Finally, in terms of the explicit notation for the

covariant derivative, the integrability condition takes the form

The

covariant derivative operator associated with the connection [theta] is defined by

where [f.sub.G](G, T) = [partial derivative]f(G, T)/[partial derivative]G, [f.sub.T](G, T) = [partial derivative]f(G, T)/[partial derivative]T, [[nabla].sup.2] = [[nabla].sub.[mu]][[nabla].sup.[mu]] ([[nabla].sub.[mu]] is a

covariant derivative), and [[THETA].sub.[mu][nu]] has the following expression [31]:

where D is the

covariant derivative on (N, h) and [nabla] is the

covariant derivative on (M,g).

or d[THETA] = [[nabla].sub.j][[theta].sub.i][u.sup.i.sub.1][u.sup.j.sub.2] + [[theta].sub.i][U.sup.i.sub.12] with the

covariant derivative [[nabla].sub.j][[theta].sub.i] = [[theta].sub.i,j] - [[theta].sub.k][[GAMMA].sup.k.sub.ij].

By virtue of

covariant derivative of (7) with respect to Z we get

Continue the above set up, define a

covariant derivative on free loop space as follows.

As usual, a BRST operator is formed as Q = [[lambda].sup.A[alpha]][D.sub.A[alpha]], D being the fermionic

covariant derivative. The nilpotency of Q demands that

The local derivative operators "[sub./1]", "[sub.|p]"and "[|.sup.(1).sub.(p)])" are called the R-horizontal

covariant derivative, the M-horizontal

covariant derivative and the vertical

covariant derivative associated to the [GAMMA]-linear connection [nabla][gamma].

In parallel to SME exists an alternative procedure that consists in modifying the interaction part using a nonminimal coupling via

covariant derivative. In this paper, the nonminimal coupling term, CPT-odd term, is used to calculate the Lorentz-violating corrections to Moller scattering at finite temperature.

Let [h.sub.ijk] denote the

covariant derivative of [h.sub.ij] so that

Consequently, the

covariant derivative of the world-metric tensor fails to vanish in the present theory, as we obtain the following non-metric expression: