Green functions of 2-dimensional Yang-Mills theories on nonorientable surfaces

By using the path integral method , we calculate the Green functions of field strength of Yang-Mills theories on arbitrary nonorientable surfaces in Schwinger-Fock gauge. We show that the non-gauge invariant correlators consist of a free part and an almost x-independent part. We also show that the gauge invariant n-point functions are those corresponding to the free part , as in the case of orientable surfaces. e-mail:[email protected] e-mail:[email protected] It has been long known that the two-dimensional Yang-Mills theory is exactly soluble and indeed locally trivial [1]. The reason for this , at a fundamental level , is that in the two dimensions the Yang-Mills action depends only on the measure μ determined by the metric g. In the recent years these theories has been studied more carefully again. In refs.[2] and [3] the partition function and the expectation value of Wilson loops has been calculated by means of lattice gauge theory for arbitrary two-dimensional closed Riemann surfaces. These quantities also have been derived in the context of path integral in [4] and [5] and the abelianization technique were used to study this theory in refs.[6] and [7]. There are also some efforts to calculate the explicit expression of the partition function on sphere in small area limit [8,9] and the Wilson loops for SU(N) gauge group [10]. The other interesting quantities that must be calculated are the Green functions of field strength. In ref.[11] some of the correlators have been calculated by the abelianization method and in ref.[12] all n-point functions have been derived by path integral method for arbitrary closed orientable Riemann surfaces. There we have shown that the gauge invariant Green functions correspond to a free field theory. In this paper we are going to complete our investigation about the correlators of 2d YangMills theories by calculating them on arbitrary closed nonorientable surfaces. Such surfaces are connected sums of an orientable suface of genus g with s copies of Klein bottle and r copies of the projective plane ; Σg,s,r. Σg,s,r can also be regarded as the connected sum of r + 2(s+ g) projective planes , provided r and s are not both zero. The procedure that we follow are along that of ref.[12]. To begin , let us first rederive the partition function of Yang-Mills theories on Σg,s,r by path integral method. This quantity has been derived in ref.[3] in the context of lattice gauge theory by using the Migdal’s suggestion about the local factor of plaquettes. First we consider r = 0 case. Consider a genus-g Riemann surfaces with n = 2s boundaries. The boundary condition of each boundary loop γi is specified by a group element gi , such that Pexp ∮ γi A = gi ∈ G , where A is the gauge field and G is an arbitrary non-abelian compact semisimple gauge group. We take the specific case of boundary condition in which the gi’s are g1, g −1 1 , ..., gs, g −1 s . The wave function corresponding to this situation is [4] : ψ(Σg,n=2s, g1, ..., g −1 s ) = ∑ λ d(λ)χλ(g1)...χλ(g −1 s )e − ǫ 2 2. (1) In this relation λ labels the irreducible unitary representation of G , χλ is the character , d(λ) the dimension and c2(λ) the quadratic Casimir of the representation. A is the area of Σg,n and ǫ is the coupling constant. Note that the Schwinger-Fock gauge was used in calculation of the above wave function [4,12] : Aaμ(x) = ∫ 1 0 dssxF a νμ(sx). (2)