Communications in Mathematical Physics最新文献

We extend the cohomological setting developed by Batalin, Fradkin and Vilkovisky (BFV), which produces a resolution of coisotropic reduction in terms of hamiltonian dg manifolds, to the case of nested coisotropic embeddings (Chookrightarrow C_circ hookrightarrow F) inside a symplectic manifold F. To this, we naturally assign (underline{C}) and (underline{C_circ }), as well as the respective BFV dg manifolds. We show that the data of a nested coisotropic embedding defines a natural graded coisotropic embedding inside the BFV dg manifold assigned to (underline{C}), whose reduction can further be resolved using the BFV prescription. We call this construction double BFV resolution, and we use it to prove that “resolution commutes with reduction” for a large class of nested coisotropic embeddings. We then deduce a quantisation of (underline{C}), from the (graded) geometric quantisation of the double BFV Hamiltonian dg manifold (when it exists), following the quantum BFV prescription. As an application, we provide a well defined candidate space of (physical) quantum states of three-dimensional Einstein–Hilbert theory, which is thought of as a partial reduction of the Palatini–Cartan model for gravity.

We investigate the extent to which two particles can be maximally entangled when they are also similarly entangled with other particles on a complete graph, focusing on Werner, isotropic, and Brauer states. To address this, we formulate and solve optimization problems that draw on concepts from many-body physics, computational complexity, and quantum cryptography. We approach the problem by formalizing it as a semi-definite program (SDP), which we solve analytically using tools from representation theory. Notably, we determine the exact maximum values for the projection onto the maximally entangled state and the antisymmetric Werner state, thereby resolving long-standing open problems in the field of quantum extendibility. Our results are achieved by leveraging SDP duality, the representation theory of symmetric, unitary and orthogonal groups, and the Brauer algebra.

This paper concerns the Couette flow for 2-D compressible Navier-Stokes equations (N-S) in an infinitely long flat torus (mathbb {T}times mathbb {R}). Compared to the incompressible flow, the compressible Couette flow has a stronger lift-up effect and weaker dissipation. To the best of our knowledge, there has been no work on the nonlinear stability in the cases of high Reynolds number until now and only linear stability was known in Antonelli et al. (Ann PDE 7(2):24, 2021) and Zeng et al. (SIAM J Math Anal 54(5):5698–5741, 2022). In this paper, we study the nonlinear stability of 2-D compressible Couette flow in Sobolev space at high Reynolds numbers. Moreover, we also show the enhanced dissipation phenomenon and stability threshold for the compressible Couette flow. First, We decompose the perturbation into zero and non-zero modes and obtain two systems for these components, respectively. Different from Antonelli et al. (Ann PDE 7(2):24, 2021) and Zeng et al. (SIAM J Math Anal 54(5):5698–5741, 2022), we use the anti-derivative technique to study the zero-mode system. We introduce a kind of diffusion wave to remove the excessive mass of the zero-modes and construct coupled diffusion waves along characteristics to improve the resulting time decay rates of error terms, and derive a new integrated system (2.25). Secondly, we observe a cancellation with the new system (2.25) so that the lift-up effect is weakened. Thirdly, the large time behavior of the zero-modes is obtained by the weighted energy method and a weighted inequality on the heat kernel (Huang et al. in Arch Ration Mech Anal 197:89–116, 2010). In addition, with the help of the Fourier multipliers method, we can show the enhanced dissipation phenomenon for the non-zero modes by commutator estimates to avoid loss of derivatives. Finally, we complete the higher-order derivative estimates to close the a priori assumptions by the energy method and show the stability threshold.

We construct a topological space (mathcal {B}) consisting of translation invariant injective matrix product states (MPS) of all physical and bond dimensions and show that it has the weak homotopy type (K(mathbb {Z}, 2) times K(mathbb {Z}, 3)). The implication is that the phase of a family of such states parametrized by a space X is completely determined by two invariants: a class in (H^2(X;mathbb {Z})) corresponding to the Chern number per unit cell and a class in (H^3(X;mathbb {Z})), the so-called Kapustin–Spodyneiko (KS) number. The space (mathcal {B}) is defined as the quotient of a contractible space (mathcal {E}) of MPS tensors by an equivalence relation describing gauge transformations of the tensors. We prove that the projection map (p:mathcal {E}rightarrow mathcal {B}) is a quasifibration, and this allows us to determine the weak homotopy type of (mathcal {B}). As an example, we review the Chern number pump—a family of MPS parametrized by (S^3)—and prove that it generates (pi _3(mathcal {B})).

