Sewing Yang–Mills Measures and Moduli Spaces over Compact Surfaces

Consider a closed surfaceΣobtained by pasting two surfacesΣ1andΣ2along some of their boundary components. A formula is derived expressing quantum Yang–Mills functional integrals forΣin terms of the corresponding integrals for the surfacesΣi. Using this, a formula is derived which expresses the symplectic volume measure on the moduli space of flatSU(2) connections over a closed Riemann surfaceΣin terms of the corresponding volume measures for the surfacesΣ1andΣ2, when the latter have one boundary component each.