A Monoidal Category for Perturbed Defects in Conformal Field Theory

Starting from an abelian rigid braided monoidal category C{\mathcal{C}} we define an abelian rigid monoidal category C_F{\mathcal{C}_F} which captures some aspects of perturbed conformal defects in two-dimensional conformal field theory. Namely, for V a rational vertex operator algebra we consider the charge-conjugation CFT constructed from V (the Cardy case). Then C = Rep(V){\mathcal{C} = {\rm Rep}(V)} and an object in C_F{\mathcal{C}_F} corresponds to a conformal defect condition together with a direction of perturbation. We assign to each object in C_F{\mathcal{C}_F} an operator on the space of states of the CFT, the perturbed defect operator, and show that the assignment factors through the Grothendieck ring of C_F{\mathcal{C}_F}. This allows one to find functional relations between perturbed defect operators. Such relations are interesting because they contain information about the integrable structure of the CFT.     

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.     

Cardy Condition for Open-Closed Field Algebras

Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that , the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over a certain -extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation on the space of intertwining operators of V. We then derive a graphical representation of S in the modular tensor category . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy -algebra. In the end, we give a categorical construction of the Cardy -algebra in the Cardy case.     

A Rigidity Result for Extensions of Braided Tensor <Emphasis Type="Italic">C</Emphasis>*–Categories Derived from Compact Matrix Quantum Groups

Let G be a classical compact Lie group and G _μ the associated compact matrix quantum group deformed by a positive parameter μ (or \({\mu\in{\mathbb R}\setminus\{0\}}\) in the type A case). It is well known that the category of unitary representations of G _μ is a braided tensor C*–category. We show that any braided tensor *–functor \({\rho: \text{Rep}(G_\mu)\to\mathcal{M}}\) to another braided tensor C*–category with irreducible tensor unit is full if |μ| ≠ 1. In particular, the functor of restriction RepG _μ → Rep(K) to a proper compact quantum subgroup K cannot be made into a braided functor. Our result also shows that the Temperley–Lieb category \({\mathcal{T}_{\pm d}}\) for d > 2 can not be embedded properly into a larger category with the same objects as a braided tensor C*–subcategory.     

Asymptotic Properties of Solvable $\mathcal{PT}$-Symmetric Potentials

The asymptotic region of potentials has strong impact on their general properties. This problem is especially interesting for PT\mathcal{PT}-symmetric potentials, the real and imaginary components of which allow for a wider variety of asymptotic properties than in the case of purely real potentials. We consider exactly solvable potentials defined on an infinite domain and investigate their scattering and bound states with special attention to the boundary conditions determined by the asymptotic regions. The examples include potentials with asymptotically vanishing and non-vanishing real and imaginary potential components (Scarf II, Rosen-Morse II, Coulomb). We also compare the results with the asymptotic properties of some exactly non-solvable PT\mathcal{PT}-symmetric potentials. These studies might be relevant to the experimental realization of PT\mathcal{PT}-symmetric systems.     

Rigidity and Defect Actions in Landau-Ginzburg Models

Studying two-dimensional field theories in the presence of defect lines naturally gives rise to monoidal categories: their objects are the different (topological) defect conditions, their morphisms are junction fields, and their tensor product describes the fusion of defects. These categories should be equipped with a duality operation corresponding to reversing the orientation of the defect line, providing a rigid and pivotal structure. We make this structure explicit in topological Landau-Ginzburg models with potential x ^d , where defects are described by matrix factorisations of x ^d − y ^d . The duality allows to compute an action of defects on bulk fields, which we compare to the corresponding N = 2{\mathcal N = 2} conformal field theories. We find that the two actions differ by phases.     

Topology of a parity-time symmetric non-Hermitian rhombic lattice

We investigate the topological properties of a trimerized parity–time(PT)symmetric non-Hermitian rhombic lattice.Although the system is PT-symmetric,the topology is not inherited from the Hermitian lattice;in contrast,the topology can be altered by the non-Hermiticity and depends on the couplings between the sublattices.The bulk–boundary correspondence is valid and the Bloch bulk captures the band topology.Topological edge states present in the two band gaps and are predicted from the global Zak phase obtained through the Wilson loop approach.In addition,the anomalous edge states compactly localize within two diamond plaquettes at the boundaries when all bands are flat at the exceptional point of the lattice.Our findings reveal the topological properties of the??PT-symmetric non-Hermitian rhombic lattice and shed light on the investigation of multi-band non-Hermitian topological phases.     

