Integrals of motion in $WE_6$ CFT and the ODE/IM correspondence

This paper establishes the ODE/IM correspondence for the affine Lie algebra $E_6^{(1)}$ by demonstrating that the period integrals of the WKB expansion of the associated ordinary differential equation match the eigenvalues of the integrals of motion in the corresponding $WE_6$ conformal field theory up to the sixth order.

The Gist

Imagine the universe as a giant, complex machine. Physicists love finding the "secret rules" that make this machine run smoothly. Two of the most fascinating rules they look for are Integrability (things that never change, like a perfect clock) and Symmetry (patterns that repeat).

This paper is like a detective story where the authors try to prove that two very different-looking clues actually point to the same hidden treasure.

Here is the breakdown of their investigation, translated into everyday language:

1. The Two Worlds: The "Math Map" and the "Physics Machine"

The authors are studying a specific, very complex type of symmetry called $E_6$. Think of $E_6$ as a rare, intricate crystal structure that exists in high-dimensional math.

• World A: The ODE (The Math Map)
  Imagine you are trying to navigate a very strange, twisting road. You have a map (a differential equation) that tells you how to drive. To understand the road, you use a technique called WKB expansion.

  • The Analogy: Think of this like a hiker trying to cross a mountain range. They can't see the whole path at once, so they take small steps, estimating the terrain ahead. They calculate the "periods" of their journey—how long it takes to go around a specific loop on the map.
  • In this paper, the "road" is defined by a complex equation involving the $E_6$ crystal structure. The authors calculated the "travel time" (period integrals) for this road up to the 6th level of detail.

• World B: The CFT (The Physics Machine)
  Now, imagine a 2D universe (like a flat sheet of paper) filled with quantum fields. This is a Conformal Field Theory (CFT). In this world, there are "conserved quantities" (Integrals of Motion).

  • The Analogy: Think of these as the "vital signs" of the machine. Just as your heart rate and temperature stay constant even if you move around, these mathematical quantities stay the same no matter how the system evolves.
  • The authors calculated the "vital signs" (eigenvalues) for the highest-energy state of this machine, also up to the 6th level of detail.

2. The Big Connection: The ODE/IM Correspondence

For a long time, physicists have suspected that World A (the Math Map) and World B (the Physics Machine) are actually the same thing seen from different angles. This is called the ODE/IM correspondence.

• The Hypothesis: The "travel time" calculated on the Math Map should match the "vital signs" calculated on the Physics Machine, provided you translate the units correctly.
• The Challenge: Most previous studies looked at simple shapes (like a square or a triangle). This paper tackles the $E_6$ crystal, which is much more complex and "exotic." It's like trying to match the weather patterns of a simple beach with those of a chaotic, multi-layered volcano.

3. The Investigation: How They Did It

The authors acted as translators between these two worlds.

1. Solving the Map (The ODE Side): They took the complex equation for the $E_6$ crystal and used a "diagonalization" trick (like untangling a knot) to break it down into simpler pieces. They then calculated the "travel times" (period integrals) around a specific loop (the Pochhammer contour) on the map.
2. Measuring the Machine (The CFT Side): They built the "vital signs" (Integrals of Motion) for the $E_6$ quantum field theory. This involved using a "free field" method, which is like building a complex sculpture out of simple, standard Lego blocks.
3. The Match: They compared the numbers from Step 1 and Step 2.

4. The Result: A Perfect Match!

The authors found that the numbers matched perfectly up to the 6th level of detail.

• The Metaphor: Imagine you have two different recipes for a cake. One is written in French (the Math Map), and the other in Japanese (the Physics Machine). You translate the ingredients and cooking times. Even though the languages are totally different, when you bake both cakes, they taste exactly the same.
• The Discovery: The "travel times" on the $E_6$ map were identical to the "vital signs" of the $E_6$ machine, once they applied the correct translation key (a specific relationship between the parameters).

5. Why Does This Matter?

This isn't just a math puzzle; it's a breakthrough for understanding the universe.

• New Tools: By proving this connection for the complex $E_6$ crystal, the authors have shown that we can use the "Math Map" to solve problems in the "Physics Machine" that were previously impossible to crack.
• Future Maps: This gives them a blueprint to study even more complex crystals (like $E_7$ and $E_8$) and other types of quantum theories.
• The "Rosetta Stone": It reinforces the idea that deep down, the laws of physics and the laws of advanced mathematics are speaking the same language.

In summary: The authors took a very difficult, abstract math problem involving a complex shape ($E_6$), solved it using a step-by-step estimation method, and proved that the answer is identical to the "energy signature" of a corresponding quantum physics system. This confirms that these two seemingly different fields of science are deeply, beautifully connected.

Technical Summary

1. Problem Statement

The paper investigates the ODE/IM (Ordinary Differential Equation / Integrable Model) correspondence for the affine Lie algebra of type $E_6^{(1)}$.

• Context: The ODE/IM correspondence establishes a deep link between the spectral problem of specific linear differential equations (ODEs) and the integrable structure of two-dimensional Conformal Field Theories (CFTs) with $W$-algebra symmetry.
• Gap: While this correspondence has been extensively studied for classical affine Lie algebras (types $A_r^{(1)}$ and $D_r^{(1)}$) and the $W_3$ algebra, the case for exceptional affine Lie algebras (specifically $E_6^{(1)}$) remains less explored. The corresponding $W$-algebra ($W_{E_6}$) has a complex structure involving higher-spin currents (spins 2, 5, 6, 8, 9, 12) that had not been fully matched with ODE spectral data in this context.
• Objective: To verify the ODE/IM correspondence for $E_6^{(1)}$ by explicitly calculating the WKB expansion of the associated linear ODE up to the sixth order and comparing the resulting period integrals with the eigenvalues of the integrals of motion (IMs) in the $W_{E_6}$ CFT.

2. Methodology

A. The ODE Side: WKB Expansion via Diagonalization

The authors analyze the linear differential equation associated with the $E_6^{(1)}$ affine Lie algebra in the 27-dimensional representation:
$$(\epsilon \partial_z + A(z)) \Psi(z) = 0$$
where the gauge connection $A(z)$ includes a polynomial potential $p(z) = z^{hM-1}$ (with $h=12$ being the Coxeter number) and parameters related to the Cartan subalgebra.

1. Diagonalization Method: Instead of reducing the system to a single higher-order scalar ODE (which is difficult for $E_6$), the authors employ a gauge transformation $g(z)$ to diagonalize the connection $A(z)$.
2. Riccati Equations: The diagonalization condition leads to a system of 26 coupled non-linear Riccati equations for the gauge parameters $g_{27,i}(z)$.
3. Recursive WKB Solution: The gauge parameters are expanded in powers of the Planck constant $\epsilon$: $g_{27,i}(z) = \sum \epsilon^k s_i^{(k)}(z)$.
   • The zeroth-order equations are solved to express all $s_i^{(0)}$ in terms of a single function $s_{26}^{(0)}$ and the potential $p(z)$.
   • Higher-order coefficients ($k \geq 1$) are determined recursively by solving linear systems involving a matrix $B$ derived from the zeroth-order solution.
4. Period Integrals: The authors calculate the WKB periods $Q_k$ by integrating the diagonalized connection component $(A_g)_{27,27}$ along a Pochhammer contour $C$ in the complex plane. The periods are expanded as $Q_k = \oint_C dz \, s_{26}^{(k)}(z)$.

B. The IM Side: Integrals of Motion in $W_{E_6}$ CFT

The authors construct the integrals of motion for the $W_{E_6}$ algebra on a cylinder.

