Stokes Matrices and Monodromy of the Quantum Cohomology of Projective Spaces

In this paper we compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a “classical” framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups. Received: 24 October 1998 / Accepted: 27 April 1999     

On Hypergeometric Functions Connected with Quantum Cohomology of Flag Spaces

A unified approach to geometric, symbol and deformation quantizations on a generalized flag manifold endowed with an invariant pseudo-K?hler structure is proposed. In particular cases we arrive at Berezin's quantization via covariant and contravariant symbols. Received: 5 August 1997 / Accepted: 8 July 1998     

Semiinfinite Cohomology of Quantum Groups

We define a new cohomology theory of associative algebras called semiinfinite cohomology in the derived categories' setting. We investigate the case of a small quantum group u, calculate semiinfinite cohomology spaces of the trivial u-module and express them in terms of local cohomology of the nilpotent cone for the corresponding semisimple Lie algebra. We discuss the connection between the semiinfinite homology of u and the conformal blocks' spaces. Received: 14 October 1996 / Accepted: 25 February 1997     

Quantum Bundle Description of Quantum Projective Spaces

We realise Heckenberger and Kolb??s canonical calculus on quantum projective (N ? 1)-space C _q [C p ^N?1] as the restriction of a distinguished quotient of the standard bicovariant calculus for the quantum special unitary group C _q [SU _N ]. We introduce a calculus on the quantum sphere C _q [S ^2N?1] in the same way. With respect to these choices of calculi, we present C _q [C p ^N?1] as the base space of two different quantum principal bundles, one with total space C _q [SU _N ], and the other with total space C _q [S ^2N?1]. We go on to give C _q [C p ^N?1] the structure of a quantum framed manifold. More specifically, we describe the module of one-forms of Heckenberger and Kolb??s calculus as an associated vector bundle to the principal bundle with total space C _q [SU _N ]. Finally, we construct strong connections for both bundles.     

Quantum Cohomology and the Periodic Toda Lattice

We describe a relation between the periodic one-dimensional Toda lattice and the quantum cohomology of the periodic flag manifold (an infinite-dimensional K?hler manifold). This generalizes a result of Givental and Kim relating the open Toda lattice and the quantum cohomology of the finite-dimensional flag manifold. We derive a simple and explicit “differential operator formula” for the necessary quantum products, which applies both to the finite-dimensional and to the infinite-dimensional situations. Received: 20 April 1999 / Accepted: 12 April 2000     

On the Structure of Inhomogeneous Quantum Groups

We investigate inhomogeneous quantum groups G built from a quantum group H and translations. The corresponding commutation relations contain inhomogeneous terms. Under certain conditions (which are satisfied in our study of quantum Poincaré groups [12]) we prove that our construction has correct ‘size’, find the R-matrices and the analogues of Minkowski space for G. Received: 3 April 1995 / Accepted: 23 September 1996     

On the Entanglement Structure in Quantum Cloning

We study the entanglement properties of the output state of a universal cloning machine. We analyse in particular bipartite and tripartite entanglement of the clones, and discuss the classical limit of infinitely many output copies.     

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

We present several results on the geometry of the quantum projective plane. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding ‘classical’ characteristic classes (via Fredholm modules); complete diagonalization of gauged Laplacians on these line bundles; ‘quantum’ characteristic classes via equivariant K-theory and q-indices.     

A Module Structure on the Symplectic Floer Cohomology

The aim of this paper is to offer an affirmative answer to the Floer conjectures in [2, p. 589] which states that there is a module structure on the Z _{2 N} -graded symplectic Floer cohomology for monotone symplectic manifolds. By constructing a Z-graded symplectic Floer cohomology as an integral lift of the Z _{2 N} -graded symplectic Floer cohomology, we gain control of the holomorphic bubbling spheres. This makes a module structure on the Z-graded Floer cohomology. There is a spectral sequence with E ¹ _*,* given by the Z-graded symplectic Floer cohomology. Such a spectral sequence converges to the Z _{2 N} -graded symplectic Floer cohomology. Hence we induce a module structure for the Z _{2 N} -graded symplectic Floer cohomology by the spectral sequence and algebraic topology methods. Received: 2 August 1999 / Accepted: 25 October 1999     

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.     

Influence of Majorana Bound States in Quantum Rings

A quantum ring coupled to a 1D topological superconductor hosting Majorana bound states (MBSs) is investigated. The MBSs effects over the spectrum and persistent current along the quantum ring are studied. The spectra of the system are obtained by an exact numerical diagonalization of the Bogoliubov-de Gennes Hamiltonian in the Majorana representation. In addition, Green's function formalism is implemented for analytical calculations and obtained a switching condition in the MBSs fermionic parity. Three different patterns that could be obtained for the spatial separation of the MBSs, named: bowtie, diamond, and asymmetric, are reported here, which are present only in odd parity in the quantum ring, while only a single pattern (bowtie) is obtained for even parity. Those patterns are subject strictly to the switching condition for the MBSs. Besides, quantum ring with the presence of a Majorana zero mode presents gapped/gapless spectra in odd/even parity showing in the even case a subtle signature in the persistent current.     

On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion

We study the small mass limit (or: the Smoluchowski–Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz–Drude cutoff, we derive the Heisenberg–Langevin equations for the particle’s observables using a quantum stochastic calculus approach. We set the mass of the particle to equal \(m = m_{0} \epsilon \), the reduced Planck constant to equal \(\hbar = \epsilon \) and the cutoff frequency to equal \(\varLambda = E_{\varLambda }/\epsilon \), where \(m_0\) and \(E_{\varLambda }\) are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as \(\epsilon \rightarrow 0\). We study the limit as \(\epsilon \rightarrow 0\) of the rescaled model and derive a limiting equation for the (slow) particle’s position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.     

On the Spectroscopic Structure of Two Interacting Electrons in a Quantum Dot

The shifted 1/N expansion technique used by El-Said [Phys. Rev. B 61 (2000) 13026], to study the relative Hamiltonian of two interacting electrons confined in a quantum dot, is investigated. El-Said's results from shifted large-N (or 1/N) expansion technique are revised and results from an alternative method are also reported. The distinctive role of the central spike term, (m ²-1/4)/q ², in determining spectral properties of the above problem is shown, moreover.     

On the Structure of Pseudo BL-algebras and Pseudo Hoops in Quantum Logics

The main aim of the paper is to solve a problem posed in Di Nola et al. (Multiple Val. Logic 8:715–750, 2002) whether every pseudo BL-algebra with two negations is good, i.e. whether the two negations commute. This property is intimately connected with possessing a state, which in turn is essential in quantum logical applications. We approach the solution by describing the structure of pseudo BL-algebras and pseudo hoops as important families of quantum structures. We show when a pseudo hoop can be embedded into the negative cone of the reals. We give an equational base characterizing representable pseudo hoops. We also describe some subvarieties: normal-valued, and varieties where each maximal filter is normal. We produce some noncommutative covers and extend the area where each algebra is good. Finally, we show that there are uncountably many subvarieties of pseudo BL-algebras having members that are not good.     

On the Lie Group of Projective Canonical Transformations

The transformation properties of the projective canonical group are considered. It is shown that the group acts transitiv on functions defined on configuration space but not transitiv on those defined on the phase space. We prove moreover that this group is the largest finite-dimensional Lie group of point transformations determined on the configuration space.     

Interaction of Twisted Light with Electrons in Two-Dimensional Quantum Rings

We investigate the dynamics of charge carriers propagating in a ring being induced by twisted light: The exciting laser beam is assumed to have nonzero orbital angular momentum. The selection rules for the transitions between the eigenstates of the two-dimensional ring are determined with the aid of analytic and numerical methods. Using these results, we gain an insight into the physical process that leads to the transfer of the angular momentum of the laser beam to the electrons in the quantum ring.     

Deformations of Vertex Algebras,Quantum Cohomology of Toric Varieties,and Elliptic Genus

 We use previous work on the chiral de Rham complex and Borisov's deformation of a lattice vertex algebra to give a simple linear algebra construction of quantum cohomology of toric varieties. Somewhat unexpectedly, the same technique allows to compute the formal character of the cohomology of the chiral de Rham complex on even dimensional projective spaces. In particular, we prove that the formal character of the space of global sections equals the equivariant signature of the loop space, a well-known example of the Ochanine-Witten elliptic genus. Received: 15 July 2000 / Accepted: 17 August 2002 Published online: 8 January 2003 RID="*" ID="*" Partially supported by an NSF grant Communicated by R. H. Dijkgraaf