---
title: E-string Quantum Curve & van Diejen Operator
url: https://www.emergentmind.com/topics/e-string-quantum-curve
type: topic
---

# E-string Quantum Curve & van Diejen Operator

Searching arXiv for the cited E-string quantum curve and closely related E-string Seiberg–Witten/Nekrasov papers.
The E-string quantum curve is the quantisation of the Seiberg–Witten curve for the E-string theory compactified on a two-torus, formulated as an elliptic difference operator whose classical limit reproduces the Seiberg–Witten geometry and whose quantum data are encoded by defect partition functions, Wilson surfaces, and refined BPS amplitudes [2103.16996]. In the formulation developed for the rank-1 E-string, the resulting operator belongs to the class of elliptic quantum curves and can be identified with a generalised rank-one van Diejen operator; its eigenfunctions are codimension-2 defect partition functions, while its eigenvalues are codimension-4 Wilson surfaces wrapping the elliptic curve [2103.16996]. Earlier work on the E-string Seiberg–Witten curve and Nekrasov-type formulas supplied the classical input for this construction by deriving the Seiberg–Witten geometry from elliptic instanton partition functions and by clarifying the dependence on Wilson lines [1312.1050] [1207.5739] [1506.05582].

## 1. Classical geometric origin

The underlying six-dimensional theory is the rank-1 \((1,0)\) E-string SCFT describing a single M5-brane ending on an M9-plane, equivalently the theory of a small \(E_8\) instanton, with a single tensor multiplet scalar controlling the tension of self-dual strings [2103.16996]. Compactification on a torus \(T^2\) with complex modulus \(\tau\) yields a Seiberg–Witten description whose curve is fibered over the torus and whose moduli are related to holonomies of an \(SU(2)\), or \(Sp(1)\), bundle over the elliptic curve [2103.16996].

A convenient route to the classical curve passes through the dual description as a 6d \(Sp(1)\) theory with 10 flavors, Higgsed down to the E-string, and restricted to the \(D_4\) conformal matter limit preserving an \(SO(8)\times SO(8)\subset E_8\) flavor subgroup [2103.16996]. With the constraint
\[
\mu_l\mapsto -\mu_{l-4}, \quad l=5,\dots,8,
\]
the Seiberg–Witten curve takes the form
\[
\frac{\prod_{l=1}^4 \theta_1(x \pm \mu_l)}{\theta_1(2x)^2}\, t^2
 - \left(2 \frac{\prod_{l=1}^4\theta_1(x\pm \mu_l)}{\theta_1(2x)^2} + c \right) t + \frac{\prod_{l=1}^4 \theta_1(x \pm \mu_l)}{\theta_1(2x)^2} =0,
\]
with \(t=e^{2\pi i y}\), elliptic coordinate \(x\), and \(c\) an \(x\)-independent parameter [2103.16996]. Dividing by \(t\) gives
\[
\frac{\prod_{l=1}^4 \theta_1(x \pm \mu_l)}{\theta_1(2x)^2}\,(t+t^{-1}) = 2 \frac{\prod_{l=1}^4\theta_1(x\pm \mu_l)}{\theta_1(2x)^2} + c,
\]
and the canonical Seiberg–Witten differential is
\[
\lambda_{\rm SW}= y\,dx = \log t\ \frac{dX}{X}.
\]
This curve is genus two in the \((x,t)\) variables [2103.16996].

Earlier analyses of the compactified E-string derived equivalent Seiberg–Witten data from Nekrasov-type formulas. In the no-Wilson-line case, the standard Seiberg–Witten curve is
\[
y^2 = 4x^3 - E_4(\tau)\,u\,x - E_6(\tau)\,u^2,
\]
or, in the normalization obtained directly from the thermodynamic limit of the partition function,
\[
y^2 = 4x^3 - E_4(\tau) u^4 x - E_6(\tau) u^6 + 4u^5,
\]
with period relation
\[
\frac{\partial F_0}{\partial \varphi} = 8\pi i (\varphi_D - \tau\varphi) + \text{const.}
\]
[1312.1050] [1506.05582]. Those results established that the E-string Seiberg–Witten geometry emerges from an elliptic instanton sum and supplied the semiclassical curve that is quantised in the quantum-curve construction [1312.1050] [1506.05582].

