Harold Steinacker

Finite gauge theory on fuzzy CP2

Abstract We give a non-perturbative definition of U ( n ) gauge theory on fuzzy C P 2 as a multi-matrix model. The degrees of freedom are 8 Hermitian matrices of finite size, 4 of which are tangential gauge fields and 4 are auxiliary variables. The model depends on a non-commutativity parameter 1 N , and reduces to the usual U ( n ) Yang–Mills action on the 4-dimensional classical C P 2 in the limit N → ∞ . We explicitly find the monopole solutions, and also certain U ( 2 ) instanton solutions for finite N. The quantization of the model is defined in terms of a path integral, which is manifestly finite. An alternative formulation with constraints is also given, and a scaling limit as R θ 4 is discussed.

Field theory on the q-deformed fuzzy sphere I

We study the q–deformed fuzzy sphere, which is related to D-branes on SU(2) WZW models, for both real q and q a root of unity. We construct for both cases a differential calculus which is compatible with the star structure, study the integral, and find a canonical frame of one–forms. We then consider actions for scalar field theory, as well as for Yang–Mills and Chern–Simons–type gauge theories. The zero curvature condition is solved.

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.

A non-perturbative approach to non-commutative scalar field theory

Non-commutative euclidean scalar field theory is shown to have an eigenvalue sector which is dominated by a well-defined eigenvalue density, and can be described by a matrix model. This is established using regularizations of 2n? via fuzzy spaces for the free and weakly coupled case, and extends naturally to the non-perturbative domain. It allows to study the renormalization of the effective potential using matrix model techniques, and is closely related to UV/IR mixing. In particular we find a phase transition for the 4 model at strong coupling, to a phase which is identified with the striped or matrix phase. The method is expected to be applicable in 4 dimensions, where a critical line is found which terminates at a non-trivial point, with nonzero critical coupling. This provides evidence for a non-trivial fixed-point for the 4-dimensional NC 4 model.

Field theory on the q-deformed fuzzy sphere II: quantization

Abstract We study the second quantization of field theory on the q -deformed fuzzy sphere for q∈ R . This is performed using a path integral over the modes, which generate a quasi-associative algebra. The resulting models have a manifest U q ( su (2)) symmetry with a smooth limit q →1, and satisfy positivity and twisted bosonic symmetry properties. A systematic way to calculate n -point correlators in perturbation theory is given. As examples, the 4-point correlator for a free scalar field theory and the planar contribution to the tadpole diagram in φ 4 theory are computed. The case of gauge fields is also discussed, as well as an operator formulation of scalar field theory in 2 q +1 dimensions. An alternative, essentially equivalent approach using associative techniques only is also presented. The proposed framework is not restricted to two dimensions.

A quantum algebraic description of D-branes on group manifolds

Abstract We propose an algebraic description of (untwisted) D-branes on compact group manifolds G using quantum algebras related to U q ( g ) . It reproduces the known characteristics of stable branes in the WZW models, in particular their configurations in G, energies as well as the set of harmonics. Both generic and degenerate branes are covered.

Unbraiding the braided tensor product

We show that the braided tensor product algebra A1⊗_A2 of two module algebras A1,A2 of a quasitriangular Hopf algebra H is isomorphic to the ordinary tensor product A1⊗A2, provided there exists a realization of H within A1. In other words, under this assumption we construct a transformation of generators which decouples A1,A2 (i.e., makes them commuting). We apply the theorem to the braided tensor product algebras of two or more quantum group covariant quantum spaces, deformed Heisenberg algebras and q-deformed fuzzy spheres.

Matrix description of D-branes on 3-spheres ⁄

We discuss a matrix model for D0-branes on S3 × M7 based on quantum group symmetries. For finite radius of S3, it gives results beyond the reach of the ordinary matrix model. For large radius of S3 all known static properties of branes on S3 are reproduced.

Covariant 4-dimensional fuzzy spheres, matrix models and higher spin

We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions. These spheres can be viewed as

Algebraic brane dynamics on SU(2): excitation spectra

SO(5)