The Hitchin System and Geometric Langlands Mirror Symmetry¶

Job: develop the Hitchin moduli space and its integrable-system fibration, the mirror symmetry of Hitchin moduli for $G$ and the Langlands dual $\check G$ (Hausel–Thaddeus, Donagi–Pantev), the geometric Langlands correspondence realized by this mirror symmetry, and the BAA / BBB brane correspondence (Kapustin–Witten) — with the $\mathrm{SL}_2$ Hitchin mirror as the worked example.

Core¶

The Hitchin fibration and the mirror moduli¶

Let $X$ be a smooth projective curve of genus $g\ge2$ and $G$ a complex reductive group. The Hitchin moduli space is $$ \mathcal{M}_{\mathrm{Hit}}(G)={(E,\Phi):E\text{ a principal }G\text{-bundle on }X,\ \Phi\in H^0(X,\mathrm{ad}(E)\otimes K_X)}/\sim, $$ the moduli of $G$-Higgs bundles $(E,\Phi)$ with the Higgs field $\Phi$ (a section of the adjoint-valued canonical bundle). For $G=\mathrm{GL}_n$, $\Phi$ is an endomorphism-valued 1-form; for $G=\mathrm{SL}_n$, $\Phi\in H^0(\mathrm{ad}(E)\otimes K_X)$ trace-free.

The Hitchin fibration is $$ h:\mathcal{M}{\mathrm{Hit}}(G)\to\mathcal{A}_G=\bigoplus), $$ where }^{r}H^0(X,K_X^{d_i+1$d_1<\cdots<d_r$ are the exponents of $G$ (the degrees of the invariant polynomials generating $\mathbb{C}[\mathfrak{g}]^G$) and $\mathcal{A}_G$ is the Hitchin base. For $G=\mathrm{SL}_n$, $\mathcal{A}=\bigoplus_{i=1}^{n-1}H^0(K_X^{i+1})$ (the coefficients of the characteristic polynomial of $\Phi$).

The generic fiber $h^{-1}(a)$ for generic $a\in\mathcal{A}_G$ is an abelian variety (the Prym / the Jacobian of the spectral curve): the spectral curve $$ \Sigma_a={\xi\in\mathrm{Tot}(K_X):\det(\xi-\Phi)=0}\subset\mathrm{Tot}(K_X) $$ (the eigenvalues of $\Phi$, a degree-$n$ cover of $X$ for $G=\mathrm{SL}_n$), and the fiber is the Prym variety $\mathrm{Prym}(\Sigma_a/X)$ (the anti-invariant part of the Jacobian $\mathrm{Jac}(\Sigma_a)$, $\dim=\dim\mathcal{A}$). The Hitchin fibration is an algebraically completely integrable system: a Lagrangian fibration of the holomorphic symplectic $\mathcal{M}_{\mathrm{Hit}}$ (the symplectic form $\omega_{\mathrm{Hit}}$ from the pairing $H^0(\mathrm{ad}\otimes K)\times H^1(\mathrm{ad})\to H^1(K)\cong\mathbb{C}$).

The Hitchin system as a mirror pair¶

Theorem / Conjecture (mirror symmetry of Hitchin moduli; Hausel–Thaddeus, Donagi–Pantev). The Hitchin moduli for $G$ and for the Langlands dual $\check G$ (the dual reductive group, e.g. $\mathrm{SL}_n\leftrightarrow\mathrm{PGL}_n$) are mirror Calabi–Yau integrable systems: $$ \mathcal{M}{\mathrm{Hit}}(G)\quad\longleftrightarrow\quad\mathcal{M}(\check G), $$ with the SYZ fibration being the Hitchin fibration: the bases }$\mathcal{A}_G\cong\mathcal{A}_{\check G}$ are identified (the invariant polynomials of $G$ and $\check G$ have the same degrees), and the dual abelian fibers are the Langlands-dual Prym: the $\check G$ Hitchin fiber is the dual abelian variety of the $G$ Hitchin fiber.

