Master spaces and the coupling principle: From geometric invariant theory to gauge theory.

*(English)*Zbl 0959.14028Many problems in geometric invariant theory or gauge theory lead to the computation of certain natural top cohomology classes on moduli spaces that are very often described as integrals of products of natural classes on the product of the original manifold by the relevant moduli space (the correlation functions of the theory). Gromov invariants of Grassmannians or Donaldson polynomials are of this kind.

The authors develop a coupling principle that enable them to compute the correlation functions of a moduli space \({\mathcal M}_0\) in terms of the correlation functions of two other spaces, a so-called final moduli space \({\mathcal M}_\infty\) and a “variety of reductions” \({\mathcal M}_R\) . In all the examples worked out, correlation functions of these two auxiliary manifolds are much easier to compute and understand. This is achieved by means of a master space \(\mathcal M\) with a \(\mathbb C^\ast\)-action whose fixed point locus is the disjoint union of \({\mathcal M}_0\), \({\mathcal M}_\infty\) and \({\mathcal M}_R\). The coupling principle provides a unified way for understanding the relationship between different kind of invariants. It is used to prove that the Gromov invariants of the moduli of stable bundles on a curve can be expressed in terms of the Gromov invariants of the symmetric powers of the curve or that the gauge theoretical invariants of moduli spaces of monopoles are determined by Seiberg-Witten and Donaldson invariants.

The coupling principle is a very enlightening tool that provides new and fundamental insight to invariant theory.

The authors develop a coupling principle that enable them to compute the correlation functions of a moduli space \({\mathcal M}_0\) in terms of the correlation functions of two other spaces, a so-called final moduli space \({\mathcal M}_\infty\) and a “variety of reductions” \({\mathcal M}_R\) . In all the examples worked out, correlation functions of these two auxiliary manifolds are much easier to compute and understand. This is achieved by means of a master space \(\mathcal M\) with a \(\mathbb C^\ast\)-action whose fixed point locus is the disjoint union of \({\mathcal M}_0\), \({\mathcal M}_\infty\) and \({\mathcal M}_R\). The coupling principle provides a unified way for understanding the relationship between different kind of invariants. It is used to prove that the Gromov invariants of the moduli of stable bundles on a curve can be expressed in terms of the Gromov invariants of the symmetric powers of the curve or that the gauge theoretical invariants of moduli spaces of monopoles are determined by Seiberg-Witten and Donaldson invariants.

The coupling principle is a very enlightening tool that provides new and fundamental insight to invariant theory.

Reviewer: Daniel Hernandez Ruiperez (Salamanca)

##### MSC:

14L24 | Geometric invariant theory |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14J80 | Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) |