular Quantum Groups at Even Roots of Unity and the Quantum Frobenius I

Abstract: We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type $$A_1$$ , and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1, p)-central charge. For example, we explain how one can (conjecturally) employ known linear equivalences between the triplet vertex algebra and quantum $$\mathfrak {sl}_2$$ , in conjunction with a natural $${{\,\mathrm{PSL}\,}}_2$$ -action on quantum $$\mathfrak {sl}_2$$ provided by our de-equivariantization construction, in order to deduce linear equivalences between “extended” quantum groups, the singlet vertex operator algebra, and the (1, p)-Virasoro logarithmic minimal model. We assume some restrictions on the order of our root of unity outside of type $$A_1$$ , which we intend to eliminate in a subsequent paper.

Random Matrix Spectral Form Factor of Dual-Unitary Quantum Circuits

Abstract: We investigate a class of local quantum circuits on chains of d-level systems (qudits) that share the so-called ‘dual unitarity’ property. In essence, the latter property implies that these systems generate unitary dynamics not only when propagating in time, but also when propagating in space. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over $$\mathrm{SU}(d)$$ , e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time t in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of t sites. For general dual unitary circuits of qubits $$(d=2)$$ and a family of their extensions to higher $$d>2$$ , we provide an explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all t, if only the local 2-qubit gates are different from a SWAP (non-interacting gate).

A PDE Construction of the Euclidean $$\Phi ^4_3$$ Quantum Field Theory

Abstract: We present a new construction of the Euclidean $$\Phi ^4$$ quantum field theory on $${\mathbb {R}}^3$$ based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on $${\mathbb {R}}^3$$ defined on a periodic lattice of mesh size $$\varepsilon $$ and side length M. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as $$\varepsilon \rightarrow 0$$ , $$M \rightarrow \infty $$ . Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder–Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models with O(N) symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson–Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a distribution on the space of Euclidean fields.

Classifying Calabi–Yau Threefolds Using Infinite Distance Limits

Abstract: We present a novel way to classify Calabi–Yau threefolds by systematically studying their infinite volume limits Each such limit is at infinite distance in Kahler moduli space and can be classified by an associated limiting mixed Hodge structure We then argue that such structures are labeled by a finite number of degeneration types that combine into a characteristic degeneration pattern associated to the underlying Calabi–Yau threefold These patterns provide a new invariant way to present crucial information encoded in the intersection numbers of Calabi–Yau threefolds For each pattern, we also introduce a Hasse diagram with vertices representing each, possibly multi-parameter, decompactification limit and explain how to read off properties of the Calabi–Yau manifold from this graphical representation In particular, we show how it can be used to count elliptic, K3, and nested fibrations and determine relations of elliptic fibrations under birational equivalence We exemplify this for hypersurfaces in toric ambient spaces as well as for complete intersections in products of projective spaces

Deformations and Homotopy Theory of Relative Rota–Baxter Lie Algebras

Abstract: We determine the $$L_\infty $$ -algebra that controls deformations of a relative Rota–Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying $$\mathsf {Lie}\mathsf {Rep}$$ pair by the dg Lie algebra controlling deformations of the relative Rota–Baxter operator. Consequently, we define the cohomology of relative Rota–Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota–Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota–Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota–Baxter Lie algebras is intimately related to pre-Lie $$_\infty $$ -algebras.

Topology in Shallow-Water Waves: A Violation of Bulk-Edge Correspondence

Abstract: We study the two-dimensional rotating shallow-water model describing Earth’s oceanic layers. It is formally analogue to a Schrodinger equation where the tools from topological insulators are relevant. Once regularized at small scale by an odd-viscous term, such a model has a well-defined bulk topological index. However, in presence of a sharp boundary, the number of edge modes depends on the boundary condition, showing an explicit violation of the bulk-edge correspondence. We study a continuous family of boundary conditions with a rich phase diagram, and explain the origin of this mismatch. Our approach relies on scattering theory and Levinson’s theorem. The latter does not apply at infinite momentum because of the analytic structure of the scattering amplitude there, ultimately responsible for the violation.

