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An Introduction to Resurgence, Trans-Series and Alien Calculus
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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.
Forward citations
Cited by 10 Pith papers
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On the $1/c$ expansion in $2d$ CFTs with degenerate operators
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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Critical dimensions and small cycle dominance from all-orders asymptotics of $d$-matrix theory
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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Resurgent Lambert series with characters
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.
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Black Hole Entropy, Quantum Corrections and EFT Transitions
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...
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The complete trans-series for conserved charges in integrable field theories
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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From Fredholm Determinants to AdS/CFT Observables: A Universal Strong-Coupling Framework
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.
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Strong coupling structure of $\mathcal{N}=4$ SYM observables with matrix Bessel kernel
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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Resurgence for large $c$ expansion in Coulomb gas formalism
The two conformal blocks of the phi21 four-point function are each other's Stokes corrections in the large-C expansion, and their z-monodromy is governed by alien calculus in the Borel plane.
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Nonperturbative features in the Lie-algebraic K\"ahler sigma model with fermions
In the deformed CP1 quantum mechanics with fermions, the nonperturbative ambiguity structure of the ground state energy persists, with the elongation parameter k conjectured to enter at three loops through g^4(k^2-1).
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Coherent states versus Glauber-Sudarshan States: Bootstrapping, Schwinger-Keldysh Contours and Lefschetz Thimbles
Quasi-de Sitter spacetimes can be realized as Glauber-Sudarshan excited states built over supersymmetric Minkowski vacua, held consistent by a bootstrap self-consistency equation rather than by a potential minimum.
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