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A framework of connections, gauge transformations, and more, is developed over cell complexes. The relationship between this and the continuum theory is explained. A 2-form over the space of flat connections over certain types of cell complexes is defined and its significant properties proved; this 2-form is induced by the Atiyah–Bott symplectic structure on the space of connections over closed, oriented surfaces.

We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang–Mills measure defined in terms of the heat-kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space.

A 2-form is constructed on the space of connections on a principal bundle over an oriented surface with boundary. This induces a symplectic structure for the moduli space of flat connections with boundary holonomies lying in prescribed conjugacy classes. The Yang–Mills quantum field measure is described for this situation. This measure converges to the normalized symplectic volume measure in the “classical” limit.

A formula is derived which expresses the symplectic volume measure on the moduli space of flat connections over a closed Riemann surface Σ in terms of the corresponding volume measures for the surfaces Σ 1 and Σ 2 , when the latter have one boundary component each.

All the connected components of the moduli space of flat connections on SU (2) and SO (3) (trivial and non-trivial) bundles over closed oriented surfaces are determined. The symplectic structure and volumes of the non-maximal strata of the moduli space are also determined.

The quantum field measure for gauge fields over a compact surface with boundary, with holonomy around the boundary components specified, is constructed. Loop expectation values for general loop configurations are computed. For a compact oriented surface with one boundary component, let \(\) be the moduli space of flat connections with boundary holonomy lying in a conjugacy class \(\) in the gauge group G. We prove that a certain natural closed 2-form on \(\), introduced in an earlier work by C. King and the author, is a symplectic structure on the generic stratum of \(\) for generic \(\). We then prove that the quantum Yang-Mills measure, with the boundary holonomy constrained to lie in \(\), converges in a natural sense to the corresponding symplectic volume measure in the classical limit. We conclude with a detailed treatment of the case \(\), and determine the symplectic volume of this moduli space.

Yang–Mills connections over closed oriented surfaces of genus ≥1, for compact connected gauge groups, are constructed explicitly. The resulting formulas for Yang–Mills connections are used to carry out a Marsden–Weinstein type procedure. An explicit formula is obtained for the resulting 2-form on the moduli space. It is shown that this 2-form provides a symplectic structure on appropriate subsets of the moduli space.

For compact surfaces with one boundary component, and semisimple gauge groups, we construct a closed gauge invariant 2-form on the space of flat connections whose boundary holonomy lies in a fixed conjugacy class. This form descends to the moduli space under the action of the full gauge group, and provides an explicit description of a symplectic structure for this moduli space.

A natural gauge-invariant 2-form is introduced on the space of connections over a compact oriented surface with boundary. It is shown that this 2-form descends to the moduli space of flat connections, under a group of based gauge transformations. An explicit expression (in terms of holonomy variations) is given for the resulting 2-form, and it is related to the symplectic structure of the extended moduli space introduced by L. C. Jeffrey.

We prove that for $SU(2)$ and $SO(3)$ quantum gauge theory on a torus, holonomy expectation values with respect to the Yang-Mills measure $d\mu_T(\o) =N_T^{-1}e^{-S_{YM}(\o)/T}[{\cal D}\o]$ converge, as $T\downarrow 0$, to integrals with respect to a symplectic volume measure $\mu_0$ on the moduli space of flat connections on the bundle. These moduli spaces and the symplectic structures are described explicitly., Comment: 18 pages

The moduli space of flat connections on a principal G‐bundle over a compact oriented surface of genus g≥1 is considered herein. Using the holonomies around noncontractible loops, the moduli space is described as a quotient of a submanifold of G2g. An explicit expression is obtained for the symplectic form on the smooth part of moduli space, and several properties of this form are established.

Though not obvious at first glance, these ideas are all connected with each other. In the following sections we shall give an account of some of these ideas and indicate how everything ties together. The relation between the algebraic problem of counting solutions of equations in a group with two-dimensional Yang–Mills theory has been explored in several works by Mulase et al. [6, 7]. (We shall not describe relations with Chern–Simons theory.) Ideas from quantum Yang–Mills may be useful in understanding representation varieties of finitely generated groups. For the case of surface groups, it still remains to carry out a complete study of the topological and symplectic structure of the various strata of the moduli space of flat connections from the quantum Yang–Mills point of view, though much progress has been made (initiated by Witten’s works [11,12]).

A short overview is presented of: (i) the quantum field measure associated with pure Yang-Mills gauge theory on a compact surface, (ii) an approach to the symplectic structure on the [moduli] space of [flat] connections over compact oriented surfaces (with and without boundary), and (iii) the relation between the classical Yang-Mills measure and the volume measure on the moduli space of flat connections.

We relate the semiclassical limit of the quantum Yang-Mills partition function on a compact oriented surface to the symplectic volume of the moduli space of flat connections, by using an explicit expression for the symplectic form. This gives an independent proof of some recent results of Witten and Forman., 10 pages

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.