Abstract
Abstract We further the study of the Donaldson–Thomas theory of the banana 3-folds which were recently discovered and studied by Bryan [3]. These are smooth proper Calabi–Yau 3-folds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a ‘banana configuration’. In [3], the Donaldson–Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article, we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the Pandharipande–Thomas theory for a rational elliptic surface and present new Gopakumar–Vafa invariants for the banana 3-fold.
Highlights
Donaldson–Thomas partition functions Donaldson–Thomas theory provides a virtual count of curves on a 3-fold
Remark 1.2.3 We note that the values given only depend on the quadratic form d := 2d1d2 + 2d1d3 + 2d2d3 − d12 − d22 − d33 appearing in the rank 3 Donaldson–Thomas partition function of [3, Theorem 4]
We will closely follow the method of [5] developed for studying the Donaldson–Thomas theory of local elliptic surfaces
Summary
Donaldson–Thomas partition functions Donaldson–Thomas theory provides a virtual count of curves on a 3-fold. It gives us valuable information about the structure of the 3-fold and has strong links to high-energy physics. We can define the (β, n) Donaldson–Thomas invariant of Y by DTβ,n(Y) = 1 ∩ [Hilbβ,n(Y)]vir. Behrend proved the surprising result in [1] that the Donaldson–Thomas invariants are weighted Euler characteristics of the Hilbert scheme: DTβ,n(Y) = e(Hilbβ,n(Y), ν) := k · e(ν−1(k)). Donaldson–Thomas invariants to be DTβ,n(Y) = e(Hilbβ,n(Y)) These are often closely related to Donaldson–Thomas invariants and their calculation provides insight to the structure of the 3-fold. We define the analogous partition function Z for the unweighted Donaldson–Thomas invariants. We will employ computational techniques developed in [5] for studying Donaldson–Thomas theory of local elliptic surfaces
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