Talks

Matrix models yield a theory of random two dimensional surfaces. They support a 1/N expansion dominated by planar graphs (corresponding to planar surfaces) and undergo a phase transition to a continuum theory. In higher dimensions matrix models generalize to tensor models. In the absence of a viable 1=N expansion, tensor models have for a long time been less successful in providing a theory of random higher dimensional spaces. This situation has drastically changed recently. Models for a non-symmetric complex tensor have been shown to admit a 1/N expansion dominated by graphs of spherical topology in arbitrary dimensions and to undergo a phase transition to a continuum theory. I will present an overview of these results and discuss their implications.

This talk will give a survey of some recent developments on the construction and classification of compact manifolds with holonomy G2 and their calibrated submanifolds. After reviewing previous work we concentrate on the following three developments: (a) the construction of many new compact G2 manifolds using weak Fano 3-folds; (b) the construction of many compact G2 manifolds containing rigid associative 3-folds; (c) the diffeomorphism classification of (2-connected) G2 manifolds obtained by twisted connect sums. If time permits we will mention new questions
suggested by our results. This work is joint with Alessio Corti, Johannes Nordstrom, and Tommaso Pacini.

In this talk, I will discuss my recent works with J. Streets on curvature ows on Hermitian manifolds and show how they can be used to study generalized Kahler manifolds. I will also show how they are related to the renormalization group ow coupled with B- elds. Some open problems will be discussed. In the end, I will also discuss briey a new ow which preserves symplectic structures.

Asymptotically conical (AC) Calabi-Yau manifolds are Ricci-at Kahler manifolds that resemble a Ricci-at Kahler cone at infinity. I will describe an existence theorem for AC Calabi-Yau manifolds which, in particular, yields a refinement of an existence theorem of Tian and Yau for such manifolds. I will also discuss some examples. This is ongoing work with Hans-Joachim Hein.

I will discuss some joint work with K. Uhlenbeck. There is a general method for constructing soliton hierarchies from a splitting of Lie algebras. We explain how formal scattering and inverse scattering, Hamiltonian structures, commuting conservation laws, Backlund transformations, tau functions, and Virasoro actions on tau functions can all be constructed in a uni ed way from such splittings.

When considering flat unitary bundles on a punctured Riemann surface, it is often convenient to have a space that includes all possible holonomies around the punctures; such a space is provided by the extended moduli space of Jeffrey. On the other hand, there are certain inconveniences, in particular no clear link to complex geometry via a Narasimhan-Seshadri type theorem. It turns out that the situation can be remedied quite nicely by considering bundles with framings taking values in a Grassmannian. Analogs for general structure groups, and in particular links with recent work of Martens and Thaddeus, will also be discussed. (This is joint work with U. Bhosle and I. Biswas.)

After introducing Killing-Yano tensors and their basic properties, I will concentrate on their applications to black hole physics. Namely, I will focus on two topics: i) symmetries of the Dirac operator in curved background and ii) generalized Killing-Yano tensors in the presence of skew-symmetric torsion and the classification of corresponding Euclidean metrics.

Generalized complex manifolds, like complex manifolds, admit a decomposition of the bundle of di
erential forms. When an analogue of the @ @ lemma holds there is a corresponding Hodge decomposition in twisted cohomology. We look at some aspects of this decomposition, in particular its behavior under deformations of generalized complex structure. We de ne period maps and show a Griths transversality result. We use Courant algebroids to develop the notion of a holomorphic family of generalized complex structures and show the period maps for such families are holomorphic.

In classical terms, an SKT structure is a Hermitian structure for which the Hermitian 2-form is closed with respect to the second order operator ddc. These structures arise naturally in the study of sigma models with (2; 0) or (2; 1)-supersymmetries, much like generalized Kahler structures arise in the (2; 2)-supersymmetric sigma model. While the introduction of generalized complex geometry has provided the correct framework to study generalized Kahler structures and great progress has been made in this area in the last few years, SKT structures laid forgotten. We will take a look at what the generalized complex framework can do for SKT structures and in the process dispel some misconceptions that have arisen over the years.

Spin liquid phases in frustrated magnets may arise in a variety of forms. Here we discuss the possibility of topological insulators of spinons or the fractionalized excitations in spin liquids. These phases should be characterized by "both" of the two popular and different definitions of topological orders, namely the long-range entanglement and the symmetry-protected topological order. We show an explicit construction of such a state in frustrated magnets on the pyrochlore lattice and discuss novel properties such as the finite surface thermal conductivity.

We report on the ab-plane optical properties of the magnetic field inducednormal state of underdoped La1.94Sr0.06CuO4  (Tc=5.5 K), the first such study. We apply strong magnetic fields (4 T and 16 T) along the c-axis. We find that a 4 T field is strong enough to destroy the superconducting condensate.  However at higher fields we observed a gap-like depression in the optical conductivity at low frequency along with parallel growth of a broad absorption peak at higher frequency just above the 5 meV gap. The loss of low frequency conductivity in the gap region is in good agreement with dc magneto resistance measurements on samples from the same batch.  The spectral weight loss in the depression at low frequency is recovered by the spectral weight in the broad peak. Significantly, this spectral weight equals the spectral weight of the superconducting condensate.  The broad peak tracks the SDW order seen by neutron scattering[1] and we suggest offers an optical signature of magnetism.