%Paper: hep-th/9206034
%From: nakanisi@rabbit.math.nagoya-u.ac.jp (Tomoki Nakanishi)
%Date: Tue, 9 Jun 92 17:22:15 JST


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\noindent
\null\hskip10cm MRR-009-92 \hfill\break
\null\hskip10cm SMS-042-92 \hfill\break
\null\hskip10cm ANU preprint \hfill\break
\null\hskip10cm May, 1992 \hfill\break
\vskip1cm
\centerline{\bf Spectra in Conformal Field Theories from the Rogers
Dilogarithm}\par
\vskip1cm
\noindent
\centerline{by}\par
\vskip1cm
\centerline{{\bf A. KUNIBA}\footnote\dag{Postal address:
 Department of Mathematics, Kyusyu University, Fukuoka 812}
 \footnote\null{E-mail: kuniba@math.sci.kyushu-u.ac.jp}
{\bf and T. NAKANISHI}\footnote\ddag{E-mail:
nakanisi@rabbit.math.nagoya-u.ac.jp}}
\vskip1.5cm
\noindent
\hskip1.2cm \dag\hskip0.2cm Mathematics and Theoretical Physics, \par\noindent
\hskip1.6cm Institute of Advanced Studies, \par \noindent
\hskip1.6cm Australian National University,\par\noindent
\hskip1.6cm  GPO Box 4 Canberra ACT 2601 Australia\par\noindent
\hskip1.6cm (On leave from Department of Mathematics, Kyushu University,\par
\noindent\hskip1.8cm Fukuoka 812 Japan)\par
%
\vskip0.4cm
\noindent
\hskip1.2cm \ddag \hskip0.2cm Department of Mathematics,
Nagoya University,\par\noindent
\hskip1.6cm Chikusa-ku, Nagoya 464 Japan\par\noindent
%
\vskip0.4cm\noindent
%
\vskip1.9cm\noindent
\centerline{\bf ABSTRACT}\par\vskip0.2cm
%
We propose a system of functional relations having a
universal form connected to the
$U_q(X^{(1)}_r)$ Bethe ansatz equation.
%
Based on the analysis of it, we conjecture
a new sum formula for the Rogers dilogarithm function in
terms of the scaling dimensions of the $X^{(1)}_r$ parafermion
conformal field theory.
\vfill\eject

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%    Main Text
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\baselineskip0.6cm
\def\xr{X_r}
\def\axr{X^{(1)}_r}
\def\uqxr{U_q(X_r)}
\def\uqaxr{U_q(X^{(1)}_r)}
\def\Log{{\rm Log}}
%
\beginsection 1. Introduction

It has been noticed for some time that
the Rogers dilogarithm function curiously emerges in the calculation of
physical quantities in various integrable systems.
%
Besides the outstanding result in three dimensions [1],
one finds extensive examples in two dimensions such as in
finite size corrections to the ground state energy
[2,3],
low-temperature asymptotics of the specific heat capacity
[4-8] and
ultraviolet Casimir energies [9-13]
in perturbed conformal field theories (CFTs) [14].
%
All these quantities are essentially identifiable with the
central charges of the relevant CFTs [4,15]
and have often been evaluated
in terms of the Rogers dilogarithm function [16].
%
It is therefore tempting to seek a universal background for the
common appearance of
the dilogarithm in the various calculational techniques
exploited in these works, e.g.,
functional relations (FRs) among row-to-row transfer
matrices, string hypotheses, thermodynamic Bethe ansatz (TBA) [17] and so
forth.
%
\par
%
This Letter is our first step toward understanding
the interrelation among these approaches.
%
We shall propose a
FR having a universal form related to the
$\uqaxr$ Bethe ansatz equation [18].
%
It may be viewed as a ``spectral parameter" dependent version of the recursion
relation among the ``Yangian characters" [19] whose combinations appear in
the arguments of the dilogarithm [7,8,20].
%
When $q = $ exp$({2\pi i \over \ell + g})$, we especially consider
a restricted version of the FR which closes among finitely many unknown
functions.
%
Here $\ell$ is a positive integer and
$g$ is the dual Coxeter number.
