Opers and Non-Abelian Hodge: Numerical Studies

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 non-Abelian Hodge correspondence, and the restriction of the hyperkähler L² 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 metric tensor. Overall, the results of our experiments support the conjecture.

Keywords:

• Opers
• non-Abelian Hodge theory
• Hitchin system
• spectral coordinates
• numerical experiments

Declaration of Interest

No potential conflict of interest was reported by the author(s).

Notes

1 In more general rank 3 cases it would be necessary to consider other types of “degenerations” of the spectral network; an algorithm for defining and computing $\Omega (w,\mu )$ in general is given in [28].

2 To verify this it is convenient to rewrite the sum of integrals by combining the terms for μ and $-\mu$, as we do in Section 7.5.

3 Here we define the relative difference between real or complex quantities a and b to be $\text{reldiff}(a,b)=\frac{2|a-b|}{\left|a\right|+\left|b\right|}$, that is, $\text{reldiff}(a,b)$ describes the difference as a fraction of the average of $\left|a\right|,\hspace{0.17em}\left|b\right|$.

4 We take $\sigma (z)=s(1-|z|/(0.9r))$ where s(t) is the smoothed step function s(t) = 0 for t < 0, s(t) = 1 for t > 1, and $s(t)=\frac{1}{2}(1-\hspace{0.17em}\text{cos}\hspace{0.17em}(\pi t))$ for $t\in [0,1]$.