Resurgent Duality in Asymptotic Analysis

• Resurgent duality is a framework linking asymptotic series with nonperturbative sectors through resurgent transseries, Borel singularities, and Stokes automorphisms.
• It provides a systematic method to decode large-order perturbative growth into low-order instanton data and resolves ambiguities via median resummation.
• Applications span topological string theory, quantum spectral problems, and classical soliton systems, offering unified insights across diverse analytical regimes.

Resurgent duality denotes a family of precise correspondences in which asymptotic perturbative data and nonperturbative sectors determine one another through resurgent transseries, Borel singularities, alien derivatives, and Stokes automorphisms. In current usage, the term appears in several closely related settings: the dual encoding between large-order perturbative growth and instanton sectors; the transport of resurgent structures across distinct limits of moduli space in B-model topological string theory; holographic matching between matrix-model and topological-string transseries; explicit weak–strong coefficient maps between asymptotic regimes; natural-boundary crossing for $q$-series and complex Chern–Simons theory; and electric–magnetic or BPS/non-BPS transseries correspondences in classical soliton problems. Taken together, these usages suggest a structurally unified theme rather than a single theorem: resurgence turns apparently separate asymptotic sectors into mutually constrained descriptions of the same analytic object (Li et al., 17 Mar 2025, Couso-Santamaría et al., 2013, Dunne et al., 25 Jun 2026, Adams et al., 20 May 2025, Dunne et al., 19 Feb 2026).

1. Formal structure and defining mechanisms

The standard formal setting is a transseries, obtained by augmenting an asymptotic power series with exponentially small sectors. A canonical one-action form is

$$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$

with $\sigma$ an instanton-counting parameter and $A$ an instanton action. In multi-parameter problems, sectors are indexed by $\boldsymbol{n}\in\mathbb{N}_0^\kappa$, actions are additive, and resonance can force logarithmic sectors. A formal series is $\Omega$-resurgent when its Borel transform has positive radius of convergence and admits analytic continuation along paths avoiding a discrete singular set $\Omega$; in the simple-resurgent class, Borel singularities have the “simple pole + logarithm” form and are stable under convolution and alien calculus (Couso-Santamaría et al., 2013, Li et al., 17 Mar 2025).

The operative objects are alien derivatives $\Delta_\omega$, which probe the singularity at a Borel point $\omega$, and the Stokes automorphism, which exponentiates alien data along a Stokes direction. In Écalle’s framework,

$$\mathfrak{S}_\theta=\exp\Bigg(\sum_{\omega \in \text{rays at }\theta} e^{-\omega z}\Delta_\omega\Bigg),$$

while bridge equations identify alien derivatives with ordinary derivatives in transseries parameters. In the multi-parameter setting this takes the schematic form

$$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$0

Resurgent duality is therefore not merely asymptotic similarity: it is an algebraic identification between Borel-plane singularity data and differential motion on the transseries manifold (Aniceto et al., 2013, Aniceto et al., 2018).

A recurrent notion in recent topological-string work is the formal integral: a multi-parameter resurgent transseries solution to a nonlinear ODE, built by exponentiating alien derivations acting on the unique perturbative solution. In this formulation, the perturbative sector is not an isolated approximation but the seed from which the full nonperturbative completion is generated (Li et al., 17 Mar 2025).

2. Large-order/nonperturbative equivalence and ambiguity cancellation

The most common meaning of resurgent duality is the equivalence between large-order growth in one sector and low-order data in adjacent nonperturbative sectors. In topological-string normalization, a representative large-order formula is

$$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$1

This formula exhibits the characteristic duality: factorial growth of perturbative coefficients is governed by instanton actions $$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$2, Stokes constants $$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$3, and low-loop data $$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$4. Conversely, once the nonperturbative sectors are known, the Stokes data and asymptotic behavior of the perturbative coefficients are fixed (Couso-Santamaría et al., 2013).