%
The restricted FR originates in the
equilibrium condition on the ratio of the string and hole densities in the
thermodynamic analysis of the $\uqaxr$ Bethe equation in [7,8].
%
On the other hand, the special case $\axr = A^{(1)}_1$ of the FR possesses
essentially the same form as that for certain combinations of
the actual transfer matrices [3,6] in
the restricted solid-on-solid (RSOS) models [21].
%
Although the full content of the FR is yet under investigation,
our heuristic analysis extending the $A^{(1)}_1$ case [3] leads to a new
conjecture that connects an analytically continued
Rogers dilogarithm function and the
scaling dimensions in parafermion CFTs [22].
%
It is basically labeled by a triad $(\axr, \ell, \Lambda)$ where
$\Lambda$ is the classical part of a level $\ell$ dominant integral weight of
$\axr$.
%
The earlier conjecture [20] (see also [8]) involving the central charges
concerns the special case $(\axr, \ell, 0)$ in our new one.
%
We expect further
extensions should be possible related to various coset CFTs by using
our FR.
%
The common structure of the FRs between the RSOS transfer matrices and
the TBA has also been observed in [3,23] for $\axr = A^{(1)}_1$.
%
%
\beginsection 2. $\uqaxr$ functional relation

Let $\axr$ denote one of the rank $r$ non-twisted affine Lie algebras [24]
$A^{(1)}_r (r \ge 1),
B^{(1)}_r (r \ge 2), C^{(1)}_r (r \ge 1), D^{(1)}_r (r \ge 3),
E^{(1)}_6, E^{(1)}_7,
E^{(1)}_8, F^{(1)}_4$ and $G^{(1)}_2$ and
$\uqaxr$ be the $q-$deformation of
its universal enveloping algebra [25].
%
Let $\alpha_a, \, \Lambda_a$ $(1 \le a \le r)$ be the simple roots and the
fundamental weights of the classical part $\xr$, respectively.
%
We set
$\rho = \sum_{a=1}^r \Lambda_a,\,
Q = \sum_{a=1}^r{\bf Z}\alpha_a,\,
P = \sum_{a=1}^r {\bf Z}\Lambda_a,\,
P_\ell = \{\Lambda \in P \vert
0 \le (\Lambda \vert \hbox{maximal root}) \le \ell \},\,
{\frak H}^\ast = \sum_{a=1}^r {\bf C}\Lambda_a$ and
${\frak H}^\ast_{\bf R} = \sum_{a=1}^r {\bf R}\Lambda_a$.
%
Introduce the bilinear form $(\, \vert \, )$ on ${\frak H}^\ast$ in
a standard way and normalize the roots as $\vert$ long root $\vert^2 = 2$.
%
Fix an integer $\ell \ge 1$ and put $\ell_a = t_a\ell, \,
G = \{ (a,m) \vert 1 \le a \le r,\, 1 \le m \le \ell_a - 1 \}$ throughout.
%
The following notation will be used in the sequel $(1 \le a, b \le r)$
%
$$\eqalignno{
&t_a = {2 \over (\alpha_a \vert \alpha_a) }, \quad
t_{a b} = \hbox{max}(t_a, t_b),&(1\rm a)\cr
%
&A^{m\, k}_{a\, b} = t_{a b}\bigl(
{\rm min}({m \over t_a}, {k \over t_b}) - {m k \over t_a t_b \ell} \bigr)
\quad\,{\rm for }\,\, (a,m), (b,k) \in G,
&(1\rm b)\cr
%
&B_{a b} = {t_b \over t_{a b}} C_{a b},\quad
C_{a b} = {2(\alpha_a \vert \alpha_b) \over (\alpha_a \vert \alpha_a) },
\quad I_{a b} = 2\delta_{a b} - B_{a b}.
&(1\rm c)\cr}
$$
%
Here $C_{a b}$ and
$B_{a b} = B_{b a}$ are the Cartan and the symmetrized Cartan matrix,
respectively.
%
The nodes in the Dynkin diagrams are numerated according to [24].
