In this article the cohomological Donaldson-Thomas theory of local multibanana threefolds is computed using Descombes' hyperbolic localisation formula. The resulting expression is precisely given by the elliptic genus of the moduli space of framed instanton sheaves. As part of the proof, an infinite-wedge-space trace formula is computed for the elliptic genus of the moduli space of framed instanton sheaves.

We prove a blow-up formula for the generating series of virtual χ-genera for moduli spaces of sheaves on projective surfaces, which is related to a conjectured formula for topological χ-genera of Göttsche. Our formula is a refinement of one by Vafa-Witten relating to S-duality. We prove the formula simultaneously in the setting of Gieseker stable sheaves on polarised surfaces and also in the setting of framed sheaves on ℙ². The proof is based on the blow-up algorithm of Nakajima-Yoshioka for framed sheaves on ℙ², which has recently been extend to the setting of Gieseker -stable sheaves on -polarised surfaces by Kuhn-Tanaka.

In this article we give an explicit construction of the moduli space of trigonal superelliptic curves with level 3 structure. The construction is given in terms of point sets on the projective line and leads to a closed formula for the number of connected (and irreducible) components of the moduli space. The results of the article generalise the description of the moduli space of hyperelliptic curves with level 2 structure, due to Dolgachev and Ortland, Runge and Tsuyumine.

The moduli space of stable maps with divisible ramification uses -th roots of a canonical ramification section to parametrise stable maps whose ramification orders are divisible by a fixed integer . In this article, a virtual fundamental class is constructed while letting domain curves have a positive genus; hence removing the restriction of the domain curves being genus zero. We apply the techniques of virtual localisation and obtain the -ELSV formula as an intersection of the virtual class with a pullback via the branch morphism.

We further the study of the Donaldson-Thomas theory of the banana threefolds which were recently discovered and studied by Bryan in [Bryan'19]. These are smooth proper Calabi-Yau threefolds 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 [Bryan'19] 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 threefold.

We develop a theory for stable maps to curves with divisible ramification. For a fixed integer , we show that the condition of every ramification locus being divisible by is equivalent to the existence of an -th root of a canonical section. We consider this condition in regards to both absolute and relative stable maps and construct natural moduli spaces in these situations. We construct an analogue of the Fantechi-Pandharipande branch morphism and when the domain curves are genus zero we construct a virtual fundamental class. This theory is anticipated to have applications to r-spin Hurwitz theory. In particular it is expected to provide a proof of the -spin ELSV formula [SSZ'15, Conj. 1.4] when used with virtual localisation.

We prove that a generalisation of simple Hurwitz numbers due to Johnson, Pandharipande and Tseng satisfies the topological recursion of Eynard and Orantin. This generalises the Bouchard-Mariño conjecture and places Hurwitz-Hodge integrals, which arise in the Gromov-Witten theory of target curves with orbifold structure, in the context of the Eynard-Orantin topological recursion.