Die elliptischen Funktionen und ihre Anwendungen

Fricke, R.

doi:10.1007/978-3-642-19561-7

Cited by 8 publications

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“…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]. These number theoretic functions are also associated with topological c = 3 Landau-Ginzburg models [104,105] and certain superconformal quantum field theories [106].…”

Section: Jhep05(2017)087mentioning

confidence: 99%

“…This modular structure can be formulated within Ramanujan's theory of elliptic functions in alternative bases [93][94][95][96][97][98][99][100][101][102][103] and it plays an important role in the corresponding quantum theories, as we discuss below in section 4. But here we first review some number theoretic results concerning the classical theories.…”

Section: Ramanujan's Theory Of Elliptic Functions In Alternative Basementioning

confidence: 99%

“…For M = 3, 4, 6 (i.e., m = 3, 2, 3/2) Ramanujan found that much of this classical modular structure generalizes in a straightforward manner. Fricke independently developed the M = 6 (m = 3/2) case [95]. The special modular signature values, M = 2, 3, 4, 6, correspond to the potentials shown in figure 3, and listed in tables 1 and 2.…”

Section: Ramanujan's Generalized Classical Modular Structurementioning

confidence: 99%

“…In the number theory literature, these values of M (or r) for which the Hecke group is commensurable with the modular group are referred to as the "arithmetic cases" [96]. For these arithmetic cases, Jacobi's inversion formula also generalizes [94][95][96][97][98][99][100][101][102]. Physically, this means that one can invert and express the energy u, and also the classical period ω 0 (u) and classical action a 0 (u), as explicit number-theoretic functions of the modular parameter τ 0 defined in (3.46)…”

Section: Jhep05(2017)087mentioning

confidence: 99%

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Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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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 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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Section: Jhep05(2017)087mentioning

confidence: 99%

Section: Ramanujan's Theory Of Elliptic Functions In Alternative Basementioning

confidence: 99%

Section: Ramanujan's Generalized Classical Modular Structurementioning

confidence: 99%

Section: Jhep05(2017)087mentioning

confidence: 99%

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Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

J. High Energ. Phys.

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“…We use the following class equations (see Fricke [11], III, pp. 401, 405, 420 for D = −24, −36, −64, −91; and Fricke [12], III, p. 201 for D = −51):…”

Section: )mentioning

confidence: 98%

Periodic points of algebraic functions and Deuring’s class number formula

Morton

2019

Ramanujan J

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Holomorphicity and modularity in Seiberg-Witten theories with matter

Huang

Klemm²

2010

J. High Energ. Phys.

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Abstract:We calculate the gravitational corrections to the effective action of N=2 SU(2) Seiberg-Witten theory with matter using modularity, the holomorphic anomaly equation and expected behavior at the boundaries of the moduli space. As in pure gauge theory we show that the gap condition at the dyon singularities completely fixes the gravitational corrections. We discuss the behavior of the gravitational corrections at the conformal points. We compare our results with the recursive solution of the loop equation in the matrix model approach, which provides in addition open amplitudes.

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Die elliptischen Funktionen und ihre Anwendungen

Cited by 8 publications

References 0 publications

Quantum geometry of resurgent perturbative/nonperturbative relations

Quantum geometry of resurgent perturbative/nonperturbative relations

Periodic points of algebraic functions and Deuring’s class number formula

Holomorphicity and modularity in Seiberg-Witten theories with matter

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