Communications in Mathematical Physics最新文献

The holomorphic twist provides a powerful framework to study minimally protected sectors in supersymmetric quantum field theories. We investigate the algebraic structure underlying the holomorphic twist of (mathcal {N}=1) superconformal field theories in four dimensions. In particular, in holomorphically twisted theories the flavour and conformal symmetry algebras are enhanced to infinite-dimensional higher Kac Moody and higher Virasoro symmetry algebras respectively. We explicitly compute the binary and ternary (lambda )-brackets and clarify their relation with the underlying infinite-dimensional symmetry algebra. Doing so we show that the central extensions of said symmetry algebras precisely encode the conformal anomalies a and c as well as the flavour central charges of the physical four-dimensional theory. This parallels the familiar story in two dimensions where the conformal anomaly c is encoded in the central extension of the Virasoro algebra.

A discrete weak KAM solution is a potential function that highlights the ground state configurations at zero temperature of an infinite chain of atoms interacting with a periodic or quasi-periodic substrate. It is well known that weak KAM solutions exist for periodic substrates as in the Frenkel–Kontorova model. Weak solutions may not exist in the almost periodic setting as in the theory of stationary ergodic Hamilton–Jacobi equations (where they are called correctors). For linearly repetitive quasi-periodic substrates, we show that equivariant interactions that fulfill a twist condition and a non-degenerate property always admit sublinear weak KAM solutions. We moreover classify all possible types of weak KAM solutions and calibrated configurations according to an intrinsic prefered order. The notion of prefered order is new even in the classical periodic case.

How can detector click probabilities respond to spatial rotations around a fixed axis, in any possible physical theory? Here, we give a thorough mathematical analysis of this question in terms of “rotation boxes”, which are analogous to the well-known notion of non-local boxes. We prove that quantum theory admits the most general rotational correlations for spins 0, 1/2, and 1, but we describe a metrological game where beyond-quantum resources of spin 3/2 outperform all quantum resources of the same spin. We prove a multitude of fundamental results about these correlations, including an exact convex characterization of the spin-1 correlations, a Tsirelson-type inequality for spins 3/2 and higher, and a proof that the general spin-J correlations provide an efficient outer SDP approximation to the quantum set. Furthermore, we review and consolidate earlier results that hint at a wealth of applications of this formalism: a theory-agnostic semi-device-independent randomness generator, an exact characterization of the quantum (2, 2, 2)-Bell correlations in terms of local symmetries, and the derivation of multipartite Bell witnesses. Our results illuminate the foundational question of how space constrains the structure of quantum theory, they build a bridge between semi-device-independent quantum information and spacetime physics, and they demonstrate interesting relations to topics such as entanglement witnesses, spectrahedra, and orbitopes.

Much of the analysis of F-theory-based Standard Models boils down to computing cohomologies of line bundles on matter curves. By varying parameters one can degenerate such matter curves to singular ones, typically with many nodes, where the computation is combinatorial and straightforward. The question remains to relate the (a priori possibly smaller) value on the original curve to the singular one. In this work, we introduce some elementary techniques (pruning trees and removing interior edges) for simplifying the resulting nodal curves to a small collection of terminal ones that can be handled directly. When applied to the QSMs, these techniques yield optimal results in the sense that obtaining more precise answers would require currently unavailable information about the QSM geometries. This provides us with an opportunity to enhance the statistical bounds established in earlier research regarding the absence of vector-like exotics on the quark-doublet curve.

Conformal blocks of the Virasoro algebra have a Coulomb-gas representation as Dotsenko-Fateev integrals over the positions of screening charges. In q-deformed Virasoro, the conformal blocks on a sphere with an arbitrary number of punctures are manifestly the same, when written in Dotsenko-Fateev representation, as the partition functions of a class of 3d U(N) gauge theories with ({{mathcal {N}}}=2) supersymmetry, in the (Omega )-background. Coupling the 3d gauge theory to a flavor in fundamental representation corresponds to inserting a Virasoro vertex operator; the two real mass parameters determine the momentum and position of the puncture. The Dotsenko-Fateev integrals can be computed by residues. The result is the instanton sum of a five dimensional ({{mathcal {N}}}=1) gauge theory. The positions of the poles are labeled by tuples of partitions, the residues of the integrand are the Nekrasov summands.

We study the dynamics of the collision of two solitary waves for the 2 and 3-dimensional Zakharov–Kuznetsov equation, a high-dimensional non-integrable version of the Korteweg-de Vries equation that appears as an asymptotic model in plasma physics. We describe the evolution of the solution behaving as a sum of 2-solitary waves of nearly equal speeds at time (t=-infty ) up to time (t=+infty ). We show that this solution behaves as the sum of two modulated solitary waves and an error term which is small in (H^1) for all time (t in {mathbb {R}}). Finally, we also prove the stability of this solution for large times around the collision. The proofs are a non-trivial extension of the ones of Martel and Merle for the quartic generalized Korteweg-de Vries equation to higher dimensions. First, despite the non-explicit nature of the solitary wave, we construct an approximate solution in an intrinsic way by canceling the error to the equation only in the natural directions of scaling and translation. Then, to control the difference between a solution and the approximate solution, we use a modified energy functional and a refined modulation estimate in the transverse variable. Moreover, we rely on the Hamiltonian structure of the ODE governing the distance between the waves, which cannot be approximated by explicit solutions, to close the bootstrap estimates on the parameters. We hope that the techniques introduced here are robust and will prove useful in studying the collision phenomena for other focusing non-linear dispersive equations with non-explicit solitary waves.