## 2. Quantisation and elliptic difference form

In the Nekrasov–Shatashvili framework, one considers the \(\Omega\)-background with \(\epsilon_2\to 0\) and \(\epsilon_1=:\hbar\) finite, so that the Seiberg–Witten curve becomes the classical Hamiltonian of a quantum system [2103.16996]. Quantisation promotes \(x,p\) to operators with
\[
[\hat x,\hat p]=i\hbar, \quad Y:=e^{-\hat p},\quad X=e^{\hat x},
\]
and in multiplicative variables
\[
Y\cdot X = p^{-1} X\cdot Y,\quad p=e^{\hbar}.
\]
Accordingly, \(Y\) acts as a finite-difference operator on functions of \(x\) [2103.16996].

For rank-1 6d SCFTs with effective 5d gauge group \(SU(2)\) or \(Sp(1)\), the expected elliptic quantum curve has the form
\[
\bigl(Y^{-1} + \mathcal P(x)\,Y\bigr)\,\widetilde\Psi(x) \;\sim\; \mathcal W^S_\hbar(x)\,\widetilde\Psi(x),
\]
or, after dividing by the wavefunction,
\[
\mathcal Y(x+\hbar)^{-1} + \mathcal P(x)\,\mathcal Y(x) \sim \mathcal W^S_\hbar, \quad \mathcal Y(x) := \frac{\widetilde\Psi(x-\hbar)}{\widetilde\Psi(x)}.
\]
For the E-string this expectation is realised explicitly, and the quantised curve is identified with a van Diejen operator [2103.16996].

A central building block is the meromorphic Jacobi-form coefficient
\[
\mathcal{V}(x)= -p\,X^{-2}\, \frac{\prod_{n=1}^8 \theta_1\bigl(x-\mu_n-\tfrac{\epsilon_1}{2}\bigr)}
     {\eta^6\,\theta_1(2x)\,\theta_1(2x-\epsilon_1)}.
\]
The perturbative codimension-2 defect partition function satisfies
\[
\frac{\widetilde Z^{\rm E-str+def}_{\rm pert}(X p^{1/2})}
     {Y\cdot\widetilde Z^{\rm E-str+def}_{\rm pert}(X p^{1/2})}
 = q_\phi\,\mathcal{V}(x),
\]
which identifies \(\mathcal{V}(x)\,Y+\mathcal{V}(-x)\,Y^{-1}\) as the quantisation of the classical \(t+t^{-1}\) term in the Seiberg–Witten curve [2103.16996].

The full E-string quantum curve is then
\[
\frac{1}{z}\,\Psi(x) = \mathcal{V}(x)\,\Psi(x-\epsilon_1) +\mathcal{V}(-x)\,\Psi(x+\epsilon_1) +V_0(x)\,\Psi(x),
\]
equivalently
\[
\bigl(\mathcal{V}(x) Y + \mathcal{V}(-x) Y^{-1} + V_0(x)\bigr)\Psi(x) = \frac{1}{z}\,\Psi(x),
\]
with \(V_0(x)\) an \(x\)-dependent external potential and \(1/z\) the eigenvalue [2103.16996]. After absorbing \(V_0\) into the right-hand side, one may write
\[
\bigl(\mathcal{V}(x) Y + \mathcal{V}(-x) Y^{-1}\bigr)\Psi(x) = q_\phi^{-1}\,\mathcal W^S_\hbar(x)\,\Psi(x),
\]
where \(\mathcal W^S_\hbar(x)\) is the Wilson surface vev in the NS limit [2103.16996].