Because Borel resummation along a Stokes ray is ambiguous, resurgent duality also identifies contour-choice ambiguity with transseries-parameter ambiguity. The canonical resolution is median resummation,

$$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$5

which produces real, ambiguity-free observables when the Stokes data and transseries parameters satisfy the appropriate reality constraints. In one- and multi-parameter examples, the half-Stokes jump in parameters cancels the imaginary ambiguities of lateral Borel sums sector by sector (Aniceto et al., 2013).

In genus-one quantum spectral problems this correspondence acquires a geometric form. For Schrödinger systems whose classical spectral curve is elliptic, the all-orders quantum action $$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$6 determines the all-orders quantum dual action $$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$7. The relation can be written as a quantum Wronskian, for example in the Mathieu normalization,

$$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$8

For the periodic cosine, symmetric double-well, symmetric degenerate triple-well, and cubic oscillator, perturbative fluctuations about the vacuum therefore encode the fluctuations about higher instanton sectors to all orders. This is a constructive perturbative/nonperturbative dictionary rather than a heuristic analogy (Basar et al., 2017).

3. Holographic and moduli-space dualities in topological strings

A major topological-string realization of resurgent duality originates in large-$$F(x,\sigma)=\sum_{n\ge 0}\sum_{g\ge 0}\sigma^n \exp\!\Big(-\frac{nA}{x}\Big)\,F^{(n)}_g\,x^g,$$9 duality. In the Dijkgraaf–Vafa setting, Hermitian matrix models map to spectral curves and then to local Calabi–Yau threefolds; on the closed-string side, the Bershadsky–Cecotti–Ooguri–Vafa holomorphic anomaly equations reproduce the perturbative genus expansion and, in their nonperturbative extension, determine the full resurgent transseries. Instanton actions are holomorphic functions of the moduli, identified with special-geometry periods, while anti-holomorphic dependence has a universal propagator structure. The result is a holographically dual resurgent transseries on the B-model side that mirrors the matrix-model transseries, including multi-instanton sectors, resonance, logarithms, and large-order relations (Couso-Santamaría et al., 2013).

In the 2025 analysis of B-model universal structures, resurgent duality is formulated in two intertwined senses. The first is perturbative/nonperturbative dual encoding, implemented by alien calculus and explicit bridge equations. The second is dual transport across moduli-space limits. Two universal regimes are studied: the Alim–Yau–Zhou double-scaling limit, governed by the Airy equation, and Couso–Santamaría’s large-radius limit, governed by a universal ODE in an antiholomorphic variable $\sigma$0. In the double-scaling variable $\sigma$1, the bridge equation reads

$\sigma$2

while the large-radius formal integral satisfies the same structure with sign changes induced by the map between variables. The two regimes are related by

$\sigma$3

with

$\sigma$4

This establishes that the Airy resurgent structure is stably transported to the large-radius regime, so that universal Stokes data and bridge equations persist across distinct limits of moduli space (Li et al., 17 Mar 2025).

These topological-string constructions also sharpen a common misconception. The nonperturbative completion is not obtained by adjoining arbitrary exponential corrections to the BCOV hierarchy. Rather, the data are constrained simultaneously by the holomorphic anomaly equations, by holomorphicity of instanton actions, by Stokes automorphisms, and by the compatibility of large-order behavior with boundary conditions such as conifold gap structure (Couso-Santamaría et al., 2013, Li et al., 17 Mar 2025).

4. Refined, Yang–Mills, and BPS enhancements

In refined topological strings, resurgent duality acquires a genuine two-family structure. With

$\sigma$5

the Borel transform of the refined genus expansion at fixed $\sigma$6 has two families of simple poles. In the conifold sector these lie at

$\sigma$7

leading to dual instanton sectors

$\sigma$8

The associated Stokes automorphism is expressed in terms of Faddeev’s non-compact quantum dilogarithm and its spin-refined generalizations, and the Stokes constants are conjecturally the refined Donaldson–Thomas invariants $\sigma$9. The $A$0 exchange is therefore the refined analogue of perturbative/nonperturbative duality, realized directly in the Borel plane and in the wall-crossing action on dual partition functions (Alexandrov et al., 2023).

