Communications in Mathematical Physics最新文献

We extend the theory of topological recursion by considering Airy ideals (also known as Airy structures) whose partition functions are highest weight vectors of particular (mathcal {W})-algebra representations. Such highest weight vectors arise as partition functions of Airy ideals only under certain conditions on the representations. In the spectral curve formulation of topological recursion, we show that this generalization amounts to adding specific terms to the correlators ( omega _{g,1}), which leads to a “shifted topological recursion” formula. We then prove that the wave-functions constructed from this shifted version of topological recursion are WKB solutions of families of quantizations of the spectral curve with ( hslash )-dependent terms. In the reverse direction, starting from an (hslash )-connection, we find that it is of topological type if the exact same conditions that we found for the Airy ideals are satisfied. When this happens, the resulting shifted loop equations can be solved by the shifted topological recursion obtained earlier.

We consider a four-dimensional globally hyperbolic and asymptotically flat spacetime (M, g) conformal to Minkowski spacetime, together with a massless, conformally coupled scalar field. Using a bulk-to-boundary correspondence, one can establish the existence of an injective (*)-homomorphism (Upsilon _M) between (mathcal {W}(M)), the Weyl algebra of observables on M and a counterpart which is defined intrinsically on future null infinity (Im ^+simeq mathbb {R}times mathbb {S}^2), a component of the conformal boundary of (M, g). Using invariance under the asymptotic symmetry group of (Im ^+), we can individuate thereon a distinguished two-point correlation function whose pull-back to M via (Upsilon _M) identifies a quasi-free Hadamard state for the bulk algebra of observables. In this setting, if we consider (textsf{V}^+_x), a future light cone stemming from (xin M) as well as (mathcal {W}(textsf{V}^+_x)=mathcal {W}(M)|_{textsf{V}^+_x}), its counterpart at the boundary is the Weyl subalgebra generated by suitable functions localized in (textsf{K}_x), a positive half strip on (Im ^+). To each such cone, we associate a standard subspace of the boundary one-particle Hilbert space, which coincides with the one associated naturally to (textsf{K}_x). We extend such correspondence replacing (textsf{K}_x) and (textsf{V}^+_x) with deformed counterparts, denoted by (textsf{S}_C) and (textsf{V}_C). In addition, since the one particle Hilbert space at the boundary decomposes as a direct integral on the sphere of U(1)-currents defined on the real line, we prove that also the generator of the modular group associated to the standard subspace of (textsf{V}_C) decomposes as a suitable direct integral. This result allows us to study the relative entropy between coherent states of the algebras associated to the deformed cones (textsf{V}_C) establishing the quantum null energy condition.

Standard subspaces are closed real subspaces of a complex Hilbert space that appear naturally in Tomita–Takesaki modular theory and its applications to quantum field theory. In this article, inclusions of standard subspaces are studied independently of von Neumann algebras. Several new methods for their investigation are developed, related to polarizers, Gelfand triples defined by modular data, and extensions of modular operators. A particular class of examples that arises from the fundamental irreducible building block of a conformal field theory on the line is analyzed in detail.

In 1939, Oppenheimer and Snyder showed that the continued gravitational collapse of a self-gravitating matter distribution can result in the formation of a black hole, cf. Oppenheimer and Snyder (Phys Rev 56:455–459, 1939). In this paper, which has greatly influenced the evolution of ideas around the concept of a black hole, matter was modeled as dust, a fluid with pressure equal to zero. We prove that when the corresponding initial data are suitably approximated by data for a collisionless gas as modeled by the Vlasov equation, then a trapped surface forms before the corresponding solution to the Einstein–Vlasov system can develop a singularity and again a black hole arises. As opposed to the dust case the pressure does not vanish for such solutions. As a necessary starting point for the analysis, which is carried out in Painlevé–Gullstrand coordinates, we prove a local existence and uniqueness theorem for regular solutions together with a corresponding extension criterion. The latter result will also become useful when one perturbs dust solutions containing naked singularities in the Vlasov framework.