## 3. Identification with the van Diejen operator

The operator obtained in the E-string construction is a generalised version of the rank-one elliptic van Diejen Hamiltonian, the canonical \(BC_1\) relativistic integrable operator with eight couplings [2103.16996]. In van Diejen conventions, one introduces
\[
R_+(x) = \prod_{k=1}^\infty \bigl(1-e^{2irx-(2k-1)ra_+}\bigr)\, \bigl(1-e^{-2irx-(2k-1)ra_+}\bigr),
\]
and
\[
V(x) = \frac{\prod_{n=1}^8 R_+(x-h_n -i a_-/2)}
     {R_+(2x + i a_+/2)\,R_+(2x- i a_+/2+i a_-)} ,
\]
with Hamiltonian
\[
\hat H = V(x)\,e^{-i a_-} + V(-x)\,e^{i a_-} + V_b(x).
\]
Under the parameter identification
\[
h_n = \frac{\mu_n}{2ir} - \frac{i a_+}{2},\quad Q=e^{-2 r a_+},\quad \epsilon_1=i a_-,\quad x = 2 i r z,
\]
the E-string coefficient functions match the van Diejen potentials up to trivial prefactors [2103.16996].

The external potential can be written as
\[
V_0(X)=\frac{1}{2}\sum_{I=1}^4 \frac{\prod_{n=1}^8 \theta_I(\mu_n)}
     {\eta^6\theta_1'(0)\theta_1(\epsilon_1)}\, \left[ \frac{\theta_I'(x-\epsilon_1/2)}{\theta_I(x-\epsilon_1/2)} - \frac{\theta_I'(x+\epsilon_1/2)}{\theta_I(x+\epsilon_1/2)} \right] + c,
\]
and is identified with minus the 1-instanton Wilson-surface contribution,
\[
V_0(x) = -\widetilde W_1(x).
\]
This gives a direct gauge-theoretic interpretation of the van Diejen external potential in terms of a codimension-4 observable of the E-string theory [2103.16996].

The integrable-systems meaning is therefore explicit: the E-string quantum curve is the van Diejen Hamiltonian acting on a defect wavefunction,
\[
\hat H\,\Psi = \frac{1}{z}\,\Psi,
\]
with the spectrum encoded in Wilson surface and Wilson loop data [2103.16996]. A plausible implication is that the E-string provides a 6d realisation of an elliptic relativistic integrable system whose spectral problem is governed by the same operator.

## 4. Defects, eigenfunctions, and eigenvalues

The eigenfunctions of the quantum curve are codimension-2 defect partition functions. In the type-IIA brane picture, the defect is engineered by inserting a D4 brane extended along \(T^2\times\mathbb{R}^2_{\epsilon_1}\) and localised in \(\mathbb{R}^2_{\epsilon_2}\), which modifies both perturbative and stringy contributions of the 6d BPS partition function [2103.16996]. The normalised defect partition function is
\[
\widetilde Z^{\rm def} = \frac{Z^{6d+{\rm def}}}{Z^{6d}},
\]
and, after stripping off the perturbative factor, the instanton wavefunction is
\[
\widetilde\Psi(x) = \widetilde Z_{\rm str}^{\rm E-str+def}(x+\epsilon_1/2).
\]
The full wavefunction is
\[
\Psi(x)=\widetilde Z_{\rm pert}^{\rm E-str+def}\,\widetilde\Psi(x).
\]
The difference operator acts on either form and yields the same spectral content [2103.16996].

The eigenvalues are codimension-4 Wilson surfaces. In the brane construction they arise from a D4′ brane along \(T^2\times\mathbb{R}^3_{789}\), with one D2 stretched between the NS5 and the D4′, producing a Wilson surface in the fundamental representation [2103.16996]. The normalised Wilson surface in the NS limit is expanded as
\[
\mathcal W^S_{\hbar=\epsilon_1} = \sum_{k=0}^\infty q_\phi^k\,\widetilde W_k,
\]
with \(\widetilde W_1(x)\) equal to \(-V_0(x)\) and higher \(\widetilde W_k\) providing the quantum-corrected eigenvalue [2103.16996].

