Opers and nonabelian Hodge: numerical studies

TL;DR

The paper conducts extensive numerical experiments to probe the twistor Riemann-Hilbert conjecture, linking the NAHC on the Hitchin section and the oper monodromy via conjectural integral equations expressed through spectral coordinates and Stokes data. It develops and compares two computational paradigms—the direct differential-equation approach and the integral-equation method—for opers and Hitchin-section Higgs bundles on the complex plane, including the restricted hyperkähler metric. Across 13 one-parameter families and multiple ranks, the results show strong qualitative and quantitative agreement between methods in regimes where numerical errors are controlled, with discrepancies consistent with expected error growth, and no counterexamples detected. The work also provides detailed implementation, error estimation, and sample calculations, along with public code and data to enable replication and future extensions. Overall, the findings support the conjectured integral-equation framework and illustrate a practical, rapidly convergent computational approach to a deep link between flat connections, Higgs bundles, and hyperkähler geometry in noncompact settings.

Abstract

We present numerical experiments that test the predictions of a conjecture of Gaiotto-Moore-Neitzke and Gaiotto concerning the monodromy map for opers, the nonabelian Hodge correspondence, and the restriction of Hitchin's hyperkähler metric to the Hitchin section. These experiments are conducted in the setting of polynomial holomorphic differentials on the complex plane, where the predictions take the form of conjectural formulas for the Stokes data and the Hitchin metric tensor. Overall, the results of our experiments support the conjecture.

Figures (21)

• Figure 1: Spectral network (left) and triangulated polygon $T$ (right) for $(A_1,A_2)$ ($P_2(z) = z^3-1$) at $\vartheta=0$. The same triangulation arises for all $|\vartheta| < \frac{\pi}{6}$. The labeled saddle connections correspond to $-\gamma_1$ (red) and $-\gamma_3$ (blue), and the dual edges in $T$ are correspondingly colored.
• Figure 2: Spectral network (left) and triangulated polygon (right) for $(A_1,A_3)$ ($P_2(z) = z^4-1$) at $\vartheta=0.4$. The same triangulation arises for all $0 < \vartheta < \frac{\pi}{4}$. At left, the colored segments are the saddle connections corresponding to $\gamma_1$ (red), $-\gamma_2$ (green), and $-\gamma_3$ (blue), and the dual edges in $T$ are correspondingly colored.
• Figure 3: Spectral network (left) and projected homology basis (right) for $(A_2,A_1)$, with $P_3(z) = \frac{1}{2}(1-z^2)$).
• Figure 4: Spectral network (left) and projected homology basis (right) for $(A_2,A_2)$, with $P_3(z) = \frac{1}{2}(z^3 - 3 z^2 - 2)$. The cycles $\gamma_1$ and $\gamma_2$ intersect over the indicated point.
• Figure 5: Oper comparison: $(A_1,A_2)$ family at base point ($\Lambda=0$, $c=1$) with $\vartheta=0$.