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**Algebraic Fermi curves [after Gieseker, Trubowitz and Knörrer].**
*(English)*
Zbl 0749.14027

Sémin. Bourbaki, Vol. 1989/90, 42ème année, Astérisque 189-190, Exp. No. 723, 239-258 (1990).

[For the entire collection see Zbl 0722.00001.]

This is an overview of the work of Gieseker, Trubowitz and Knörrer on the theory of algebraic Fermi curves. Fermi curves are the 1-dimensional analogs of the usual Fermi surfaces from solid state physics. Given a potential function you have one curve for each energy-level. The density of states function plays a central role in the theory and it can be measured experimentally. This is explained in some detail in section 2 of the paper.

If one goes over to a certain discrete analogon, the Fermi curves form the real points of certain algebraic curves varying in a family parametrized by the complex projective line. Together these form an algebraic surface, the Bloch variety. The density of states function reveals itself as the period integral of a certain canonical 1-form over the cycle formed by the real Fermi curve. The Bloch variety admits a natural compactification, first described by Bättig and recalled in section 5. Deep methods from algebraic geometry then can be applied to show that generically the Bloch variety essentially uniquely determines the potential and that the density of states function essentially determines the Bloch variety (for generic potentials). In view of the preceding one might say that the experimentally determinable density of states function generically determines the potential and hence tells you all of the physics. But—of course one would need the density of states function also for complex parameters, i.e. for complex values of the energy and this does not seem to have physical meaning.

This is an overview of the work of Gieseker, Trubowitz and Knörrer on the theory of algebraic Fermi curves. Fermi curves are the 1-dimensional analogs of the usual Fermi surfaces from solid state physics. Given a potential function you have one curve for each energy-level. The density of states function plays a central role in the theory and it can be measured experimentally. This is explained in some detail in section 2 of the paper.

If one goes over to a certain discrete analogon, the Fermi curves form the real points of certain algebraic curves varying in a family parametrized by the complex projective line. Together these form an algebraic surface, the Bloch variety. The density of states function reveals itself as the period integral of a certain canonical 1-form over the cycle formed by the real Fermi curve. The Bloch variety admits a natural compactification, first described by Bättig and recalled in section 5. Deep methods from algebraic geometry then can be applied to show that generically the Bloch variety essentially uniquely determines the potential and that the density of states function essentially determines the Bloch variety (for generic potentials). In view of the preceding one might say that the experimentally determinable density of states function generically determines the potential and hence tells you all of the physics. But—of course one would need the density of states function also for complex parameters, i.e. for complex values of the energy and this does not seem to have physical meaning.

Reviewer: C.Peters (Leiden)

### MSC:

14H99 | Curves in algebraic geometry |

14J99 | Surfaces and higher-dimensional varieties |

35J10 | Schrödinger operator, Schrödinger equation |

35Q60 | PDEs in connection with optics and electromagnetic theory |

81V99 | Applications of quantum theory to specific physical systems |