Communications in Mathematical Physics最新文献

Let V be a (C_2)-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-q-traces ((q=e^{2pi itau })) shifted by (-frac{c}{24}) of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized V-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-q-traces were proved by Fiordalisi (Logarithmic intertwining operator and genus-one correlation functions, 2015) and Fiordalisi (Commun Contemp Math 18:1650026, 2016) using the method developed in Huang (Commun Contemp Math 7:649–706, 2005). The method that we use to prove this conjecture is based on the theory of the associative algebras (A^{N}(V)) for (Nin mathbb {N}), their graded modules and their bimodules introduced and studied by the author in Huang (Associative algebras and the representation theory of grading-restricted vertex algebras, 2020) and Huang (Commun Math Phys 396:1–44, 2022). This modular invariance result gives a construction of (C_2)-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.

We introduce a trilinear functional of differential one-forms for a finitely summable regular spectral triple with a noncommutative residue. We demonstrate that for a canonical spectral triple over a closed spin manifold it recovers the torsion of the linear connection. We examine several spectral triples, including Hodge-de Rham, Einstein-Yang-Mills, almost-commutative two-sheeted space, conformally rescaled noncommutative tori, and quantum SU(2) group, showing that the third one has a nonvanishing torsion if nontrivially coupled.

The singular behaviour of quantum fields in Minkowski space can often be bounded by polynomials of the Hamiltonian H. These so-called H-bounds and related techniques allow us to handle pointwise quantum fields and their operator product expansions in a mathematically rigorous way. A drawback of this approach, however, is that the Hamiltonian is a global rather than a local operator and, moreover, it is not defined in generic curved spacetimes. In order to overcome this drawback we investigate the possibility of replacing H by a component of the stress tensor, essentially an energy density, to obtain analogous bounds. For definiteness we consider a massive, minimally coupled free Hermitean scalar field. Using novel results on distributions of positive type we show that in any globally hyperbolic Lorentzian manifold M for any (f,Fin C_0^{infty }(M)) with (Fequiv 1) on (textrm{supp}(f)) and any timelike smooth vector field (t^{mu }) we can find constants (c,C>0) such that (omega (phi (f)^*phi (f))le C(omega (T^{textrm{ren}}_{mu nu }(t^{mu }t^{nu }F^2))+c)) for all (not necessarily quasi-free) Hadamard states (omega ). This is essentially a new type of quantum energy inequality that entails a stress tensor bound on the smeared quantum field. In (1+1) dimensions we also establish a bound on the pointwise quantum field, namely (|omega (phi (x))|le C(omega (T^{textrm{ren}}_{mu nu }(t^{mu }t^{nu }F^2))+c)), where (Fequiv 1) near x.

In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder (Sigma ), which gives rise to the quasi Poisson bracket of G. Massuyeau and V. Turaev on the group algebra (textbf{k}pi _1(Sigma ,p)) of the fundamental group of a surface based at (pin partial Sigma ). This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space (mathcal C_natural ), which is a ({textbf{k}})-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r-matrix formalism. This gives a more conceptual proof of the result of Ovenhouse (Adv Math 373:107309, 2020) that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on (mathcal C_natural ).

We propose a study of multipartite entanglement through persistent homology, a tool used in topological data analysis. In persistent homology, a 1-parameter filtration of simplicial complexes called a persistence complex is used to reveal persistent topological features of the underlying data set. This is achieved via the computation of homological invariants that can be visualized as a persistence barcode encoding all relevant topological information. In this work, we apply this technique to study multipartite quantum systems by interpreting the individual systems as vertices of a simplicial complex. To construct a persistence complex from a given multipartite quantum state, we use a generalization of the bipartite mutual information called the deformed total correlation. Computing the persistence barcodes of this complex yields a visualization or ‘topological fingerprint’ of the multipartite entanglement in the quantum state. The barcodes can also be used to compute a topological summary called the integrated Euler characteristic of a persistence complex. We show that in our case this integrated Euler characteristic is equal to the deformed interaction information, another multipartite version of mutual information. When choosing the linear entropy as the underlying entropy, this deformed interaction information coincides with the n-tangle, a well-known entanglement measure. The persistence barcodes thus provide more fine-grained information about the entanglement structure than its topological summary, the n-tangle, alone, which we illustrate with examples of pairs of states with identical n-tangle but different barcodes. Furthermore, a variant of persistent homology computed relative to a fixed subset yields an interesting connection to strong subadditivity and entropy inequalities. We also comment on a possible generalization of our approach to arbitrary resource theories.

