Resurgence matches quantization

We show that the late-time expansion of the energy density of N = 4 supersymmetric Yang-Mills plasma at infinite coupling undergoing Bjorken flow takes the form of a multiparameter transseries. Using the AdS/CFT correspondence we find a gravity solution which supplements the well known large proper-time expansion by exponentially-suppressed sectors corresponding to quasinormal modes of the AdS black-brane. The full solution also requires the presence of further sectors which have a natural interpretation as couplings between these modes. The exponentially-suppressed sectors represent nonhydrodynamic contributions to the energy density of the plasma. We use resurgence techniques on the resulting transseries to show that all the information encoded in the nonhydrodynamic sectors can be recovered from the original hydrodynamic gradient expansion.

Section: Discussionmentioning

confidence: 99%

The large proper-time expansion of Yang-Mills plasma as a resurgent transseries

Aniceto

Jankowski

Meiring

et al. 2019

“…This is a quantum-mechanical example of how a Borel-resummable perturbative series fails to reproduce the exact results due to the appearance of complex instantons, as emphasized in[65,66] in a more general context.…”

mentioning

confidence: 90%

TBA equations and resurgent Quantum Mechanics

Ito

Mariño

Shu

2019

Self Cite

124

We derive a system of TBA equations governing the exact WKB periods in onedimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in N = 2 gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a powerful method to solve spectral problems in Quantum Mechanics. We illustrate our general analysis with a detailed study of PT-symmetric cubic oscillators and quartic oscillators. arXiv:1811.04812v3 [hep-th] 31 Jan 2019Contents -1 -periods involved in the problem. However, this flourishing of ideas did not provide alternative computational methods in Quantum Mechanics. One was supposed to calculate the WKB series and their Borel resummations with traditional tools.Partially inspired by the results of [3], a correspondence between certain quantum mechanical models in one dimension and integrable models (or ODE/IM correspondence) was proposed by Dorey and Tateo in [7]. This correspondence was originally based on functional relations discovered in the context of resurgent Quantum Mechanics, which are similar to Baxter-type equations appearing in integrable systems. The ODE/IM correspondence makes it possible to write down TBA equations which calculate efficiently and exactly some of the quantities appearing in the quantum mechanical models, like spectral determinants and Borel resummations of WKB periods. An important limitation of the ODE/IM correspondence is that it applies to very special quantum-mechanical models, namely, monic potentials of the form V (x) = x M (and a limited amount of perturbations thereof).In this paper we will present TBA equations governing the WKB periods for general polynomial potentials in one dimension, providing in this way a generalization of the ODE/IM correspondence. The basic idea was already pointed out by Voros in his seminal paper [3], where he called it the "analytic bootstrap" for the exact WKB method. In this approach, the fundamental objects are the WKB periods, which are (Borel resummed) perturbative series in the Planck constant. These periods can be characterized by two types of data: their classical limit and their discontinuity structure, which has been known since the work of [3,4,6]. In the theory of resurgence, this discontinuity structure is encoded in the action of the so-called Stokes automorphisms. As Voros explained, we can think about these data as defining a Riemann-Hilbert problem. Building on recent developments in seemingly very different contexts [8][9][10][11][12][13][14][15], we show that Voros' Riemann-Hilbert problem has a solution in terms of a TBA-like system, which determines the exact dependence of the WKB periods on the Planck constant 1 . The TBA system of [7] is recove...

“…Further motivation along these lines comes from the ODE/Integrable Model correspondence [66][67][68][69][70], which provides explicit mappings between monodromy operators in certain Schrödinger systems and Yang-Baxter operators in integrable models. We are also strongly motivated by the geometric relation between supersymmetric gauge theories, matrix models and topological strings [71][72][73][74][75], for which a rich web of resurgent structures has been comprehensively established both analytically and numerically [76][77][78][79][80][81][82][83][84][85][86]. There are surprisingly close parallels between the resurgent structures found in such theories for the partition function (or free energy) as a function of (at least) two parameters, g s and N , and the resurgent structure of the Schrödinger energy eigenvalue u( , N ), as a function of and the perturbative level number N .…”

Section: Jhep05(2017)087mentioning

confidence: 99%

Quantum geometry of resurgent perturbative/nonperturbative relations

Başar

Dunne

Ünsal

2017

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.