A Fedosov Star Product of the Wick Type for Kähler Manifolds

In this Letter we compute some elementary properties of the Fedosov star product of Weyl type, such as symmetry and order of differentiation. Moreover, we define the notion of a star product of the Wick type on every Kähler manifold by a straightforward generalization of the corresponding star product in Cn: the corresponding sequence of bidifferential operators differentiates its first argument in holomorphic directions and its second argument in antiholomorphic directions. By a Fedosov type procedure, we give an existence proof of such star products for any Kähler manifold.     

Para-Grassmann Star Product Calculation

In this Letter, we construct the star product for polynomials over the para-Grassmann variable and we present as an example a para-Grassmann version of the model of q–quantum mechanics. Moreover, using the structural relations of the q–deformed algebra generated by annihilation and creation operators, we decompose the Jacobi matrix in the product of three matrices: the diagonal matrix, the upper and lower diagonal matrices, mutually adjoint.     

Star Product Formula of Theta Functions

As a noncommutative generalization of the addition formula of theta functions, we construct a class of theta functions which are closed with respect to the Moyal star product of a fixed noncommutative parameter. These theta functions can be regarded as bases of the space of holomorphic homomorphisms between holomorphic line bundles over noncommutative complex tori.     

An Explicit Formula for the Berezin Star Product

We prove an explicit formula of the Berezin star product on K?hler manifolds. The formula is expressed as a summation over certain strongly connected digraphs. The proof relies on a combinatorial interpretation of Engli??? work on the asymptotic expansion of the Laplace integral.     

Kontsevich Star Product on the Dual of a Lie Algebra

We show that on the dual of a Lie algebra g of dimension d, the star product recently introduced by M. Kontsevich is equivalent to the Gutt star product on g*. We give an explicit expression for the operator realizing the equivalence between these star products.     

Open Superstring Star as a Continuous Moyal Product

The matter part of the open string superstar is identified with the continuous Moyal product of functions of anti-commuting variables by diagonalizing the three-string vertex and using a special coordinate representation. We show that in this representation the identity and the sliver have a simple form.     

The Group of Hamiltonian Automorphisms of a Star Product

We deform the group of Hamiltonian diffeomorphisms into a group of Hamiltonian automorphisms, Ham(M,?), of a formal star product ? on a symplectic manifold (M,ω). We study the geometry of that group and deform the Flux morphism in the framework of deformation quantization.     

Star Product and q-Deformation of Grassmann and Symmetric Algebras

We review the relation between the star producton a Poisson-Lie group and the quantum Yang-Baxterequation. We define the q-deformed Grassmann andq-deformed symmetric algebras on a vector space V, and prove that a star product on a triangular(simple quasitriangular) Poisson-Lie group G determinesa q-deformation of both the symmetric and Grassmannalgebras over a dual of a g-module where g is thecorresponding triangular (simple quasitriangular) Liebialgebra of the Poisson-Lie group G.     

Operator-Norm Convergence of the Trotter Product Formula for Sectorial Generators

The operator-norm convergence of the Trotter product formula is known for self-adjoint semigroups with compactness or smallness conditions on the generators involved in this formula. We generalize these two types of results to sectorial generators.     

Note on the Norm Convergence of the Unitary Trotter Product Formula

We discuss when the unitary Trotter product formula converges in operator norm and apply the results obtained to the unitary group generated by the Dirac operator and the relativistic Schrödinger operator with suitable potentials.     

Convergent Star Product Algebras on ‘ax+b’

We define nontempered (exponential growth) function spaces on the Lie group ax+b which are stable under some left-invariant (convergent) star product. The techniques used to achieved the latter come from symmetric spaces geometry and star representation theory.     

Nonassociative Star Product Deformations for D-Brane World-Volumes in Curved Backgrounds

We investigate the deformation of D-brane world-volumes in curved backgrounds. We calculate the leading corrections to the boundary conformal field theory involving the background fields, and in particular we study the correlation functions of the resulting system. This allows us to obtain the world-volume deformation, identifying the open string metric and the noncommutative deformation parameter. The picture that unfolds is the following: when the gauge invariant combination ω=B+F is constant one obtains the standard Moyal deformation of the brane world-volume. Similarly, when dω= 0 one obtains the noncommutative Kontsevich deformation, physically corresponding to a curved brane in a flat background. When the background is curved, H=dω≠ 0, we find that the relevant algebraic structure is still based on the Kontsevich expansion, which now defines a nonassociative star product with an A _∞ homotopy associative algebraic structure. We then recover, within this formalism, some known results of Matrix theory in curved backgrounds. In particular, we show how the effective action obtained in this framework describes, as expected, the dielectric effect of D-branes. The polarized branes are interpreted as a soliton, associated to the condensation of the brane gauge field. Received: 22 March 2001 / Accepted: 13 July 2001     

Wick Rotation for D-modules

We extend the classical Wick rotation to D-modules and higher codimensional submanifolds.     

Open Superstring Star as a Continuous Moyal Product with B Field

In this paper, we recast the matter part of the open superstring star in the present ofa constant B field. Byusing a different coordinate representation the matter part of the open superstring star is identified with the continuousMoyal product of functions of anti-commuting variables. Fortunately we find it does not depend on the value of the Bfield.     

Higher-derivative Lee–Wick unification

We consider gauge coupling unification in Lee–Wick extensions of the Standard Model that include higher-derivative quadratic terms beyond the minimally required set. We determine how the beta functions are modified when some Standard Model particles have two Lee–Wick partners. We show that gauge coupling unification can be achieved in such models without requiring the introduction of additional fields in the higher-derivative theory and we comment on possible ultraviolet completions.     

On Gammelgaard’s Formula for a Star Product with Separation of Variables

We show that Gammelgaard’s formula expressing a star product with separation of variables on a pseudo-Kähler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal Fock space. We prove this inversion formula directly and thus offer an alternative approach to Gammelgaard’s formula which gives more insight into the question why the directed graphs in his formula have no cycles.