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An Introduction to Resurgence, Trans-Series and Alien Calculus

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arxiv 1411.3585 v2 pith:TXZS2ODF submitted 2014-11-13 hep-th

classification hep-th
keywords alienresurgencetheorycalculusfunctionsphysicalphysicsresurgent
verification ladder T0 review T1 audit T2 compute T3 formal

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In these notes we give an overview of different topics in resurgence theory from a physics point of view, but with particular mathematical flavour. After a short review of the standard Borel method for the resummation of asymptotic series, we introduce the class of simple resurgent functions, explaining their importance in physical problems. We define the Stokes automorphism and the alien derivative and discuss these objects in concrete examples using the notion of trans-series expansion. With all the tools introduced, we see how resurgence and alien calculus allow us to extract non-perturbative physics from perturbation theory. To conclude, we apply Morse theory to a toy model path integral to understand why physical observables should be resurgent functions.

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Cited by 10 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the $1/c$ expansion in $2d$ CFTs with degenerate operators

    hep-th 2024-12 accept novelty 8.0 of 10

    The all-order 1/c expansion of Virasoro blocks for degenerate operators is Borel resummable except across Stokes lines, and the resurgent analysis reconstructs the full correlator and resolves forbidden singularities.

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    The weighted partition numbers of d-matrix theory admit an all-orders asymptotic expansion that switches from divergent to convergent at d=13 (bosonic) or 7 (fermionic).

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    A complete transseries expansion is derived for doubly Dirichlet-twisted Lambert series near q=1, with applications to quantum modularity and topological string spectral traces.

  4. Black Hole Entropy, Quantum Corrections and EFT Transitions

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    Quantum corrections to BPS black hole entropy in 4d N=2 string compactifications can be resummed into a finite formula that interpolates between four- and five-dimensional EFT descriptions and matches exact 5d microst...

  5. The complete trans-series for conserved charges in integrable field theories

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    A unified Wiener-Hopf construction gives all-order trans-series for conserved charges in O(N), SU(N), Lieb-Liniger, Gaudin-Yang and capacitor models, with numerically checked Borel resummation.

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    The O(6) mass gap's strong-coupling trans-series is generated from Fredholm-determinant data via a conjectured alien calculus, yielding an all-orders relation to the cusp anomalous dimension.

  7. Strong coupling structure of $\mathcal{N}=4$ SYM observables with matrix Bessel kernel

    hep-th 2026-02 unverdicted novelty 6.0 of 10

    The strong-coupling transseries for matrix Bessel determinant observables is generated from its perturbative part by shifting a→a−Δ and replacing moments I_n, with all Stokes constants fixed by two recurrences.

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