A distinct but related enhancement occurs in the topological-string dual of $A$1d $A$2 Yang–Mills theory on a torus. There the dominant real instanton tower has actions

$A$3

and the authors derive closed-form instanton amplitudes to arbitrary instanton order, construct the corresponding Stokes automorphisms, and propose a nonperturbative partition function that is real for $A$4 and $A$5. The real-instanton Stokes transform is expressed through complete Bell polynomials, while the Borel plane also exhibits two infinite towers of complex instantons with actions

$A$6

The paper interprets these complex towers as organized by BPS data in type II string theory and argues that its real nonperturbative completion improves on the Okuyama–Sakai proposal because the latter introduces extra sectors not seen in the perturbative sector’s resurgent structure and fails to be exactly real at positive $A$7 and $A$8 (Chen et al., 26 Jul 2025).

5. Weak–strong coefficient duality and Mellin-based transmutation

A different use of the term concerns explicit weak–strong correspondences between asymptotic regimes. In the Mellin-based analysis of catastrophe integrals, the central phenomenon is a duality between a weak expansion with infinite radius of convergence and a strong expansion with zero radius of convergence. The large-order decay of the convergent series and the large-order growth of the divergent series are tied by exact coefficient-level maps obtained from complementary residue sets in inverse Mellin integrals. For the Pearcey reductions the maps are

$A$9

with generalizations

$\boldsymbol{n}\in\mathbb{N}_0^\kappa$0

In Airy, Pearcey, zero-dimensional $\boldsymbol{n}\in\mathbb{N}_0^\kappa$1 Dyson–Schwinger equations, and the Gross–Neveu kink–antikink crystal heat-kernel trace, the same rational subleading coefficients appear on both sides. The duality is therefore not limited to matching leading exponential scales; it matches the full inverse-factorial structure that encodes the Stokes data of the divergent regime (Dunne et al., 25 Jun 2026).

A related Mellin–Barnes framework appears in the analytic realization of indicial umbral calculus. There, resurgent duality is formulated as a topological duality between umbral kernels, encoded by spectral jump functions $\boldsymbol{n}\in\mathbb{N}_0^\kappa$2, and admissible ground states $\boldsymbol{n}\in\mathbb{N}_0^\kappa$3, encoded by Mellin–Barnes pairings. The central spectral transmutation law is

$\boldsymbol{n}\in\mathbb{N}_0^\kappa$4

which lifts to the pairing identity

$\boldsymbol{n}\in\mathbb{N}_0^\kappa$5

In this setting, entire and rational kernels are dual under $\boldsymbol{n}\in\mathbb{N}_0^\kappa$6-regularization, and formal umbral identities become local expansions of a global analytic correspondence controlled by Borel–Laplace and Mellin–Barnes theories (Ricci, 9 Jun 2026).

6. Natural boundaries, $\boldsymbol{n}\in\mathbb{N}_0^\kappa$7-series, and complex Chern–Simons theory

In complex Chern–Simons theory, resurgent duality is formulated as crossing a natural boundary by preservation of resurgent relations. The “resurgent bridge” starts from perturbative resurgent series $\boldsymbol{n}\in\mathbb{N}_0^\kappa$8, extracts their Borel singularities and Stokes data, and reconstructs dual $\boldsymbol{n}\in\mathbb{N}_0^\kappa$9-series across $\Omega$0. In the Mordell–Borel class, one constructs a unique decomposition on the Stokes line into a $\Omega$1-series and a “dual” $\Omega$2-series, then continues to the other side by preserving algebraic and functional relations. For unary $\Omega$3-series, this yields a bijection $\Omega$4 with the same perturbative expansion near $\Omega$5 up to the sign flip $\Omega$6 on odd powers (Costin et al., 2023).

