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Exponential Decay for the Asymptotic Geometry of the Hitchin Metric

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Abstract

We consider Hitchin’s hyperkähler metric \(g_{L^2}\) on the SU(n)-Hitchin moduli space over a compact Riemann surface. We prove that the difference between the metric \(g_{L^2}\) and a simpler “semiflat” hyperkähler metric \(g_{\mathrm {sf}}\) is exponentially-decaying along generic rays in the Hitchin moduli space, as conjectured by Gaiotto–Moore–Neitzke.

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Notes

  1. For SU(2), we take \(({\overline{\partial }}_E, \varphi )\) in any non-degenerate fiber. For SU(n), the genericity condition is defined in Definition 4.1.

  2. Here, the compatibility condition is \(F_{D({\overline{\partial }}_{{{\,\mathrm{Det}\,}}E}, h_{{{\,\mathrm{Det}\,}}E})}=0\).

  3. The hyperkähler metric \(g_{L^2}\) on \({\mathcal {M}}\) determines and is completely determined by a twisted fiber-wise holomorphic symplectic structure on the twistor space \({\mathcal {Z}}={\mathcal {M}}\times {{\mathbb {C}}}{{\mathbb {P}}}^1\). The integral relation is actually formulated in terms of certain “holomorphic Darboux coordinates” \({\mathcal {X}}_\gamma \) on the twistor space \({\mathcal {Z}}={\mathcal {M}}\times {{\mathbb {C}}}{{\mathbb {P}}}^1\)

  4. In the SU(2) case, \(\Gamma _b=H^{\mathrm {odd}}_1(\Sigma _b, {\mathbb {Z}})\), the sublattice of \(H_1(\Sigma _b,{\mathbb {Z}})\) which is odd under the exchange of the two sheets of \(\Sigma \). For SU(n), \(\Gamma _b = H_1(\Sigma _b, {\mathbb {Z}})_{\sigma }\), a sublattice defined in (2.36).

  5. The function \(K_0(x)\) solves the modified Bessel differential equation \(x^2 y''(x) + x y'(x) - x^2 y(x) =0\) on \((0, \infty )\). Within the two-dimensional family of solutions, the function \(K_0(x)\) is determined (up to multiplication by constant) by the property that \(\lim _{x \rightarrow \infty } K_0(x)=0\). It is defined by the integral \(K_0(x) = \int _0^\infty \frac{\cos (xt)}{\sqrt{t^2+1}} \mathrm {d}t\).

  6. Let \(\Gamma \rightarrow {\mathcal {B}}'\) be the lattice bundle with fiber \(\Gamma _b\). Then, \(\Omega : \Gamma \rightarrow {\mathbb {Z}}\). Given a section \(\gamma \) of \(\Gamma \rightarrow {\mathcal {B}}'\), the function \(\Omega (\gamma ; \; \cdot \;): {\mathcal {B}}' \rightarrow {\mathbb {Z}}\) is typically not continuous. The value jumps at real-codimension-1 walls in \({\mathcal {B}}'\) and satisfies the Kontsevich–Soibelman wall-crossing formula.

  7. This special Kähler metric is not unique; we have the freedom to rescale it by a constant.

  8. The statement \(\nabla ^{GM}_{{\dot{b}}} \lambda = {{\dot{\tau }}}\) appears in [HHP10, Proposition 8.2], where \(\nabla ^{GM}\) is the Gauss–Manin connection.

  9. Given a Higgs bundle \(({\overline{\partial }}_E, \varphi )\), let \({\mathcal {L}}\rightarrow \Sigma _b\) be the associated spectral data. To see the difficulty in passing from \(({\dot{\eta }}, {\dot{\varphi }})\) to \({\dot{\xi }}\), note that the holomorphic line bundle \({\mathcal {L}}\) arises from \(({\overline{\partial }}_E, \varphi )\) as follows: the underlying complex line bundle \(L \subset \pi ^*E\) is the eigenline bundle of the tautological eigenvalue of \(\pi ^* \varphi \) acting on \(\pi ^*E \rightarrow \Sigma _b\). The holomorphic structure on \({\mathcal {L}}=(L, {\overline{\partial }}_L)\) is from the holomorphic structure of the ambient holomorphic bundle \(\pi ^*{{\mathcal {E}}}\rightarrow \Sigma _b\). Now, given a deformation \(({\dot{\eta }}, {\dot{\varphi }})\) note that the section \(L_\epsilon \subset \pi ^*E\) moves in the family \(\varphi +\epsilon {\dot{\varphi }}\), consequently, we first need to identify \(L_\epsilon \) with a common fixed complex line bundle L. We can avoid this in the case where \({\dot{\varphi }}=0\). In this case, the deformation \({\dot{\xi }}\) is given in terms of \({\dot{\eta }}\) by \(\pi ^*{\dot{\eta }} \cdot s = {\dot{\xi }} s\), where \(\pi ^*{\dot{\eta }}\) is locally a (0, 1)-form valued in \({{\mathfrak {s}}}{{\mathfrak {l}}}(n,{\mathbb {C}})\) acting by matrix multiplication on any section s of L, while \({\dot{\xi }}\) is locally a (0, 1)-form acting by scalar multiplication on s.

