This situation contrasts with the four-dimensional case, where a rigorous construction of the theory as a measure is currently unknown.
, the norm-squared in the integrand comes from the metric on the Lie algebra and the one on the base manifold, and
More precisely, the probability measure is more likely to be meaningful on the space of orbits of connections under gauge transformations.
[1] Some formulas appearing in Migdal's work can, in retrospect, be seen to be connected to the heat kernel on the structure group of the theory.
by Bruce Driver[3] and by Leonard Gross, Christopher King, and Ambar Sengupta.
Wilson loop variables (certain important variables on the space) were defined using stochastic differential equations and their expected values computed explicitly and found to agree with the results of the heat kernel action.
Dana S. Fine[9][10][11] used the formal Yang–Mills functional integral to compute loop expectation values.
This formula was proved by Driver[3] and by Gross et al.[3] using the Gaussian measure construction of the Yang–Mills measure on the plane and by defining parallel transport by interpreting the equation of parallel transport as a Stratonovich stochastic differential equation.
This, and counterparts in higher genus as well as for surfaces with boundary and for bundles with nontrivial topology, were proved by Sengupta.
[6][8] There is an extensive physics literature on loop expectation values in two-dimensional Yang–Mills theory.
[18][19][20][21][22][23][24][25] Many of the above formulas were known in the physics literature from the 1970s, with the results initially expressed in terms of a sum over the characters of the gauge group rather than the heat kernel and with the function
[2] The role of the convolution property of the heat kernel was used in works of Sergio Albeverio et al.[26][27] in constructing stochastic cosurface processes inspired by Yang–Mills theory and, indirectly, by Makeenko and Migdal[22] in the physics literature.
The Yang–Mills partition function is, formally, In the two-dimensional case we can view this as being (proportional to) the denominator that appears in the loop expectation values.
whose Lie algebra is equipped with an Ad-invariant metric, over a compact two-dimensional orientable manifold of genus
This was proved rigorous by Sengupta[33] (see also the works by Lisa Jeffrey and by Kefeng Liu[34]).
There is a large literature[35][36][37][38][39] on the symplectic structure on the moduli space of flat connections, and more generally on the moduli space itself, the major early work being that of Michael Atiyah and Raoul Bott.
Thierry Lévy and James R. Norris [41] established a large deviations principle for this convergence, showing that the Yang–Mills measure encodes the Yang–Mills action functional even though this functional does not explicitly appear in the rigorous formulation of the measure.
There is a large physics literature on this subject, including major early works by Gerardus 't Hooft.
In two dimensions, the Makeenko–Migdal equation takes a special form developed by Kazakov and Kostov.
In the large-N limit, the 2-D form of the Makeenko–Migdal equation relates the Wilson loop functional for a complicated curve with multiple crossings to the product of Wilson loop functionals for a pair of simpler curves with at least one less crossing.
In the case of the sphere or the plane, it was the proposed that the Makeenko–Migdal equation could (in principle) reduce the computation of Wilson loop functionals for arbitrary curves to the Wilson loop functional for a simple closed curve.
In dimension 2, some of the major ideas were proposed by I. M. Singer,[42] who named this limit the master field (a general notion in some areas of physics).
limit of 2-dimensional Yang–Mills loop expectation values using ideas from random matrix theory.
Sengupta[44] computed the large-N limit of loop expectation values in the plane and commented on the connection with free probability.
Confirming one proposal of Singer,[42] Michael Anshelevich and Sengupta[45] showed that the large-N limit of the Yang–Mills measure over the plane for the groups
An extensive study of the master field in the plane was made by Thierry Lévy.
[46][47] Several major contributions have been made by Bruce K. Driver, Brian C. Hall, and Todd Kemp,[48] Franck Gabriel,[49] and Antoine Dahlqvist.
In spacetime dimension larger than 2, there is very little in terms of rigorous mathematical results.
Sourav Chatterjee has proved several results in large-N gauge theory for dimension larger than 2.
Chatterjee[52] established an explicit formula for the leading term of the free energy of three-dimensional