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E-string Quantum Curve
High Energy Physics - Theory · Submitted 31 Mar 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We find that the resulting operator expression belongs to the class of elliptic quantum curves.
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The abstract, sentence by sentence
In this work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. We find that the resulting operator expression belongs to the class of elliptic quantum curves. It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. Although the microscopic representation of the co-dimension 4 defect only furnishes an $\mathrm{SO}(16)$ flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine $E_8$ characters. Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. | contribution 0.95 | 0.87 | 0.06 | asserted 0.90 |
| s1 | We find that the resulting operator expression belongs to the class of elliptic quantum curves. | result 1.00 | 0.96 | 0.35 | asserted 1.00 |
| s2 | It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. | interpretation 0.86 | 0.80 | 0.23 | asserted 0.98 |
| s3 | Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. | result 0.85 | 0.79 | 0.13 | asserted 1.00 |
| s4 | Although the microscopic representation of the co-dimension 4 defect only furnishes an $\mathrm{SO}(16)$ flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine $E_8$ characters. | result 0.99 | 0.94 | 0.14 | asserted 1.00 |
| s5 | Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve. | method 0.79 | 0.96 | 0.12 | asserted 1.00 |
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The paper
Original abstract
In this work we study the quantisation of the Seiberg-Witten curve for the E-string theory compactified on a two-torus. We find that the resulting operator expression belongs to the class of elliptic quantum curves. It can be rephrased as an eigenvalue equation with eigenvectors corresponding to co-dimension 2 defect operators and eigenvalues to co-dimension 4 Wilson surfaces wrapping the elliptic curve, respectively. Moreover, the operator we find is a generalised version of the van Diejen operator arising in the study of elliptic integrable systems. Although the microscopic representation of the co-dimension 4 defect only furnishes an $\mathrm{SO}(16)$ flavour symmetry in the UV, we find an enhancement in the IR to representations in terms of affine $E_8$ characters. Finally, using the Nekrasov-Shatashvili limit of the E-string BPS partition function, we give a path integral derivation of the quantum curve.
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References
How the literature uses it
18 arXiv papers cite this in 44 sentences
Papers citing it · 25 sentences without a confident label
Builds on it 1 paper
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These theories are a direct generalization of the E-string theory, as studied in [ref] .
D-type Minimal Conformal Matter: Quantum Curves, Elliptic Garnier Systems, and the 5d Descendants · Introduction · 0.84
Uses its method or tool 3 papers
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For this, one varies [ref] with respect to $ρ(u)$, which leads to [equation] As in [ref] , it is useful to introduce the following quantities: [equation] The saddle point equation [ref] can be rewritten as
Elliptic Quantum Curves of 6d SO(N) theories · Path integral representation · 0.96
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In the same fashion as [ref] , we will first discuss how to introduce codimension two and four defects under $Ω$-background and compute the partition functions with defects.
D-type Minimal Conformal Matter: Quantum Curves, Elliptic Garnier Systems, and the 5d Descendants · D-type Minimal Conformal Matters · 0.92
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To mitigate the singularity at infinity, we can make use of an antisymmetric hypermultiplet [ref] .
Freezing and BPS jumping · E-string to M-string · 0.83
Compares with it 1 paper
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We checked that the solution agrees with the result in [ref] based on the ADHM construction up to 2-instanton order.
5d/6d Wilson loops from blowups · E-string theory · 1.00
Cites it as background 7 papers
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Seiberg–Witten curve with this configuration has been studied [ref] .
E-strings, $F_4$, and $D_4$ triality · Another $W(F_4)$ automorphism · 0.90
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It is an active field of research to examine $=1$ theories of class $𝒮_k$ and the broader theories of class $𝒮_Γ$ where the $ℤ_k$ singularity is generalized to any ADE orbifold singularity $Γ$ [ref] .
$\mathcal{N}=1$ Curves on Generalized Coulomb Branches · Introduction · 0.90
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Various ways of obtaining 5d/6d Seiberg-Witten curves are developed [ref] which includes thermodynamic limit [ref] , Nekrasov-Shatashvili limit of the partition function leading to quantum curves [ref] as well as 5-brane webs [ref] .
