Communications in Mathematical Physics最新文献

We introduce and study a new family of commuting Baxter operators in the Ruijsenaars hyperbolic system, different from that considered by us earlier. Using a degeneration of Rains integral identity we verify the commutativity between the two families of Baxter operators and explore this fact for the proof of the coupling constant symmetry of the wave function. We also establish a connection between new Baxter operators and Noumi–Sano difference operators.

We prove that all Galerkin truncations of the 2d stochastic Navier–Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies (Nge 392). By “chaotic” we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander’s condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using (a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and (b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.

We study the dynamics of a collisionless kinetic gas whose particles follow future-directed timelike and spatially bound geodesics in the exterior of a sub-extremal Kerr black hole spacetime. Based on the use of generalized action-angle variables, we analyze the large time asymptotic behavior of macroscopic observables associated with the gas. We show that, as long as the fundamental frequencies of the system satisfy a suitable non-degeneracy condition, these macroscopic observables converge in time to the corresponding observables determined from an averaged distribution function. In particular, this implies that the final state is characterized by a distribution function which is invariant with respect to the full symmetry group of the system, that is, it is stationary, axisymmetric and Poisson-commutes with the integral of motion associated with the Carter constant. As a corollary of our result, we demonstrate the validity of the strong Jeans theorem in our setting, stating that the distribution function belonging to a stationary state must be a function which is independent of the generalized angle variables. An analogous theorem in which the assumption of stationarity is replaced with the requirement of invariance with respect to the Carter flow is also proven. Finally, we prove that the aforementioned non-degeneracy condition holds if the black hole is rotating. This is achieved by providing suitable asymptotic expansions for the energy and Carter constant in terms of action variables for orbits having sufficiently large radii, and by exploiting the analytic dependency of the fundamental frequencies on the integrals of motion.

Using computations in the bidual of ({mathbb {B}}(L^2M)) we develop a new technique at the von Neumann algebra level to upgrade relative proper proximality to full proper proximality. This is used to structurally classify subalgebras of (LGamma ) where (Gamma ) is an infinite group that is biexact relative to a finite family of subgroups ({Lambda _i}_{iin I}) such that each (Lambda _i) is almost malnormal in (Gamma ). This generalizes the result of Ding et al. (Properly proximal von Neumann algebras, 2022. arXiv:2204.00517) which classifies subalgebras of von Neumann algebras of biexact groups. By developing a combination with techniques from Popa’s deformation-rigidity theory we obtain a new structural absorption theorem for free products and a generalized Kurosh type theorem in the setting of properly proximal von Neumann algebras.

We introduce a category of q-oscillator representations over the quantum affine superalgebras of type D and construct a new family of its irreducible representations. Motivated by the theory of super duality, we show that these irreducible representations naturally interpolate the irreducible q-oscillator representations of type (X_n^{(1)}) and the finite-dimensional irreducible representations of type (Y_n^{(1)}) for ((X,Y)=(C,D),(D,C)) under exact monoidal functors. This can be viewed as a quantum (untwisted) affine analogue of the correspondence between irreducible oscillator and irreducible finite-dimensional representations of classical Lie algebras arising from Howe’s reductive dual pairs ((mathfrak {g},G)), where (mathfrak {g}=mathfrak {sp}_{2n}, mathfrak {so}_{2n}) and (G=O_ell , Sp_{2ell }).

We construct a series of examples of Calabi-Yau manifolds in an arbitrary dimension and compute the main invariants. In particular, we give higher dimensional generalization of Borcea-Voisin Calabi-Yau threefolds. We give a method to compute a local zeta function using the Frobenius morphism for orbifold cohomology introduced by Rose. We compute Hodge numbers of the constructed examples using orbifold Chen-Ruan cohomology.

We derive the Weil–Petersson measure on the moduli space of hyperbolic surfaces with defects of arbitrary opening angles and use this to compute its volume. We conjecture a matrix integral computing the corresponding volumes and confirm agreement in simple cases. We combine this mathematical result with the equivariant localization approach to Jackiw–Teitelboim gravity to justify a proposed exact solution of pure 2d dilaton gravity for a large class of dilaton potentials.

Associated to any finite graph (Lambda ) is a closed surface ({textbf{S}}={textbf{S}}_Lambda ), the boundary of a regular neighbourhood of an embedding of (Lambda ) in any three manifold. The surface retracts to the graph, mapping loops on the surface to loops on the graph. The (SU(2)) character variety ({{mathcal {M}}}) of ({textbf{S}}) has a symplectic structure and associated Liouville measure; on the other hand, the character variety ({textbf{M}}) of (Lambda ) carries a natural measure inherited from the Haar measure. Loops on ({textbf{S}}) define functions on the character varieties, the Wilson loops. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over ({{mathcal {M}}}). We develop a calculus for calculating correlations of Wilson loops on ({{mathcal {M}}}) w.r.to the normalised Liouville measure, and present evidence that they approximate—for large graphs—the corresponding integrals over ({textbf{M}}). Lattice field theory involves integrals over ({textbf{M}}); we present “symplectic” analogues of expressions for partition functions, Wilson loop expectations, etc., in two and three space-time dimensions.

We propose a general method to produce orthogonal polynomial dualities from the (^*)-bialgebra structure of Drinfeld–Jimbo quantum groups. The (^*)-structure allows for the construction of certain unitary symmetries, which imply the orthogonality of the duality functions. In the case of the quantum group (mathcal {U}_q(mathfrak {gl}_{n+1})), the result is a nested multivariate q-Krawtchouk duality for the n-species ASEP((q,varvec{theta }) ). The method also applies to other quantized simple Lie algebras and to stochastic vertex models. As a probabilistic application of the duality relation found, we provide the explicit formula of the q-shifted factorial moments (namely the q-analogue of the Pochhammer symbol) for the two-species q-TAZRP (totally asymmetric zero range process).

We study a class (widehat{{mathfrak {F}}}) of one-dimensional full branch maps introduced in Coates et al. (Commun Math Phys 402(2):1845–1878, 2023), admitting two indifferent fixed points as well as critical points and/or singularities with unbounded derivative. We show that (widehat{{mathfrak {F}}}) can be partitioned into 3 pairwise disjoint subfamilies (widehat{{mathfrak {F}}} = {mathfrak {F}} cup {mathfrak {F}}_pm cup {mathfrak {F}}_*) such that all (g in {mathfrak {F}}) have a unique physical measure equivalent to Lebesgue, all (g in {mathfrak {F}}_{pm }) have a physical measure which is a Dirac-(delta ) measure on one of the (repelling) fixed points, and all (g in {mathfrak {F}}_{*}) are non-statistical and in particular have no physical measure. Moreover we show that these subfamilies are intermingled: they can all be approximated by maps in the other subfamilies in natural topologies.