Tensor functors between Morita duals of fusion categories

Given a fusion category \({\mathcal C}\) and an indecomposable \({\mathcal C}\)-module category \({\mathcal M}\), the fusion category \({\mathcal C}^*_{_{{\mathcal M}}}\) of \({\mathcal C}\)-module endofunctors of \({\mathcal M}\) is called the (Morita) dual fusion category of \({\mathcal C}\) with respect to \({\mathcal M}\). We describe tensor functors between two arbitrary duals \({\mathcal C}^*_{_{{\mathcal M}}}\) and \({\mathcal D}^*_{\mathcal N}\) in terms of data associated to \({\mathcal C}\) and \({\mathcal D}\). We apply the results to G-equivariantizations of fusion categories and group-theoretical fusion categories. We describe the orbits of the action of the Brauer–Picard group on the set of module categories and we propose a categorification of the Rosenberg–Zelinsky sequence for fusion categories.     

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra \({\mathcal{D}}\) of regular differential operators on the circle has a universal central extension \({\hat{\mathcal{D}}}\). The invariant subalgebra \({\hat{\mathcal{D}}^+}\) under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum \({\hat{\mathcal{D}}^+}\)-module with central charge \({c \in \mathbb{C}}\), and its irreducible quotient \({\mathcal{V}_c}\), possess vertex algebra structures, and \({\mathcal{V}_c}\) has a nontrivial structure if and only if \({c \in \frac{1}{2}\mathbb{Z}}\). We show that for each integer \({n > 0}\), \({\mathcal{V}_{n/2}}\) and \({\mathcal{V}_{-n}}\) are \({\mathcal{W}}\)-algebras of types \({\mathcal{W}(2, 4,\dots,2n)}\) and \({\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}\), respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group \({{\rm O}(n)}\) and the symplectic group \({{\rm Sp}(2n)}\), respectively. Based on Sergeev’s theorems on the invariant theory of \({{\rm Osp}(1, 2n)}\) we conjecture that \({\mathcal{V}_{-n+1/2}}\) is of type \({\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}\), and we prove this for \({n = 1}\). As an application, we show that invariant subalgebras of \({\beta\gamma}\)-systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.     

Noncommutative products of Euclidean spaces

We present natural families of coordinate algebras on noncommutative products of Euclidean spaces \({\mathbb {R}}^{N_1} \times _{\mathcal {R}} {\mathbb {R}}^{N_2}\). These coordinate algebras are quadratic ones associated with an \(\mathcal {R}\)-matrix which is involutive and satisfies the Yang–Baxter equations. As a consequence, they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have eight-dimensional noncommutative euclidean spaces \({\mathbb {R}}^{4} \times _{\mathcal {R}} {\mathbb {R}}^{4}\). Among these, particularly well behaved ones have deformation parameter \(\mathbf{u} \in {\mathbb {S}}^2\). Quotients include seven spheres \({\mathbb {S}}^{7}_\mathbf{u}\) as well as noncommutative quaternionic tori \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u} = {\mathbb {S}}^3 \times _\mathbf{u} {\mathbb {S}}^3\). There is invariance for an action of \({{\mathrm{SU}}}(2) \times {{\mathrm{SU}}}(2)\) on the torus \({\mathbb {T}}^{{\mathbb {H}}}_\mathbf{u}\) in parallel with the action of \(\mathrm{U}(1) \times \mathrm{U}(1)\) on a ‘complex’ noncommutative torus \({\mathbb {T}}^2_\theta \) which allows one to construct quaternionic toric noncommutative manifolds. Additional classes of solutions are disjoint from the classical case.     

The Exoticness and Realisability of Twisted Haagerup–Izumi Modular Data

The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data \({\mathcal{D}{\rm Hg}}\) fits into a family \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\) , where n ≥  0 and \({\omega\in \mathbb{Z}_{2n+1}}\) . We show \({\mathcal{D}^0 {\rm Hg}_{2n+1}}\) is related to the subfactors Izumi hypothetically associates to the cyclic groups \({\mathbb{Z}_{2n+1}}\) . Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions, etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of Izumi type \({\mathbb{Z}_7, \mathbb{Z}_9}\) and \({\mathbb{Z}_3^2}\) , and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type \({\mathbb{Z}_{11},\mathbb{Z}_{13},\mathbb{Z}_{15},\mathbb{Z}_{17},\mathbb{Z}_{19}}\) (previously, Izumi had shown uniqueness for \({\mathbb{Z}_3}\) and \({\mathbb{Z}_5}\)), and we identify their modular data. We explain how \({\mathcal{D}{\rm Hg}}\) (more generally \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\)) is a graft of the quantum double \({\mathcal{D} Sym(3)}\) (resp. the twisted double \({\mathcal{D}^\omega D_{2n+1}}\)) by affine so(13) (resp. so\({(4n^2+4n+5)}\)) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data \({\mathcal{D}^\omega {\rm Hg}_{2n+1}}\) . For example we show there are exactly 2 possible character vectors (giving graded dimensions of all modules) for the Haagerup VOA at central charge c = 8. It seems unlikely that any of this twisted Haagerup-Izumi modular data can be regarded as exotic, in any reasonable sense.     

