Nilpotent Higgs Bundles and Families of Flat Connections

Abstract

We investigate ${{\mathbb {C}}}^\times$-families of flat connections whose leading term is a nilpotent Higgs field. Examples of such families include real twistor lines and families arising from the conformal limit. We show that these families have the same monodromy as families whose leading term is a regular Higgs bundle and use this to deduce that traces of holonomies are asymptotically exponential in rational powers of the parameter of the family.

Appendix A WKB Curves via Translation Surfaces

In this appendix we will discuss the existence of WKB curves on Riemann surfaces (with holomorphic quadratic differentials). For this, we will remind the reader of the interpretation of such surfaces as half-translation surfaces and construct a few explicit examples such as Fig. 2. Afterwards we will discuss the general theory. We do not claim any originality but were unable to find this discussed in the literature.

Fig. 2

The trajectory structure for a generic quadratic differential in the toy model, and a WKB loop. The four first-order poles are at the marked locations 0, 1, i, $1+i$ and each emit a single ray. The genericity condition concerns the phase at which these rays are emitted

First, recall our definition of a WKB curve: Start with a connected Riemann surface C with an $\textrm{SL}(2,{{\mathbb {C}}})$-Higgs bundle $(\overline{\partial }_E, \varphi )$ such that $\varphi$ is not nilpotent, then $\varphi$ has eigenvalues $\pm \mu (z) dz$ that are generically nonzero. These define a branched double-covering $\Sigma \subset T^*C$ of C known as the Spectral Curve that keeps track of the two eigenvalues. Of course, $( \mu (z) dz )^2 = 1/2 {\text {Tr}}(\varphi ^2)$, so the eigenvalues only depend on the quadratic differential $q_2$ associated to $(\overline{\partial }_E, \varphi )$ via the Hitchin map, and then so does $\Sigma = \Sigma _{q_2}$. Around any point $z_0 \in C$ where $\mu (z_0) \ne 0,\infty$ one can define a local coordinate $w = w(z)$ via

$$\begin{aligned} w(z):= \int _{z_0}^z \mu (z) dz = \int _{z_0}^z \sqrt{\frac{1}{2} {\text {Tr}}(\varphi (z)^2)} \hspace{0.2cm}, \end{aligned}$$

(A.1)

and a change in base point corresponds to a constant translation of this local coordinate, while a different choice of square root (i.e. the exchange $\mu \leftrightarrow -\mu$) corresponds to an overall change of sign of w.

These local coordinates give C the structure of a half-translation surface: Roughly speaking, a half-translation surface is a finite or countable collection of polygons in the Euclidean plane whose edges are identified by maps of the form $w \mapsto \pm w + a$ called half-translations. Given such a half-translation surface, the inherited metric induces a complex structure on the glued surface, and the quadratic differential that is locally defined as $(dw)^2$ is invariant under half-translations so descends to a globally well-defined quadratic differential.

There is one more subtlety that is worth pointing out for our discussion, namely the role of zeroes or poles of the (meromorphic) quadratic differential. In the description above, we allow zeroes of arbitrary order but poles only of order 1. While the neighborhood of a regular point is Euclidean as described above, zeroes or poles correspond to conical singularities on the side of half-translation surfaces: An opening angle larger than $(2+k) \pi$ corresponds to a zero of order k ($k \geqslant 1$), while an opening angle of $\pi$ signifies a simple pole.

A special type of half-translation surface is a translation surface: All edge identifications preserve the orientation and are of the form $w \mapsto w + a$. These surfaces correspond to the case that the quadratic differential is globally the square of an Abelian differential $A \in H^0(C, K_C)$. We give some examples of translation and half-translation surfaces in Figs. 3 resp. 4. Determining the genus of the corresponding surface is an easy exercise using the Euler characteristic: For example, the left surface in Fig. 3 contains 2 vertices, 12 edges (7 external and 5 internal) and 6 faces, yielding a surface of Euler characteristic $2-2g=2-12+6=-4$ and hence of genus 3. Equivalently, one can count the order of zeroes and poles and determine this way the degree of the canonical bundle. Of course, the individual rectangles are not required to be of the same size (though sides that are to be identified need to be of the same length), and the complex structure of the resulting surface depends on these choices.

Fig. 3

Two translation surfaces. The labelling of the edges prescribes the gluing and induces an identification of the vertices which we have included for the convenience of the reader. A staircase composed of 2n rectangles as on the left and one with $2n-1$ as on the right both yield a surface of genus n

Now we can come back to our original intention of constructing examples of WKB curves on Riemann surfaces C equipped with a quadratic differential. Upon translating the problem to the language of half-translation surfaces, a curve $\gamma : [0,1] \rightarrow C$ becomes a collection of curves $\gamma _i: [t_i, t_{i+1}] \rightarrow P$ where P is the set of polygons defining the half-translation structure, and $\gamma _i (t_i), \gamma _i (t_{i+1}) \in \partial P$ such that $\gamma _i (t_{i+1})$ and $\gamma _{i+1} (t_{i+1})$ are identified when gluing the edges of P in the prescribed fashion. With this understood, a WKB curve is one for which ${\text {Im}}\gamma _i$ is a strictly increasing function on $[t_i, t_{i+1}]$ for all i. It is now easy to see that the examples we have provided in Figs. 3 and 4 have WKB curves given by vertical paths connecting two edges labeled by $A_i$.