The operator equation may be written in the form
\[
\bigl(\mathcal{V}(x)Y + \mathcal{V}(-x)Y^{-1}\bigr)\Psi(x) = \bigl(-V_0(x) + \mathcal W^L_\hbar\bigr)\Psi(x) = q_\phi^{-1} \mathcal W^S_\hbar(x)\,\Psi(x),
\]
where \(\mathcal W^L_\hbar\) is the 5d Wilson loop obtained after circle reduction [2103.16996]. The six-dimensional and five-dimensional observables are matched by the parameter map
\[
Q = U^2,\quad q_\phi = \alpha U e^{-m_8},\quad M_8=e^{\mu_8}=e^{m_8}U^{-2},\quad \mu_l=m_l,\ l=1,\dots,7,
\]
and the mirror-map expansion of the eigenvalue has the form
\[
\frac{1}{z}  = q_\phi + \frac{1}{q_\phi} + \text{const} + \sum_{n\geq1} \text{(characters)}\,Q^{n/2}.
\]
This character expansion is a central feature of the E-string quantum spectrum [2103.16996].

A further structural property is parity invariance. The Hamiltonian
\[
\widehat H := \mathcal{V}(x)Y + \mathcal{V}(-x)Y^{-1} - q_\phi^{-1}\mathcal W^S_\hbar(x)
\]
commutes with \(x\mapsto -x\), so the ground states are twofold degenerate and may be represented by
\[
\psi_1(x)=\Phi(x),\qquad \psi_2(x)=\Phi(x)^{-1},
\]
with even and odd combinations
\[
\Phi_\pm(x) = \Phi(x)\pm\Phi(x)^{-1}.
\]
[2103.16996]

## 5. Relation to Nekrasov-type formulas and thermodynamic limits

The quantum-curve construction rests on a substantial classical foundation provided by Nekrasov-type partition functions for the compactified E-string. In the elliptic instanton formalism, the partition function is expressed as a sum over \(N\)-tuples of Young diagrams,
\[
Z \;=\; \sum_{R} (-e^{2\pi i \varphi})^{|R|}\,\prod_{k=1}^N\prod_{(i,j)\in R_k} \frac{\displaystyle \prod_{n=1}^{2N} \vartheta_1\big(2(a_k - m_n + (j-1)\hbar),\tau\big)}
     {\displaystyle \prod_{l=1}^N \vartheta_1\big(2(a_k - a_l + h_{k,l}(i,j)\hbar),\tau\big)^2},
\]
with specialisations of \(a_k\) and \(m_n\) appropriate to the E-string and its Wilson lines [1506.05582]. In the thermodynamic limit \(\hbar\to 0\), the sum over partitions is replaced by a path integral over a density \(f''\),
\[
Z \sim \int \mathcal{D}f''\, d\lambda\, \exp\left( \frac{1}{2\hbar^2}F_0[f'',\lambda] + O(\hbar^0)\right),
\]
and the saddle-point equation determines an elliptic resolvent function \(H(z)\) whose associated Riemann surface is the Seiberg–Witten curve [1506.05582].

The same logic was developed in a slightly different normalisation in the proof of the thermodynamic limit, where the antiderivative of the resolvent \(\omega(z)\) and the elliptic function
\[
H(z) := \cosh^2\left(\frac{\omega(z)-2\pi i\varphi}{2}\right)
\]
encode the saddle-point solution [1312.1050]. In the \(E_8\)-symmetric case the thermodynamic limit first yields a genus-4 hyperelliptic curve, which is mapped to the standard elliptic Seiberg–Witten curve by a simple change of variables involving the Weierstrass function \(\wp(z)\) [1312.1050]. This classical higher-genus-to-elliptic reduction is conceptually close to the later quantum construction, where the finite-\(\hbar\) difference operator is the noncommutative deformation of the Seiberg–Witten geometry [1312.1050] [2103.16996].

Wilson-line dependence was later extended to three and four Wilson lines. In those cases the Nekrasov-type expression reproduces Seiberg–Witten curves with coefficients depending on \(\wp(2\pi m_i)\), their symmetric combinations \(\sigma_1,\sigma_2,\sigma_3,\sigma_4\), and prefactors built from \(\vartheta_1(m_i)\) and \(\eta(\tau)\) [1506.05582]. This dependence is precisely the semiclassical data required for a Wilson-line-dependent E-string quantum curve, and it clarifies how mass deformations enter the quantised operator through elliptic functions and Jacobi forms [1506.05582].

