Ursula Carow-Watamura

A QUANTUM LORENTZ GROUP

We examine the properties of the quantum Lorentz group SOq(3, 1) using the matrix given in Ref. 14. We show that this matrix together with the q-deformed metric C provide a representation of a BWM algebra. Using the projection operators which decompose the matrix into irreducible components, we give the general definition of the corresponding quantum space, i.e. the q-deformed Minkowski space and the q-deformed Clifford algebra. We also construct the q analog of Dirac matrices and show that they form a matrix representation of the q-deformed Clifford algebra.

Monopole bundles over fuzzy complex projective spaces

Abstract We give a construction of the monopole bundles over fuzzy complex projective spaces as projective modules. The corresponding Chern classes are calculated. They reduce to the monopole charges in the N → ∞ limit, where N labels the representation of the fuzzy algebra.

Chiral bosonization of superconformal ghosts on the Riemann surface and path-integral measure

Abstract The bosonization of the bosonic ghosts β and γ using a pair of scalar fields φ and χ is formulated for the case of the Riemann surface with higher genus. The measure of the path integral over φχ fields is defined with a projector which specifies the picture of each loop and removes the zero-modes. Nevertheless, the resulting correlation functions are independent of the chosen pictures. The correspondence between the anomalous ghost number conservation laws and the Riemann-Roch theorem becomes also manifest.

Complex Quantum Group, Dual Algebra and Bicovariant Differential Calculus

The method used to construct the bicovariant bimodule in ref. [CSWW] is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group. The complex quantum group Funq(SL(N, C)) is defined by requiring that it contains Funq(SU(N)) as a subalgebra analogously to the quantum Lorentz group. Analyzing the properties of the fundamental bimodule, we show that the dual algebra has the structure of the twisted product Funq(SU(N))⊗Funq(SU(N))reg*. Then the bicovariant differential calculi on the complex quantum group are constructed.

The q deformed Schrodinger equation of the harmonic oscillator on the quantum Euclidian space

We consider the q-deformed Schrodinger equation of the harmonic oscillator on the N-dimensional quantum Euclidean space. The creation and annihilation operators are found, which systematically produce all energy levels and eigenfunctions of the Schrodinger equation. In order to get the q series representation of the eigenfunction, we also give an alternative way to solve the Schrodinger equation which is based on the q analysis. We represent the Schrodinger equation by the q difference equation and solve it by using q polynomials and q exponential functions.

N-string vertex, canonical forms and bosonization

We give the field representation of the canonical forms for the operators which generate the projective transformation. This representation is also applicable in formulating the N-string vertex of Lovelace and Olive. It shows that they have the properties of a transition operator. Requiring the same properties for the ghost sector in bosonized form we construct the BRST invariant canonical form and the N-string vertex. We consider the (N − 3) b-ghost insertion. We show that the resulting vertex with (N − 3) parameters corresponding to the location of the ghosts is symmetric with respect to the N strings and it gives the proper integration measure.

Operator construction of ghost g-vacuum, handle operator and BRST-invariant multiloop amplitude of the bosonic string

Abstract Applying the Feynman-like rules in the operator formalism with bosonized bc -ghost, we construct the g -loop, N -string vertex out of three-string vertices and propagators. The measure factor is analyzed in detail. It is shown that a part of the ghost contribution can be represented by using g 2 -differential cand Riemann constants. The resulting measure factor is represented by the ghost g -vacuum with the b -ghost insertion accompanied by a contour integration. This contour integration can be performed explicitly in the one-loop case, in which we reproduce the well-known formula of the one-loop N -tachyon amplitude.

N-Superstring vertex and the vertex operator for the emission of the superstring

Abstract The BRST invariant, supersymmetric N-string vertex which applies to both, the Neveu-Schwarz and Ramond sector of the superstring is formulated using the vertex operator for the emission of a superstring. It is shown that the N-superstring vertex thus obtained is cyclic symmetric when GSO projected on-shell states and operators for the external strings are applied. The constraint equations of this vertex and their singularity structure are examined and we show that this vertex also has the required property of a transition operator. We also give a proof of its BRST invariance and supersymmetry.

Feynman-like rules for bosonic strings; Tree vertices, b-ghost insertion and factorization

Abstract Feynman-like rules for constructing string vertices are formulated in the operator formalism in such a way that the underlyinh geometrical picture becomes manifest. Using Olives as well as Lovelaces parametrization for the matter part we complete Feynman-like rules by including the ghost fields. The twisted propagator is defined with a b-ghost insertion together with a contour integration, in order to take the measure properly into account. The measure part of the string vertex is analyzed in detail. We formulate the BRST-invariant N -string tree vertex which enjoys the exact factorization property.

Higher Gauge Theories from Lie n-algebras and Off-Shell Covariantization

A bstractWe analyze higher gauge theories in various dimensions using a supergeometric method based on a differential graded symplectic manifold, called a QP-manifold, which is closely related to the BRST-BV formalism in gauge theories. Extensions of the Lie 2-algebra gauge structure are formulated within the Lie n-algebra induced by the QP-structure. We find that in 5 and 6 dimensions there are special extensions of the gauge algebra. In these cases, a restriction of the gauge symmetry by imposing constraints on the auxiliary gauge fields leads to a covariantized theory. As an example we show that we can obtain an off-shell covariantized higher gauge theory in 5 dimensions, which is similar to the one proposed in [1].