Week 25.10.2009 – 31.10.2009

Using methods of integrability, enormous progress has been done in calculating the spectrum of operators in N=4 Super Yang-Mills theory as well as of energies of strings. I will discuss advances in finding the spectrum of the Konishi operator, the simplest nontrivial operator of finite size, which is already a hard problem. A further mathematical problem is to prove integrability. I will discuss the algebraic objects appearing in the AdS/CFT correspondence, namely Yangians, and also show how they are related to the spectral equations.

We will discuss a direct prescription for computing the mixing among gauge invariant operators in N=4 SYM. This approach is based on the action of the superalgebra on the states of the theory and circumvents the problem of diagonalising the dilatation matrix. As an example, we will derive the leading and subleading planar quantum corrections to the BMN-like operators belonging to the supermultiplet with 2 impurities. Furthermore, we will discuss the relation to Beisert's SU(2,2) dynamic S-matrix approach. The same method can be applied in the BMN limit of the dual string description. The string theory expressions for the states match nicely the field theory ones up to order lambda'. We will also focus on the implications of the mixing terms for 3-point functions of 2 BPS and one non-BPS operator. Namely, should the mixing terms were not there, constraints following from the superconformal and the bonus U(1)Y symmetries would be violated. Subsequently, we will utilise the BMN limit of the AdS5xS5 string theory to analyse the string amplitudes corresponding to the 3-point functions studied above. The mixing is again crucial to ensure the cancellation of various amplitudes that would otherwise violate symmetry constaints. However,on the string side, we find examples of interactions which violate expectations coming from superconformal Ward identities and the U(1)Y symmetry. Finally, we will comment on the possible resolution of this mismatch.

Sigma models endowed with target space supersymmetry have an array of peculiar properties and important applications. A new method is presented, whereby the computation of certain correlation functions in these theories is drastically simplified. In particular, this allows for the identification of new families of conformally invariant 2d sigma models.

In this talk, I will discuss a new family of supersymmetric Wilson loop operators in N=4 SYM. These operators can be defined for any loop on a three-sphere in space-time and generically preserve two super-conformal charges. On the string side, via AdS/CFT duality, these loops map to surfaces which are 'pseudo-holomorphic' with respect to a novel almost-complex structure defined on a suitable subspace of AdS5xS5. Of particular interest is the subclass of loops lying on a two-sphere in space-time whose expectation value is conjectured to be computed exactly in terms of the analogous observables in bosonic 2d Yang-Mills on S2. Several evidences for this conjecture, both on the gauge theory and on the string theory side will be discussed.