Asymptotic Geometry of the Hitchin Metric

Abstract

We study the asymptotics of the natural L 2 metric on the Hitchin moduli space with group G = SU(2). Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore [GMN], is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric from [GMN]. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. New work by Dumas and Neitzke shows that the convergence is actually exponential in directions tangent to the Hitchin section.

Exponential Decay for the Asymptotic Geometry of the Hitchin Metric

“…The lengthy §2 contains two more notable results: In Proposition 2.14 we give a practical characterization of the semiflat metric. In Theorem 2.15, we prove that the L 2 -metric on the moduli space of limiting configurations M ∞ agrees with the semiflat metric for SU(n), generalizing Mazzeo-Swoboda-Weiss-Witt's proof for SU(2) in [MSWW19]. Our proof is simpler, making use of the spectral cover and the analog of (1.22) for M ∞ .…”

“…Additionally, we review Gaiotto-Moore-Neitzke's conjecture [GMN10,GMN09] for Hitchin's hyperkähler metric g L 2 and the progress made towards this conjecture in the rank 2 case ( §1.4). Of particular note, Mazzeo-Swoboda-Weiss-Witt [MSWW19] have shown that g L 2 − g sf decays polynomially in t along a generic ray in M SU(2) ; Dumas-Neitzke [DN19] have shown that-restricted to the image of the Hitchin section of M SU(2) -g L 2 − g sf decays exponentially in t. We comment on some of the important ingredients in their respective proofs that we gladly borrow, and describe our strategy of proof in §1.5. We also highlight some results that we prove in the process about the semiflat metric and Hitchin's hyperkähler metric that we believe are of independent interest and utility.…”

“…Thus, it is polynomially-decaying even though it appears in (1.14) to be polynomially-growing.) The result in [MSWW19] is built on the description of h t near the ends of M given in [MSWW16]. They build an approximate solution of Hitchin's equations h app t close to h t [MSWW16, Theorem 6.7] by desingularizing the singular metric h ∞ on disks D around the zeros of q 2 .…”

“…(In the extension to SU(n), we make use of the construction of approximate solutions h app t in [Fre18].) As in [MSWW19], we break the integral into two parts…”

“…Using the analogy of the clean expression for the hyperkähler metric in (2.18) for g L 2 (M ∞ ), we give a proof that g sf = g L 2 (M ∞ ) that works for any SU(n). Our initial exposition follows [MSWW19].…”