Annales Henri Poincaré最新文献

Interpreting the GNVW index for 1D quantum cellular automata (QCA) in terms of the Jones index for subfactors leads to a generalization of the index defined for QCA on more general abstract spin chains. These include fusion spin chains, which arise as the local operators invariant under a global (categorical/MPO) symmetry, and as the boundary operators of 2D topological codes. We show that for the fusion spin chains built from the fusion category (textbf{Fib}), the index is a complete invariant for the group of QCA modulo finite depth circuits.

We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not be synchronous. We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron involving a matrix related to the cost matrix. We give an example of a game such that the synchronous value of repeated products of the game is strictly increasing. We show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative. Finally, we derive geometric and algebraic conditions that a set of projections that yields the synchronous value of a game must satisfy.

We study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio N, we identify the first Dirichlet mode to feature a crossing in its nodal set and perturb one of the sides of the rectangle by a close to flat, smooth curve. Such perturbations will often “open” the crossing in the nodal set, splitting it into two curves, and we study the separation between these curves and their regularity. The main technique used is an approximate separation of variables that allows us to restrict study to the first two Fourier modes in an eigenfunction expansion. We show how the nature of the boundary perturbation provides conditions on the orientation of the opening and estimates on its size. In particular, several features of the perturbed nodal set are asymptotically independent of the aspect ratio, which contrasts with prior works. Numerical results supporting our findings are also presented.

We introduce the idea of constructing recursion operators for full-fledged nonlocal symmetries and apply it to the reduced quasi-classical self-dual Yang–Mills equation. It turns out that the discovered recursion operators can be interpreted as infinite-dimensional matrices of differential functions which act on the generating vector functions of the nonlocal symmetries simply by matrix multiplication. To the best of our knowledge, there are no other examples of such recursion operators in the literature so far, so our approach is completely innovative. Further, we investigate the algebraic properties of the discovered operators and discuss the ({mathbb {R}})-algebra structure on the set of all recursion operators for full-fledged nonlocal symmetries of the equation in question. Finally, we illustrate the action of the obtained recursion operators on particularly chosen full-fledged symmetries and emphasize their advantages compared to the action of traditionally used recursion operators for shadows.

Anyons with a statistical phase parameter (alpha in (0,2)) are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave guide. More specifically, we study the 2D magnetic gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the y-direction than in the x-direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the x-direction is given by the impenetrable Tonks–Girardeau Bose gas, which has no dependency on (alpha ), and no trace left of the long-range interactions of the 2D model.

We show that quantum graph parameters for finite, simple, undirected graphs encode winning strategies for all possible synchronous non-local games. Given a synchronous game (mathcal {G}=(I,O,lambda )) with (|I|=n) and (|O|=k), we demonstrate what we call a weak (*)-equivalence between (mathcal {G}) and a 3-coloring game on a graph with at most (3+n+9n(k-2)+6|lambda ^{-1}({0})|) vertices, strengthening and simplifying work implied by Ji [16] for winning quantum strategies for synchronous non-local games. As an application, we obtain a quantum version of Lovász’s reduction [21] of the k-coloring problem for a graph G with n vertices and m edges to the 3-coloring problem for a graph with (3+n+9n(k-2)+6mk) vertices. Moreover, winning strategies for a synchronous game (mathcal {G}) can be transformed into winning strategies for an associated graph coloring game, where the strategies exhibit perfect zero knowledge for an honest verifier. We also show that, for “graph of the game” (X(mathcal {G})) associated with (mathcal {G}) from Atserias et al. [1], the independence number game (text {Hom}(K_{|I|},overline{X(mathcal {G})})) is hereditarily (*)-equivalent to (mathcal {G}), so that the possibility of winning strategies is the same in both games for all models, except the game algebra. Thus, the quantum versions of the chromatic number, independence number and clique number encode winning strategies for all synchronous games in all quantum models.

We consider a deformation of a family of hyperelliptic refined spectral curves and investigate how deformation effects appear in the hyperelliptic refined topological recursion as well as the (mathcal {Q})-top recursion. We then show a coincidence between a deformation condition and a quantisation condition in terms of the (mathcal {Q})-top recursion on a degenerate elliptic curve. We also discuss a relation to the corresponding Nekrasov–Shatashivili effective twisted superpotential.

We consider the Laplace–Beltrami operator with Dirichlet boundary conditions on convex domains in a Riemannian manifold ((M^n,g)) and prove that the product of the fundamental gap with the square of the diameter can be arbitrarily small whenever (M^n) has even a single tangent plane of negative sectional curvature. In particular, the fundamental gap conjecture strongly fails for small deformations of Euclidean space which introduce any negative curvature. We also show that when the curvature is negatively pinched, it is possible to construct such domains of any diameter up to the diameter of the manifold. The proof is adapted from the argument of Bourni et al. (in: Annales Henri Poincaré, Springer, 2022), which established the analogous result for convex domains in hyperbolic space, but requires several new ingredients.

We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh–Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Araki (Helv Phys Acta 36:132–139, 1963) and Borchers (Nuovo Cim (10) 19:787–793, 1961) to curved spacetimes.

We pursue a uniform quantization of all twists of 4-dimensional (mathcal N=4) supersymmetric Yang–Mills theory, using the BV formalism, and we explore consequences for factorization algebras of observables. Our central result is the construction of a one-loop exact quantization on (mathbb {R}^4) for all such twists and for every point in a moduli of vacua. When an action of the group (textrm{SO}(4)) can be defined—for instance, for Kapustin and Witten’s family of twists—the associated framing anomaly vanishes. It follows that the local observables in such theories can be canonically described by a family of framed (mathbb E_4) algebras; this structure allows one to take the factorization homology of observables on any oriented 4-manifold. In this way, each Kapustin–Witten theory yields a fully extended, oriented 4-dimensional topological field theory à la Lurie and Scheimbauer.