The 2025 orientation-reversal analysis reformulates this construction in terms of Mordell integrals that are analytic across the natural boundary. Writing

$\Omega$7

the unit circle $\Omega$8 corresponds to the imaginary axis in $\Omega$9, while orientation reversal of the $\Omega$0-manifold acts as $\Omega$1, equivalently $\Omega$2. On the Stokes line, the Mordell integrals admit a unique decomposition into real and imaginary parts, identified respectively with perturbative and nonperturbative sectors; these parts are combinations of unary false theta functions in $\Omega$3 and $\Omega$4. Preservation of relations then uniquely transports the same mixing matrices and rational exponents to the other side of the boundary, defining the dual $\Omega$5-series of the orientation-reversed manifold. This procedure recovers order $\Omega$6, $\Omega$7, $\Omega$8, and $\Omega$9 mock theta identities and produces new duals for Brieskorn spheres such as $\Delta_\omega$0 and $\Delta_\omega$1 (Adams et al., 20 May 2025).

The complex Chern–Simons literature also emphasizes algorithmic extraction of nonperturbative data from perturbative coefficients. High-order perturbative coefficients are analyzed by ratio tests, Padé–Borel and Padé–conformal–Borel transforms, and the method of singularity elimination. In surgery examples on hyperbolic twist knots, this procedure identifies Chern–Simons values, adjoint Reidemeister torsions, and Stokes constants from perturbative data alone, and reveals phenomena such as phantom saddles, lattice-like arrays of Borel singularities, and a double-scaling limit that probes $\Delta_\omega$2-surgeries from $\Delta_\omega$3-surgeries (Costin et al., 2023).

7. Classical solitons, BPS limits, and broader scope

In non-BPS monopole and dyon sectors of $\Delta_\omega$4 Yang–Mills–Higgs theory with adjoint Higgs, resurgent duality is realized directly in radial transseries. For $\Delta_\omega$5, the far-field transseries takes the form

$\Delta_\omega$6

with factorially divergent fluctuation factors in $\Delta_\omega$7. A key structural result is that all higher exponential sectors are fixed explicitly in terms of the leading homogeneous solutions, through linear inhomogeneous ODEs and variation-of-parameters integral formulas. In the BPS limit $\Delta_\omega$8, these fluctuation towers truncate: the second-order system reduces to first-order Bogomolny equations, the transseries collapse to convergent sums, and exact electric–magnetic duality is restored. In the dyon problem, electric and magnetic sectors organize into intertwined transseries families with weights $\Delta_\omega$9 and $\omega$0; electric–magnetic interchange reshuffles the weights and homogeneous solutions but preserves the overall resurgent skeleton, while $\omega$1-angle shifts act by redefining the electric transseries parameter and leave the magnetic tower invariant (Dunne et al., 19 Feb 2026).

This usage clarifies that resurgent duality need not be confined to quantum perturbation theory or string-theoretic genus expansions. It can also describe the structural exchange between asymptotic sectors of classical nonlinear ODEs, provided the fields admit transseries, their fluctuation factors are factorially divergent, and the Stokes geometry controls the cancellation of ambiguities. The BPS/non-BPS distinction then appears as a structural change in the transseries itself: first-order integrability suppresses the sources that generate factorial growth, while non-BPS deformations restore the infinite resurgent hierarchy (Dunne et al., 19 Feb 2026).

Across the literature, several open directions recur. They include direct determination of Stokes constants and alien calculus in broader B-model backgrounds, extensions beyond gamma-factorizable Mellin transforms, higher-genus generalizations of the quantum-geometry picture, multidimensional extensions of Mellin–Barnes resurgent pairings, and systematic application of resurgent duality to wider classes of classical solitons, refined wall-crossing problems, and compact Calabi–Yau geometries. These programs are technically distinct, but they share the same guiding idea: perturbative and nonperturbative descriptions are linked by exact resurgent relations, and duality acts at the level of the full transseries rather than at the level of a single expansion (Couso-Santamaría et al., 2013, Dunne et al., 25 Jun 2026, Basar et al., 2017, Dunne et al., 19 Feb 2026).