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Seiberg-Witten integrable systems. (English) Zbl 0896.58057

Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 3-43 (1997).
This is a concisely written survey for pure mathematicians, comprising many aspects of mathematical theories related to the so-called supersymmetric versions of Yang-Mills (YM) quantum gauge-field-theory models of strong interactions, as well as of the algebraic-geometry integrability of these and other models. The supersymmetry (SUSY) is a special form of an intrinsic invariance of a field model which includes bosonic and fermionic fields. The SUSY transformation mixes the bosonic and fermionic components. It is characterized by a number \(N\) (\(1\leq N\leq 4\)) of independent generators. The central point in this survey is a review of a recent discovery of Seiberg and Witten, who put forward an exact solution to the \(N=2\) SUSY version of the YM quantum field-theory model with the \(SU(2)\) gauge group in four dimensions. Their exact solution is based on algebraic-geometry techniques (involving families of elliptic curves), the key point being to find the needed monodromy transformation.
The survey also reviews some other quantum-field models that can be solved exactly along similar lines. In particular, setting the mass constant of the model’s matter component equal to zero upgrades the SUSY order from \(N=2\) to \(N=4\).
For the entire collection see [Zbl 0882.00033].

MSC:

37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
81R12 Groups and algebras in quantum theory and relations with integrable systems
81T60 Supersymmetric field theories in quantum mechanics
14H70 Relationships between algebraic curves and integrable systems
81T13 Yang-Mills and other gauge theories in quantum field theory
14D20 Algebraic moduli problems, moduli of vector bundles
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