Communications in Mathematical Physics最新文献

We study the mapping properties of a large class of elliptic operators (P_T) in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length 2T. In the limit where (T rightarrow infty ), we reduce the question of constructing approximate solutions of (P_T u = f) to a finite-dimensional linear system, and provide a geometric interpretation of the obstructions to solving this system. Under some assumptions on the real roots of the model operator (P_0) on the cylinder, we construct a Fredholm inverse for (P_T) with good control on the growth of its norm. As applications of our method, we study the decay rate and density of the low eigenvalues of the Laplacian acting on differential forms, and give improved estimates for compact (G_2)-manifolds constructed by twisted connected sum. We relate our results to the swampland distance conjectures in physics.

In this article, we establish sharp endpoint (L_p) estimates of Schrödinger groups on general measure spaces which may not be equipped with good metrics but admit submarkovian semigroups satisfying purely algebraic assumptions. One of the key ingredients of our proof is to introduce and investigate a new noncommutative high-cancellation BMO space by constructing an abstract form of P-metric codifying some sort of underlying metric and position. This provides the first form of Schrödinger group theory on arbitrary von Neumann algebras and can be applied to many models, including Schrödinger groups associated with non-negative self-adjoint operators satisfying purely Gaussian upper bounds on doubling metric spaces, standard Schrödinger groups on quantum Euclidean spaces, matrix algebras, and group von Neumann algebras with finite dimensional cocycles.

In this paper, we build the Hamiltonian system and the corresponding Lax pairs associated to a twisted connection in (mathfrak {gl}_2(mathbb {C})) admitting an irregular and ramified pole at infinity of arbitrary degree, hence corresponding to the Painlevé 1 hierarchy. We provide explicit formulas for these Lax pairs and Hamiltonians in terms of the irregular times and standard 2g Darboux coordinates associated to the twisted connection. Furthermore, we obtain a map that reduces the space of irregular times to only g non-trivial isomonodromic deformations. In addition, we perform a symplectic change of Darboux coordinates to obtain a set of symmetric Darboux coordinates in which Hamiltonians and Lax pairs are polynomial. Finally, we apply our general theory to the first cases of the hierarchy: the Airy case ((g=0)), the Painlevé 1 case ((g=1)) and the next two elements of the Painlevé 1 hierarchy.

We prove a universality result that reduces the free energy of rank-one matrix estimation problems in the setting of mismatched prior and noise to the computation of the free energy for a modified Sherrington–Kirkpatrick spin glass. Our main result is an almost sure large deviation principle for the overlaps between the true signal and the estimator for both the Bayes-optimal and mismatched settings. Through the large deviations principle, we recover the limit of the free energy in mismatched inference problems and the universality of the overlaps.

We construct a polynomial family of semisimple left module categories over the representation category of the Drinfeld-Jimbo deformation, with the fusion rule of the representation category of each Levi subalgebra. In this construction we perform a kind of generalized parabolic induction using a deformed quantum enveloping algebra, whose definition depends on an arbitrary choice of a positive system and corresponds to De Commer’s definition for the standard positive system. These algebras define a sheaf of algebras on the toric variety associated to the root system, which contains the moduli of equivariant Poisson brackets. This fact finally produces the family of 2-cocycle. We also obtain a comparison theorem between our module categories and module categories induced from our construction for intermediate Levi subalgebras. The construction of deformed quantum enveloping algebras and the comparison theorem are discussed in the integral setting of Lusztig’s sense.

Quiver theories constitute an important class of supersymmetric gauge theories with well-defined holographic duals. Motivated by holographic duality, we use localisation on ({mathbb {S}}^d) to study long linear quivers at large-N. The large-N solution shows a remarkable degree of universality across dimensions, including (d=4) where quivers are genuinely superconformal. In that case we upgrade the solution of long quivers to quivers of any length.

We show that the classic Verlinde numbers on the moduli space ({{textsf{M}}}(r,d)) of rank r and degree d semistable vector bundles over a smooth projective curve can also be regarded as Segre numbers of natural universal complexes over ({{textsf{M}}}(r,d).) This leads to interesting identities among universal integrals on ({{textsf{M}}}(r,d).)

Quantum and classical systems evolving under the same formal Hamiltonian H may exhibit dramatically different behavior after the Ehrenfest timescale (t_E sim log (hbar ^{-1})), even as (hbar rightarrow 0). Coupling the system to a Markovian environment results in a Lindblad equation for the quantum evolution. Its classical counterpart is given by the Fokker–Planck equation on phase space, which describes Hamiltonian flow with friction and diffusive noise. The quantum and classical evolutions may be compared via the Wigner-Weyl representation. Due to decoherence, they are conjectured to match closely for times far beyond the Ehrenfest timescale as (hbar rightarrow 0). We prove a version of this correspondence, bounding the error between the quantum and classical evolutions for any sufficiently regular Hamiltonian H(x, p) and Lindblad functions (L_{k}(x,p)). The error is small when the strength of the diffusion D associated to the Lindblad functions satisfies (D gg hbar ^{4/3}), in particular allowing vanishing noise in the classical limit. Our method uses a time-dependent semiclassical mixture of variably squeezed Gaussian states. The states evolve according to a local harmonic approximation to the Lindblad dynamics constructed from a second-order Taylor expansion of the Lindbladian. Both the exact quantum trajectory and its classical counterpart can be expressed as perturbations of this semiclassical mixture, with the errors bounded using Duhamel’s principle. We present heuristic arguments suggesting the 4/3 exponent is optimal and defines a boundary in the sense that asymptotically weaker diffusion permits a breakdown of quantum-classical correspondence at the Ehrenfest timescale. Our presentation aims to be comprehensive and accessible to both mathematicians and physicists. In a shorter companion paper, we treat the special case of Hamiltonians that decompose into kinetic and potential energy with linear Lindblad operators, with explicit bounds that can be applied directly to physical systems.

We prove local (in space and time) well-posedness for a mildly regularised version of the stochastic quantisation of the (hbox {Yukawa}_{{2}}) Euclidean field theory with a self-interacting boson. Our regularised dynamic is still singular but avoids non-local divergences, allowing us to use a version of the Da Prato–Debussche argument (Da Prato and Debussche in Ann Probab 31(4):1900–1916, 2003. https://doi.org/10.1214/aop/1068646370). This model is a test case for a non-commutative probability framework for formulating the kind of singular SPDEs arising in the stochastic quantisation of field theories mixing both bosons and fermions.