Final Report Summary - AN D-T INVARIANTS (Analytic Donaldson-Thomas invariants)
In the first year of the project, rather limited progress was made towards the original goal of defining analytic Donaldson-Thomas type invariants of Calabi-Yau 3-folds through "counting" solutions of a gauge-theoretic equation called "Donaldson-Thomas instantons", and showing that these invariants are unchanged under deformation of the Calabi-Yau 3-fold, and (even stronger) depend only on the underlying symplectic manifold.
Something we believe we have understood is that the functional we have been using is not the right one to analyze D-T instantons and compactify the moduli space, and the Fellow continues to search for a new formulation.
In the second year of the project, it was decided to change direction, and study a related but easier problem. "Donaldson-Thomas instantons" are a partial differential equation in 6 dimensions, but they are a dimensional reduction of another partial differential equation in 8 dimensions, known as "Spin (7) instantons". These are solutions of a gauge-theoretic equation on 8-manifolds with holonomy Spin (7).
Analytic properties of Spin (7) instantons, and related equations, have been studied by Gang Tian and others. However, little was known about the existence of examples of families of Spin (7) instantons on compact 8-manifolds with holonomy Spin (7); the only reference is an unpublished 1998 Oxford PhD thesis by Christopher Lewis.
There are two known constructions of compact 8-manifolds with holonomy Spin (7), both due to the Host Scientist Dominic Joyce. The first begins with a special kind of torus orbifold T^8/G, with non-isolated singularities, and resolves the singularities using algebraic geometry techniques. The second begins with a Calabi-Yau 4-orbifold with isolated singularities, divides it by an antiholomorphic involution fixing only these singularities, and then resolves the singularities.
Christopher Lewis (PhD thesis, unpublished) studied the problem of constructing Spin (7) instantons on Spin (7) manifolds coming from Joyce's first construction. Many difficulties were caused by the extended nature of the singularties of the torus orbifold T^8/G.
During the second half of the fellowship, the Fellow studied the problem of constructing examples of families of Spin (7) instantons on compact 8-manifolds coming from Joyce's second construction. This is a nicer problem than that attempted by Lewis, largely because the singularities being resolved during Joyce's second construction are isolated.
The project was successful: the Fellow produced a general construction of Spin (7) instantons on Joyce's examples by a method of "simultaneous gluing": Joyce's compact Spin (7) manifolds are made by "gluing" together several noncompact pieces coming from Kahler geometry. To make examples of Spin (7) instantons on these, the Fellow started with examples (coming from of Kahler geometry, and results on solutions of Hermitian-Einstein equations) of Spin (7) instantons on each piece, with matching conditions on the gluing annuli, and showed that these Spin (7) instantons can be "glued" together to make a Spin (7) instanton on Joyce's compact 8-manifold. The Fellow also gave simple examples of the construction, yielding families of Spin (7) instantons with gauge group SU (2).
The project was written up in a paper "A construction of Spin (7) instantons", submitted to a journal, and available on the mathematics archive at http://arXiv. org as arXiv: 1201. 3150.
Something we believe we have understood is that the functional we have been using is not the right one to analyze D-T instantons and compactify the moduli space, and the Fellow continues to search for a new formulation.
In the second year of the project, it was decided to change direction, and study a related but easier problem. "Donaldson-Thomas instantons" are a partial differential equation in 6 dimensions, but they are a dimensional reduction of another partial differential equation in 8 dimensions, known as "Spin (7) instantons". These are solutions of a gauge-theoretic equation on 8-manifolds with holonomy Spin (7).
Analytic properties of Spin (7) instantons, and related equations, have been studied by Gang Tian and others. However, little was known about the existence of examples of families of Spin (7) instantons on compact 8-manifolds with holonomy Spin (7); the only reference is an unpublished 1998 Oxford PhD thesis by Christopher Lewis.
There are two known constructions of compact 8-manifolds with holonomy Spin (7), both due to the Host Scientist Dominic Joyce. The first begins with a special kind of torus orbifold T^8/G, with non-isolated singularities, and resolves the singularities using algebraic geometry techniques. The second begins with a Calabi-Yau 4-orbifold with isolated singularities, divides it by an antiholomorphic involution fixing only these singularities, and then resolves the singularities.
Christopher Lewis (PhD thesis, unpublished) studied the problem of constructing Spin (7) instantons on Spin (7) manifolds coming from Joyce's first construction. Many difficulties were caused by the extended nature of the singularties of the torus orbifold T^8/G.
During the second half of the fellowship, the Fellow studied the problem of constructing examples of families of Spin (7) instantons on compact 8-manifolds coming from Joyce's second construction. This is a nicer problem than that attempted by Lewis, largely because the singularities being resolved during Joyce's second construction are isolated.
The project was successful: the Fellow produced a general construction of Spin (7) instantons on Joyce's examples by a method of "simultaneous gluing": Joyce's compact Spin (7) manifolds are made by "gluing" together several noncompact pieces coming from Kahler geometry. To make examples of Spin (7) instantons on these, the Fellow started with examples (coming from of Kahler geometry, and results on solutions of Hermitian-Einstein equations) of Spin (7) instantons on each piece, with matching conditions on the gluing annuli, and showed that these Spin (7) instantons can be "glued" together to make a Spin (7) instanton on Joyce's compact 8-manifold. The Fellow also gave simple examples of the construction, yielding families of Spin (7) instantons with gauge group SU (2).
The project was written up in a paper "A construction of Spin (7) instantons", submitted to a journal, and available on the mathematics archive at http://arXiv. org as arXiv: 1201. 3150.