On the Convexity of the KdV Hamiltonian

Motivated by perturbation theory, we prove that the nonlinear part \({H^{*}}\) of the KdV Hamiltonian \({H^{kdv}}\), when expressed in action variables \({I = (I_{n})_{n \geqslant 1}}\), extends to a real analytic function on the positive quadrant \({\ell^{2}_{+}(\mathbb{N})}\) of \({\ell^{2}(\mathbb{N})}\) and is strictly concave near \({0}\). As a consequence, the differential of \({H^{*}}\) defines a local diffeomorphism near 0 of \({\ell_{\mathbb{C}}^{2}(\mathbb{N})}\). Furthermore, we prove that the Fourier-Lebesgue spaces \({\mathcal{F}\mathcal{L}^{s,p}}\) with \({-1/2 \leqslant s \leqslant 0}\) and \({2 \leqslant p < \infty}\), admit global KdV-Birkhoff coordinates. In particular, it means that \({\ell^{2}_+(\mathbb{N})}\) is the space of action variables of the underlying phase space \({\mathcal{F}\mathcal{L}^{-1/2,4}}\) and that the KdV equation is globally in time \({C^{0}}\)-well-posed on \({\mathcal{F}\mathcal{L}^{-1/2,4}}\).     

Random Repeated Interaction Quantum Systems

We consider a quantum system \({\mathcal{S}}\) interacting sequentially with independent systems \({\mathcal{E}_m}\) , m = 1,2,... Before interacting, each \({\mathcal{E}_m}\) is in a possibly random state, and each interaction is characterized by an interaction time and an interaction operator, both possibly random. We prove that any initial state converges to an asymptotic state almost surely in the ergodic mean, provided the couplings satisfy a mild effectiveness condition. We analyze the macroscopic properties of the asymptotic state and show that it satisfies a second law of thermodynamics. We solve exactly a model in which \({\mathcal{S}}\) and all the \({\mathcal{E}_m}\) are spins: we find the exact asymptotic state, in case the interaction time, the temperature, and the excitation energies of the \({\mathcal{E}_m}\) vary randomly. We analyze a model in which \({\mathcal{S}}\) is a spin and the \({\mathcal{E}_m}\) are thermal fermion baths and obtain the asymptotic state by rigorous perturbation theory, for random interaction times varying slightly around a fixed mean, and for small values of a coupling constant.     

Universal Probability Distribution for the Wave Function of a Quantum System Entangled with its Environment

A quantum system (with Hilbert space \({\mathcal {H}_{1}}\)) entangled with its environment (with Hilbert space \({\mathcal {H}_{2}}\)) is usually not attributed to a wave function but only to a reduced density matrix \({\rho_{1}}\). Nevertheless, there is a precise way of attributing to it a random wave function \({\psi_{1}}\), called its conditional wave function, whose probability distribution \({\mu_{1}}\) depends on the entangled wave function \({\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}\) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of \({\mathcal {H}_{2}}\) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about \({\mu_{1}}\), e.g., that if the environment is sufficiently large then for every orthonormal basis of \({\mathcal {H}_{2}}\), most entangled states \({\psi}\) with given reduced density matrix \({\rho_{1}}\) are such that \({\mu_{1}}\) is close to one of the so-called GAP (Gaussian adjusted projected) measures, \({GAP(\rho_{1})}\). We also show that, for most entangled states \({\psi}\) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval \({[E, E+ \delta E]}\)) and most orthonormal bases of \({\mathcal {H}_{2}}\), \({\mu_{1}}\) is close to \({GAP(\rm {tr}_{2} \rho_{mc})}\) with \({\rho_{mc}}\) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then \({\mu_{1}}\) is close to \({GAP(\rho_\beta)}\) with \({\rho_\beta}\) the canonical density matrix on \({\mathcal {H}_{1}}\) at inverse temperature \({\beta=\beta(E)}\). This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

For a Hopf algebra B, we endow the Heisenberg double \({\mathcal{H}(B^*)}\) with the structure of a module algebra over the Drinfeld double \({\mathcal{D}(B)}\). Based on this property, we propose that \({\mathcal{H}(B^*)}\) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan–Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) quantum group that is Kazhdan–Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair \({(\mathcal{D}(B),\mathcal{H}(B^*))}\) is “truncated” to \({(\overline{\mathcal{U}}_{\mathfrak{q}} s\ell2,\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2))}\), where \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)}\) is a \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) module algebra that turns out to have the form \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)=\mathbb{C}_{\mathfrak{q}}[z,\partial]\otimes\mathbb{C}[\lambda]/(\lambda^{2p}-1)}\), where \({\mathbb{C}_{\mathfrak{q}}[z,\partial]}\) is the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\)-module algebra with the relations z ^p  = 0, ? ^p  = 0, and \({\partial z = \mathfrak{q}-\mathfrak{q}^{-1} + \mathfrak{q}^{-2} z\partial}\).     