Theorem (Hausel–Thaddeus; Donagi–Pantev). The stringy Hodge numbers / the E-polynomial of $\mathcal{M}_{\mathrm{Hit}}(G)$ and $\mathcal{M}_{\mathrm{Hit}}(\check G)$ are related by the mirror Hodge-number swap: $$ E(\mathcal{M}{\mathrm{Hit}}(G);u,v)=E(\mathcal{M}. $$ Proven for }}(\check G);u,v)\quad\text{with the Hodge-weight swap }(u,v)\leftrightarrow\text{ (the SYZ mirror of the stringy Hodge numbers)$G=\mathrm{SL}_n$ and $\check G=\mathrm{PGL}_n$ on a generic curve (the Hausel–Thaddeus conjecture; Donagi–Pantev for the smooth / generic-fiber duality).

This is the SYZ / Hodge-number mirror symmetry for the Hitchin system: the mirror of $\mathcal{M}_{\mathrm{Hit}}(G)$ is $\mathcal{M}_{\mathrm{Hit}}(\check G)$, the duality being the abelian-fiber duality over the same base.

Geometric Langlands via mirror symmetry¶

The geometric Langlands correspondence (the Betti / de Rham version) is the conjectural equivalence of derived categories $$ D^b\mathrm{Coh}(\mathrm{LocSys}_{\check G})\simeq D^b\mathrm{Coh}(\mathrm{DMod}(\mathrm{Bun}_G)), $$ where $\mathrm{LocSys}_{\check G}=\mathrm{Hom}(\pi_1 X,\check G)/\check G$ is the moduli of $\check G$-local systems (the Betti character variety) and $\mathrm{DMod}(\mathrm{Bun}_G)$ is the category of $D$-modules on the moduli of $G$-bundles $\mathrm{Bun}_G$.

The Donagi–Pantev program (and the Arinkin–Gaitsgory framework) realizes the geometric Langlands correspondence as the mirror symmetry of the Hitchin system: the mirror $\mathcal{M}_{\mathrm{Hit}}(G)\leftrightarrow\mathcal{M}_{\mathrm{Hit}}(\check G)$ (the SYZ mirror above) gives, via the non-abelian Hodge correspondence (the $\mathbb{C}^*$-family from $\mathcal{M}_{\mathrm{Hit}}$ to $\mathrm{LocSys}$) and the categorical interpretation of the branes, the geometric Langlands duality of the moduli of Higgs bundles / local systems.

Object	Langlands dual
$G$	$\check G$ (the dual reductive group)
$\mathcal{M}_{\mathrm{Hit}}(G)$	$\mathcal{M}_{\mathrm{Hit}}(\check G)$ (the mirror Hitchin moduli)
$\mathrm{Bun}_G$ (moduli of $G$-bundles)	$\mathrm{LocSys}_{\check G}$ (moduli of $\check G$-local systems)
$D$-modules on $\mathrm{Bun}_G$	Coherent sheaves on $\mathrm{LocSys}_{\check G}$
Hitchin base $\mathcal{A}_G$	$\mathcal{A}_{\check G}\cong\mathcal{A}_G$ (the same base)
Hitchin fiber: $\mathrm{Prym}(\Sigma_a/X)$ for $G$	dual abelian variety for $\check G$

The BAA / BBB branes and the hyperkähler mirror¶

The Hitchin moduli $\mathcal{M}_{\mathrm{Hit}}$ is a hyperkähler manifold (the $\mathbb{P}^1$ of complex structures $(I,J,K)$, the $L^2$-metric from non-abelian Hodge theory — the Corlette–Donaldson / Simpson hyperkähler metric). The branes on the Hitchin system are classified by their behavior in the three complex structures:

BBB branes (B-type in all three): holomorphic in $(I,J,K)$ — coherent sheaves / Higgs bundles with a holomorphic structure in all three complex structures. These are the "algebraic" branes (e.g. the structure sheaf, the skyscraper sheaves on the Hitchin base).
BAA branes (B-type in $I$, A-type in $J,K$): holomorphic in $I$ and Lagrangian in the complex-symplectic structures $(J,K)$ — these are the "symplectic" branes (e.g. the Lagrangian fibers of the Hitchin fibration, the nilpotent cone, the components of the Hitchin system).

