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Quantum Periods and TBA-like Equations for a Class of Calabi-Yau Geometries
High Energy Physics - Theory · Submitted 15 Sep 2020
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.
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The abstract, sentence by sentence
We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev's quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. | contribution 0.94 | 0.89 | 0.29 | asserted 0.97 |
| s1 | We give two methods to derive the TBA-like equations. | contribution 0.73 | 0.94 | 0.11 | asserted 1.00 |
| s2 | One method uses only elementary functions while the other method uses Faddeev's quantum dilogarithm function. | method 0.94 | 0.87 | 0.08 | asserted 1.00 |
| s3 | The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period. | result 0.68 | 0.89 | 0.37 | asserted 1.00 |
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The paper
Original abstract
We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev's quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.
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References
How the literature uses it
4 arXiv papers cite this in 7 sentences
Papers citing it · 5 sentences without a confident label
Cites it as background 2 papers
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The relations between quantum periods and TBA-like equations for these Calabi-Yau geometries are also recently studied in [ref] .
Bootstrapping Calabi-Yau Quantum Mechanics · Introduction · 0.90
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This may lead us to understand the relations between the ABJM matrix model and the Painlevé equation [ref] , the integrable hierarchy [ref] , the chiral projection [ref] or the TBA system [ref] more systematically.
Brane Transitions from Exceptional Groups · Conclusion · 0.82
What it cites · 43 sentences without a confident label
Uses the method or tool of 5 papers
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After a change of coordinates $x,p$, as well as a similarity transformation of the inverse of the operator ([ref]), which were discussed in details in [ref] , the integral kernel becomes the following expression ⟨x|_1 | ρ_m,n | x_2|=⟩ u_m,n(x_1)^-1/2 u_m,n(x_2)^-1/2 / 2(m+n+1)ħcosh(π(x_1-x_2)/(m+n+1)ħ+i (m-n+1)π/2(m+n+1) ), where the potential $u_m,n(x)$ can be described by Faddeev's quantum dilogarithm u_m,n(x)=e^-m/m+n+1x |Φ_b(x-iħ(m+1)/2/2πb)|^2 , with the parameter $b:=√((m+n+1)ħ/2π)$.
TBA-like integral equations from quantized mirror curves · The $𝒪_m,1$ operators: second method · 0.91
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After a change of coordinates $x,p$, as well as a similarity transformation of the inverse of the operator ([ref]), which were discussed in details in [ref] , the integral kernel becomes the following expression ⟨x|_1 | ρ_m,n | x_2|=⟩ u_m,n(x_1)^-1/2 u_m,n(x_2)^-1/2 / 2(m+n+1)ħcosh(π(x_1-x_2)/(m+n+1)ħ+i (m-n+1)π/2(m+n+1) ), where the potential $u_m,n(x)$ can be described by Faddeev's quantum dilogarithm u_m,n(x)=e^-m/m+n+1x |Φ_b(x-iħ(m+1)/2/2πb)|^2 , with the parameter $b:=√((m+n+1)ħ/2π)$.
Operators from mirror curves and the quantum dilogarithm · The $𝒪_m,1$ operators: second method · 0.91
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Now we will build up the TBA difference equations, from the Lemma 2 in [ref] , we have the difference equation about $Ψ_k$
Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz · The $𝒪_m,1$ operators: second method · 0.92
Compares with 2 papers
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We should note that unlike the examples in [ref] , in our case the integral $∫_-∞^∞ |u(x)|^-1 dx$ is actually divergent, which implies the corresponding integral kernel ([ref]) may not be a trace class operator.
Operators from mirror curves and the quantum dilogarithm · The $𝒪_m,1$ operators: first method · 0.91
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In the class of geometries of $𝒪_m,1$ operators, the TBA-like equations ([ref]) in Section [ref] are similar to those of $ℙ^1×ℙ^1$ geometry in [ref] .
Note on Quantum Periods and a TBA-like System · Discussions · 0.89
Cites as background 17 papers
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In the 1990's, it was conjectured in [ref] and proved in [ref] that the spectral equation ([ref]) is related to some TBA-like equations which first appeared in the context of two-dimensional $𝒩 = 2$ supersymmetric theories [ref] .
N=2 Supersymmetry, Painleve III and Exact Scaling Functions in 2D Polymers · Quantum periods and TBA-like equations · 0.91
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In the 1990's, it was conjectured in [ref] and proved in [ref] that the spectral equation ([ref]) is related to some TBA-like equations which first appeared in the context of two-dimensional $𝒩 = 2$ supersymmetric theories [ref] .
Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz · Quantum periods and TBA-like equations · 0.91
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In the 1990's, it was conjectured in [ref] and proved in [ref] that the spectral equation ([ref]) is related to some TBA-like equations which first appeared in the context of two-dimensional $𝒩 = 2$ supersymmetric theories [ref] .
A New Supersymmetric Index · Quantum periods and TBA-like equations · 0.91
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