Talks

The tt* equations define a flat connection on the moduli spaces of 2d, N=2 quantum field theories. For conformal theories with c=3d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. I will show that the non-holomorphic content of the tt* equations in the cases d=1,2,3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with generators of a Lie algebra. This Lie algebra replaces the anti-holomorphic derivatives of tt* and provides these with a finer and algebraic meaning. The generators of the differential rings of special functions are given by quasi-modular forms for d=1 and their generalizations in d=2,3. For d=3, this can be used to provide a purely Lie algebraic formulation of the higher genus topological string theory amplitudes and of the BCOV holomorphic anomaly equations which govern them.

We study a contracting universe composed of cold dark matter and radiation, and with a positive cosmological constant. Assuming that loop quantum cosmology captures the correct high-curvature dynamics of the space-time, we calculate the spectrum of scalar and tensor perturbations after the bounce, assuming initial quantum vacuum fluctuations. We find that the modes that exit the (sound) Hubble radius during matter-domination when the effective equation of state is slightly negative due to the cosmological constant will be nearly scale-invariant with a slight red tilt, in agreement with observations. The tensor perturbations are also nearly scale-invariant, and the predicted tensor-to-scalar ratio is small.  Finally, as this scenario predicts a positive running of the scalar index, it can be differentiated from inflationary models.

Symmetry protected topological (SPT) states are bulk gapped states with gapless edge excitations. The SPT phases in free fermion systems, like topological insulators, can be classified by K-theory. However, it is not yet known what SPT phases exist in general interacting systems. In this talk, I will first present a systematic way to construct SPT phases in interacting bosonic systems, which allows us to identify many new SPT phases. Just as group theory allows us to construct 230 crystal structures in three dimensions, we find that group cohomology theory allows us to construct many interacting bosonic SPT phases. In my talk, I shall show how topological terms in the path integral description of the system can be constructed from nontrivial group cohomology classes, giving rise to exactly soluble Hamiltonians with explicit ground state wavefunctions. Next, I will discuss the generalization of the classifying scheme to interacting fermionic systems and a new mathematical framework – group supercohomology theory, which predicts a fermionic SPT phase that can neither be realized in free fermionic nor interacting bosonic systems.

Finally, I will briefly mention the deep relationship between SPT phases and chiral anomalies in high energy physics.