Quantized mirror curves and resummed WKB
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Springer
Received: February 11, 2019 Accepted: May 6, 2019 Published: May 21, 2019
Szabolcs Zakany D´epartement de Physique Th´eorique, Universit´e de Gen`eve, Gen`eve, CH-1211 Switzerland
E-mail: [email protected] Abstract: Based on previous insights, we present an ansatz to obtain quantization conditions and eigenfunctions for a family of difference equations which arise from quantized mirror curves in the context of local mirror symmetry of toric Calabi-Yau threefolds. It is a first principles construction, which yields closed expressions for the quantization conditions and the eigenfunctions when ~/2π ∈ Q, the so-called rational case. The key ingredient is the modular duality structure of the underlying quantum integrable system. We use our ansatz to write down explicit results in some examples, which are successfully checked against purely numerical results for both the spectrum and the eigenfunctions. Concerning the quantization conditions, we also provide evidence that, in the rational case, this method yields a resummation of conjectured quantization conditions involving enumerative invariants of the underlying toric Calabi-Yau threefold. Keywords: Topological Strings, Lattice Integrable Models, Nonperturbative Effects ArXiv ePrint: 1711.01099
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)114
JHEP05(2019)114
Quantized mirror curves and resummed WKB
Contents 1 Introduction
1
2 About the difference equation 2.1 The difference equation and its dual 2.2 WKB eigenfunction at small ~ 2.3 Resummed WKB from recursion
4 4 7 8
rational case Pole cancellation and modular duality Finite contribution Exact expressions for the building blocks Relations between the parameters The “fully on-shell” eigenfunctions
10 10 12 15 20 22
4 Examples 4.1 Local P2 4.2 Local P1 × P1 4.3 Resolved C3 /Z5
23 23 26 30
5 Comparison with conjectured quantization condition
34
6 Conclusion
38
1
Introduction
Finding eigenfunctions and eigenvalues of differential or difference operators is an ubiquitous problem in physics. Finding exact expressions for them is most of the time rather difficult, even in the one dimensional case where the operator acts on functions on the real line (the setup often considered in standard quantum mechanics). The operators considered in this work are built as Laurent polynomials of the exponentials u = ex ,
v = ey ,
(1.1)
where the operators x and y satisfy the canonical commutation relation [x, y] = i~ for ~ ∈ R. The operators u and v can be represented as multiplication and shift operators acting on functions on R. Polynomials of u and v and their inverses correspond to more general difference operators acting on functions. Such operators arise in several areas of theoretical physics. One example is the quantization of the spectral curves of some particular integrable systems, yielding the Baxter equation which is central in the study
–1–
JHEP05(2019)114
3 The 3.1 3.2 3.3 3.4 3.5
of those systems. Another is the quantization of mir
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