We investigate the massive sine-Gordon model in the finite ultraviolet regime on the two-dimensional Minkowski spacetime (({mathbb {R}}^2,eta )) with an additive Gaussian white noise. In particular we construct the expectation value and the correlation functions of a solution of the underlying stochastic partial differential equation (SPDE) as a power series in the coupling constant, proving ultimately uniform convergence. This result is obtained combining an approach first devised in Dappiaggi et al. (Commun Contemp Math 24(07):2150075, 2022. arXiv:2009.07640 [math-ph]) to study SPDEs at a perturbative level with the one discussed in Bahns and Rejzner (Commun Math Phys 357(1):421, 2018. arXiv:1609.08530 [math-ph]) to construct the quantum sine-Gordon model using techniques proper of the perturbative, algebraic approach to quantum field theory (pAQFT). At a formal level the relevant expectation values are realized as the evaluation of suitably constructed functionals over (C^infty ({mathbb {R}}^2)). In turn, these are elements of a distinguished algebra whose product is a deformation of the pointwise one, by means of a kernel which is a linear combination of two components. The first encompasses the information of the Feynmann propagator built out of an underlying Hadamard, quantum state, while the second encodes the correlation codified by the Gaussian white noise. In our analysis, first of all we extend the results obtained in Bahns et al. (J Math Anal Appl 526:127249, 2023. arXiv:2103.09328 [math-ph]) and Bahns and Rejzner (Commun Math Phys 357(1):421, 2018. arXiv:1609.08530 [math-ph]) proving the existence of a convergent modified version of the S-matrix and of an interacting field as elements of the underlying algebra of functionals. Subsequently we show that it is possible to remove the contribution due to the Feynmann propagator by taking a suitable (hbar rightarrow 0^+)-limit, hence obtaining the sought expectation value of the solution and of the correlation functions of the SPDE associated to the stochastic sine-Gordon model.

We consider a system of random autonomous ODEs introduced by Cugliandolo et al. (Phys Rev Lett 78: 350–353, 1997), which serves as a non-relaxational analog of the gradient flow for the spherical p-spin model. The asymptotics for the expected number of equilibria in this model was recently computed by Fyodorov (J Stat Mech Theory Exp 12: 124003–21, 2016) in the high-dimensional limit, followed a similar computation for the expected number of stable equilibria by Garcia (Garcia: On the number of equilibria with a given number of unstable directions. arXiv:1709.04021, 2017). We show that for (p>9), the number of equilibria, as well as the number of stable equilibria, concentrate around their respective averages, generalizing recent results of Subag and Zeitouni (Ann Probab 45: 3385–3450, 2017) and (J Math Phys 62: 123301–15, 2021) in the relaxational case. In particular, we confirm that this model undergoes a transition from relative to absolute instability, in the sense of Ben Arous, Fyodorov, and Khoruzhenko (Proc Natl Acad Sci U.S.A. 118: 2023719118–8 2021).

We study the SK model at inverse temperature (beta >0) and strictly positive field (h>0) in the region of ((beta ,h)) where the replica-symmetric formula is valid. An integral representation of the partition function derived from the Hubbard-Stratonovitch transformation combined with a duality formula is used to prove that the infinite volume free energy of the SK model can be expressed as a variational formula on the space of magnetisations, m. The resulting free energy functional differs from that of Thouless, Anderson and Palmer (TAP) by the term ( -frac{beta ^2}{4}left( q-q_{text {EA}}(m)right) ^2 ) where (q_{text {EA}}(m)) is the Edwards-Anderson parameter and q is the minimiser of the replica-symmetric formula. Thus, both functionals have the same critical points and take the same value on the subspace of magnetisations satisfying (q_{text {EA}}(m)=q). This result is based on an in-depth study of the global maximum of this near-TAP free energy functional using Bolthausen’s solutions of the TAP equations, Bandeira & van Handel’s bounds on the spectral norm of non-homogeneous Wigner-type random matrices, and Gaussian comparison techniques. It holds for ((beta ,h)) in a large subregion of the de Almeida and Thouless high-temperature stability region.

We consider the Boltzmann equation with the soft potential and angular cutoff. Inspired by the methods from dispersive PDEs, we establish its sharp local well-posedness and ill-posedness in (H^{s}) Sobolev space. We find the well/ill-posedness separation at regularity (s=frac{d-1}{2}), strictly (frac{1}{2})-derivative higher than the scaling-invariant index (s=frac{d-2}{2}), the usually expected separation point.