A distinct but complementary development expressed the four-Wilson-line E-string Seiberg–Witten curve in terms of the \(SU(2)\) Seiberg–Witten curve with \(N_f=4\), using affine \(D_4\) theta functions \(\Theta_R(\tau,\mathbf m)\) and a specific map between the \(SU(2)\) masses \(M_i\) and the E-string Wilson lines \(\mathbf m=(m_1,m_2,m_3,m_4)\) [1207.5739]. This established an explicit bridge between E-string geometry and a better-studied four-dimensional Seiberg–Witten system, and it suggests a structural relation between the E-string quantum curve and quantisations of the \(SU(2)\), \(N_f=4\) curve [1207.5739].

## 6. Modular structure, symmetry enhancement, and broader interpretations

A defining feature of the E-string quantum curve is the enhancement from the ultraviolet \(SO(16)\) flavour symmetry of the defect worldsheet theory to affine \(E_8\) structure in the infrared [2103.16996]. Although the codimension-2 and codimension-4 defects are naturally formulated in a 6d \(Sp(1)\) description with 10 flavors, and thus only manifest \(SO(16)\), the Wilson-surface eigenvalues and the mirror-map coefficients organise into \(E_8\)-invariant Jacobi forms and affine \(E_8\) characters [2103.16996].

At 2-instanton order, for example, the Wilson surface expansion is written in terms of \(E_8\) characters such as \(\chi_{\bf 248}\), \(\chi_{\bf 3875}\), and \(\chi_{\bf 30380}\), tensored with \(SU(2)\) characters \(\chi_n^{\mathfrak{su}_2}\) [2103.16996]. This is consistent with earlier studies of E-string elliptic genera, where the one-string and two-string partition functions were expressed in terms of the \(E_8\) theta function \(\Theta(\tau,m_l)\), the Jacobi forms \(\phi_{-2,1}\), \(\phi_{0,1}\), and the \(E_8\)-invariant generators \(A_1,A_2,B_2\) [1411.2324]. The broader refined topological-string partition function on \(K_{\frac12 K3}\) is likewise organised by modular and Jacobi structures [1411.2324].

The modular properties of E-string partition functions are also central. The elliptic genera obey an \(S\)-modular transformation with an anomaly containing a term quadratic in the string number \(n\),
\[
Z_n\left(-\frac{1}{\tau},\frac{\epsilon_{1,2}}{\tau},\frac{m_l}{\tau}\right) = Z_n(\tau,\epsilon_{1,2},m_l)\,\varepsilon^{-6n} \exp\left[\frac{\pi i}{\tau} \left(2\epsilon_1\epsilon_2 n^2 - \left(\sum_{l=1}^8m_l^2 -4\epsilon_+^2\right)n\right) \right],
\]
which signals non-Hecke behaviour and interactions among E-strings [1411.2324]. A plausible implication is that any nonperturbative formulation of the E-string quantum curve must respect this anomalous modular structure rather than the simpler symmetric-product behaviour of free-string systems.

In a broader topological-string and Painlevé context, genus-one quantum mirror curves have been related to \(q\)-difference Painlevé equations via Fredholm determinants of the associated quantum operators [1710.11603]. That work explicitly identified elliptic Painlevé with E-strings and half K3 in Sakai’s classification, and proposed that the tau-function of the elliptic Painlevé equation should be computed by the grand canonical topological string partition function of the corresponding geometry [1710.11603]. This does not construct the E-string operator directly, but it places the E-string quantum curve within a wider nonperturbative framework in which quantum curves, spectral determinants, integrable systems, and Painlevé tau-functions are different realisations of the same underlying structure [1710.11603].

A related but more general viewpoint reconstructs topological-string partition functions from quantum curves using Riemann–Hilbert and isomonodromic methods, with the partition function appearing as a generalised theta series in appropriately normalised Fenchel–Nielsen coordinates [1811.01978]. Although that analysis does not treat the E-string explicitly, it suggests that an E-string quantum curve may also admit an isomonodromic interpretation once the mirror curve of local half K3 is recast as an \(SL(2)\) oper. This suggests a possible nonperturbative reformulation of the E-string quantum curve in terms of tau-functions and monodromy data.