On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

Let K be a field of characteristic zero. For \({n \in \mathbb{N}^{*}}\) , let \({\mathcal{T}^{\,\prime}_{n}}\) be the vector space of non-planar rooted trees with n vertices (Foissy in Bull Sci Math 126, no. 3, 193–239; no. 4, 249–288, 2002). Let \({\vartriangleright}\) be the left pre-Lie product of insertion of a tree inside another defined on infinitesimal characters of the graded Hopf algebra \({\mathcal{H}}\) introduced by Calaque, Ebrahimi-Fard and Manchon. Let \({\mathcal{T}^{\,\prime}=\oplus_{n\geq 2}\mathcal{T}^{\,\prime}_{n}}\) . In this work, we first prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) a pre-Lie algebra generated by the two ladders E ₁ and E ₂ where E ₁ is the ladder with one edge and E ₂ is the ladder with two edges. Second, we prove that \({(\mathcal{T}^{\,\prime}, \vartriangleright)}\) is not a free pre-Lie algebra, and we exhibit a family of relations.     

SU(3) symmetry breaking in charmed baryon decays

We explore the breaking effects of the SU(3) flavor symmetry in the singly Cabibbo-suppressed anti-triplet charmed baryon decays of \(\mathbf{B}_c\rightarrow \mathbf{B}_n M\), with \(\mathbf{B}_c=(\Xi _c^0,\Xi _c^+,\Lambda _c^+)\) and \(\mathbf{B}_n(M)\) the baryon (pseudo-scalar) octets. We find that these breaking effects can be used to account for the experimental data on the decay branching ratios of \({\mathcal {B}}(\Lambda _c^+\rightarrow \Sigma ^{0} K^{+},\Lambda ^{0} K^{+})\) and \(R'_{K/\pi }={\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- K^+)\)/\({\mathcal {B}}(\Xi ^0_c \rightarrow \Xi ^- \pi ^+)\). In addition, we obtain that \({\mathcal {B}}(\Xi _{c}^{0} \rightarrow \Xi ^{-} K^{+},\Sigma ^{-} \pi ^{+})=(4.6 \pm 1.7,12.8 \pm 3.1)\times 10^{-4}\), \({\mathcal {B}}(\Xi _c^0\rightarrow pK^-,\Sigma ^+\pi ^-)=(3.0 \pm 1.0, 5.2 \pm 1.6)\times 10^{-4}\) and \({\mathcal {B}}(\Xi _c^+\rightarrow \Sigma ^{0(+)} \pi ^{+(0)})=(10.3 \pm 1.7)\times 10^{-4}\), which all receive significant contributions from the breaking effects, and can be tested by the BESIII and LHCb experiments.     

Eigenvalue Order Statistics for Random Schrödinger Operators with Doubly-Exponential Tails

We consider random Schrödinger operators of the form \({\Delta+\xi}\), where \({\Delta}\) is the lattice Laplacian on \({\mathbb{Z}^{d}}\) and \({\xi}\) is an i.i.d. random field, and study the extreme order statistics of the Dirichlet eigenvalues for this operator restricted to large but finite subsets of \({\mathbb{Z}^{d}}\). We show that, for \({\xi}\) with a doubly-exponential type of upper tail, the upper extreme order statistics of the eigenvalues falls into the Gumbel max-order class, and the corresponding eigenfunctions are exponentially localized in regions where \({\xi}\) takes large, and properly arranged, values. The picture we prove is thus closely connected with the phenomenon of Anderson localization at the spectral edge. Notwithstanding, our approach is largely independent of existing methods for proofs of Anderson localization and it is based on studying individual eigenvalue/eigenfunction pairs and characterizing the regions where the leading eigenfunctions put most of their mass.     

A Note on Expansiveness and Hyperbolicity for Generic Geodesic Flows

In this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥?2. We prove that there exists a C²-residual subset \(\mathscr{R}\) of metrics on a given compact Riemannian manifold such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if the closure of the set of periodic orbits of \({\varphi ^{t}_{g}}\) is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C²-residual \(\mathscr{R}\) such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}\) is expansive if and only if \({\varphi ^{t}_{g}}\) is an Anosov flow.     

