Mathematical structures of non-perturbative topological string theory: from GW to DT invariants

Abstract

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.

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

References in corpus (14)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• The quantum dilogarithm and representations quantum cluster varieties
• Non-commutative Donaldson-Thomas theory and the conifold
• Nonperturbative corrections to 4D string theory effective actions from SL(2,Z) duality and supersymmetry
• Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2
• D-instantons and twistors
• Wall crossing in local Calabi Yau manifolds
• Peacock patterns and new integer invariants in topological string theory
• D-instantons in Type IIA string theory on Calabi-Yau threefolds
• Hypermultiplet metric and D-instantons
• Conformal TBA for resolved conifolds
• Quaternionic Kähler metrics associated to special Kähler manifolds with mutually local variations of BPS structures
• A Hyperkähler geometry associated to the BPS structure of the resolved conifold
• Opers and TBA

Cited by in corpus (13)

• Exact multi-instantons in topological string theory
• On the resurgent structure of quantum periods
• Quantum Curves, Resurgence and Exact WKB
• Exponential Networks, WKB and Topological String
• Resurgent Structure of the Topological String and the First Painlevé Equation
• Resurgence of Refined Topological Strings and Dual Partition Functions
• Resurgence in complex Chern-Simons theory at generic levels
• Resurgence, Stokes constants, and arithmetic functions in topological string theory
• Resurgence and Riemann-Hilbert problems for elliptic Calabi-Yau threefolds
• Symplectic cuts and open/closed strings I
• Les Houches lectures on non-perturbative Seiberg-Witten geometry
• Gromov-Witten/Hilbert versus AdS3/CFT2 Correspondence
• GW/DT invariants and 5D BPS indices for strips from topological recursion