Seiberg-Witten curves with O7$^\pm$-planes · Introduction · 0.92
What it cites · 133 sentences without a confident label
Builds on 3 papers
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In our work, we find that the defect partition function with unrestricted $E_8$ fugacities satisfies a generalised version of the van Diejen equation found in [ref] and can indeed be brought into the form [ref].
Surface defects in E-string compactifications and the van Diejen model · Introduction · 0.99
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In this appendix, we understand the co-dimension 2 defect partition function as an open refined topological string partition function Z^top_open=Z^pert(1+∑_k=1^∞q_ϕ^k Z_k^def), and extract the refined Ooguri-Vafa (OV) invariants, as a generalisation of the disk invariants [ref] , from the free energy $F^top_open(t,x)=logZ^top_open(t,x)$.
Mirror Symmetry, D-Branes and Counting Holomorphic Discs · Open topological strings and refined Ooguri-Vafa invariants · 0.86
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In this appendix, we understand the co-dimension 2 defect partition function as an open refined topological string partition function Z^top_open=Z^pert(1+∑_k=1^∞q_ϕ^k Z_k^def), and extract the refined Ooguri-Vafa (OV) invariants, as a generalisation of the disk invariants [ref] , from the free energy $F^top_open(t,x)=logZ^top_open(t,x)$.
Disk Instantons, Mirror Symmetry and the Duality Web · Open topological strings and refined Ooguri-Vafa invariants · 0.86
Uses the method or tool of 2 papers
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For later convenience in any comparison to 5d instanton partition functions, one may apply a flop transition [ref] [equation] for all $A^-1$ associated to the “$1/2$" terms in [ref].
Equivalence of several descriptions for 6d SCFT · Perturbative contributions · 0.96
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Sometimes, it is more convenient to work with the following function [equation] as already used in [ref] .
Elliptic Quantum Curves of Class $\mathcal{S}_k$ · Theta functions · 0.80
Uses data or values from 1 paper
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The $k=1$ elliptic genus for the E-string theory has been computed in [ref] and reads [equation] For convenience, the details of the derivation is summarised in Appendix [ref].
Elliptic Genus of E-strings · Elliptic genus · 0.93
Compares with 1 paper
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It is straightforward to verify that in the classical limit $p=1$ together with the constraints [ref] one finds X^2V(x)=X^-2V(x)=-Q^-1/2∏_n=1^4θ_1(x±μ_n)/η^6θ_1(2x)^2 such that the curve becomes Q^1/2/z=∏_n=1^4θ_1(x±μ_n)/η^6θ_1(2x)^2(Y+Y^-1+2), which matches the curve given in [ref] after some normalisation of $z$.
D-type fiber-base duality · Quantisation · 0.89
Cites as support 1 paper
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The above form of the quantum curve was confirmed for the case of two M5 branes probing a $ℂ^2/ℤ_k$ singularity in [ref] and can be motivated by dividing both sides of the equation by $Ψ(x)$, leading to an equation of the form [equation] where the function $𝒴(x)$ is specified by a suitable ratio of wave-functions.
Elliptic Quantum Curves of Class $\mathcal{S}_k$ · Introduction · 0.96
Cites as background 23 papers
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A particularly interesting class of quantum curves is the class of “elliptic quantum curves" which arise in compactifications of 6d superconformal field theories (SCFTs) with $8$ supercharges on elliptic curves [ref] .
Elliptic Quantum Curves of Class $\mathcal{S}_k$ · Introduction · 0.97
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The Seiberg-Witten curve for the E-string theory compactified on an elliptic curve was derived for the case where a $(8)×(8)$ subgroup of the flavour symmetry is preserved in [ref] .
D-type fiber-base duality · Classical curve · 0.90
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The Seiberg-Witten curve of the E-string theory was derived in [ref] .
Seiberg-Witten Curve for the E-String Theory · Introduction · 0.93
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