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Let $X$ be a Calabi-Yau threefold with an elliptic fibration. We investigate the non-linear Riemann-Hilbert problems associated to the Donaldson-Thomas theory of fibre classes, and relate them to the Borel sum of the $A$-model topological string free energy for such classes., Comment: 36 pages, 1 figure

Joyce structures were introduced by T. Bridgeland in the context of the space of stability conditions of a three-dimensional Calabi-Yau category and its associated Donaldson-Thomas invariants. In subsequent work, T. Bridgeland and I. Strachan showed that Joyce structures satisfying a certain non-degeneracy condition encode a complex hyperk\"{a}hler structure on the tangent bundle of the base of the Joyce structure. In this work we give a definition of an analogous structure over an affine special K\"{a}hler (ASK) manifold, which we call a special Joyce structure. Furthermore, we show that it encodes a real hyperk\"{a}hler (HK) structure on the tangent bundle of the ASK manifold, possibly of indefinite signature. Particular examples include the semi-flat HK metric associated to an ASK manifold (also known as the rigid c-map metric) and the HK metrics associated to certain uncoupled variations of BPS structures over the ASK manifold. Finally, we relate the HK metrics coming from special Joyce structures to HK metrics on the total space of algebraic integrable systems., Comment: 31 pages. v2: Definition of special Joyce structure generalized to allow for a complex-valued J. Main result unchanged. Appendix B and references added

Given a conical affine special K\"{a}hler (CASK) manifold together with a compatible mutually local variation of BPS structures, one can construct a quaternionic-K\"{a}hler (QK) manifold. We call the resulting QK manifold an instanton corrected c-map space. Our main aim is to study the isometries of a subclass of instanton corrected c-map spaces associated to projective special real (PSR) manifolds with a compatible mutually local variation of BPS structures. We call the latter subclass instanton corrected q-map spaces. In the setting of Calabi-Yau compactifications of type IIB string theory, instanton corrected q-map spaces are related to the hypermultiplet moduli space metric with perturbative corrections, together with worldsheet, D(-1) and D1 instanton corrections. In the physics literature, it has been shown that the hypermultiplet metric with such corrections must have an $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries, related to S-duality. We give a mathematical treatment of this result, specifying under which conditions instanton corrected q-map spaces carry an action by isometries by $\mathrm{SL}(2,\mathbb{Z})$ or some of its subgroups. We further study the universal isometries of instanton corrected q-map spaces, and compare them to the universal isometries of tree-level q-map spaces. Finally, we give an explicit example of a non-trivial instanton corrected q-map space with full $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries and admitting a quotient of finite volume by a discrete group of isometries., Comment: 56 pages

We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov-Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson-Thomas invariants of the underlying Calabi-Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed setting. This guides us to a five-dimensional extension of the Nekrasov-Rosly-Shatashvili proposal, in which the NS free energy is computed as a generating function of $q$-difference opers in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system.

The tree-level q-map assigns to a projective special real (PSR) manifold of dimension $n-1\geq 0$, a quaternionic K\"{a}hler (QK) manifold of dimension $4n+4$. It is known that the resulting QK manifold admits a $(3n+5)$-dimensional universal group of isometries (i.e. independently of the choice of PSR manifold). On the other hand, in the context of Calabi-Yau compactifications of type IIB string theory, the classical hypermultiplet moduli space metric is an instance of a tree-level q-map space, and it is known from the physics literature that such a metric has an $\mathrm{SL}(2,\mathbb{R})$ group of isometries related to the $\mathrm{SL}(2,\mathbb{Z})$ S-duality symmetry of the full 10d theory. We present a purely mathematical proof that any tree-level q-map space admits such an $\mathrm{SL}(2,\mathbb{R})$ action by isometries, enlarging the previous universal group of isometries to a $(3n+6)$-dimensional group $G$. As part of this analysis, we describe how the $(3n+5)$-dimensional subgroup interacts with the $\mathrm{SL}(2,\mathbb{R})$-action, and find a codimension one normal subgroup of $G$ that is unimodular. By taking a quotient with respect to a lattice in the unimodular group, we obtain a quaternionic K\"ahler manifold fibering over a projective special real manifold with fibers of finite volume, and compute the volume as a function of the base. We furthermore provide a mathematical treatment of results from the physics literature concerning the twistor space of the tree-level q-map space and the holomorphic lift of the $(3n+6)$-dimensional group of universal isometries to the twistor space., Comment: 43 pages. Typos fixed, references added, and improved presentation based on the comments of the reviewer

We study the Borel summation of the Gromov-Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson-Thomas invariants of the resolved conifold, having a direct relation to the Riemann-Hilbert problem formulated by T. Bridgeland. There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann-Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by D. Jafferis and G. Moore., Comment: 65 pages, 1 figure. Typos fixed and references added. Discussion in Section 2.2 and 6.1 expanded. Definition 4.7 (Def. 28 in v1) slightly modified, strengthening the previous Prop. 40 of v1 to the new Prop. 4.20 of v2. Corollary 3.13, 4.8 and Lemma B.2 added

We associate to the resolved conifold an affine special K\"{a}hler (ASK) manifold of complex dimension 1, and an instanton corrected hyperk\"{a}hler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland., Comment: 49 pages, references added and typos fixed. Previous theorem 43 and corollary 44 revised, and the rest of the results unchanged

We construct a quaternionic-K\"ahler manifold from a conical special K\"ahler manifold with a certain type of mutually-local variation of BPS structures. We give global and local explicit formulas for the quaternionic-K\"ahler metric, and specify under which conditions it is positive-definite. Locally, the metric is a deformation of the 1-loop corrected Ferrara-Sabharval metric obtained via the supergravity c-map. The type of quaternionic-K\"ahler metrics we obtain are related to work in the physics literature by S. Alexandrov and S. Banerjee, where they discuss the hypermultiplet moduli space metric of type IIA string theory, with mutually local D-instanton corrections., Comment: 31 pages. Typos fixed and improved presentation based on the suggestions by the referees. Final version to appear in Annales Henri Poincar\'e

The Ooguri-Vafa space is a 4-dimensional incomplete hyperk\"{a}hler manifold, defined on the total space of a singular torus fibration with one singular nodal fiber. It has been proposed that the Ooguri-Vafa hyperk\"ahler metric should be part of the local model of the hyperk\"{a}hler metric of the Hitchin moduli spaces, near the most generic kind of singular locus of the Hitchin fibration. In order to relate the Ooguri-Vafa space with the Hitchin moduli spaces, we show that the Ooguri-Vafa space can be interpreted as a set of rank 2, framed wild harmonic bundles over $\mathbb{C}P^1$, with one irregular singularity. Along the way we show that a certain twistor family of holomorphic Darboux coordinates, which describes the hyperk\"{a}hler geometry of the Ooguri-Vafa space, has an interpretation in terms of Stokes data associated to our framed wild harmonic bundles., Comment: 61 pages, 7 figures. Improved introduction. Typos fixed and references added. Based on PhD thesis