We consider a random walk among a Poisson cloud of moving traps on (mathbb {Z}^d), where the walk is killed at a rate proportional to the number of traps occupying the same position. In dimension (d=1), we have previously shown that under the annealed law of the random walk conditioned on survival up to time t, the walk is sub-diffusive. Here we show that in (dgeqslant 6) and under diffusive scaling, this annealed law satisfies an invariance principle with a positive diffusion constant if the killing rate is small. Our proof is based on the theory of thermodynamic formalism, where we extend some classic results for Markov shifts with a finite alphabet and a potential of summable variation to the case of an uncountable non-compact alphabet.

We use the newly developed technique of inverse quantum hamiltonian reduction to investigate the representation theory of the simple affine vertex algebra (textsf{A}_2(textsf{u},2)) associated to (mathfrak {sl}_{3}) at level (textsf{k}= -3+frac{textsf{u}}{2}), for (textsf{u}geqslant 3) odd. Starting from the irreducible modules of the corresponding simple Bershadsky-Polyakov vertex operator algebras, we show that inverse reduction constructs all irreducible lower-bounded weight (textsf{A}_2(textsf{u},2))-modules. This proceeds by first constructing a complete set of coherent families of fully relaxed highest-weight (textsf{A}_2(textsf{u},2))-modules and then noting that the reducible members of these families degenerate to give all remaining irreducibles. Using this fully relaxed construction and the degenerations, we deduce modular S-transforms for certain natural generalised characters of these irreducibles and their spectral flows. With this modular data in hand, we verify that the (conjectural) standard Verlinde formula predicts Grothendieck fusion rules with nonnegative-integer multiplicities.

We study traveling waves of the KPP equation in the half-space with Dirichlet boundary conditions. We show that minimal-speed waves are unique up to translation and rotation but faster waves are not. We represent our waves as Laplace transforms of martingales associated to branching Brownian motion in the half-plane with killing on the boundary. We thereby identify the waves’ asymptotic behavior and uncover a novel feature of the minimal-speed wave (Phi ). Far from the boundary, (Phi ) converges to a logarithmic shift of the 1D wave w of the same speed: (displaystyle lim _{y rightarrow infty } Phi big (x + tfrac{1}{sqrt{2}}log y, ybig ) = w(x)).

We work out a generalization of the Szegö limit theorems on the determinant of large matrices. We focus on matrices with nonzero leading principal minors and elements that decay to zero exponentially fast with the distance from the main diagonal, but we relax the constraint of the Toeplitz structure. We obtain an expression for the asymptotic behaviour of the determinant written in terms of the factors of a left and right Wiener–Hopf type factorization of an appropriately defined symbol. For matrices with elements varying slowly along the diagonals (e.g., in locally Toeplitz sequences), we propose to apply the analogue of the semiclassical expansion of the Moyal star product in phase-space quantum mechanics. This is a systematic method that provides approximations up to any order in the typical scale of the inhomogeneity and allows us to obtain explicit asymptotic formulas.

We construct exponentially growing mode solutions to the uncharged and charged Klein–Gordon equations on the (3+1)-dimensional sub-extremal Reissner–Nordström-anti-de-Sitter (AdS) spacetime under reflecting (Dirichlet or Neumann) boundary conditions. Our result applies to a range of Klein–Gordon masses above the so-called Breitenlohner–Freedman bound, notably including the conformal mass case. The mode instability of the Reissner–Nordström-AdS spacetime for some black hole parameters is in sharp contrast to the Schwarzschild-AdS spacetime, where the solution to the Klein–Gordon equation is known to decay in time. Contrary to other mode instability results on the Kerr and Kerr-AdS spacetimes, our exponentially growing mode solutions of the uncharged and weakly charged Klein–Gordon equation exist independently of the occurrence or absence of superradiance. We discover a novel mechanism leading to an exponentially growing mode solution, namely, a near-extremal instability for the Klein–Gordon equation. Our result seems to be the first rigorous mathematical realization of this instability.

We define twisted versions of the classical and quantum double ramification hierarchy construction based on intersection theory of the strata of meromorphic differentials in the moduli space of stable curves and k-twisted double ramification cycles for (k=1), respectively, we prove their integrability and tau symmetry and study their connection. We apply the construction to the case of the trivial cohomological field theory to find it produces the KdV hierarchy, although its relation to the untwisted case is nontrivial. The key role of the KdV hierarchy in controlling the intersection theory of several natural tautological classes translates this relation into a series of remarkable identities between intersection numbers involving psi-classes, Hodge classes, Norbury’s theta class and the strata of meromorphic differentials.