Fig. 4

Two half-translation surfaces: Notice that the vertices 2 and 3 in the left figure (resp. 3 and 4 in the right one) have opening angle $\pi$ and correspond to first-order poles of the quadratic differential. A configuration such as the one on the left with $2n+1$ rectangles yields a surface of genus n, the same is true for a configuration such as the one on the right consisting of 2n rectangles

Let $D \subset C$ be the set of zeroes and poles of $q_2$, and $C^o:= C-D$. We are interested in curves $\gamma$ that evade the zeroes and poles of $q_2$ and so we are really interested in homotopy classes of loops $[\gamma ] \in \pi _1 (C^o)$. In particular, this shows that two of examples considered in the previous paragraph are homotopic if and only if they connect the same two edges.

WKB loops have the special feature that their lift to the spectral curve $\Sigma$ consists of two disjoint loops. The reason is that ${\text {Re}}(\mu (z)) > 0$ along a WKB curve, so the two sheets of the covering $\Sigma \rightarrow C$ cannot mix in a neighborhood of $\gamma$. It is therefore convenient to consider the notion of WKB curves on $\Sigma$ which is by itself a translation surface: The Liouville 1-form $\lambda$ on $T*C$ naturally restricts to an abelian differential on $\Sigma$.

Let us now discuss the general existence of WKB curves on half-translation surfaces where we will restrict our attention to quadratic differentials with at most simple poles. Notice that the foliation induced by $-q_2$ is equivalent to the one coming from $q_2$ (the segments are unoriented) but that for any $\theta \in \mathbb {R}/[0,\pi ] \simeq \mathbb{R}\mathbb{P}^1$ the horizontal foliation $\mathcal {F}_\theta$ coming from $e^{i \theta } q_2$ is transverse to the original $\mathcal {F} = \mathcal {F}_0$.

Now let $p \in C^o$, we want to construct a homologically non-trivial WKB loop that passes through p. It is a well-known fact (see e.g. [2, 45]) that there are at most countably many angles $\theta$ for which the leaf of $\mathcal {F}_\theta$ passing through p is non-recurrent so that generically the leaf forms a dense subset on C. It is then clear that in a small neighborhood of p, a leaf at a generic angle consists of parallel line segments and that one can smoothly connect a different line segment from the one that contains p via a small path that remains transverse to $\mathcal {F}$ to form a loop. A sketch of this can be found in Fig. 5.

Fig. 5

A minor detour connects two strands of a generic leaf of $\mathcal {F}_\theta$ to create a closed loop that is everywhere transverse to the foliation $\mathcal {F}$

This carries a natural orientation when interpreted via its lift to $\Sigma$. Observe that $\gamma$ is necessarily homologically non-trivial as we shall explain now. If $\gamma$ were to close up without the detour, one could easily compute its period

$$\begin{aligned} Z_\gamma = l_\gamma \times e^{i\theta } \end{aligned}$$

(A.2)

where $l_\gamma$ is the length of $\gamma$. Since the period is nonzero and depends linearly on the homology class, this implies that it is homologically nontrivial. If we include the detour then the computation of the period is slightly more cumbersome but only differs marginally from the previous outcome, so in particular remains non-zero. This proves the following

Proposition A.1

Let C be a compact Riemann surface and $q_2$ a meromorphic quadratic differential on C that is non-zero and has at most simple poles. Then $(C, q_2)$ admits a WKB loop.

Lastly, let us comment on the existence of WKB curves in the case of a non-nilpotent ${\textrm{SL}}(n, {{\mathbb {C}}})$-Higgs bundle $(\overline{\partial }_E, \varphi )$. Let $u:= h(\overline{\partial }_E, \varphi )$ denote its projection to the Hitchin base and $\Sigma = \Sigma _u$ the associated spectral curve. Recall that $\Sigma _u \rightarrow C$ is a branched (n:1)-cover and that $(\Sigma _u, \lambda )$ is a translation surface.

Pick a class $[\gamma ] \in \pi _1 (C)$ that admits a representative $\gamma : [0,1] \rightarrow C$ with the following properties:

• 1. Its lift to $\Sigma _u$ is the disjoint union of n loops;

• 2. It has extension $\tilde{\gamma }$ to a tubular neighborhood U of $[0,1] \subset {{\mathbb {C}}}$ such that $\tilde{\gamma }: U \rightarrow C$ is holomorphic.

U is contractible and so $\tilde{\gamma }^*\mathcal {E}$ is trivializable in a global coordinate z. The first condition implies that $\tilde{\gamma }^*\varphi$ is diagonalizable on U with distinct eigenvalues $\mu _i (z) dz$. A WKB curve is one for which there is an ordering

$$\begin{aligned} {\text {Re}}(\mu _1)> {\text {Re}}(\mu _2)> \dots > {\text {Re}}( \mu _n ) \end{aligned}$$

(A.3)

for all $z \in U$. Every ordered pair $\{i,j | i > j \} \subset \{1, \dots ,,n \}$ determines an abelian differential $\lambda _{ij} = (\mu _i - \mu _j) dz$ on U and $\gamma$ is a WKB curve if and only if it is a WKB curve with respect to each of the $\lambda _{ij}$. In other words, there are $\left( {\begin{array}{c}n\\ 2\end{array}}\right)$ different foliations $\mathcal {F}_{ij}$ on U and $\gamma$ has to be transverse to all of their leaves.

Effectively, the asymptotics considered around (4.23) should be governed by the largest (by real part) eigenvalue $\mu _1$, so even if some of the smaller eigenvalues were to exchange order along $\gamma$ it should not effect the asymptotic behavior. Consequently, the condition on $\gamma$ being eligible for the WKB method would require it to be transverse to $(n-1)$ foliations on C.