Polylogarithm Identities in a Conformal Field Theory in Three Dimensions

TL;DR

This paper uses polylogarithm identities to analytically determine universal constants in a three-dimensional conformal field theory, revealing rational ratios for key quantities and highlighting differences from two-dimensional theories.

Contribution

It derives exact rational values for universal constants in a 3D conformal field theory using polylogarithm identities, extending the understanding of conformal invariants.

Findings

/N = 3/4, a rational number

/N = 4/5, a rational number

Results highlight differences between 2D and 3D conformal theories

Abstract

The $N=\infty$ vector $O(N)$ model is a solvable, interacting field theory in three dimensions ($D$). In a recent paper with A. Chubukov and J. Ye~\cite{self}, we have computed a universal number, $\tilde{c}$, characterizing the size dependence of the free energy at the conformally-invariant critical point of this theory. The result~\cite{self} for $\tilde{c}$ can be expressed in terms of polylogarithms. Here, we use non-trivial polylogarithm identities to show that $\tilde{c}/N = 4/5$, 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 $c = \tilde{c}$; thus such a result is clearly not valid in $D=3$.