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Spectral determinants and quantum theta functions
High Energy Physics - Theory · Submitted 22 Apr 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find that they are modular invariant, order by order, and we give explicit expressions for the very first terms of the expansion.
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The abstract, sentence by sentence
It has been recently conjectured that the spectral determinants of operators associated to mirror curves can be expressed in terms of a generalization of theta functions, called quantum theta functions. In this paper we study the symplectic properties of these spectral determinants by expanding them around the point hbar=2pi, where the quantum theta functions become conventional theta functions. We find that they are modular invariant, order by order, and we give explicit expressions for the very first terms of the expansion. Our derivation requires a detailed understanding of the modular properties of topological string free energies in the Nekrasov-Shatashvili limit. We derive these properties in a diagrammatic form. Finally, we use our results to provide a new test of the duality between topological strings and spectral theory.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | It has been recently conjectured that the spectral determinants of operators associated to mirror curves can be expressed in terms of a generalization of theta functions, called quantum theta functions. | background 1.00 | 0.04 | 0.08 | speculative 0.98 |
| s1 | In this paper we study the symplectic properties of these spectral determinants by expanding them around the point hbar=2pi, where the quantum theta functions become conventional theta functions. | contribution 0.75 | 0.89 | 0.07 | asserted 0.97 |
| s2 | We find that they are modular invariant, order by order, and we give explicit expressions for the very first terms of the expansion. | result 1.00 | 0.97 | 0.46 | asserted 1.00 |
| s3 | Our derivation requires a detailed understanding of the modular properties of topological string free energies in the Nekrasov-Shatashvili limit. | method 0.86 | 0.41 | 0.04 | asserted 0.98 |
| s4 | We derive these properties in a diagrammatic form. | method 0.93 | 0.95 | 0.03 | asserted 1.00 |
| s5 | Finally, we use our results to provide a new test of the duality between topological strings and spectral theory. | contribution 0.62 | 0.95 | 0.32 | asserted 0.96 |
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The paper
Original abstract
It has been recently conjectured that the spectral determinants of operators associated to mirror curves can be expressed in terms of a generalization of theta functions, called quantum theta functions. In this paper we study the symplectic properties of these spectral determinants by expanding them around the point hbar=2pi, where the quantum theta functions become conventional theta functions. We find that they are modular invariant, order by order, and we give explicit expressions for the very first terms of the expansion. Our derivation requires a detailed understanding of the modular properties of topological string free energies in the Nekrasov-Shatashvili limit. We derive these properties in a diagrammatic form. Finally, we use our results to provide a new test of the duality between topological strings and spectral theory.
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References
How the literature uses it
18 arXiv papers cite this in 23 sentences
Papers citing it · 15 sentences without a confident label
Uses its method or tool 1 paper
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One important assumption needed in a derivation along the lines of [ref] is that the quantum periods can be obtained from the classical periods by acting with differential operators depending on the energy (which is the modulus of this problem).
Holomorphic Anomaly and Quantum Mechanics · The double well and the quartic oscillator · 0.90
Compares with it 2 papers
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In contrast, the wavefunction ([ref]) can be effectively computed as an expansion at large radius or as an expansion around $ħ=2π$ [ref] .
Exact eigenfunctions and the open topological string · The maximally supersymmetric case · 0.95
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The resulting expansion in the $ϵ$ $Ω$-background parameter can be possibly compared with the results of [ref] .
Surface observables in gauge theories, modular Painlev\'e tau functions and non-perturbative topological strings · Open problems · 0.99
Cites it as background 4 papers
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This correspondence was built on previous work in [ref] formulated in detail in [ref] (see [ref] for a review), and further developed in [ref] .
The complex side of the TS/ST correspondence · Introduction · 0.96
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This correspondence, called topological string/spectral theory (TS/ST) correspondence [ref] , is believed to hold for more general local Calabi-Yau threefolds and matrix models of Fermi gas form and has been tested through various non-trivial examples [ref] .
Large N expansion of mass deformed ABJM matrix model: M2-instanton condensation and beyond · Introduction and Summary · 0.92
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Other aspects of the correspondence have been discussed in [ref] .
Operators and higher genus mirror curves · Introduction · 0.98
What it cites · 131 sentences without a confident label
Uses the method or tool of 4 papers
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There is a part which involves only standard topological strings Ξ(κ, 2π) ∏_i^k ∂^n_i_T F_g_i Θ^(n_1+⋯+n_k)/ Θ, g_i ≤n.By using the combinatorial rules of [ref] it follows that this part is modular invariant.
A holomorphic and background independent partition function for matrix models and topological strings · The general mechanism · 0.94
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By using the transformation properties of the standard free energy [ref] together with [ref] and [ref] we find 16∂_T^3 F_0(∂_T F_1^NS) Θ^(4)/Θ→∂_T ^3 F_0(∂_T F_1^NS)( 16 ΘΘ^(4)- 𝒦2π ΘΘ^(2)+12( 𝒦2 π)^2 ).Similarly ∂_T F_1 ^NS∂_T F_1 Θ^(2) /Θ→ ∂_T F_1 ^NS∂_T F_1 (ΘΘ^(2)- 𝒦 2π) +12 𝒦2 π ∂_T F_1 ^NS∂_T^3 F_0 (ΘΘ^(2)- 𝒦 2π).
Topological Strings and (Almost) Modular Forms · Second order · 0.83
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By using the transformation properties of the standard free energy [ref] together with [ref] and [ref] we find 16∂_T^3 F_0(∂_T F_1^NS) Θ^(4)/Θ→∂_T ^3 F_0(∂_T F_1^NS)( 16 ΘΘ^(4)- 𝒦2π ΘΘ^(2)+12( 𝒦2 π)^2 ).Similarly ∂_T F_1 ^NS∂_T F_1 Θ^(2) /Θ→ ∂_T F_1 ^NS∂_T F_1 (ΘΘ^(2)- 𝒦 2π) +12 𝒦2 π ∂_T F_1 ^NS∂_T^3 F_0 (ΘΘ^(2)- 𝒦 2π).