Long-Time Asymptotics for the Integrable Nonlocal Focusing Nonlinear Schrödinger Equation for a Family of Step-Like Initial Data

Abstract: We study the Cauchy problem for the integrable nonlocal focusing nonlinear Schrodinger (NNLS) equation $$iq_{t}(x,t)+q_{xx}(x,t)+2 q^{2}(x,t)\bar{q}(-x,t)=0 $$ with the step-like initial data close, in a certain sense, to the “shifted step function” $$\chi _R(x)=AH(x-R)$$ , where H(x) is the Heaviside step function, and $$A>0$$ and $$R>0$$ are arbitrary constants. Our main aim is to study the large-t behavior of the solution of this problem. We show that for $$R\in \left( \frac{(2n-1)\pi }{2A},\frac{(2n+1)\pi }{2A}\right) $$ , $$n=1,2,\ldots $$ , the (x, t) plane splits into $$4n+2$$ sectors exhibiting different asymptotic behavior. Namely, there are $$2n+1$$ sectors, where the solution decays to 0, whereas in the other $$2n+1$$ sectors (alternating with the sectors with decay), the solution approaches (different) constants along each ray $$x/t=const$$ . Our main technical tool is the representation of the solution of the Cauchy problem in terms of the solution of an associated matrix Riemann–Hilbert problem and its subsequent asymptotic analysis following the ideas of the nonlinear steepest descent method.

Asymptotic performance of port-based teleportation

Abstract: Quantum teleportation is one of the fundamental building blocks of quantum Shannon theory. While ordinary teleportation is simple and efficient, port-based teleportation (PBT) enables applications such as universal programmable quantum processors, instantaneous non-local quantum computation and attacks on position-based quantum cryptography. In this work, we determine the fundamental limit on the performance of PBT: for arbitrary fixed input dimension and a large number N of ports, the error of the optimal protocol is proportional to the inverse square of N. We prove this by deriving an achievability bound, obtained by relating the corresponding optimization problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered simplex. We also give an improved converse bound of matching order in the number of ports. In addition, we determine the leading-order asymptotics of PBT variants defined in terms of maximally entangled resource states. The proofs of these results rely on connecting recently-derived representation-theoretic formulas to random matrix theory. Along the way, we refine a convergence result for the fluctuations of the Schur-Weyl distribution by Johansson, which might be of independent interest.

Microlocal Analysis of the Bulk-Edge Correspondence

Abstract: The bulk-edge correspondence predicts that interfaces between topological insulators support robust currents. We prove this principle for PDEs that are periodic away from an interface. Our approach relies on semiclassical methods. It suggests novel perspectives for the analysis of topologically protected transport.

The Generalized TAP Free Energy II

Abstract: In a recent paper (Chen et al. in The generalized TAP free energy, to appear in Comm. Pure Appl. Math.), we developed the generalized TAP approach for mixed p-spin models with Ising spins at positive temperature. Here we extend these results in two directions. We find a simplified representation for the energy of the generalized TAP states in terms of the Parisi measure of the model and, in particular, show that the energy of all states at a given distance from the origin is the same. Furthermore, we prove the analogues of the positive temperature results at zero temperature, which concern the ground-state energy and the organization of ground-state configurations in space.

Rigorous Asymptotics of a KdV Soliton Gas

Abstract: We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann–Hilbert problem which we show arises as the limit $$N\rightarrow + \infty $$ of a gas of N-solitons. We show that this gas of solitons in the limit $$N\rightarrow \infty $$ is slowly approaching a cnoidal wave solution for $$x \rightarrow - \infty $$ up to terms of order $$\mathcal {O} (1/x)$$ , while approaching zero exponentially fast for $$x\rightarrow +\infty $$ . We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.

Determining a Random Schrödinger Operator: Both Potential and Source are Random

Abstract: We study an inverse scattering problem associated with a Schrodinger system where both the potential and source terms are random and unknown. The well-posedness of the forward scattering problem is first established in a proper sense. We then derive two unique recovery results in determining the rough strengths of the random source and the random potential, by using the corresponding far-field data. The first recovery result shows that a single realization of the passive scattering measurements uniquely recovers the rough strength of the random source. The second one shows that, by a single realization of the backscattering data, the rough strength of the random potential can be recovered. The ergodicity is used to establish the single realization recovery. The asymptotic arguments in our study are based on techniques from the theory of pseudodifferential operators and microlocal analysis.

The Focusing NLS Equation with Step-Like Oscillating Background: Scenarios of Long-Time Asymptotics

Abstract: We consider the Cauchy problem for the focusing nonlinear Schrodinger equation with initial data approaching two different plane waves $$A_j\mathrm {e}^{\mathrm {i}\phi _j}\mathrm {e}^{-2\mathrm {i}B_jx}$$ , $$j=1,2$$ as $$x\rightarrow \pm \infty $$ . Using Riemann–Hilbert techniques and Deift–Zhou steepest descent arguments, we study the long-time asymptotics of the solution. We detect that each of the cases $$B_1B_2$$ , and $$B_1=B_2$$ deserves a separate analysis. Focusing mainly on the first case, the so-called shock case, we show that there is a wide range of possible asymptotic scenarios. We also propose a method for rigorously establishing the existence of certain higher-genus asymptotic sectors.

