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.
Highlights
One of the most intriguing aspects of resurgent asymptotics [1,2,3,4,5], as applied to quantum theories, is that the characteristic divergence of fluctuations about certain saddle points, such as the perturbative vacuum, may encode detailed information about the global nonperturbative structure of the system
For a special class of genus one potentials this relation is 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 N = 2 supersymmetric quantum field theories, to mirror maps for hypersurfaces in projective spaces, and to topological c = 3 Landau-Ginzburg models and ‘special geometry’
Another aspect of our approach is that we find a deep connection between the allorders P/NP relations (1.5) and (1.6), and Ramanujan’s theory of elliptic functions with respect to alternative bases [93,94,95,96,97,98,99,100,101,102], and extensions to modular functions [103]
Summary
One of the most intriguing aspects of resurgent asymptotics [1,2,3,4,5], as applied to quantum theories, is that the characteristic divergence of fluctuations about certain saddle points, such as the perturbative vacuum, may encode detailed information about the global nonperturbative structure of the system. We stress that this manifestation of resurgence is completely constructive: given a certain number of terms of the expansion of upert( , N ), the expression (1.4) generates a similar number of terms in the fluctuations about the one-instanton sector, Pinst( , N ) These relations propagate throughout the entire trans-series, so that perturbation theory encodes the fluctuations about each nonperturbative sector [17,18,19,20,21,22,23,24,25, 27]. This is because for these genus 1 cases, all expressions for the coefficient functions an(u) and aDn (u) in (2.8), (2.9) reduce to elliptic functions, and we have complete control over the analytic continuations and modular transformations connecting different spectral regions
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