1. Free Field Realization: The $W_{E_6}$ algebra is constructed using the quantum Miura transformation applied to the $A_5$ subalgebra of $E_6$ and an additional free boson.
2. Currents and Charges: The algebra is generated by currents $W_s$ with spins $s \in \{2, 5, 6, 8, 9, 12\}$. The authors explicitly construct the spin-6 primary field $\tilde{W}_6$ and the conserved currents $j_k$ up to spin 6.
3. Eigenvalues: The eigenvalues $I_k$ of the integrals of motion (zero modes of the conserved currents) are calculated for the highest-weight state $|\Delta\rangle$. These eigenvalues are expressed in terms of the Casimir invariants $D_k$ of the weight vector $\mu = \lambda + a\rho$, where $\lambda$ is the highest weight and $a$ is the coupling constant.

3. Key Contributions and Results

A. Explicit WKB Coefficients up to Order 6

The paper provides the first explicit recursive solution for the WKB coefficients of the $E_6^{(1)}$ linear problem up to the sixth order ($s_{26}^{(k)}$ for $k=1, \dots, 6$).

• Vanishing Orders: It is found that $Q_1 = Q_3 = Q_4 = Q_7 = 0$, consistent with the symmetry and structure of the algebra.
• Non-trivial Orders: Explicit formulas for $Q_2, Q_5,$ and $Q_6$ are derived. These are expressed in terms of the Casimirs $C_k$ of $E_6$ (specifically $C_2, C_5, C_6$) and the parameters $M$ (related to the potential) and $l$ (related to the Cartan subalgebra).
• Integral Evaluation: The period integrals are evaluated using Gamma functions and recurrence relations for the contour integrals $J(a,b)$.

B. Matching with CFT Integrals of Motion

The core result is the exact agreement between the ODE periods and the CFT eigenvalues under specific parameter mappings:

1. Parameter Identification: The correspondence holds if the ODE parameters $(l, M)$ are related to the CFT parameters $(\lambda, a)$ via:
   $$\frac{l + \rho}{h\sqrt{M+1}} = \lambda + a\rho$$
   and
   $$a^2 = \frac{M^2}{M+1}$$
   where $\rho$ is the Weyl vector.
2. Casimir Relation: Under these mappings, the Casimirs of the ODE side ($C_k$) and the CFT side ($D_k$) satisfy:
   $$\text{tr}(q \cdot H)^k = h^k (M+1)^{k/2} \text{tr}(\mu \cdot H)^k$$
3. Verification:
   • Spin 2: $Q_2 \propto I_2$ (verified).
   • Spin 5: $Q_5 \propto I_5$ (verified).
   • Spin 6: $Q_6 \propto I_6$ (verified). The complex expression for $Q_6$ involving $C_6, C_2^3, C_2^2$, etc., matches the eigenvalue $I_6$ derived from the complicated normal-ordered products in the CFT.

4. Significance

• Extension to Exceptional Algebras: This work successfully extends the ODE/IM correspondence to the exceptional affine Lie algebra $E_6^{(1)}$, a non-trivial case due to the high dimensionality of the representation (27D) and the complexity of the $W_{E_6}$ algebra.
• Validation of the Correspondence: The agreement up to the sixth order provides strong evidence that the ODE/IM correspondence is universal across simply-laced affine Lie algebras, not just the classical series ($A, D$).
• Reconstruction of W-Algebras: The method demonstrates that the eigenvalues of the integrals of motion (which encode the structure of the $W$-algebra) can be recovered purely from the WKB expansion of the linear ODE. This offers a powerful tool for reconstructing the free field realization and structure of $W$-algebras for other exceptional or non-simply laced cases where the algebraic structure is not yet fully understood.
• Future Applications: The authors suggest this approach is crucial for understanding the Nekrasov partition functions for arbitrary gauge groups and the quantum Seiberg-Witten curves for Argyres-Douglas theories, particularly those associated with exceptional singularities. It also opens the door to studying wall-crossing phenomena and Thermodynamic Bethe Ansatz (TBA) equations for $E_6$-type theories.

In summary, the paper rigorously establishes the ODE/IM correspondence for $E_6^{(1)}$ by bridging the gap between the spectral geometry of a 27-dimensional linear system and the integrable structure of the $W_{E_6}$ conformal field theory.