Averaging Principle and Normal Deviations for Multiscale Stochastic Systems

Abstract: We study the asymptotic behavior for an inhomogeneous multiscale stochastic dynamical system with non-smooth coefficients. Depending on the averaging regime and the homogenization regime, two strong convergences in the averaging principle of functional law of large numbers type are established. Then we consider the small fluctuations of the system around its average. Nine cases of functional central limit type theorems are obtained. In particular, even though the averaged equation for the original system is the same, the corresponding homogenization limit for the normal deviation can be quite different due to the difference in the interactions between the fast scales and the deviation scales. We provide quite intuitive explanations for each case. Furthermore, sharp rates both for the strong convergences and the functional central limit theorems are obtained, and these convergences are shown to rely only on the regularity of the coefficients of the system with respect to the slow variable, and do not depend on their regularity with respect to the fast variable, which coincide with the intuition since in the limit equations the fast component has been totally averaged or homogenized out.

The vertex algebras $\mathcal R^{(p)}$ and $\mathcal V^{(p)}$

Abstract: The vertex algebras $$\mathcal {V}^{(p)}$$ and $$\mathcal R^{(p)}$$ introduced in Adamovic (Transform Groups 21(2):299–327, 2016) are very interesting relatives of the well-known triplet algebras of logarithmic CFT. The algebra $$\mathcal {V}^{(p)}$$ (respectively, $$\mathcal {R}^{(p)}$$ ) is a large extension of the simple affine vertex algebra $$L_{k}(\mathfrak {sl}_{2})$$ (respectively, $$L_{k}(\mathfrak {sl}_{2})$$ times a Heisenberg algebra), at level $$k=-2+1/p$$ for positive integer p. Particularly, the algebra $$\mathcal {V}^{(2)}$$ is the simple small $$N=4$$ superconformal vertex algebra with $$c=-9$$ , and $$\mathcal {R}^{(2)}$$ is $$L_{-3/2}(\mathfrak {sl}_3)$$ . In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $$\mathcal {V}^{({p})}$$ and we decompose $$\mathcal {V}^{({p})}$$ as an $$L_{k}(\mathfrak {sl}_{2})$$ -module and $$\mathcal {R}^{({p})}$$ as an $$L_k(\mathfrak {gl}_2)$$ -module. The decomposition of $$\mathcal {V}^{({p})}$$ shows that $$\mathcal {V}^{({p})}$$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $$\mathcal {V}^{({p})}$$ is the logarithmic doublet algebra $$\mathcal {A}^{({p})}$$ introduced in Adamovic and Milas (Contemp Math 602:23–38, 2013), while the reduction of $$\mathcal {R}^{({p})}$$ yields the $$\mathcal {B}^{({p})}$$ -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014). Conversely, we realize $$\mathcal {V}^{({p})}$$ and $$\mathcal {R}^{({p})}$$ from $$\mathcal {A}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $$KL_{k}$$ of ordinary $$L_{k}(\mathfrak {sl}_{2})$$ -modules at level $$k=-2+1/p$$ is a rigid vertex tensor category equivalent to a twist of the category $$\text {Rep}(SU(2))$$ . This finally completes rigid braided tensor category structures for $$L_{k}(\mathfrak {sl}_{2})$$ at all complex levels k. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $$\mathcal {R}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ are certain non-principal $$\mathcal {W}$$ -algebras of type A at boundary admissible levels. The same uniqueness result also shows that $$\mathcal {R}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ are the chiral algebras of Argyres-Douglas theories of type $$(A_1, D_{2p})$$ and $$(A_1, A_{2p-3})$$ .

Stochastic Lagrangian Path for Leray’s Solutions of 3D Navier–Stokes Equations

Abstract: In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray’s solution of 3D Navier–Stokes equations. More precisely, for any Leray’s solution $$\mathbf{u }$$ of 3D-NSE and each $$(s,x)\in \mathbb {R}_+\times \mathbb {R}^3$$ , we show the existence of weak solutions to the following SDE, which have densities $$\rho _{s,x}(t,y)$$ belonging to $$\mathbb {H}^{1,p}_q$$ with $$p,q\in [1,2)$$ and $$\frac{3}{p}+\frac{2}{q}>4$$ : $$\begin{aligned} \text {d} X_{s,t}=\mathbf{u } (s,X_{s,t})\text {d} t+\sqrt{2 u }\text {d} W_t,\ \ X_{s,s}=x,\ \ t\geqslant s, \end{aligned}$$ where W is a three dimensional standard Brownian motion, $$ u >0$$ is the viscosity constant. Moreover, we also show that for Lebesgue almost all (s, x), the solution $$X^n_{s,\cdot }(x)$$ of the above SDE associated with the mollifying velocity field $$\mathbf{u }_n$$ weakly converges to $$X_{s,\cdot }(x)$$ so that X is a Markov process in almost sure sense.