In this paper, we develop a framework for quantum integrable systems on an integrable classical background. We call them hybrid quantum integrable systems (hybrid integrable systems), and we show that they occur naturally in the semiclassical limit of quantum integrable systems. We start with an outline of the concept of hybrid dynamical systems. Then, we give several examples of hybrid integrable systems. The first series of examples is a class of hybrid integrable systems that appear in the semiclassical limit of quantum spin chains. Then, we look at the semiclassical limit of the quantum spin Calogero–Moser–Sutherland (CMS) system. The result is a hybrid integrable system driven by usual classical Calogero–Moser–Sutherland dynamics. This system at the fixed point of the multi-time classical dynamics CMS system gives the commuting spin Hamiltonians of Haldane–Shastry model.

For a basic classical Lie superalgebra (mathfrak {s}), let (mathfrak {g}) be the central extension of the Takiff superalgebra (mathfrak {s}otimes Lambda (theta )), where (theta ) is an odd indeterminate. We study the category of (mathfrak {g})-Whittaker modules associated with a nilcharacter (chi ) of (mathfrak {g}) and show that it is equivalent to the category of (mathfrak {s})-Whittaker modules associated with a nilcharacter of (mathfrak {s}) determined by (chi ). In the case when (chi ) is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite W-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite W-superalgebra associated to (mathfrak {s}). Here, a supersymmetric finite W-algebra is conjecturally the Zhu algebra of a supersymmetric affine W-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric W-algebra.

Zeta generators are derivations associated with odd Riemann zeta values that act freely on the Lie algebra of the fundamental group of Riemann surfaces with marked points. The genus-zero incarnation of zeta generators are Ihara derivations of certain Lie polynomials in two generators that can be obtained from the Drinfeld associator. We characterize a canonical choice of these polynomials, together with their non-Lie counterparts at even degrees (wge 2), through the action of the dual space of formal and motivic multizeta values. Based on these canonical polynomials, we propose a canonical isomorphism that maps motivic multizeta values into the f-alphabet. The canonical Lie polynomials from the genus-zero setup determine canonical zeta generators in genus one that act on the two generators of Enriquez’ elliptic associators. Up to a single contribution at fixed degree, the zeta generators in genus one are systematically expanded in terms of Tsunogai’s geometric derivations dual to holomorphic Eisenstein series, leading to a wealth of explicit high-order computations. Earlier ambiguities in defining the non-geometric part of genus-one zeta generators are resolved by imposing a new representation-theoretic condition. The tight interplay between zeta generators in genus zero and genus one unravelled in this work connects the construction of single-valued multiple polylogarithms on the sphere with iterated-Eisenstein-integral representations of modular graph forms.

We prove global well-posedness and scattering for the massive Dirac-Klein-Gordon system with small and low regularity initial data in dimension two; a non-resonance condition on the masses is imposed. To achieve this we introduce new resolutions spaces which act as an effective replacement of the normal form transformation.

We develop a general theory of 3-dimensional “orbifold completion”, to describe (generalised) orbifolds of topological quantum field theories as well as all their defects. Given a semistrict 3-category (mathcal {T}) with adjoints for all 1- and 2-morphisms (more precisely, a Gray category with duals), we construct the 3-category ({mathcal {T}}_{text {orb}}) as a Morita category of certain (E_1)-algebras in (mathcal {T}) which encode triangulation invariance. We prove that in ({mathcal {T}}_{text {orb}}) again all 1- and 2-morphisms have adjoints, that it contains (mathcal {T}) as a full subcategory, and we argue, but do not prove, that it satisfies a universal property which implies ({({mathcal {T}}_{text {orb}})}_{text {orb}} cong {mathcal {T}}_{text {orb}}). This is a categorification of the work in Carquevill and Runkel (Quantum Topol 7(2):203–279, 2016). Orbifold completion by design allows us to lift the orbifold construction from closed TQFT to the much richer world of defect TQFTs. We illustrate this by constructing a universal 3-dimensional state sum model with all defects from first principles, and we explain how recent work on defects between Witt equivalent Reshetikhin–Turaev theories naturally appears as a special case of orbifold completion.

We introduce a formalism for constructing cohomological field theories (CohFTs), inspired by Witten’s paradigm of “symmetries, fields, and equations”. We apply this formalism to various first-order nonlinear PDEs and show that the resulting CohFTs agree with those previously proposed by physicists. In particular, applying it to the generalized Seiberg–Witten equations provides a unified perspective on the supersymmetric action functionals of the Donaldson–Witten, Seiberg–Witten, and Kapustin–Witten theories.