## 7. Classical limit, applications, and open directions

The classical limit of the E-string quantum curve is obtained by imposing the \(D_4\times D_4\) symmetry conditions and sending \(p=e^{\epsilon_1}\to1\). In this limit
\[
X^2 V(x) = X^{-2} V(-x) = -Q^{-1/2}\frac{\prod_{n=1}^4 \theta_1(x\pm\mu_n)}{\eta^6\theta_1(2x)^2},
\]
and
\[
V_b(x) = -\Sigma(x) + (\text{$X$-independent}),\quad \Sigma(x) := X^2 V(x)+X^{-2}V(-x).
\]
The quantum curve reduces to
\[
Q^{1/2}\,z^{-1} = \frac{\prod_{n=1}^4 \theta_1(x\pm\mu_n)}{\eta^6\theta_1(2x)^2}\,(Y+Y^{-1}+2),
\]
which reproduces the classical Seiberg–Witten curve after the identification \(t\leftrightarrow Y^{-1}\) [2103.16996]. The quantisation is therefore exact in the sense that the difference operator is a deformation of a known algebraic Seiberg–Witten relation.

The NS-limit path-integral derivation strengthens this interpretation. Introducing
\[
\omega(x) := \exp\left(\int du'\,\rho(u')\,\frac{\theta_1'(u'+x)}{\theta_1(u'+x)}\right),\quad \Phi(x) := \frac{\omega(x)}{\omega(-x)},
\]
the saddle-point equation for the D2-brane density becomes
\[
\mathcal Y(u) + \frac{\prod_{l=1}^8\theta_1(u\pm\mu_l+\hbar/2)}{\eta^{12}\theta_1(2u)\theta_1(2u+\hbar)^2\theta_1(2u+2\hbar)}\,\frac{q_\phi^2}{\mathcal Y(u+\hbar)} = 0,
\]
with
\[
\mathcal Y(u) = \frac{\Phi(u-\hbar/2)}{\Phi(u+\hbar/2)}.
\]
Adding the Wilson-surface term yields the full van Diejen equation,
\[
\mathcal Y(x)+\frac{\prod_{l=1}^8\theta_1(x\pm\mu_l+\hbar/2)}{\eta^{12}\theta_1(2x)\theta_1(2x+\hbar)^2\theta_1(2x+2\hbar)}\,\frac{q_\phi^2}{\mathcal Y(x+\hbar)} + q_\phi V_0(x) = q_\phi \mathcal W^L_\hbar,
\]
and the defect partition function is identified directly as
\[
\widetilde Z_{\rm str}^{\rm E-str+def}(x) = \Phi(x).
\]
[2103.16996]

This derivation shows that the E-string quantum curve is not merely an abstract quantisation of a classical algebraic curve. It is an operator equation extracted from the BPS path integral, with wavefunction and spectrum realised by concrete defect observables [2103.16996]. Earlier work had already suggested that the Nekrasov partition function should play the role of a wavefunction or \(\tau\)-function for a quantum curve whose semiclassical limit reproduces the Seiberg–Witten curve [1506.05582]. The 2021 construction makes that suggestion explicit for the E-string by identifying the operator, the wavefunction, and the eigenvalue problem [2103.16996].

Several directions remain structurally natural. The literature on elliptic genera of E-strings provides exact low-string-number data and the refined topological-string partition function on \(K_{\frac12 K3}\), which can be used to test more general quantisations and higher-rank extensions [1411.2324] [1406.0850]. The relation to elliptic Painlevé and half-K3 geometry suggests a nonperturbative spectral-determinant interpretation [1710.11603]. The extension from rank one to higher-rank E-strings or other 6d SCFTs plausibly leads to higher-rank van Diejen or Inozemtsev-type operators, though this is an implication rather than an explicit result of the cited works [2103.16996].

Source: https://www.emergentmind.com/topics/e-string-quantum-curve