nLab spectral curve

Idea

Roughly speaking, a spectral curve is a locus of zeros of the characteristic polynomial of the Lax matrix of a classical integrable system.

Hitchin has taken a deeper geometric point of view in terms of the moduli space of stable vector bundles of fixed rank $r$ and degree $d$ over a fixed projective nonsingular curve $\Sigma$ of genus $g$. A point in the cotangent bundle of this bundle is a stable vector bundle equipped with a 1-form valued endomorphism; to each such datum one associates a spectral curve.

Related concepts

• topological recursion

References

• N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126, MR887284 doi; Stable bundles and integrable systems, Duke Math. J. , 54 (1987) pp. 91–114 MR89a:32021 doi euclid

• O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge Univ. Press 2003.

• Eyal Markman, Spectral curves and integrable systems Compositio Math. 93 no. 3 (1994), p. 255-290, numdam

• Ron Donagi, Eyal Markman, Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, pp. 1-119 in Integrable systems and quantum groups (expanded CIME lectures, Montecatini Terme, 1993), Lecture Notes in Math. 1620 (1996) arXiv:alg-geom/9507017, MR97h:14017

• Ron Donagi, Spectral covers, MSRI series 28, pdf

• A. Beauville, Jacobiennes des courbes spectrales et systèmes Hamiltoniens complètement intégrables, Acta Math. 164 (1990), 211-235, MR86c:32030, doi

• A. Beauville, M. S. Narasimhan, S. Ramanan, Spectral curves and the generalized theta divisor, J. Reine Angew. Math. 398 (1989), 169-179.

• Carlos T. Simpson, Higgs bundles and local systems, Publ. Mathématiques de l’IHÉS, 75 (1992), p. 5-95, numdam

• J. C. Hurtubise, Integrable systems and algebraic surfaces, Duke Math. J. 83:1 (1996), 19-50, euclid; erratum Duke Math. J. 84:3 (1996), 815, euclid

• Anton Izosimov, Singularities of integrable systems and algebraic curves, arxiv/1509.08996

Quantum spectral curve:

• V. Talalaev, The quantum Gaudin system, Funct. Anal. Appl. 40, 73–77 (2006) ( Квантовая система Годена, Функц. анализ и его прил. 40, вып. 1, 2006, с. 86–91, pdf)

• Д. Талалаев, А. Червов, Уравнение КЗ, G-оперы, квантовая редукция Дринфельда–Соколова и квантовое тождество Гамильтона–Кэли, Zapiski Naučnyh Seminarov POMI 360, 2008, 246–259, pdf; Engl, transl. in J. Math, Sci. (New York), 2009, 158:6, 904–911, doi