Further studies on the exclusive productions of $$J/\psi +\chi _{cJ}$$ ($$J=0,1,2$$J=0,1,2) via $$e^+e^-$$e+e- annihilation at the B factories

By including the interference effect between the QCD and the QED diagrams, we carry out a complete analysis on the exclusive productions of \(e^+e^- \rightarrow J/\psi +\chi _{cJ}\) (\(J=0,1,2\)) at the B factories with \(\sqrt{s}=10.6\) GeV at the next-to-leading-order (NLO) level in \(\alpha _s\), within the nonrelativistic QCD framework. It is found that the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms that represent the tree-level interference are comparable with the usual NLO QCD corrections, especially for the \(\chi _{c1}\) and \(\chi _{c2}\) cases. To explore the effect of the higher-order terms, namely \({\mathcal {O}} (\alpha ^3\alpha _s^2)\), we perform the QCD corrections to these \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order terms for the first time, which are found to be able to significantly influence the \({\mathcal {O}} (\alpha ^3\alpha _s)\)-order results. In particular, in the case of \(\chi _{c1}\) and \(\chi _{c2}\), the newly calculated \({\mathcal {O}} (\alpha ^3\alpha _s^2)\)-order terms can to a large extent counteract the \({\mathcal {O}} (\alpha ^3\alpha _s)\) contributions, evidently indicating the indispensability of the corrections. In addition, we find that, as the collision energy rises, the percentage of the interference effect in the total cross section will increase rapidly, especially for the \(\chi _{c1}\) case.     

Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential \({V \in C^{\infty} (\mathbb{S}^{d})}\) to the Laplacian on the sphere results in the existence of geodesics \({\gamma}\) such that the uniform measure supported on \({\gamma}\) cannot be obtained as a weak-\({\star}\) accumulation point of the densities \({(|\psi_{n}|^{2} {vol}_{\mathbb{S}^d})}\) for any sequence of eigenfunctions \({(\psi_n)}\) of \({\Delta_{\mathbb{S}^{d}} - V}\). We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.     

Quantum Electrodynamics of Atomic Resonances

A simple model of an atom interacting with the quantized electromagnetic field is studied. The atom has a finite mass m, finitely many excited states and an electric dipole moment, \({\vec{d}_0 = -\lambda_{0} \vec{d}}\), where \({\| d^{i}\| = 1, i = 1, 2, 3,}\) and \({\lambda_0}\) is proportional to the elementary electric charge. The interaction of the atom with the radiation field is described with the help of the Ritz Hamiltonian, \({-\vec{d}_0 \cdot \vec{E}}\), where \({\vec{E}}\) is the electric field, cut off at large frequencies. A mathematical study of the Lamb shift, the decay channels and the life times of the excited states of the atom is presented. It is rigorously proven that these quantities are analytic functions of the momentum \({\vec{p}}\) of the atom and of the coupling constant \({\lambda_0}\), provided \({\vert\vec{p} \vert < mc}\) and \({\vert \Im \vec{p} \vert}\) and \({\vert \lambda_{0} \vert}\) are sufficiently small. The proof relies on a somewhat novel inductive construction involving a sequence of ‘smooth Feshbach–Schur maps’ applied to a complex dilatation of the original Hamiltonian, which yields an algorithm for the calculation of resonance energies that converges super-exponentially fast.     

Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum

We study a spatial birth-and-death process on the phase space of locally finite configurations \({\varGamma }^+ \times {\varGamma }^-\) over \({\mathbb {R}}^d\). Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator \(L^+(\gamma ^-) + \frac{1}{\varepsilon }L^-\), \(\varepsilon > 0\). Here \(L^-\) describes the environment process on \({\varGamma }^-\) and \(L^+(\gamma ^-)\) describes the system process on \({\varGamma }^+\), where \(\gamma ^-\) indicates that the corresponding birth-and-death rates depend on another locally finite configuration \(\gamma ^- \in {\varGamma }^-\). We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states \(\mu _t^{\varepsilon }\) on \({\varGamma }^+ \times {\varGamma }^-\). Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let \(\mu _{\mathrm {inv}}\) be the invariant measure for the environment process on \({\varGamma }^-\). In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of \(\mu _t^{\varepsilon }\) onto \({\varGamma }^+\) converges weakly to an evolution of states on \({\varGamma }^+\) associated with the averaged Markov birth-and-death operator \({\overline{L}} = \int _{{\varGamma }^-}L^+(\gamma ^-)d \mu _{\mathrm {inv}}(\gamma ^-)\).     

A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥T _c,k =J/arctan?(1/k) the limiting Gibbs measure is unique, and for T<T _c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if \(T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})\), with \(\sqrt{k} then there is a new uncountable set \({\mathcal{G}}_{k,k_{0}}\) of Gibbs measures. Moreover \({\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}\), for k ₀≠k′₀. Therefore if \(T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})\), \(T_{c,\sqrt{k}+1} then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}\(\cup(\bigcup_{k_{0}:\sqrt{k}.