The workshop is supported by the <link http: www.mis.mpg.de internal>Max-Planck-Institute for Mathematics in the Sciences, the <link http: www.uni-leipzig.de external>Graduiertenkolleg Quantenfeldtheorie (DFG) and the <link http: www.uni-leipzig.de zhs external>Center for Advanced Studies

We extend the gauge coupling of Super-Yang-Mills theories to an external superfield and find an anomalous breaking of supersymmetry in one-loop order. The anomaly is the variation of a gauge invariant field monomial depending on the logarithm of the coupling. Since the perturbative expansion and all loop diagrams are power series in the coupling, the anomaly cannot be absorbed as a counterterm to the action. We prove that the coefficient of the anomaly is gauge- and scheme-independent and strictly of one-loop order.
With local coupling the symmetric counterterms to the Yang-Mills part are exhausted in one-loop order and it is the anomaly, which gives rise to the two-loop order of the beta-function in pure Super-Yang-Mills theories.

A consistent formulation of quantum field theory on noncommutative spacetimes is possible provided time ordering and Wick rotation are done in the appropriate way. The resulting Feynman rules differ from those obtained by formal analogy to the commutative case. The problem of renormalization is discussed.

It has been conjectured by A.Sen that at the stationary point of the tachyon potential for the non-BPS D-brane or brane-anti-D-brane pair, the negative energy density cancels the brane tension. The results of study this conjecture within the frameworks of a cubic superstring field theory (SSFT) will be presented. We compute the tachyon potential at levels (1/2,1) and (2,6). In the first case we obtain that the value of the potential at the minimum is 97.5% of the non BPS D-brane tension. Using a special gauge in the second case we get 105.8 of the tension. We also study the cubic open SSFT expanded around the perturbatively stable vacuum including all scalar fields at levels 0,1/2 and 2. We check that kinetic terms is absent with good accuracy. This gives an evidence for the absence of physical open string states in this vacuum.

In commutative gauge theory one finds two representations of the conformal transformations, a `primitive' and a `covariant' related to the canonical and the symmetrical energy-momentum tensor, respectively. In noncommutative Yang-Mills theory one finds the same two representation provided one also transforms the noncommutative tensor theta. We find a surprising relation between covariant coordinates, the Seiberg-Witten map and the noncommutative, covariant representation of the (rigid) conformal transformations.

Being an example for the intersection of noncommutative geometry with quantum field theory, Yang-Mills theory on noncommutative space-time characterized by a constant field \theta became recently very attractive. Reasons are the possibility to perform explicit calculations, their very strange results and the even more mysterious Seiberg-Witten map between noncommutative and commutative field theories. The talk will cover the infrared problems when treating \theta nonperturbatively, the ultraviolet problems when expanding the action in \theta and the noncommutative Lorentz transformations as the origin of the Seiberg-Witten map.

Deformation Quantization according to Bayen, Flato, Fronsdal, Lichnerowicz and Sternheimer (1978) consists in a formal associative deformation (a so-called star-product) of the commutative algebra $A$ of all smooth complex-valued functions on a Poisson manifold such that the formal parameter corresponds to Planck's constant and the first order commutator is proportional to the Poisson bracket. Existence and classification of star-products have been shown by DeWilde-Lecomte (1983, symplectic case), Nest-Tsygan (1995, symplectic case) and M.Kontsevitch (1997, general Poisson case). We show how to quantize a morphism of a finite-dimensional Lie algebra in $A$ (a so-called momentum map) in the case of symplectic manifolds. A Liouville-integrable Hamiltonian system is a particular case of this for an abelian Lie algebra.

For quantizable compact Kähler manifolds $(M,\omega)$ with associated quantum line bundle $(L,h,\nabla)$ the Berezin transform $I$ for $C^{\infty}$ functions is introduced. This is a generalization of the original transform between contravariant and covariant Berezin symbols. If one considers all positve tensor powers $L^{\otimes m}$ of $L$ and the Berezin transform $I^{(m)}$ then it admits a complete asymptotic expansion in powers of $1/m$ , e.g. $$I^{(m)}f(x)\sim \sum_{k=0}^\infty (1/m)^kI_kf(x)$$ with differential operators $I_k$. It turns out that $I_0=id$ and $I_1=\Delta$, the Laplace-Beltrami operator. Consequences of this expansion for the Berezin-Toeplitz operator quantization and the Berezin-Toeplitz deformation quantization are discussed.

Global properties of abelian noncommutative gauge theories based on star products which are deformation quantizations of arbitrary Poisson structures are studied. The consistency condition for finite noncommutative gauge transformations and its explicit solution in the abelian case are given. It is shown that the local existence of invertible covariantizing maps (which are closely related to the Seiberg-Witten map) leads naturally to the notion of a noncommutative line bundle with noncommutative transition functions. We introduce the space of sections of such a line bundle and explicitly show that it is a projective module. The local covariantizing maps define a new Morita equivalent star product.

So called trialgebras, i.e. algebraic structures with two associative products and a coassociative coproduct, all joined in a pairwise compatible way, have been suggested in the context of four dimensional topological field theory by Crane and Frenkel. We show how to construct explicit examples of trialgebras by applying once more deformation theory to the algebra of functions on the Manin plane and to some of the classical examples of quantum groups and quantum algebras. For a certain class of trialgebras we show how to formulate the relevant data in terms of a system of coupled matrix equations. For one of the trialgebras which we construct, we will see that it is realized as a symmetry of a two dimensional spin system, generalizing the quantum group symmetry of the XXZ model (with suitable boundary conditions) known from the one dimensional case. The same trialgebra is also realized as a symmetry of a system of infinitely many q-oscillators. Finally, we give a result which shows that with trialgebras the final level of quantum deformations is reached. There are no nontrivial deformations of trialgebras to algebraic structures involving two associative products and two coassociative coproducts in a compatible way.

The two-point function for scalar field theory on the fuzzy sphere is calculated at one loop, including planar and nonplanar contributions. It turns out that there is no UV/IR mixing on the fuzzy sphere. The limit of the commutative sphere is regular without UV/IR mixing; however quantization does not commute with the commutative limit, and a finite ``noncommutative anomaly'' survives in the commutative limit. In a different limit, the noncommutative plane R^2_theta is obtained, and the UV/IR mixing appears. This provides an explanation of the UV/IR mixing as an infinite variant of the ``noncommutative anomaly''.

The notion of quantum groups has, in recent years, been generalised by parametrizing the corresponding generators with some continuously varying `colour' variables and the associated algebra and the coalgebra are defined in a way that all Hopf algebraic properties remain preserved. This results in the coloured extension of a quantum group. Focussing on the most intuitive example of GL_q(2), I will establish a picture of duality and construct the corresponding differential calculus. The quantum plane covariant under the action of coloured GL_q(2) will also be exhibited.

A method for the classification of covariant first order differential calculus on a certain class of quantum spaces is devised. The main tool is a notion of tangent space in analogy to Woronowicz' tangent space for differential calculi on quantum groups. The method is applied to quantum complex Grassmann manifolds Oq(Gr(r,N)). It is shown that there exists a unique covariant first order differential calculus of dimension 2r(N-r)=dim(Gr(r,N)) over Oq(Gr(r,N)).