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Physics and geometry: A look at the last twenty years. (English) Zbl 0945.14026

Pragacz, Piotr (ed.) et al., Algebraic geometry: Hirzebruch 70. Proceedings of the algebraic geometry conference in honor of F. Hirzebruch’s 70th birthday, Stefan Banach International Mathematical Center, Warszawa, Poland, May 11-16, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 241, 1-8 (1999).
The paper summarizes the development of interactions between geometry and physics in the last two decades and the role played by F. Hirzebruch in these interactions. Among the main ideas mentioned is the concept of duality in quantum field theory (QFT), a topic that involves infinite-dimensional analogues of groups and spaces of classical geometry. There are two special mentions about duality: Poisson summation formula between a lattice and its dual lattice related to the modular group \(SL(2,\mathbb{Z})\) that appears in many aspects of QFT and the formula that connects the Seiberg-Witten theory which counts solutions of nonlinear equations, essentially elliptic, and the Donaldson theory which counts solutions of the Yang-Mills equations.
Other topics commented are quantum cohomology and mirror symmetry, Hirzebruch-Riemann-Roch theorem, conformal field theory, a branch of 2-dimensional QFT that lives naturally on complex Riemann surfaces and elliptic genus.
For the entire collection see [Zbl 0924.00033].

MSC:

14J81 Relationships between surfaces, higher-dimensional varieties, and physics
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14H81 Relationships between algebraic curves and physics
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
14-03 History of algebraic geometry
01A60 History of mathematics in the 20th century
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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