Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations

Basar, Gokce; Dunne, Gerald V.; Unsal, Mithat

doi:10.1007/JHEP05(2017)087

Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations

Gokce Basar
(
- Maryland U., College Park
)
,
Gerald V. Dunne
(
- Connecticut U.
)
,
Mithat Unsal
(
- NCSU, Raleigh
)

Jan 23, 2017

50 pages

Published in:

JHEP 05 (2017) 087

Published: May 16, 2017

e-Print:

1701.06572 [hep-th]

DOI:

10.1007/JHEP05(2017)087

View in:

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Abstract: (Springer)

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.

Note:

50 pages, 3 figures

Nonperturbative Effects
Solitons Monopoles and Instantons
Topological Strings
potential: quantum
field theory
supersymmetry: 2
Landau-Ginzburg model
WKB approximation
quantum geometry
quantization

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