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UBC Theses and Dissertations

An introduction to modern enumerative geometry with applications to the banana manifold Pietromonaco, Stephen


In this masters thesis we survey some foundational material needed for the presentation (in the final chapter) of new results on the Donaldson-Thomas and Gromov-Witten theories of the formal banana manifold. The first half is geared towards understanding stable vector bundles and instantons in Yang-Mills theory, stability conditions on coherent sheaves, applications to D-branes, as well as the standard curve-counting invariants in modern enumerative geometry and their physical interpretations. Much of the second half is devoted to introducing equivariant localization, automorphic forms, and arithmetic lifts, all of which appear in our new results. These are vast topics, but we are particularly interested in equivariant elliptic genera, Jacobi forms and Siegel modular forms, as well as the Maass and Borcherds lifts. In the final chapter, using a theorem of J. Bryan, we show that the Donaldson-Thomas partition function of the formal banana manifold is the formal Borcherds lift of a weight zero weak Jacobi form of matrix index. This weak Jacobi form is the equivariant elliptic genus of ℂ², and the formal Borcherds lift is closely related to a partition function of rank one instantons on ℂ². Assuming the GW/DT correspondence, we then prove that for g ≥ 2, the Gromov-Witten potentials Fg are genus two meromorphic Siegel modular forms of weight 2g - 2 arising as the Maass lift of an explicit weak Jacobi form. With this identification, we observe that the invariances of such Siegel modular forms precisely encode geometric symmetries of the formal banana manifold, as well as a flop symmetry. The Gopakumar-Vafa invariants are encoded non-trivially into the equivariant elliptic genus of ℂ², and depend only on a quadratic form. We observe that they share a number of features with those coming from the Katz-Klemm-Vafa formula.

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