Congruence Subgroups and Generalized Frobenius-Schur Indicators

We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C{\mathcal {C}} , an equivariant indicator of an object in C{\mathcal {C}} is defined as a functional on the Grothendieck algebra of the quantum double Z(C){Z(\mathcal {C})} via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay’s second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert.     

Quantum conformal superspace

For a compact connected orientablen-manifoldM, n 3, we study the structure ofclassical superspace ,quantum superspace ,classical conformal superspace , andquantum conformal superspace . The study of the structure of these spaces is motivated by questions involving reduction of the usual canonical Hamiltonian formulation of general relativity to a non-degenerate Hamiltonian formulation, and to questions involving the quantization of the gravitational field. We show that if the degree of symmetry ofM is zero, thenS,S ₀,C, andC ₀ areilh orbifolds. The case of most importance for general relativity is dimensionn=3. In this case, assuming that the extended Poincaré conjecture is true, we show that quantum superspaceS ₀ and quantum conformal superspaceC ₀ are in factilh-manifolds. If, moreover,M is a Haken manifold, then quantum superspace and quantum conformal superspace arecontractible ilh-manifolds. In this case, there are no Gribov ambiguities for the configuration spacesS ₀ andC ₀. Our results are applicable to questions involving the problem of thereduction of Einstein's vacuum equations and to problems involving quantization of the gravitational field. For the problem of reduction, one searches for a way to reduce the canonical Hamiltonian formulation together with its constraint equations to an unconstrained Hamiltonian system on a reduced phase space. For the problem of quantum gravity, the spaceC ₀ will play a natural role in any quantization procedure based on the use of conformal methods and the reduced Hamiltonian formulation.     

$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Generalized PT\mathcal{PT} symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT\mathcal{PT}-symmetric Hamiltonian. There is a corresponding class of PT\mathcal{PT}-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit PT\mathcal{PT} symmetry and also the ANNNI model, which has a hidden PT\mathcal{PT} symmetry. For both quantum and classical models, the class of models with generalized PT\mathcal{PT} symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT\mathcal{PT} symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT\mathcal{PT} symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT\mathcal{PT} symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with PT\mathcal{PT} symmetry which can be simulated for all parameter values, including cases where PT\mathcal{PT} symmetry is broken.     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

A Categorification of the Boson–Fermion Correspondence via Representation Theory of sl(∞)

In recent years different aspects of categorification of the boson–fermion correspondence have been studied. In this paper we propose a categorification of the boson–fermion correspondence based on the category of tensor modules of the Lie algebra sl(∞) of finitary infinite matrices. By \({\mathbb{T}^{+}}\) we denote the category of “polynomial” tensor sl(∞)-modules. There is a natural “creation” functor \({{\mathcal{T}_{N}} : {\mathbb{T}^{+}} \to {\mathbb{T}^{+}}}\), \({M \mapsto N \otimes M, \quad M,N \in \mathbb{T}^{+}}\). The key idea of the paper is to employ the entire category \({\mathbb{T}}\) of tensor sl(∞)-modules in order to define the “annihilation” functor \({{\mathcal{D}_{N}} : {\mathbb{T}^{+}} \to {\mathbb{T}^{+}}}\) corresponding to \({{\mathcal{T}_{N}}}\). We show that the relations allowing one to express fermions via bosons arise from relations in the cohomology of complexes of linear endofunctors on \({{\mathbb{T}^{+}}}\).     

Discrete Symmetries and Generalized Fields of Dyons

We have studied the different symmetric properties of the generalized Maxwell’s–Dirac equation along with their quantum properties. Applying the parity (℘), time reversal ( T\mathcal{T} ), charge conjugation (C\mathcal{C}) and their combined effect like parity time reversal (PT\mathcal{PT}), charge conjugation and parity (CP\mathcal{CP}) and CPT\mathcal{CP}T transformations to various equations of generalized fields of dyons, it is shown that the corresponding dynamical quantities and equations of dyons are invariant under these discrete symmetries.     

Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics

This paper deals with topos-theoretic truth-value valuations of quantum propositions. Concretely, a mathematical framework of a specific type of modal approach is extended to the topos theory, and further, structures of the obtained truth-value valuations are investigated. What is taken up is the modal approach based on a determinate lattice , which is a sublattice of the lattice of all quantum propositions and is determined by a quantum state e and a preferred determinate observable R. Topos-theoretic extension is made in the functor category of which base category is determined by R. Each true atom, which determines truth values, true or false, of all propositions in , generates also a multi-valued valuation function of which domain and range are and a Heyting algebra given by the subobject classifier in , respectively. All true propositions in are assigned the top element of the Heyting algebra by the valuation function. False propositions including the null proposition are, however, assigned values larger than the bottom element. This defect can be removed by use of a subobject semi-classifier. Furthermore, in order to treat all possible determinate observables in a unified framework, another valuations are constructed in the functor category . Here, the base category includes all ’s as subcategories. Although has a structure apparently different from , a subobject semi-classifier of gives valuations completely equivalent to those in ’s.     

Random Time-Dependent Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on the lattice \mathbb Z^d{\mathbb {Z}^d} performed by a quantum particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in \mathbb Z^d{\mathbb {Z}^d} when the sequence of unitary updates is given by an i.i.d. sequence of random matrices. When averaged over the randomness, this distribution is shown to display a drift proportional to the time and its centered counterpart is shown to display a diffusive behavior with a diffusion matrix we compute. A moderate deviation principle is also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. A generalization to unitary updates distributed according to a Markov process is also provided.     

On the Lattice Structure of Probability Spaces in Quantum Mechanics

Let $\mathcal{C}$ be the set of all possible quantum states. We study the convex subsets of $\mathcal{C}$ with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of quantum mechanics, including entanglement and Jaynes’ Max-Ent principle. We also encounter links with entanglement witnesses, which leads to a new separability criteria expressed in lattice language. We also provide an extension of a separability criteria based on convex polytopes to the infinite dimensional case and show that it reveals interesting facets concerning the geometrical structure of the convex subsets. It is seen that the above mentioned framework is also capable of generalization to any statistical theory via the so-called convex operational models’ approach. In particular, we show how to extend the geometrical structure underlying entanglement to any statistical model, an extension which may be useful for studying correlations in different generalizations of quantum mechanics.     

$\mathcal{CPT}$-Symmetric Discrete Square Well

In an addendum to the recent systematic Hermitization of certain N by N matrix Hamiltonians H ^(N)(λ) (Znojil in J. Math. Phys. 50:122105, 2009) we propose an amendment H ^(N)(λ,λ) of the model. The gain is threefold. Firstly, the updated model acquires a natural mathematical meaning of Runge-Kutta approximant to a differential PT\mathcal{PT}-symmetric square well in which P\mathcal{P} is parity. Secondly, the appeal of the model in physics is enhanced since the related operator C\mathcal{C} of the so called “charge” (the requirement of observability of which defines the most popular Bender’s metric Q = PC\Theta=\mathcal{PC}) becomes also obtainable (and is constructed here) in an elementary antidiagonal matrix form at all N. Last but not least, the original phenomenological energy spectrum is not changed so that the domain of its reality (i.e., the interval of admissible couplings λ∈(−1,1)) remains the same.     

Topological Open Strings on Orbifolds

We use the remodeling approach to the B-model topological string in terms of recursion relations to study open string amplitudes at orbifold points. To this end, we clarify modular properties of the open amplitudes and rewrite them in a form that makes their transformation properties under the modular group manifest. We exemplify this procedure for the \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3} orbifold point of local \mathbb P²{{\mathbb P}^2}, where we present results for topological string amplitudes for genus zero and up to three holes, and for the one-holed torus. These amplitudes can be understood as generating functions for either open orbifold Gromov–Witten invariants of \mathbb C³/\mathbb Z₃{{\mathbb C}^3/{\mathbb Z}_3}, or correlation functions in the orbifold CFT involving insertions of both bulk and boundary operators.