Communications in Mathematical Physics最新文献

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.

Consider the generalized Korteweg–de Vries (gKdV) equations with integer power nonlinearities (qge 2) in dimension (N=1), and the Zakharov–Kuznetsov (ZK) model with integer power nonlinearities (qge 2) in higher dimensions (Nge 2). Among these power-type models, the only conjectured equation with space localized time periodic breathers is the modified KdV (mKdV), corresponding to the case (q=3) and (N=1). Quasimonochromatic solutions were introduced by Mandel (Partial Differ Equ Appl 2:8, 2021) to show that sine-Gordon is the only scalar field model with breather solutions in this class. In this paper we consider smooth generalized quasimonochromatic solutions of arbitrary size for gKdV and ZK models and provide a rigorous proof that mKdV is the unique power-like model among them with spatially localized breathers of this type. In particular, we show the nonexistence of breathers of this class in the ZK models. The method of proof involves the use of the naturally coherent algebra of Bell’s polynomials to obtain particularly distinctive structural elliptic PDEs satisfied by breather-like quasimonochromatic solutions. A reduction of the problem to the classification of solutions of these elliptic PDEs in the entire space is performed, and de Giorgi type uniqueness results are proved in this particular case, concluding the uniqueness of the mKdV breather, and the nonexistence of localized smooth breathers in the ZK case. No assumption on well-posedness is made, and the size of the nonlinearity is arbitrary.

Livine and Bonzom recently proposed a geometric formula for a certain set of complex zeros of the partition function of the Ising model defined on planar graphs Livine and Bonzom (Phys Rev D 111(4):046003, 2025). Remarkably, the zeros depend locally on the geometry of an immersion of the graph in three dimensional Euclidean space (different immersions give rise to different zeros). When restricted to the flat case, the weights become the critical weights on circle patterns Lis (Commun Math Phys 370(2):507–530, 2019).We rigorously prove the formula by geometrically constructing a null eigenvector of the Kac–Ward matrix whose determinant is the squared partition function. The main ingredient of the proof is the realisation that the associated Kac–Ward transition matrix gives rise to an (text {SU}(2)) connection on the graph, creating a direct link with rotations in three dimensions. The existence of a null eigenvector turns out to be equivalent to this connection being flat.

We quantize Hamiltonian structures with hydrodynamic leading terms using the Heisenberg vertex algebra. As an application, we construct the quantum dispersionless KdV hierarchy via a non-associative Weyl quantization procedure and compute the corresponding eigenvalue problem.