YangâMills theory in two dimensions with a well-defined measure
In mathematical physics, two-dimensional YangâMills theory is the special case of
YangâMills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously defined YangâMills measure, meaning that the (Euclidean) path integral can be interpreted as a
measure on the set of connections modulo gauge transformations. This situation contrasts with the four-dimensional case, where a rigorous construction of the theory as a measure is currently unknown.
An aspect of the subject of particular interest is the
large-N limit, in which the structure group is taken to be the
unitary group and then the tends to infinity limit is taken. The large-N limit of two-dimensional YangâMills theory has connections to string theory.
Interest in the YangâMills measure comes from a statistical mechanical or constructive quantum field theoretic approach to formulating a quantum theory for the YangâMills field. A
gauge field is described mathematically by a 1-form on a principal -bundle over a manifold taking values in the
Lie algebra of the
Lie group. We assume that the structure group , which describes the physical symmetries of the gauge field, is a compact Lie group with a bi-invariant metric on the Lie algebra , and we also assume given a
Riemannian metric on the manifold . The YangâMills action functional is given by
where is the
curvature of the
connection form, the norm-squared in the integrand comes from the metric on the Lie algebra and the one on the base manifold, and is the Riemannian volume measure on .
The measure is given formally by
as a normalized probability measure on the space of all connections on the bundle, with a parameter, and is a formal
normalizing constant. More precisely, the probability measure is more likely to be meaningful on the space of orbits of connections under
The YangâMills measure for two-dimensional manifolds
Study of YangâMills theory in two dimensions dates back at least to work of A. A. Migdal in 1975. 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. The role of the heat kernel was made more explicit in various works in the late 1970s, culminating in the introduction of the heat kernel action in work of Menotti and Onofri in 1981.
In the continuum theory, the YangâMills measure was rigorously defined for the case where by Bruce Driver and by
Leonard Gross, Christopher King, and
Ambar Sengupta. For compact manifolds, both oriented and non-oriented, with or without boundary, with specified bundle topology, the YangâMills measure was constructed by Sengupta In this approach the 2-dimensional YangâMills measure is constructed by using a Gaussian measure on an infinite-dimensional space conditioned to satisfy relations implied by the topologies of the surface and of the bundle.
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.
The discrete YangâMills measure is a term that has been used for the
lattice gauge theory version of the YangâMills measure, especially for compact surfaces. The lattice in this case is a triangulation of the surface. Notable facts are: (i) the discrete YangâMills measure can encode the topology of the bundle over the continuum surface even if only the triangulation is used to define the measure; (ii) when two surfaces are sewn along a common boundary loop, the corresponding discrete YangâMills measures convolve to yield the measure for the combined surface.
Wilson loop expectation values in 2 dimensions
For a piecewise smooth loop on the base manifold and a point on the fiber in the principal -bundle over the base point of the loop, there is the
holonomy of any
connection on the bundle. For regular loops , all based at and any function on the function is called a
Wilson loop variable, of interest mostly when is a product of traces of the holonomies in representations of the group . With being a two-dimensional Riemannian manifold the loop expectation values
where now is the area of the region "outside" the loop , and is the total area of the sphere. This formula was proved by Sengupta using the conditioned Gaussian measure construction of the YangâMills measure and the result agrees with what one gets by using the heat kernel action of Menotti and Onofri.
As an example for higher genus surfaces, if is a
with being the total area of the torus, and a contractible loop on the torus enclosing an area . This, and counterparts in higher genus as well as for surfaces with boundary and for bundles with nontrivial topology, were proved by Sengupta.
There is an extensive physics literature on loop expectation values in two-dimensional YangâMills theory. Many of the above formulas were known in the physics literature form 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 being the trace in some representation of the group. Expressions involving the heat kernel then appeared explicitly in the form of the "heat kernel action" in work of Menotti and Onofri. The role of the
convolution property of the heat kernel was used in works of
Sergio Albeverio et al. in constructing stochastic cosurface processes inspired by YangâMills theory and, indirectly, by Makeenko and Migdal in the physics literature.
In the two-dimensional case we can view this as being (proportional to) the denominator that appears in the loop expectation values. Thus, for example, the partition function for the torus would be
where is the area of the torus. In two of the most impactful works in the field,
Edward Witten showed that as the partition function yields the volume of the
moduli space of flat connections with respect to a natural volume measure on the moduli space. This volume measure is associated to a natural
symplectic structure on the moduli space when the surface is
orientable, and is the torsion of a certain complex in the case where the surface is not orientable. Witten's discovery has been studied in different ways by several researchers. Let denote the
moduli space of flat connections on a trivial bundle, with structure group being a compact connected semi-simple Lie group whose Lie algebra is equipped with an Ad-invariant metric, over a compact two-dimensional orientable manifold of genus . Witten showed that the symplectic volume of this moduli space is given by
where the sum is over all irreducible representations of . This was proved rigorous by Sengupta (see also the works by
Lisa Jeffrey and by Kefeng Liu). There is a large literature 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
large-N limit of gauge theories refers to the behavior of the theory for gauge groups of the form , , , , and other such families, as goes to . There is a large physics literature on this subject, including major early works by
Gerardus 't Hooft. A key tool in this analysis is the MakeenkoâMigdal equation.
In two dimensions, the MakeenkoâMigdal equation takes a special form developed by Kazakov and Kostov. In the large-N limit, the 2-D form of the MakeenkoâMigdal 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 MakeenkoâMigdal 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 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 theory for dimension larger than 2. Chatterjee established an explicit formula for the leading term of the free energy of three-dimensional lattice gauge theory for any N, as the lattice spacing tends to zero. Let be the partition function of -dimensional lattice gauge theory with coupling strength in the box with lattice spacing and size being n spacings in each direction. Chatterjee showed that in dimensions d=2 and 3, is
up to leading order in , where is a limiting free-energy term. A similar result was also obtained for in dimension 4, for , , and independently.
^Migdal, A. A. (1975). "Recursion equations in gauge field theories". Soviet Physics JETP. 42: 413â418.
^King, Christopher; Sengupta, Ambar N. (1994). "An explicit description of the symplectic structure of moduli spaces of flat connections". Journal of Mathematical Physics. 35 (10): 5338?5353.
^Jeffrey, Lisa; Weitsman, Jonathan (2000). "Symplectic geometry of the moduli space of flat connections on a Riemann surface: inductive decompositions and vanishing theorems". Canadian Journal of Mathematics. 52 (3): 582â612.
^Huebschmann, Johannes (1996). "The singularities of Yang-Mills connections for bundles on a surface. II. The stratification". Mathematische Zeitschrift. 221 (1): 83â92.
^Atiyah, Michael; Bott, Raoul (1983). "The Yang-Mills equations over Riemann surfaces". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. 308 (1505): 523â615.