Resurgence Matches Quantization

Abstract

The quest to find a nonperturbative formulation of topological string theory has recently seen two unrelated developments. On the one hand, via quantization of the mirror curve associated to a toric Calabi-Yau background, it has been possible to give a nonperturbative definition of the topological-string partition function. On the other hand, using techniques of resurgence and transseries, it has been possible to extend the string (asymptotic) perturbative expansion into a transseries involving nonperturbative instanton sectors. Within the specific example of the local P2 toric Calabi-Yau threefold, the present work shows how the Borel-Pade-Ecalle resummation of this resurgent transseries, alongside occurrence of Stokes phenomenon, matches the string-theoretic partition function obtained via quantization of the mirror curve. This match is highly non-trivial, given the unrelated nature of both nonperturbative frameworks, signaling at the existence of a consistent underlying structure.

arXiv comment35 pages, 15 figures; v2: 36 pages, minor changes, final version in JPAMT

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• A Primer on Resurgent Transseries and Their Asymptotics
• TBA equations and resurgent Quantum Mechanics
• Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
• Fast Summation of Divergent Series and Resurgent Transseries in Quantum Field Theories from Meijer-G Approximants
• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
• Peacock patterns and new integer invariants in topological string theory
• The large proper-time expansion of Yang-Mills plasma as a resurgent transseries
• Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$
• Mathematical structures of non-perturbative topological string theory: from GW to DT invariants
• Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics
• Exact multi-instantons in topological string theory
• On the resurgent structure of quantum periods
• Calabi-Yau geometry and electrons on 2d lattices
• Phase structure of the twisted $SU(3)/U(1{)}^2$ flag sigma model on $\mathbb{R}\times S^1$
• The resurgent structure of quantum knot invariants
• Bands and gaps in Nekrasov partition function
• Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Topological String Geometry
• Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries
• Resurgence of one-point functions in a matrix model for 2D type IIA superstrings
• Large N instantons from topological strings
• Two-point functions at arbitrary genus and its resurgence structure in a matrix model for 2D type IIA superstrings
• Resurgence in complex Chern-Simons theory at generic levels