Talks

Attempts to go beyond the framework of local quantum field theory include scenarios in which the action of external symmetries on the quantum fields Hilbert space is deformed. A common feature of these models is that the quantum group symmetry of their Hilbert spaces induces additional structure in the multiparticle states which in turns reflects a non-trivial momentum-dependent statistics. In certain particular models which might be relevant for quantum gravity the richer structure of the deformed Fock space allows for the possibility of entanglement between the field modes and certain ''planckian'' degrees of freedom invisible to an observer that cannot probe the Planck scale.

An entirely new kind of band insulator was discovered recently. These new electronic states - called "topological insulators" - are fundamentally different from standard band insulators. They are distinguished by the fact that their edges (in the 2D case) or surfaces (in the 3D case) support gapless transport which is extremely robust. In the two dimensional case, topological insulators can be thought of as time reversal invariant analogues of integer quantum Hall states. This analogy is intriguing since integer quantum Hall states are a special case of the far richer class of fractional quantum Hall states. It is natural to wonder: can topological insulators also be generalized? In this talk, I will investigate this question. I will show that, just as in quantum Hall systems, electron interactions allow for a whole new class of states - which we call "fractional topological insulators." These states have excitations with fractional charge and statistics, in addition to protected edge modes.

I consider some of the issues we face in trying to understand dark energy. Huge fluctuations in the unknown dark energy equation of state can be hidden in distance data, so I argue that model-independent tests which signal if the cosmological constant is wrong are valuable. These can be constructed to remove degeneracies with the cosmological parameters. Gravitational effects can play an important role. Even small inhomogeneity clouds our ability to say something definite about dark energy. I discuss how the averaging problem confuses our potential understanding of dark energy by considering the backreaction from density perturbations to second-order in the concordance model: this effect leads to at least a 10\% increase in the dynamical value of the deceleration parameter, and could be significantly higher owing to a UV divergence. Large Hubble-scale inhomogeneity has not been investigated in detail, and could conceivably be the cause of apparent cosmic acceleration. I discuss void models which defy the Copernican principle in our Hubble patch can explain acceleration through inhomogeneous cosmic curvature. These can fit the small scale CMB, and can explain the observed primordial lithium abundances - a niggling 4 or 5 sigma discrepancy in the concordance model. I describe how we can potentially rule out these models, and so provide an important test for the existence of dark energy.

A plethora of Higgsless models have been proposed and we are in the peculiar situation where Fermilab & LHC results will be extremely interesting whether or not the Higgs boson is found. I present here a model where one of the sacred assumption of quantum field theory (renormalizability) is dropped. A precise prescription for the removal of the divergences guarantees both unitarity and predictivity. Interestingly the model is consistent if the Power counting criterion is enforced in a weak form (Weak Power Counting). Among the novel features one notices an extra parameter (the ratio M_W/M_Z) but also a consistent accommodation of the Majorana neutrino mass.

Two-dimensional non-linear sigma models on some supergroup manifolds are conformal field theories whether the action includes a Wess-Zumino term or not. These models are relevant for the worldsheet description of string theory in Anti-de Sitter backgrounds with Ramond-Ramond fluxes. The current algebra is an useful tool to study these theories. In these lectures I will review the construction of the current algebra. Then I will discuss some applications to the computation of the spectrum and integrability.

Loop quantum gravity and spin foams are two closely related theories of quantum gravity. There is an expectation that the sum over histories or path integral formulation of LQG will take the form of a spin foam, although a rigorous connection between the two is available only in 2+1 gravity. Understanding the relation between them will resolve many open questions of both theories. We probe the connection through an exactly soluble model of loop quantum cosmology. Beginning from the canonical theory we construct a spin foam like expansion of LQC. This construction reveals a number of insights into spin foams including the nature of the continuum limit.

In this talk (based on arXiv:1001.0354) we give a quantum statistical interpretation for the Kauffmann bracket polynomial state sum <K> for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting C(K) denote the Hilbert space for this model, there is a natural unitary transformation U from C(K) to itself such that <K> = <F|U|F> where |F> is a sum over basis states for C(K). The quantum algorithm arises directly from this formula via the Hadamard Test. We then show that the framework for our quantum model for the bracket polynomial is a natural setting for Khovanov homology. The Hilbert space C(K) of our model has basis in one-to-one correspondence with the enhanced states of the bracket state summmation and is isomorphic with the chain complex for Khovanov homology with coefficients in the complex numbers. We show that for the Khovanov boundary operator d defined on C(K) we have the relationship dU + Ud = 0. Consequently, the unitary operator U acts on the Khovanov homology, and we therefore obtain a direct relationship between Khovanov homology and this quantum algorithm for the Jones polynomial. The formula for the Jones polynomial as a graded Euler characteristic is now expressed in terms of the eigenvalues of U and the Euler characteristics of the eigenspaces of U in the homology. The quantum algorithm given here is inefficient, and so it remains an open problem to determine better quantum algorithms that involve both the Jones polynomial and the Khovanov homology.