Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations

Abstract

For a wide variety of quantum potentials, including the textbook `instanton' examples of the periodic cosine and symmetric double-well potentials, the perturbative data coming from fluctuations about the vacuum saddle encodes all non-perturbative data in all higher non-perturbative sectors. Here we unify these examples in geometric terms, arguing that the all-orders quantum action determines the all-orders quantum dual action for quantum spectral problems associated with a classical genus one elliptic curve. Furthermore, for a special class of genus one potentials this relation is particularly simple: this class includes the cubic oscillator, symmetric double-well, symmetric degenerate triple-well, and periodic cosine potential. These are related to the Chebyshev potentials, which are in turn related to certain $\mathcal{N}=2$ supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and also to topological $c=3$ Landau-Ginzburg models and `special geometry'. These systems inherit a natural modular structure corresponding to Ramanujan's theory of elliptic functions in alternative bases, which is especially important for the quantization. Insights from supersymmetric quantum field theory suggest similar structures for more complicated potentials, corresponding to higher genus. Our approach is very elementary, using basic classical geometry combined with all-orders WKB.

50 pages, 3 figures

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• Hofstadter's Butterfly in Quantum Geometry
• RG-Whitham dynamics and complex Hamiltonian systems
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• Modular anomaly equations in N=2* theories and their large-N limit
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• Calabi-Yau geometry and electrons on 2d lattices
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Cited by in corpus (52)

• A Primer on Resurgent Transseries and Their Asymptotics
• Solar structure and evolution
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• Deconstructing zero: resurgence, supersymmetry and complex saddles
• Non-perturbative approaches to the quantum Seiberg-Witten curve
• Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb{C}{P}^{N-1}$ models
• On exact-WKB analysis, resurgent structure, and quantization conditions
• Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on $S^1$
• Argyres-Douglas theories, Painlevé II and quantum mechanics
• TBA Equations and Quantization Conditions
• TBA equations for the Schrödinger equation with a regular singularity
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Bands and gaps in Nekrasov partition function
• Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories
• Quantum periods and prepotential in $\mathcal{N}=2$ SU(2) SQCD
• Exact WKB analysis and TBA equations for the Mathieu equation
• Exact WKB analysis for $\mathcal{P}\mathcal{T}$ symmetric quantum mechanics: Study of the Ai-Bender-Sarkar conjecture
• Quantum periods for $\mathcal{N}=2$ $SU(2)$ SQCD around the superconformal point
• Resurgence, Painleve Equations and Conformal Blocks
• Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice
• Non-Perturbative String Theory from AdS/CFT
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
• On Chebyshev Wells: Periods, Deformations, and Resurgence
• On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues
• The Picard-Fuchs equation in classical and quantum physics: Application to higher-order WKB method
• Resumming perturbative series in the presence of monopole bubbling effects
• Chiral trace relations in $\Omega$-deformed $\mathcal{N}=2$ theories
• Exact instanton transseries for quantum mechanics
• Resurgence and semiclassical expansion in two-dimensional large-$N$ sigma models
• Resummed Wentzel-Kramers-Brillouin Series: Quantization and Physical Interpretation
• Unified genus-1 potential and a parametric perturbative/nonperturbative relation
• Resurgence of one-point functions in a matrix model for 2D type IIA superstrings
• All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability
• Cosmological Weyl-Einsteinian-Cubic Gravity as a Gauge Theory of Gravity
• Orbital Bifurcations and Shoaling of Cnoidal Waves
• Dynamics of ion channels via non-Hermitian quantum mechanics
• Aspects of Hecke Symmetry I: Ramanujan Identities and Inversion Formulas
• Two-point functions at arbitrary genus and its resurgence structure in a matrix model for 2D type IIA superstrings
• Quantum Mechanics from General Relativity and the Quantum Friedmann Equation
• Thermodynamic Bethe ansatz and wall crossing for deformed supersymmetric quantum mechanics
• Quantum Hamilton-Jacobi Theory, Spectral Path Integrals and Exact-WKB Analysis
• TBA equations and quantum periods for D-type Argyres-Douglas theories
• Resurgence of deformed genus-1 curves: A novel perturbative/nonperturbative relation
• The Toda system and solution to the $\mathcal{N}$=2 SUSY Yang-Mills theory
• Exact WKB in all sectors I: Potentials with degenerate saddles
• Recursive Generation of The Semi-Classical Expansion in Arbitrary Dimension
• Exact-WKB Analysis of Two-level Floquet Systems
• Exact WKB in all sectors II: Potentials with non-degenerate saddles
• Pair Production in Real Proper Time and Unitarity Without Borel Ambiguity
• Triangle Groups: Automorphic Forms and Nonlinear Differential Equations