- 09/14/17: Singularities and special holonomy in physics
- 01/13/17: Degenerations of K3 surfaces, gravitational instantons, and M-theory
- 09/15/2016: Odds and ends
- 09/14/2016: Calibrations and branes
- 09/13/2016: The role of special holonomy in string/M/F compactification
- 09/8/2016: Moduli, calibrations, and singularities
September 15, 2017
TITLE: Singularities and special holonomy in physics
ABSTRACT: One route to describing a supersymmetric quantum theory of gravity in four-dimensional spacetime begins with the proposed eleven-dimensional quantum theory of gravity known as M-theory, which is then studied on the Cartesian product of a compact seven-dimensional space and a four-dimensional spacetime. To obtain a supersymmetric theory of gravity, the compact space should be equipped with a Riemannian metric admitting a covariantly constant spinor; the latter leads to special holonomy. Geometric properties of the compact space determine physical properties of the four-dimensional theory.
Two key physical properties — non-abelian gauge fields and chiral matter — cannot be realized in this setup unless the compact space has singularities. We will present some work in progress which modifies existing constructions of compact manifolds with holonomy G2 to include (some of) the relevant singularities.
January 13, 2017
TITLE: Degenerations of K3 surfaces, gravitational instantons, and M-theory
ABSTRACT: The detailed study of degenerations of K3 surfaces as complex manifolds goes back more than forty years and is fairly complete. Much less is known about the analogous problem in differential geometry of finding Gromov–Hausdorff limits for sequences of Ricci-flat metrics on the K3 manifold. I will review recent work of H.-J. Hein and G. Chen–X. Chen on gravitational instantons with curvature decay, and describe applications to the K3 degeneration problem. M-theory suggests an additional geometric structure to add, and I will give a conjectural sketch of how that structure should clarify the limiting behavior.
NOTE: The video below is AUDIO ONLY.
September 15, 2016
TITLE: Odds and ends
September 14, 2016
TITLE: Calibrations and branes
September 13, 2016
TITLE: The role of special holonomy in string/M/F compactification
September 8, 2016
TITLE: Moduli, calibrations, and singularities
ABSTRACT: A very effective way to study singularities of Calabi-Yau threefolds is to express the singular space as a limit of smooth spaces. If the complex structure is held fixed while the Kähler class is varied, there are algebraic cycles (calibrated by a power of the Kähler form) which approach zero volume in the limit. If the Kähler class is held fixed while the complex structure is varied, there are “vanishing cycles” whose volumes approach zero in the limit. These vanishing cycles are conjectured to have special Lagrangian representatives under appropriate conditions, i.e., to be calibrated by the real part of a holomorphic 3-form. Both kinds of limit can be studied with techniques from algebraic geometry.
In this talk, we will discuss the prospects for a similar picture holding in the case of manifolds. In a limit of smooth metrics in which the volume of some associative or co-associative cycle approaches zero, what happens to the space?