Polylogarithm identities in a conformal field theory in three dimensions

The N = ∞ vector O(N) model is a solvable, interacting field theory in three dimensions (D). In a recent paper with A. Chubukov and J. Ye [1], we have computed a universal number, c̃, characterizing the size dependence of the free energy at the conformally-invariant critical point of this theory. The result [1] for c̃ can be expressed in terms of polylogarithms. Here, we use non-trivial polylogarithm identities to show that c̃/N = 4/5, a rational number; this result is curiously parallel to recent work on dilogarithm identities in D = 2 conformal theories. The amplitude of the stress-stress correlator of this theory, c (which is the analog of the central charge), is determined to be c/N = 3/4, also rational. Unitary conformal theories in D = 2 always have c = c̃; thus such a result is clearly not valid in D = 3.