On the Size of Chaos via Glauber Calculus in the Classical Mean-Field Dynamics

Abstract: We consider a system of classical particles, interacting via a smooth, long-range potential, in the mean-field regime, and we optimally analyze the propagation of chaos in form of sharp estimates on many-particle correlation functions. While approaches based on the BBGKY hierarchy are doomed by uncontrolled losses of derivatives, we propose a novel non-hierarchical approach that focusses on the empirical measure of the system and exploits discrete stochastic calculus with respect to initial data in form of higher-order Poincare inequalities for cumulants. This main result allows to rigorously truncate the BBGKY hierarchy to an arbitrary precision on the mean-field timescale, thus justifying the Bogolyubov corrections to mean field. As corollaries, we also deduce a quantitative central limit theorem for fluctuations of the empirical measure, and we discuss the Lenard–Balescu limit for a spatially homogeneous system away from thermal equilibrium.

Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ velocity and boundary

Abstract: Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30, 31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $$C^{1,\alpha }$$ initial data for the velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations in the presence of boundary with $$C^{1,\alpha }$$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30, 31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $$C^{1,\alpha }$$ initial data for the velocity field. We use a dynamic rescaling formulation and follow the general framework of analysis developed by Elgindi in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler equations or the 2D Boussinesq equations with $$C^{1,\alpha }$$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time.

Regularity and Expansion for Steady Prandtl Equations

Abstract: Due to degeneracy near the boundary, the question of high regularity for solutions to the steady Prandtl equations has been a longstanding open question since the celebrated work of Oleinik. We settle this open question in affirmative in the absence of an external pressure. Our method is based on energy estimates for the quotient, $$q = \frac{v}{\bar{u}}$$ , $$\bar{u}$$ being the classical Prandtl solution, via the linear derivative Prandtl (LDP) equation. As a consequence, our regularity result leads to the construction of Prandtl layer expansion up to any order.

Long-Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions in the Presence of a Discrete Spectrum

Abstract: The long-time asymptotic behavior of solutions to the focusing nonlinear Schrodinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity is studied in the case of initial conditions that allow for the presence of discrete spectrum. The results of the analysis provide the first rigorous characterization of the nonlinear interactions between solitons and the coherent oscillating structures produced by localized perturbations in a modulationally unstable medium. The study makes crucial use of the inverse scattering transform for the focusing NLS equation with nonzero boundary conditions, as well as of the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann–Hilbert problems. Previously, it was shown that in the absence of discrete spectrum the xt-plane decomposes asymptotically in time into two types of regions: a left far-field region and a right far-field region, where to leading order the solution equals the condition at infinity up to a phase shift, and a central region where the asymptotic behavior is described by slowly modulated periodic oscillations. Here, it is shown that in the presence of a conjugate pair of discrete eigenvalues in the spectrum a similar coherent oscillatory structure emerges but, in addition, three different interaction outcomes can arise depending on the precise location of the eigenvalues: (i) soliton transmission, (ii) soliton trapping, and (iii) a mixed regime in which the soliton transmission or trapping is accompanied by the formation of an additional, nondispersive localized structure akin to a soliton-generated wake. The soliton-induced position and phase shifts of the oscillatory structure are computed, and the analytical results are validated by a set of accurate numerical simulations.

On the Modular Operator of Mutli-component Regions in Chiral CFT

Abstract: We introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

Von Neumann Entropy in QFT

Abstract: In the framework of Quantum Field Theory, we provide a rigorous, operator algebraic notion of entanglement entropy associated with a pair of open double cones $$O \subset {\widetilde{O}}$$ of the spacetime, where the closure of O is contained in $${\widetilde{O}}$$ . Given a QFT net $${\mathcal {A}}$$ of local von Neumann algebras $${\mathcal {A}}(O)$$ , we consider the von Neumann entropy $$S_{\mathcal {A}}(O, {\widetilde{O}})$$ of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras $${\mathcal {A}}(O)\subset {\mathcal {A}}({\widetilde{O}})$$ (split property). We show that this canonical entanglement entropy $$S_{\mathcal {A}}(O, {\widetilde{O}})$$ is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). To this end, we first study the notion of von Neumann entropy of a closed real linear subspace of a complex Hilbert space, that we then estimate for the local free fermion subspaces. We further consider the lower entanglement entropy $${\underline{S}}_{\mathcal {A}}(O, {\widetilde{O}})$$ , the infimum of the vacuum von Neumann entropy of $${\mathcal {F}}$$ , where $${\mathcal {F}}$$ here runs over all the intermediate, discrete type I von Neumann algebras. We prove that $${\underline{S}}_{\mathcal {A}}(O, {\widetilde{O}})$$ is finite for the local chiral conformal net generated by finitely many commuting U(1)-currents.