We study dynamical systems generated by skew products:

where integer (bge 2), (gamma in mathbb {C}) are such that (0<|gamma |<1), and (phi ) is a real analytic (mathbb {Z})-periodic function. Let (Delta in [0,1) ) be such that (gamma =|gamma |e^{2pi iDelta }). For the case (Delta notin mathbb {Q}) we prove the following dichotomy for the solenoidal attractor (K^{phi }_{b,,gamma }) for T: Either (K^{phi }_{b,,gamma }) is the graph of a real analytic function, or the Hausdorff dimension of (K^{phi }_{b,,gamma }) is equal to (min {3,1+frac{log b}{log 1/|gamma |}}). Furthermore, given b and (phi ), the former alternative only happens for countably many (gamma ) unless (phi ) is constant.

We investigate the asymptotic properties of the self-intersection numbers for (mathbb {Z})-extensions of chaotic dynamical systems, including the (mathbb {Z})-periodic Lorentz gas and the geodesic flow on a (mathbb {Z})-cover of a negatively curved compact surface. We establish a functional limit theorem.

The pure spinor superfield formalism reveals that, in any dimension and with any amount of supersymmetry, one particular supermultiplet is distinguished from all others. This “canonical supermultiplet” is equipped with an additional structure that is not apparent in any component-field formalism: a (homotopy) commutative algebra structure on the space of fields. The structure is physically relevant in several ways; it is responsible for the interactions in ten-dimensional super Yang–Mills theory, as well as crucial to any first-quantised interpretation. We study the (L_infty ) algebra structure that is Koszul dual to this commutative algebra, both in general and in numerous examples, and prove that it is equivalent to the subalgebra of the Koszul dual to functions on the space of generalised pure spinors in internal degree greater than or equal to three. In many examples, the latter is the positive part of a Borcherds–Kac–Moody superalgebra. Using this result, we can interpret the canonical multiplet as the homotopy fiber of the map from generalised pure spinor space to its derived replacement. This generalises and extends work of Movshev–Schwarz and Gálvez–Gorbounov–Shaikh–Tonks in the same spirit. We also comment on some issues with physical interpretations of the canonical multiplet, which are illustrated by an example related to the complex Cayley plane, and on possible extensions of our construction, which appear relevant in an example with symmetry type (G_2 times A_1).

We study the moduli space of (G_2)-instantons on (projectively) flat bundles over torsion-free (G_2)-orbifolds. We prove that the moduli space is compact and smooth at the irreducible locus after adding small and generic holonomy perturbations. Consequently, we define the (G_2)-Casson invariant that is invariant under (C^0)-deformation of torsion-free (G_2)-structures. We compute this invariant for some orbifolds that arise in Joyce’s construction of compact (G_2)-manifolds.

We study translationally invariant Pauli stabilizer codes with qudits of arbitrary, not necessarily uniform, dimensions. Using homological methods, we define a series of invariants called charge modules. We describe their properties and physical meaning. The most complete results are obtained for codes whose charge modules have Krull dimension zero. This condition is interpreted as mobility of excitations. We show that it is always satisfied for translation invariant 2D codes with unique ground state in infinite volume, which was previously known only in the case of uniform, prime qudit dimension. For codes all of whose excitations are mobile we construct a p-dimensional excitation and a ((D-p-1))-form symmetry for every element of the p-th charge module. Moreover, we define a braiding pairing between charge modules in complementary degrees. We discuss examples which illustrate how charge modules and braiding can be computed in practice.