Holomorphic anomaly and matrix models · Second order · 0.83
Uses data or values from 4 papers
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The small $κ$ expansion [ref] of the spectral determinant at $ħ=2π$ was computed in [ref] and reads Ξ(κ, 2 π) =1+19 κ+( 112 π√(3)-181)κ^2+(5 2187 - 172 π^2 -1 216 π√(3))κ^3+𝒪(κ^4).Similarly by using [ref], [ref] we have ħ Ξ(κ, ħ)|_ħ=2π = Ξ(κ, 2 π) ( A'(2π)- 9 2 ∂_t ^3 F_0 ( 12 π∂_ξ-6 πt(κ, 2π) )^3 ϑ_2(τ_or(κ), ξ(κ)-1/4) ϑ_2(τ_or(κ), ξ(κ)-1/4) - 3 ( ∂_t F_1+∂_t F_1^NS)( 12 π∂_ξ- 6 πt(κ,2π))ϑ_2(τ_or(κ), ξ(κ)-1/4)ϑ_2(τ_or(κ)), ξ(κ)-1/4)),where we denote by $ϑ_2(τ, v)$ the Jacobi theta function of characteristic two ϑ_2(τ, v)=∑_n ∈^π(n+1/2)^2 τ+2π(n+1/2)v and we use τ_or(κ)=-9 2 π∂^2_t F_0(κ)-12, ξ(κ)=3/4 π^2( -t(2π) ∂^2_t F_0(κ)+∂_t F_0(κ) ).To compute the small $κ$ expansion of [ref] one has to analytically continue the free energy and the Kähler parameter to the orbifold point.
Topological Strings from Quantum Mechanics · Testing the topological string/quantum mechanics duality · 0.90
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It was shown in [ref] that Z_1(ħ)= 1𝖻|( (b^2+3)/6 b)|^3, Z_2(ħ)= |(((b^2-3))/6 b)|^2/√(3)b ∫_ℝsinh(πs b/3)/sinh(πbs)|AA(x)|^4 , A=b^2+3/12 b.
Operators from mirror curves and the quantum dilogarithm · Testing the topological string/quantum mechanics duality · 0.84
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We have [ These are given at $ħ$=0.] F_0(t)= -118t^3-∑_j_L, j_R ∑_w, d N^d_j_L, j_R (2 j_L+1) (2 j_R+1) (-1)^2 j_L+2 j_R ^-d w t/w^3 = log^3(z)/18+z (-log^2(z)-3)+z^2 (15 log^2(z)/2+6 log(z)+189/8)+O(z^3)F_1^NS(t)=124t+∑_j_L, j_R ∑_w, d N^d_j_L, j_R m_L m_R ( m_L^2+ m_R^2-3) ^-d wt/24 w= 1/24 log(27 z+1/z)F_2^NS(t)= - ∑_j_L, j_R ∑_w, d N^d_j_L, j_R m_L m_R w (-1)^ m_L+ m_R^-d wt)/5760 (3 m_L^4+10 m_L^2 ( m_R^2-3)+3 m_R^4-30 m_R^2+51) =-29 z/640+561 z^2/160-20091 z^3/128+ 1898037 z^4/320+𝒪(z^5)F_3^NS(t)=- ∑_j_L, j_R ∑_w, d N^d_j_L, j_R m_L m_R w^3 (-1)^ m_L+ m_R+1 ^-d w t/967680 (3 ( m_L^6+7 m_L^4 ( m_R^2-3)+7 m_L^2 ( m_R^4-10 m_R^2+17)+ m_R^4 ( m_R^2-21)) +357 m_R^2-457 ) = -101329821 z^4/17920+2678163 z^3/35840-1449 z^2/2560+137 z/322560+𝒪(z^5),where we denote $t= t^(0)$ and we use the BPS $N^d_j_L, j_R$ invariants listed in [ref] .
The refined BPS index from stable pair invariants · The large radius point · 1.00
Compares with 4 papers
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This improvement of the convergence properties is related to the fact that to compute the spectral determinant [ref] one has to sum over imaginary shifts of the moduli, whereas in [ref] the sum runs over real shifts.
Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence · Spectral determinant and topological strings · 0.97
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This improvement of the convergence properties is related to the fact that to compute the spectral determinant [ref] one has to sum over imaginary shifts of the moduli, whereas in [ref] the sum runs over real shifts.
A holomorphic and background independent partition function for matrix models and topological strings · Spectral determinant and topological strings · 0.97
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This improvement of the convergence properties is related to the fact that to compute the spectral determinant [ref] one has to sum over imaginary shifts of the moduli, whereas in [ref] the sum runs over real shifts.
Breakdown of universality in multi-cut matrix models · Spectral determinant and topological strings · 0.97
Cites as background 30 papers
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In this section we will review some aspects of the work [ref] and its generalization to mirror curves of higher genus [ref] .
Spectral Theory and Mirror Curves of Higher Genus · Spectral determinant and topological strings · 0.81
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In this section we will review some aspects of the work [ref] and its generalization to mirror curves of higher genus [ref] .
Topological Strings from Quantum Mechanics · Spectral determinant and topological strings · 0.81
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A discussion of the modular properties of the NS free energies has been done in [ref] .
On Gauge Theory and Topological String in Nekrasov-Shatashvili Limit · Change of symplectic frame · 0.81
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