Week 15.03.2009 – 21.03.2009

The speaker will show how Ulam's increasing subsequence problem is connected to a variety of classical integrable systems.

Topological orders are non-symmetry breaking phases of matter, which provide a new paradigm in condensed-matter physics. Their characterization is a major open problem, although considerable progress has been made since the proposal of the topological entanglement entropy. In this talk I will review how topological orders have been proposed to serve as robust quantum memories. Then the dynamical stability of topologically ordered quantum states will be examined. Conditions will be derived on the type of dynamical evolution that allows quantum recurrence and preservation of the initial topological order.

We present a set of functional equations defining the anomalous dimensions of arbitrary local single trace operators in planar N=4 SYM theory. It takes the form of a Y-system based on the integrability of the dual superstring sigma-model on the AdS5 x S5 background. This Y-system passes some very important tests: it reproduces the full asymptotic Bethe ansatz at large L, including the dressing factor, and it confirms all recently found wrapping corrections. We shall also describe the derivation of these equations in some detail.

The hierarchy problem can be represented as a tension between the need for a large cut-off scale suggested, for instance, by flavor physics and the need for a low cut-off scale suggested by naturalness in electroweak symmetry breaking. I will illustrate how this tension could be largely alleviated if the Standard Model flowed to an approximate CFT above the weak scale with a specific relation among the scaling dimensions of the Higgs sector fields. To investigate the viability of that scenario one is led to ask the following simple question: in an arbitrary CFT, given a scalar operator phi, and the operator S=phi phi defined as the lowest dimension scalar S which appears in the OPE phi phi, what is the bound (that is d(S) is smaller than f(d(phi))) on the scaling dimensions of the two operators? I will present a derivation of the bound based on general considerations of OPE, conformal block decomposition, and crossing symmetry. The function f(d(phi)) is computed numerically. When d(phi) goes to 1, one has f(d(phi))=2+O(sqrt(d(phi)-1)), which shows that the free theory limit is approached continuously. An analogous bound can be derived in 2D where some non-trivial consistency check can be made. I will discuss the relevance of the result for the hierarchy problem and illustrate the directions of future investigation.