N=2 superalgebra and non-commutative geometry
A construction of supersymmetric field-theoretical models in non-commutative geometry is reviewed. The underlying superstructure of the models is encoded in $osp(2,2)$ superalgebra.
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arXiv subjects
H. Grosse
Publications and source records attributed to H. Grosse.
Simple field theoretical models on noncommutative manifolds
We review recent progress in formulating two-dimensional models over noncommutative manifolds where the space-time coordinates enter in the formalism as non-commuting matrices. We describe the Fuzzy sphere and a way to approximate topological nontrivial configurations using matrix models. We obtain an ultraviolet cut off procedure, which respects the symmetries of the model. The treatment of spinors results from a supersymmetric formulation; our cut off procedure preserves even the supersymmetry.
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Topologically nontrivial field configurations in noncommutative geometry
In the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated (fuzzy) sphere. The corresponding field theory actions conserve all basic symmetries of the standard commutative version (space isometries and global chiral symmetry), but due to the noncommutativity of the space the fields are regularized and they contain only finite number of modes.
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Field Theory on a Supersymmetric Lattice
A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the differencial calculus of non-commutative geometry. The regulated theory involves only finite number of degrees of freedom and is manifestly supersymmetric.
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Towards Finite Quantum Field Theory in Non-Commutative Geometry
We describe the self-interacting scalar field on the truncated sphere and we perform the quantization using the functional (path) integral approach. The theory posseses a full symmetry with respect to the isometries of the sphere. We explicitely show that the model is finite and the UV-regularization automatically takes place.
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Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
This paper further develops the combinatorial approach to quantization of the Hamiltonian Chern Simons theory advertised in \cite{AGS}. Using the theory of quantum Wilson lines, we show how the Verlinde algebra appears within the context of quantum group gauge theory. This allows to discuss flatness of quantum connections so that we can give a mathe- matically rigorous definition of the algebra of observables $\A_{CS}$ of the Chern Simons model. It is a *-algebra of ``functions on the quantum moduli space of flat connections'' and comes equipped with a positive functional $ω$ (``integration''). We prove that this data does not depend on the particular choices which have been made in the construction. Following ideas of Fock and Rosly \cite{FoRo}, the algebra $\A_{CS}$ provides a deformation quantization of the algebra of functions on the moduli space along the natural Poisson bracket induced by the Chern Simons action. We evaluate a volume of the quantized moduli space and prove that it coincides with the Verlinde number. This answer is also interpreted as a partition partition function of the lattice Yang-Mills theory corresponding to a quantum gauge group.
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Combinatorial quantization of the Hamiltonian Chern-Simons theory I
Motivated by a recent paper of Fock and Rosly \cite{FoRo} we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *-operation and a positive inner product.
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Novel Symmetry of Non-Einsteinian Gravity in Two Dimensions
The integrability of $R^2$-gravity with torsion in two dimensions is traced to an ultralocal dynamical symmetry of constraints and momenta in Hamiltonian phase space. It may be interpreted as a quadratically deformed $iso(2,1)$-algebra with the deformation consisting of the Casimir operators of the undeformed algebra. The locally conserved quantity encountered in the explicit solution is identified as an element of the centre of this algebra. Specific contractions of the algebra are related to specific limits of the explicit solutions of this model.
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