Theorem / framework (Kapustin–Witten). Under the mirror symmetry $\mathcal{M}_{\mathrm{Hit}}(G)\leftrightarrow\mathcal{M}_{\mathrm{Hit}}(\check G)$, the BAA branes on $\mathcal{M}_{\mathrm{Hit}}(G)$ correspond to BBB branes on $\mathcal{M}_{\mathrm{Hit}}(\check G)$ (and vice versa) — the geometric Langlands as a brane mirror. The physical origin is the S-duality of $\mathcal{N}=4$ supersymmetric gauge theory (the Montonen–Olive duality $G\leftrightarrow\check G$), the Kapustin–Witten topological twist realizing the geometric Langlands as the brane correspondence.

Brane type	Hitchin ($\mathcal{M}_{\mathrm{Hit}}(G)$)	Mirror ($\mathcal{M}_{\mathrm{Hit}}(\check G)$)
BBB	coherent sheaf in $(I,J,K)$	BAA (Lagrangian in $(J,K)$, holomorphic in $I$)
BAA	Lagrangian in $(J,K)$, holomorphic in $I$	BBB (coherent sheaf)
Example	the Lagrangian Hitchin fiber (BAA)	a skyscraper sheaf on $\mathcal{A}_{\check G}$ (BBB)
Example	the structure sheaf of a component (BBB)	a Lagrangian (BAA)
Origin	$\mathcal{N}=4$ gauge theory, the Kapustin–Witten twist	the S-duality $G\leftrightarrow\check G$

The Nahm transform / the non-abelian Hodge correspondence (the $\mathbb{C}^*$-family interpolating between the Hitchin moduli and the character variety) realizes the BAA / BBB correspondence as the Fourier–Mukai transform along the Hitchin fibers.

The Fock–Goncharov cluster / higher Teichmüller connection¶

The Fock–Goncharov cluster coordinates on the higher Teichmüller / character variety $\mathrm{Rep}(\pi_1 X,G)=\mathrm{Hom}(\pi_1 X,G)/G$ (the Hitchin component for $G$ split real, the higher Teichmüller space of higher Teichmüller frontiers) are the mirror of the cluster structure of the Hitchin base: the same scattering diagrams / theta functions as the Gross–Siebert cluster (see SYZ & Gross–Siebert) appear on the Fock–Goncharov positive structure.

The higher Teichmüller mirror: the Hitchin section (the section of the Hitchin fibration from $\mathcal{A}_G$ to $\mathcal{M}_{\mathrm{Hit}}(G)$, the $\mathrm{SL}_2$ uniformization / the Fuchsian point) and the Fock–Goncharov positive structure are mirror-symmetric — the cluster coordinates on the Hitchin base / the character variety are the theta functions of the SYZ mirror of the Hitchin system.

See it¶

The $\mathrm{SL}_2$ Hitchin mirror and spectral curve duality. Let $G=\mathrm{SL}_2$ on a curve $X$ of genus $g\ge2$. The Hitchin base is $$ \mathcal{A}_{\mathrm{SL}_2}=H^0(X,K_X^2)\quad(\dim=3g-3), $$ the space of quadratic differentials $q\in H^0(K^2)$. For generic $q$ (with simple zeros), the spectral curve is the double cover $$ \Sigma_q={\xi\in\mathrm{Tot}(K_X):\xi^2=q}\subset\mathrm{Tot}(K_X), $$ a hyperelliptic curve of genus $g(\Sigma_q)=4g-3$ (by Riemann–Hurwitz, $2g(\Sigma)-2=2(2g-2)+\deg R=2(2g-2)+(4g-4)$, the ramification at the $4g-4$ zeros of $q$). The Hitchin fiber is the Prym variety $$ \mathrm{Prym}(\Sigma_q/X)=\mathrm{Jac}(\Sigma_q)^{-}={L\in\mathrm{Jac}(\Sigma_q):\sigma^*L=L^{-1}}, $$ the anti-invariant part under the involution $\sigma:\Sigma_q\to\Sigma_q$ (the deck transformation), $\dim\mathrm{Prym}=g(\Sigma_q)-g(X)=3g-3=\dim\mathcal{A}$.