Computing Spectral Measures and Spectral Types

Abstract: Spectral measures arise in numerous applications such as quantum mechanics, signal processing, resonance phenomena, and fluid stability analysis. Similarly, spectral decompositions (into pure point, absolutely continuous and singular continuous parts) often characterise relevant physical properties such as the long-time dynamics of quantum systems. Despite new results on computing spectra, there remains no general method able to compute spectral measures or spectral decompositions of infinite-dimensional normal operators. Previous efforts have focused on specific examples where analytical formulae are available (or perturbations thereof) or on classes of operators that carry a lot of structure. Hence the general computational problem is predominantly open. We solve this problem by providing the first set of general algorithms that compute spectral measures and decompositions of a wide class of operators. Given a matrix representation of a self-adjoint or unitary operator, such that each column decays at infinity at a known asymptotic rate, we show how to compute spectral measures and decompositions. We discuss how these methods allow the computation of objects such as the functional calculus, and how they generalise to a large class of partial differential operators, allowing, for example, solutions to evolution PDEs such as the linear Schrodinger equation on $$L^2({\mathbb {R}}^d)$$ . Computational spectral problems in infinite dimensions have led to the Solvability Complexity Index (SCI) hierarchy, which classifies the difficulty of computational problems. We classify the computation of measures, measure decompositions, types of spectra, functional calculus, and Radon–Nikodym derivatives in the SCI hierarchy. The new algorithms are demonstrated to be efficient on examples taken from orthogonal polynomials on the real line and the unit circle (giving, for example, computational realisations of Favard’s theorem and Verblunsky’s theorem, respectively), and are applied to evolution equations on a two-dimensional quasicrystal.

Evolution of Angular Momentum and Center of Mass at Null Infinity

Abstract: We study how conserved quantities such as angular momentum and center of mass evolve with respect to the retarded time at null infinity, which is described in terms of a Bondi–Sachs coordinate system. These evolution formulae complement the classical Bondi mass loss formula for gravitational radiation. They are further expressed in terms of the potentials of the shear and news tensors. The consequences that follow from these formulae are (1) Supertranslation invariance of the fluxes of the CWY conserved quantities. (2) A conservation law of angular momentum a la Christodoulou. (3) A duality paradigm for null infinity. In particular, the supertranslation invariance distinguishes the CWY angular momentum and center of mass from the classical definitions.

Short Time Large Deviations of the KPZ Equation

Abstract: We establish the Freidlin–Wentzell Large Deviation Principle (LDP) for the Stochastic Heat Equation with multiplicative noise in one spatial dimension. That is, we introduce a small parameter $$ \sqrt{\varepsilon } $$ to the noise, and establish an LDP for the trajectory of the solution. Such a Freidlin–Wentzell LDP gives the short-time, one-point LDP for the KPZ equation in terms of a variational problem. Analyzing this variational problem under the narrow wedge initial data, we prove a quadratic law for the near-center tail and a $$ \frac{5}{2} $$ law for the deep lower tail. These power laws confirm existing physics predictions (Kolokolov and Korshunov in Phys Rev B 75(14):140201, 2007, Phys Rev E 80(3):031107, 2009; Meerson et al. in Phys Rev Lett 116(7):070601, 2016; Le Doussal et al. in Phys Rev Lett 117(7):070403, 2016; Kamenev et al. in Phys Rev E 94(3):032108, 2016).

Thermodynamic Formalism for Random Non-uniformly Expanding Maps

Abstract: We develop a quenched thermodynamic formalism for a wide class of random maps with non-uniform expansion, where no Markov structure, no uniformly bounded degree or the existence of some expanding dynamics is required. We prove that every measurable and fibered $$C^1$$ -potential at high temperature admits a unique equilibrium state which satisfies a weak Gibbs property, and has exponential decay of correlations. The arguments combine a functional analytic approach for the decay of correlations (using Birkhoff cone methods) and Caratheodory-type structures to describe the relative pressure of not necessary compact invariant sets in random dynamical systems. We establish also a variational principle for the relative pressure of random dynamical systems.