  10. The Gauss–Manin connection is considerably more general. Consider a family of algebraic varieties \({\mathcal {X}} \rightarrow {\mathcal {B}}\) where the fibers \(\pi ^{-1}(b)=X_b\) are all diffeomorphic. Fixing \(k \in {\mathbb {N}}\), the associated deRham cohomology groups \(H^k_{dR}(X_b)\) are all isomorphic. The Gauss–Manin connection is a connection on the vector bundle \({\mathcal {H}} \rightarrow {\mathcal {B}}\) whose fiber over b is \({\mathcal {H}}_b=H^k_{dR}(X_b)\). As such, the Gauss–Manin connection allows us to identify nearby fibers of \({\mathcal {H}}\).

    For the SU(n)-Hitchin moduli space, the relevant family of varieties is the family of spectral curves \(\Sigma _b\) and the relevant cohomology groups are \({\mathcal {H}}_b=H^{0,1}(\Sigma _b)_\sigma \), since \(T \mathrm {Prym}(\Sigma _b) \simeq H^0(\Sigma _b, K_{\Sigma _b})_\sigma ^* \simeq H^{0,1}(\Sigma _b)_\sigma \). We view \({\mathcal {M}}' \simeq {\mathcal {H}}/\Gamma \) where \(\Gamma \) is a local system with fiber \(\Gamma _b\).

  11. The fact that \({{\dot{\nu }}}^X_t\) satisfies the complex variation equation reduces to the fact that

    figure a

    We can reduce this to the case \(t=1\) which appears in [DN19, Eq. 10.15] in which the deformation associated to \({\dot{P}}\) is

    figure b

    Define \(\rho _t(z) =t^{-2/3} z\). One can check that \(\rho _t^* u_t = u_1\). Because of this, by pulling back the expression (3.12) by \(\rho _t\) and dividing by \(t^{4/3}\), we get

    figure c

    Thus the function \(F_t^X\) in (3.13) is defined so that \(\rho _t^* F^X_t\) solves the complex variation for the deformation \(t^{2/3} \rho _t^*{\dot{P}}\).

  12. This is because

    figure d

  13. For a fixed value of t, once we know that

    figure e

    we immediately get that that for all \(z \in {{\mathbb {D}}}\)

    figure f

    This is because the slope of any secant line of the function \({\mathrm e}^x\) valued on \([0,C_t]\) is less than the slope of the tangent line to \({\mathrm e}^x\) at \(x=C_t\).

  14. To prove this, Mazzeo–Swoboda–Weiss–Witt use their earlier work describing the family of harmonic metrics. In [MSWW16], they prove that \(h_t(w_1,w_2)= h_t^\mathrm {app}({\mathrm e}^{-\kappa _t} w_1, {\mathrm e}^{-\kappa _t} w_2)\) for \(h_0\)-hermitian \(\kappa _t\) satisfying \(\Vert \kappa _t\Vert _{H^2\left( i {{\mathfrak {s}}}{{\mathfrak {u}}}(E)\right) } \le C {\mathrm e}^{-\delta t}\). (This formulation in terms of the hermitian metrics appears in [Fre18], but Mazzeo–Swoboda–Weiss–Witt’s \(h_0\)-unitary formulation is equivalent to it.)

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Acknowledgements

The author thanks Rafe Mazzeo, Andy Neitzke, Hartmut Weiss, and Ákos Nagy for helpful discussions related to this work, and Georgios Kydonakis for his comments on an earlier version. The author also thanks the anonymous referee for helpful comments.

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  1. Department of Mathematics, Stanford University, Stanford, USA

    Laura Fredrickson

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Correspondence to Laura Fredrickson.

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Communicated by N. Nekrasov.

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Fredrickson, L. Exponential Decay for the Asymptotic Geometry of the Hitchin Metric. Commun. Math. Phys. 375, 1393–1426 (2020). https://doi.org/10.1007/s00220-019-03547-9

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