Lunin and Maldacena recently found a simple method to generate AdS_5 supergravity backgrounds dual to exactly marginal deformations of SCFTs. In the first part of the talk, we will consider extensions to AdS_4 backgrounds, concentrating on those for which dual IR 3d field theories have been proposed. In the second part, we will concentrate on the deformation of N=4 SYM in 4d and analyze a certain Penrose limit. We will be led to consider a 'modified Landau problem' and relate it to the different set of chiral operators of the deformed N=4.

The IIB supersymmetry algebra in d=10 can be extended including 10-forms. One of these is the RR 10-form, which takes part in the tadpole cancellation mechanism of the type-I string. D9-branes are electrically charged with respect to this 10-form. IIB supergravity possesses a global SL(2,R) symmetry, which the quantum theory breaks to SL(2,Z). Even if 10-forms have no dynamics, since their field-strength vanishes identically, it is possible to study their behaviour under SL(2,Z) transformations. We determine all the possible independent 10-forms as representations of SL(2,Z), and in particular we show in which multiplet the RR 10-form lies. Finally, we perform the same analysis for the IIA supersymmetry algebra, and interpret the results in terms of T-duality.

In this seminar we will look at two places where two-dimensional conformal field theory is related to the study of matrix models. The first is that a matrix model may be thought of as a statistical model on a fluctuating lattice. In the continuum limit, that is in the limit of large matrix size N, this yields a c less than 1 CFT coupled to gravity. The second relation is that the matrix model can, even before the continuum limit, be described in terms of free bosons. The free boson CFT turns out to be useful in understanding the large N behaviour of the matrix model.

KZ equations are an essential tool in the study of conformal field theories with affine algebra symmetry. They are satisfied by all correlation functions of affine primary fields. However, string theory in AdS3 forces us to consider fields which are not affine primaries. I will explain that some generalizations of the KZ equations nevertheless still hold. I will discuss the implications of this finding for the relation with Liouville theory and the operator product expansion in the H3 model.

We propose affine Toda field theories related to the non-crystallographic Coxeter groups H_2, H_3 and H_4. The classical mass spectrum, the classical three-point couplings and the one-loop corrections to the mass renormalisation are determined. The construction is carried out by means of a reduction procedure from crystallographic to non-crystallographic Coxeter groups. The embedding structure explains for various affine Toda field theories that their particles can be organised in pairs, such that their relative masses differ by the golden ratio.

The truncated conformal space approach is a very useful tool in the numerical study of renormalisation group flows. I will discuss how one can study the effect of truncation in a standard RG manner and its relevance for studying perturbative fixed points.

(This talk is an exceptional colloquium of the Department of Mathematics.) A real-valued function, F, on an interval (a,b) is called matrix monotone if F(A) is less than F(B) whenever A and B are finite matrices of the same order with eigenvalues in (a,b) and A less than B. In 1934, Loewner proved the remarkable theorem that F is matrix monotone if and only if F is real analytic with continuations to the upper and lower half planes so that Im F is positive in the upper half plane. This deep theorem has evoked enormous interest over the years and a number of alternate proofs. There is a lovely 1954 proof that seems to have been lost in that the proof is not mentioned in various books and review article presentations of the subject, and I have found no references to the proof since 1960. The proof uses continued fractions. I'll provide background on the subject and then discuss the lost proof and a variant of that proof which I've found, which even avoids the need for estimates, and proves a stronger theorem.

The linearised curvatures of de Wit and Freedman for totally symmetric gauge fields of any spin are shown to give rise to geometric equations in which both fields and gauge parameters are not constrained a priori. These equations, which are non local, reduce to the traditional, local Fronsdal form by a suitable gauge fixing, and on-shell degrees of freedom are just the physical ones. Developments and generalisations of these results are discussed.

I will introduce the Gaudin Model. This model is described by a system of commuting Hamiltonians. I will explain how the eigenvalue equations for these Hamiltonians arise as the critical level limit of the Knizhnik-Zamolodchikov equations. In particular, some eigenvectors can be built from H3 correlators. Then I will use the H3-Liouville relation to relate these correlators to Liouville theory correlators. The critical level limit is interpreted in Liouville theory as a geometrical limit. This leads to the construction of Gaudin eigenvalues from the accessory parameters which arise in the uniformization of certain Riemann surfaces.

I will give an elementary and informal introduction to an efficient method of solving the Killing spinor equations. The method is based on the description of spinors in terms of forms and can be used to classify the supersymmetric solutions in e.g. 11D and IIB supergravity.

I am going to review the status of loop quantum gravity as it is seen from the point of view of a Lorentz covariant approach to loop quantization. I'll start from a brief review of the standard loop approach based on the SU(2) gauge group, its main results and problems. Then I'll present another approach which is based on a canonical formulation of general relativity which is explicitly covariant under the local Lorentz transformations. It allows to overcome several problems of the standard loop quantization and at the same time shows that the latter breaks the diffeomorphism invariance and is not a correct way for quantizing gravity. In the covariant framework I'll derive a new spectrum of the area operator, both for spacelike and timelike surfaces, and show that its predictions agree with the spin foam quantization.

I present results on non-perturbative effects in the c=1 string theory. First, I describe a geometric picture found in the CFT framework which gives an interpretation of D-branes in non-critical strings in terms of a complex curve associated with any closed string background. I show that its c=1 limit is degenerate and the degeneracy can be removed by considering a condensation of tachyon modes. Using the matrix model description, I calculate the leading as well as the subleading non-perturbative corrections to the string partition function. We find them by using the Toda integrable structure and from the realization of 2D string theory in terms of free fermions. Both methods give the same result which is also interpreted through correlation functions of a bosonic field. The leading corrections can be interpreted in terms of localized D-branes, whereas the sub-leading ones do not have a simple D-brane description.

(Note: this talk will have a great overlap with the April 5, 2004 talk by the same speaker.) The talk will give a fairly basic introduction to the evidence why a special class of Kac-Moody algebras, called very-extended algebras, might play a role in the formulation of supergravity theories. This evidence will contain a natural understanding of the field contents of some (super)gravity theories as well as the known dualities that usually are expected to arrive only after dimensional reduction.

This talk is part of the joint Triangular Seminars.

This part is part of the Triangular Seminars.