##
**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].

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].

Reviewer: B.A.Malomed (Tel Aviv)

### 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 |