The self-duality equations on a Riemann surface.

*(English)*Zbl 0634.53045Only some of the many beautiful ideas and results in this extraordinary paper can be mentioned in this review. Let \(A\) be a connection on a principal \(G\)-bundle \(P\) over \(\mathbb R^ 4\) and \(F(A)\in \Omega (\mathbb R^ 4; \mathrm{ad}(P))\) its curvature 2-form with values in \(\mathrm{ad}(P)\), the vector bundle associated with adjoint representation. A connection \(A\) satisfies the self duality equations if and only if \(F(A)\) is invariant under the Hodge star operator \(*: \Omega (\mathbb R^ 4;\mathrm{ad}(P))\to \Omega (\mathbb R^ 4;\mathrm{ad}(P))\). Considering only solutions invariant under two translations the equations are reduced to two dimensions. They are conformally invariant and can hence be defined on Riemann surfaces. The moduli space \({\mathcal M}\) of all solutions of these equations on a compact Riemann surface \(M\) of genus \(g\) is the central object during almost 70 pages of dense information. Its rich structure is investigated using methods from topology, Riemannian, algebraic and symplectic geometry. To simplify calculations only the cases are considered where the group \(G\) of the principal bundle \(P\) is \(\mathrm{SU}(2)\) or \(\mathrm{SO}(3)\). The results for \(\mathrm{SU}(2)\) also give information on the Teichmüller space of complex structures for the surface \(M\) itself.

The moduli space \({\mathcal M}\) is a smooth manifold of dimension \(12(g-1)\). Its natural Riemannian metric is complete and a hyper-Kähler metric. It is shown that \(M\) is non-compact, simply connected and its Betti numbers are computed. A solution of the self-duality equations defines a holomorphic rank-2 vector bundle \(V\) over \(M\), together with a holomorphic section \(\Phi\) of \(\mathrm{End}\,V\otimes K\), where \(K\) is the canonical bundle of \(M\). Such a pair \((V,\Phi)\) is called stable if any \(\Phi\)-invariant subbundle must have degree less than half the degree of \(V\). It is shown that the pairs arising from solutions are necessarily stable and conversely: to each stable pair there exists a solution of the self duality equations unique modulo unitary gauge transformations.

It is this correspondence between solutions and stable pairs which provides the basis for the study of the moduli space \({\mathcal M}\). Fixing one complex structure the hyper-Kähler manifold \({\mathcal M}\) becomes in a holomorphic manner a symplectic manifold and can be regarded as an algebraically completely integrable Hamiltonian system. Other complex structures on \({\mathcal M}\) make it a Stein manifold. Hence one can consider the map \((A,\Phi)\to A+\Phi +\Phi^*\) which associates a complex connection to the pair \((A,\Phi)\). This connection is flat if the pair arises from a solution of the self duality equation and it is irreducible if the solution is. The converse, namely that every irreducible flat connection is gauge-equivalent to a connection associated with a solution, is proved by S. K. Donaldson in ibid., 127–131 (1987; Zbl 0634.53046).

The author does not try to give explicit solutions to the duality equations. But he certainly gives a fascinating description of the moduli space of the solutions and a wonderful overview over the interesting mathematics which is embedded in it.

The moduli space \({\mathcal M}\) is a smooth manifold of dimension \(12(g-1)\). Its natural Riemannian metric is complete and a hyper-Kähler metric. It is shown that \(M\) is non-compact, simply connected and its Betti numbers are computed. A solution of the self-duality equations defines a holomorphic rank-2 vector bundle \(V\) over \(M\), together with a holomorphic section \(\Phi\) of \(\mathrm{End}\,V\otimes K\), where \(K\) is the canonical bundle of \(M\). Such a pair \((V,\Phi)\) is called stable if any \(\Phi\)-invariant subbundle must have degree less than half the degree of \(V\). It is shown that the pairs arising from solutions are necessarily stable and conversely: to each stable pair there exists a solution of the self duality equations unique modulo unitary gauge transformations.

It is this correspondence between solutions and stable pairs which provides the basis for the study of the moduli space \({\mathcal M}\). Fixing one complex structure the hyper-Kähler manifold \({\mathcal M}\) becomes in a holomorphic manner a symplectic manifold and can be regarded as an algebraically completely integrable Hamiltonian system. Other complex structures on \({\mathcal M}\) make it a Stein manifold. Hence one can consider the map \((A,\Phi)\to A+\Phi +\Phi^*\) which associates a complex connection to the pair \((A,\Phi)\). This connection is flat if the pair arises from a solution of the self duality equation and it is irreducible if the solution is. The converse, namely that every irreducible flat connection is gauge-equivalent to a connection associated with a solution, is proved by S. K. Donaldson in ibid., 127–131 (1987; Zbl 0634.53046).

The author does not try to give explicit solutions to the duality equations. But he certainly gives a fascinating description of the moduli space of the solutions and a wonderful overview over the interesting mathematics which is embedded in it.

Reviewer: S. Timmann

##### MSC:

53D18 | Generalized geometries (à la Hitchin) |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

30F30 | Differentials on Riemann surfaces |

53C05 | Connections, general theory |