- 09/18/20: Codimension seven revisited
- 09/10/19: Nonabelian gauge symmetry and charged chiral matter in G2 compactifications
- 01/07/19: Degenerations of complex structures on K3 surfaces, following Kulikov, Persson, and Pinkham
- 01/07/19: Supersymmetric torus fibrations of Calabi-Yau threefolds
- 09/09/18: spaces with K3 fibrations
- 04/13/18: Collapsing metrics in string theory and M-theory
- 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 18, 2020
TITLE: Codimension seven revisited
ABSTRACT:Riemannian manifolds with special holonomy are ideal spaces on which to compactify M-theory, since the covariantly-constant spinors typical of such spaces give rise to supersymmetry in the effective, lower-dimensional theory.
However, it has long been recognized that other important features of the effective theory (including nonabelian gauge symmetry and massless chiral matter in four dimensions) cannot be realized by compactification on manifolds, and so it has been proposed to compactly M-theory on singular spaces as well. The effective theory in such a case is not derived exclusively from supergravity, but must contain other massless fields such as nonabelian gauge fields or massless matter fields corresponding to physics localized at singularities. One of the challenges is identify the new massless fields representing the new physics, based on the geometry of the singularities.
The case of gauge fields is very well studied and is understood to be derived from ADE singularities in (real) codimension four. The other cases (codimensions six and seven for compactifications to four dimension as well as codimension eight for compactifications to three dimension) are less well understood. Many examples are known, but in examples it is often assumed that the singularity is asymptotically a metric cone, which seems to have been justified on the basis of simplicity rather than on physical grounds.
We will propose a mathematical framework for studying singular spaces which contain a manifold with a metric of special holonomy as a dense open set, and we hope that this framework will capture all the relevant physical phenomena. We also hope that this is a reasonable framework mathematically — for example, one might hope that limits of metrics which are at finite distance from the bulk in a Weil-Peterson type metric would always fall into this class (as is already known for K3 surfaces).
Our main burden will be finding a good formulation for singularities of codimension seven, but we shall also discuss other relevant codimensions (four, six, and eight).
September 10, 2019
TITLE: Nonabelian gauge symmetry and charged chiral matter in G2 compactifications
ABSTRACT: Although M-theory compactified on a manifold of holonomy G2 yields a four-dimensional physical theory with many desirable properties, two important physical properties — nonabelian gauge symmetry and charged chiral matter — require that the G2 space have singularities (rather than being a manifold). We will explain the reason for this, and describe what is known about the types of singularities required (and thus, how the construction of G2 spaces needs to be extended beyond the manifold case).
January 7, 2019
TITLE: Degenerations of complex structures on K3 surfaces, following Kulikov, Persson, and Pinkham
ABSTRACT: We shall review the classical theory of deformations and degenerations of complex structure on K3 surfaces, ignoring metrical aspects of the problem. Our talk will follow the 1977 Izvestija paper of Kulikov, the 1981 Annals paper of Persson and Pinkham, and the author’s PhD thesis, published in the Duke Math Journal in 1981.
January 7, 2019
TITLE: Supersymmetric torus fibrations of Calabi-Yau threefolds
ABSTRACT: We shall review the original Strominger-Yau-Zaslow (SYZ) conjecture, early work by Zharkov, Gross, and W.D. Ruan on the subject, and some results of Joyce. Together, these considerations lead us to a refined version of the original SYZ conjecture (which is somewhat orthogonal to the approach taken by Gross and Siebert). We explain how our refined conjecture is related to Batyrev’s combinatorial version of mirror symmetry for hypersurfaces in toric varieties, and to the Batyrev-Borisov extension of combinatorial mirror symmetry to complete intersections in toric varieties. This talk will be based on arXiv:1002.4921 and arXiv:1504.08337
September 9, 2018
TITLE: spaces with K3 fibrations
ABSTRACT: Many manifolds and singular spaces with holonomy have fibrations by K3 surfaces. These fibrations are particularly useful when the K3 surfaces are calibrated by the 4-form. We will explore a number of properties and applications of this phenomenon.
April 13, 2018
TITLE: Collapsing metrics in string theory and M-theory
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?