TBA equations and quantization conditions

Emery, Yoan

doi:10.1007/jhep07(2021)171

Cited by 14 publications

(17 citation statements)

References 30 publications

(124 reference statements)

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“…[9][10][11][12][13][14][15][16]. Some recent papers which discuss quantum periods and their relation to four dimensional N = 2 theories are [8,[17][18][19][20][21][22][23][24][25][26][27][28][29].…”

Section: Quantum Periodsmentioning

confidence: 99%

“…In addition in figure 7i, when the series of poles for finite number of terms from the two hypermultiplets seems rotated to slightly different direction from real axis, the interference still makes the vectormultiplet hard to be seen. The mixing of these poles makes it impossible to find a path of integration for the lateral Padé-Borel summation around arg( ) = 0 which is not 27 The Schrödinger operator is invariant under u → −u, m → im, Λ → Λ e −πi/6 , → i , z → e 2πi 3 z. For numerical calculation, it was more convenient to take Λ real; thus we did Borel summation at m = 1/10, u = −2, Λ = 1, = −i/2 and used the cycles related to γ [0,2,0] and γ [1,0,0] under z → e 2πi 3 z.…”

Section: Jhep01(2022)046mentioning

confidence: 99%

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Exact WKB methods in SU(2) Nf = 1

Grassi

Hao

Neitzke

2022

J. High Energ. Phys.

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We study in detail the Schrödinger equation corresponding to the four dimensional SU(2) $$ \mathcal{N} $$ N = 2 SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinger operator using the TS/ST correspondence and Zamolodchikov’s TBA, again finding good agreement. In addition, we explore two aspects of the relationship between singularities of the Borel transformed WKB series and BPS states: BPS states of the 4d theory are related to singularities in the Borel transformed WKB series for the quantum periods, and BPS states of a coupled 2d+4d system are related to singularities in the Borel transformed WKB series for local solutions of the Schrödinger equation.

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Section: Quantum Periodsmentioning

confidence: 99%

Section: Jhep01(2022)046mentioning

confidence: 99%

Exact WKB methods in SU(2) Nf = 1

Grassi

Hao

Neitzke

2022

J. High Energ. Phys.

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“…The integral equations take the form of the Thermodynamic Bethe ansatz (TBA) equations of certain integrable models, and provide the solution to Voros's Riemann-Hilbert JHEP10(2021)167 problem to determine the exact WKB period. The TBA equations also provide an efficient method to solve the spectral problems with generic polynomial potential in quantum mechanics [4][5][6].…”

Section: Introductionmentioning

confidence: 99%

WKB periods for higher order ODE and TBA equations

Ito

Kondo

Kuroda

et al. 2021

J. High Energ. Phys.

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We study the WKB periods for the (r + 1)-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the $$ {A}_r^{(1)} $$ A r 1 affine Toda field equation. We compute the quantum corrections by using the Picard-Fuchs operators. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra Ar. For the quadratic potential, we propose a formula to show the equivalence between the logarithm of the Y-function and the WKB period, which is confirmed by solving the TBA equation numerically.

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“…The method has deep connections with 4d N = 2 gauge theories in the contexts of Ω-deformation [19,20] and wall-crossing phenomena [21], as well as with ODE/IM correspondence [22,23], topological string theory, see e.g. [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. The common tread of all these different examples is the quantization of an underlying classical curve.…”

Section: Introductionmentioning

confidence: 99%

Exact-WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms

Kamata¹,

Misumi

Sueishi

et al. 2021

Preprint

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Quantum deformed potentials arise naturally in quantum mechanical systems of one bosonic coordinate coupled to N f Grassmann valued fermionic coordinates, or to a topological Wess-Zumino term. These systems decompose into sectors with a classical potential plus a quantum deformation. Using exact WKB, we derive exact quantization condition and its median resummation. The solution of median resummed form gives physical Borel-Ecalle resummed results, as we show explicitly in quantum deformed double-and triple-well potentials. Despite the fact that instantons are finite action, for generic quantum deformation, they do not contribute to the energy spectrum at leading order in semi-classics. For certain quantized quantum deformations, where the alignment of levels to all order in perturbation theory occurs, instantons contribute to the spectrum. If deformation parameter is not properly quantized, their effect disappears, but higher order effects in semi-classics survive. In this sense, we classify saddle contributions as fading and robust. Finally, for quantum deformed triple-well potential, we demonstrate the P/NP relation, by computing period integrals and Mellin transform.

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TBA equations and quantization conditions

Cited by 14 publications

References 30 publications

Exact WKB methods in SU(2) Nf = 1

Exact WKB methods in SU(2) Nf = 1

WKB periods for higher order ODE and TBA equations

Exact-WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms

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