A proposed "South-East rule" reads the directions of edges in a Borel-plane adjacency graph of critical values to decide, without steepest-descent flow computations, which complex and real saddles contribute to multidimensional oscillatory integrals.
Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations
5 Pith papers cite this work. Polarity classification is still indexing.
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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.
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Scalar-channel quasinormal modes of the planar AdS5 black brane are captured across all wave numbers by exact WKB quantisation, transseries resummation, and Seiberg–Witten analytic continuation, with resummed large-q predictions matching independent numerics to ten to thirty decimal places.
Exact saddles and finite-T Picard-Lefschetz contour integrals over quasi-zero modes encode the full resurgent structure and yield non-perturbative splittings for every energy level in the double well.
Exact WKB analysis produces median-summed spectra and an algebraic equation for the exceptional point of PT-symmetry breaking in the inverted triple-well system.
Presents explicit trans-series calculations for the double-well spectrum via exact WKB and path integral approaches up to four-instanton level.
citing papers explorer
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Which Saddles Contribute? The South-East Rule for Multidimensional Integrals
A proposed "South-East rule" reads the directions of edges in a Borel-plane adjacency graph of critical values to decide, without steepest-descent flow computations, which complex and real saddles contribute to multidimensional oscillatory integrals.
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Analytic approaches to perturbations of strongly coupled Yang-Mills plasma
Scalar-channel quasinormal modes of the planar AdS5 black brane are captured across all wave numbers by exact WKB quantisation, transseries resummation, and Seiberg–Witten analytic continuation, with resummed large-q predictions matching independent numerics to ten to thirty decimal places.
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Beyond the Dilute Instanton Gas: Resurgence with Exact Saddles in the Double Well
Exact saddles and finite-T Picard-Lefschetz contour integrals over quasi-zero modes encode the full resurgent structure and yield non-perturbative splittings for every energy level in the double well.
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Exact WKB analysis of inverted triple-well: resonance, PT-symmetry breaking, and resurgence
Exact WKB analysis produces median-summed spectra and an algebraic equation for the exceptional point of PT-symmetry breaking in the inverted triple-well system.
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The Double Well Done Doubly-Well
Presents explicit trans-series calculations for the double-well spectrum via exact WKB and path integral approaches up to four-instanton level.