# Hitchin system

An algebraically completely integrable Hamiltonian system defined on the cotangent bundle to the moduli space of stable vector bundles (of fixed rank and degree; cf. also Vector bundle) over a given Riemann surface $X$ of genus $g\geq2$. Hitchin's definition of the system [a9] greatly enhanced the theory of spectral curves [a8], which underlies the discovery of a multitude of algebraically completely integrable systems in the 1970s. Such systems are given by a Lax-pair equation: $L=[M,L]$ with $(n\times n)$-matrices $L$, $M$ depending on a parameter $\lambda$, the spectral curve is an $n$-fold covering of the parameter space and the system lives on a co-adjoint orbit in a loop algebra, by the Adler–Kostant–Symes method of symplectic reduction, cf. [a1]. N.J. Hitchin defines the curve of eigenvalues on the total space of the canonical bundle of $X$, and linearizes the flows on the Jacobi variety of this curve.

The idea gave rise to a great amount of algebraic geometry: moduli spaces of stable pairs [a12]; meromorphic Hitchin systems [a3] and [a4]; Hitchin systems for principal $G$-bundles [a5]; and quantized Hitchin systems with applications to the geometric Langlands program [a2].

Moreover, by moving the curve $X$ in moduli, Hitchin [a10] achieved geometric quantization by constructing a projective connection over the spaces of bundles, whose associated heat operator generalizes the heat equation that characterizes the Riemann theta-function for the case of rank-one bundles. The coefficients of the heat operator are given by the Hamiltonians of the Hitchin systems.

Explicit formulas for the Hitchin Hamiltonian and connection were produced for the genus-two case [a7], [a6]. A connection of Hitchin's Hamiltonians with KP-flows (cf. also KP-equation) is given in [a4] and [a11].

#### References

[a1] | M.R. Adams, J. Harnad, J. Hurtubise, "Integrable Hamiltonian systems on rational coadjoint orbits of loop algebras, Hamiltonian systems, transformation groups and spectral transform methods" , Proc. CRM Workshop, Montreal 1989 (1990) pp. 19–32 |

[a2] | A.A. Beilinson, V.G. Drinfel'd, "Quantization of Hitchin's fibration and Langlands program" A. Boutet de Monvel (ed.) et al. (ed.) , Algebraic and Geometric Methods in Math. Physics. Proc. 1st Ukrainian–French–Romanian Summer School, Kaciveli, Ukraine, Sept. 1-14 1993 , Math. Phys. Stud. , 19 , Kluwer Acad. Publ. (1996) pp. 3–7 MR1385674 |

[a3] | F. Bottacin, "Symplectic geometry on moduli spaces of stable pairs" Ann. Sci. Ecole Norm. Sup. 4 , 28 (1995) pp. 391–433 MR1334607 Zbl 0864.14004 |

[a4] | R. Donagi, E. Markman, "Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles" M. Francaviglia (ed.) et al. (ed.) , Integrable Systems and Quantum Groups. Lectures at the 1st session of the Centro Internaz. Mat. Estivo (CIME), Montecatini Terme, Italy, June 14-22 1993 , Lecture Notes Math. , 1620 , Springer (1996) pp. 1–119 MR1397273 Zbl 0853.35100 |

[a5] | G. Faltings, "Stable G-bundles and projective connections" J. Alg. Geometry , 2 (1993) pp. 507–568 MR1211997 Zbl 0790.14019 |

[a6] | B. van Geemen, A.J. de Jong, "On Hitchin's connection" J. Amer. Math. Soc. , 11 (1998) pp. 189–228 |

[a7] | B. van Geemen, E. Previato, "On the Hitchin system" Duke Math. J. , 85 : 3 (1996) pp. 659–683 MR1422361 Zbl 0879.14010 |

[a8] | N.J. Hitchin, "The self-duality equations on a Riemann surface" Proc. London Math. Soc. , 55 (1987) pp. 59–126 MR0887284 Zbl 0634.53045 |

[a9] | N.J. Hitchin, "Stable bundles and integrable systems" Duke Math. J. , 54 (1987) pp. 91–114 MR0885778 Zbl 0627.14024 |

[a10] | N.J. Hitchin, "Flat connections and geometric quantization" Comm. Math. Phys. , 131 : 2 (1990) pp. 347–380 MR1065677 Zbl 0718.53021 |

[a11] | Yingchen Li, M. Mulase, "Hitchin systems and KP equations" Internat. J. Math. , 7 : 2 (1996) pp. 227–244 MR1382724 Zbl 0863.58036 |

[a12] | C.T. Simpson, "Moduli of representations of the fundamental group of a smooth projective variety I–II" Publ. Math. IHES , 79/80 (1994/5) pp. 47–129;5–79 |

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Hitchin system.

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