The Langlands dual of $\mathrm{SL}_2$ is $\check G=\mathrm{PGL}_2=\mathrm{SL}_2/\{\pm I\}$, and the mirror Hitchin fibration $\mathcal{M}_{\mathrm{Hit}}(\mathrm{PGL}_2)\to\mathcal{A}_{\mathrm{PGL}_2}=\mathcal{A}_{\mathrm{SL}_2}=H^0(K^2)$ (the same base) has fibers the dual Prym: $$ \mathrm{Prym}(\Sigma_q/X)^{\vee}\cong\mathrm{Prym}(\Sigma_q/X)/\mathrm{Prym}[2], $$ the $\mathrm{PGL}_2$ Hitchin fiber = the $\mathrm{SL}_2$ Hitchin fiber modulo the 2-torsion (the Langlands-dual abelian variety). The SYZ mirror is the duality of the abelian fibers over the same base $\mathcal{A}=H^0(K^2)$, with the instanton corrections (the nilpotent cone / the hyperkähler metric corrections, the singular fibers of the Hitchin fibration) giving the stringy / Hodge-mirror symmetry (the Hausel–Thaddeus E-polynomial equality). The geometric Langlands correspondence $$ \mathrm{DMod}(\mathrm{Bun}{\mathrm{SL}_2})\simeq\mathrm{Coh}(\mathrm{LocSys}) $$ is the }_2categorical shadow of this mirror symmetry: the $D$-modules on $\mathrm{Bun}_{\mathrm{SL}_2}$ (the BBB branes, coherent sheaves on $\mathrm{LocSys}$) correspond to the coherent sheaves on $\mathrm{LocSys}_{\mathrm{PGL}_2}$ (the BAA branes, the Lagrangian fibers / the Hitchin section) under the mirror $\mathcal{M}_{\mathrm{Hit}}(\mathrm{SL}_2)\leftrightarrow\mathcal{M}_{\mathrm{Hit}}(\mathrm{PGL}_2)$.

Check¶

Show that the generic Hitchin fiber for $G=\mathrm{SL}_2$ is the Prym of the spectral double cover, and identify its dimension.

The spectral curve $\Sigma_q\to X$ is the double cover ramified at the zeros of $q\in H^0(K^2)$ (the simple zeros, $4g-4$ of them). Riemann–Hurwitz: $2g(\Sigma)-2=2(2g-2)+\deg R=2(2g-2)+(4g-4)$, so $g(\Sigma)=4g-3$. The Hitchin fiber for $\mathrm{SL}_2$ is the Prym $\mathrm{Jac}(\Sigma)^-=\{L:\sigma^*L=L^{-1}\}$, dimension $g(\Sigma)-g(X)=(4g-3)-g=3g-3=\dim\mathcal{A}$. The fiber is Lagrangian (the integrable system) of the right dimension.

Why is the geometric Langlands correspondence the categorical shadow of the Hitchin mirror, and what is the role of the BAA / BBB branes?

The Hitchin mirror $\mathcal{M}_{\mathrm{Hit}}(G)\leftrightarrow\mathcal{M}_{\mathrm{Hit}}(\check G)$ exchanges BAA and BBB branes (Kapustin–Witten): the BAA branes (Lagrangian Hitchin fibers, the A-type in $(J,K)$) on $\mathcal{M}_{\mathrm{Hit}}(G)$ correspond to BBB branes (coherent sheaves on $\mathrm{LocSys}_{\check G}$) on $\mathcal{M}_{\mathrm{Hit}}(\check G)$. Via the non-abelian Hodge correspondence (the $\mathbb{C}^*$-family from $\mathcal{M}_{\mathrm{Hit}}$ to $\mathrm{LocSys}$), the BBB branes on $\mathrm{LocSys}_{\check G}$ are coherent sheaves, and the BAA branes on $\mathrm{Bun}_G$ (via the Hitchin fibration of $\mathrm{Bun}_G\to\mathcal{A}_G$) are $D$-modules (the Hecke eigensheaves). The mirror exchanges these, giving $\mathrm{DMod}(\mathrm{Bun}_G)\simeq\mathrm{Coh}(\mathrm{LocSys}_{\check G})$ — the geometric Langlands.

Pitfalls¶

The Hitchin mirror is a hyperkähler / integrable-system mirror (the SYZ abelian-fiber duality, with the nilpotent-cone corrections) — not the same as the CY mirror of gromov-witten / fukaya-category. The hyperkähler metric (the $L^2$-metric of non-abelian Hodge theory) is the metric data, not a Calabi–Yau metric.
The geometric Langlands via mirror symmetry is a framework (the Donagi–Pantev program) — the Arinkin–Gaitsgory categorical GL is a separate (conjectural / announced for $\mathrm{GL}_n$) statement; the two are conjecturally related but not identical.
The BAA / BBB brane mirror is the Kapustin–Witten physics framework (the $\mathcal{N}=4$ S-duality), not a theorem in full — it is a physical framework with substantial mathematical confirmation (the Hausel–Thaddeus E-polynomial, the Donagi–Pantev generic-fiber duality) but the full categorical equivalence is the GL conjecture.
The Langlands dual of $\mathrm{SL}_n$ is $\mathrm{PGL}_n$ (not $\mathrm{SL}_n$ itself); the duality exchanges simply-connected and adjoint groups, and the Hitchin fibers differ by the finite torsion (the $\mathrm{PGL}_n$ fiber is the $\mathrm{SL}_n$ fiber modulo the $n$-torsion).
The Hitchin base is the same for $G$ and $\check G$ (the invariant polynomials have the same degrees), but the fibers are dual abelian varieties — do not confuse the base identification with the fiber duality.
The spectral curve $\Sigma_a$ for $G=\mathrm{SL}_n$ is a degree-$n$ cover of $X$; for non-simply-laced $G$ the spectral cover is more subtle (the cameral cover, the $\check G$-cover of $X$).
The non-abelian Hodge correspondence (the $\mathbb{C}^*$-family from $\mathcal{M}_{\mathrm{Hit}}$ to $\mathrm{LocSys}$) is the bridge; the Hitchin moduli and the character variety are diffeomorphic (by non-abelian Hodge) but not algebraically isomorphic — the hyperkähler rotation is essential.

Refs¶

Hitchin, "The self-duality equations on a Riemann surface" (Proc. LMS 1987); "Stable bundles and integrable systems" (Duke 1987)
Hausel–Thaddeus, "Mirror symmetry, Langlands duality, and the Hitchin system" (Invent. 2003)
Donagi–Pantev, "Langlands duality for Hitchin systems" (Annals 2012)
Kapustin–Witten, "Electric-magnetic duality and the geometric Langlands program" (Comm. Numb. Th. Phys. 2006)
Arinkin–Gaitsgory, "Singularities of genericopers and the Geometric Langlands conjecture" (Sel. Math. 2015); Ngô, "Fibration de Hitchin et endoscopie" (Annals 2010)
Fock–Goncharov, "Moduli spaces of local systems and higher Teichmüller theory" (Publ. Math. IHÉS 2006)

Next¶

MS index · Foundations · Gromov–Witten & Quantum Cohomology · Fukaya Category & HMS · SYZ & Gross–Siebert · Geometric Langlands / derived · Higher Teichmüller frontiers

Object	Langlands dual
\(G\)	\(\check G\) (the dual reductive group)
\(\mathcal{M}_{\mathrm{Hit}}(G)\)	\(\mathcal{M}_{\mathrm{Hit}}(\check G)\) (the mirror Hitchin moduli)
\(\mathrm{Bun}_G\) (moduli of \(G\)-bundles)	\(\mathrm{LocSys}_{\check G}\) (moduli of \(\check G\)-local systems)
\(D\)-modules on \(\mathrm{Bun}_G\)	Coherent sheaves on \(\mathrm{LocSys}_{\check G}\)
Hitchin base \(\mathcal{A}_G\)	\(\mathcal{A}_{\check G}\cong\mathcal{A}_G\) (the same base)
Hitchin fiber: \(\mathrm{Prym}(\Sigma_a/X)\) for \(G\)	dual abelian variety for \(\check G\)