%
Let
$Y^{(a)}_m(u),\, 1 \le a \le r,\, m \ge 1$ be functions of a
complex parameter $u$ obeying
%
$$\eqalignno{
&Y^{(a)}_m(u+{i \over t_a})Y^{(a)}_m(u-{i \over t_a}) =
%
{\prod_{b=1}^r\prod_{k=1}^3 F_k^{I_{a b}\delta_{t_a k, t_{a b}}}
\over
\bigl(1+Y_{m-1}^{(a)}(u)^{-1}\bigr)\bigl(1+Y_{m+1}^{(a)}(u)^{-1}\bigr)},
&(2\rm a)\cr
%
&F_k = \prod_{j=-k+1}^{k-1}\prod_{n=0}^{k-1-\vert j \vert}
\Bigl(1+Y^{(b)}_{t_bm/t_a+j}\bigl(
u+i(k-1-\vert j \vert - 2n)/t_b\bigr)\Bigr),&(2\rm b)\cr}
$$
%
for $1 \le a \le r,\, m \ge 1$, where by convention
$Y^{(a)}_0(u){}^{-1} = 0$ and $Y^{(a)}_m(u) = 0\,\hbox{ if } m \not\in {\bf
Z}$.
%
We call (2) the (unrestricted) $\uqaxr$ functional relation.
%
If $Y^{(a)}_{\ell_a}(u){}^{-1} = 0$ is further imposed,
the denominator of (2a) becomes
$\prod_{j=1}^{\ell_a-1}(1+Y^{(a)}_j(u){}^{-1})^{\overline{I}_{j m}}$ with
$\overline{I}$ being the matrix $I$ of (1c) for $X_r = A_{\ell_a-1}$.
%
In this case, (2) closes among $\{Y^{(a)}_m(u) \vert (a,m) \in G\}$ and will be
referred as the restricted $\uqaxr$ functional relation.
%
The next section will concern this situation (see (6) and (15)).
%
Several remarks are in order:
%
\par\noindent
(i) The restricted FR (2) is closely related to the $\uqaxr$ Bethe equation
[18] and
its thermodynamic treatment using the string hypothesis in [8].
%
Under the identification
$Y^{(a)}_m(u) =$ exp$(-\beta\epsilon^{(a)}_m(u))$,
it actually corresponds to the thermal equilibrium
condition (2.22) of [8] in the high temperature limit.
%
\par\noindent
(ii) In the simply laced cases $X_r = A_r, D_r$ and $E_{6,7,8}$,
the restricted FR (2) takes a simple form
%
$$
Y^{(a)}_m(u+i)Y^{(a)}_m(u-i) =
%
{\prod_{b=1}^r \bigl(1+Y_m^{(b)}(u)\bigr)^{I_{a b}}
\over
\prod_{j=1}^{\ell-1}(1+Y^{(a)}_j(u){}^{-1})^{\overline{I}_{j m}}},
\eqno(3)
$$
%
where $\overline{I}$ denotes the $I$ for $A_{\ell-1}$.
%
\par\noindent
(iii) Besides the trivial $\ell = 1$ case,
(3) essentially coincides with the FR in [13,26] for $\ell = 2$
and the FR in [3,6] for $X_r = A_1$ with $\ell$ general.
%
The former appeared in the TBA approach to
factorized scattering theories
whilst the latter is known to hold [3,6] for certain
combinations of the actual
transfer matrices in the RSOS models [21].
%
In these cases, the FRs have been successfully used
to determine the central charges
by combining them with the data
from actual physical systems such as ``bulk behavior" [3],
``mass term" [9,13], etc.
%
\par\noindent
%
(iv) The FR (2) for $Y^{(a)}_m(u)$'s can be viewed as a
generalization of the recursion relation
among the quantity $Q^{(a)}_m$ in [19].
%
To see this, recall the definitions
%
$$\eqalignno{
&Q^{(a)}_m = \sum_{\bf n} Z(a,m,{\bf n})\,
\chi_{\omega(a,m,{\bf n})}, \quad\,
{\bf n} = (n_1, \ldots ,n_r) \in {\bf Z}_{\ge 0}^{\otimes r},
&(4\rm a)\cr
%
&\omega(a,m,{\bf n}) = m\Lambda_a - \sum_{b=1}^r n_b \alpha_b,\quad
Z(a,m,{\bf n}) = \sum_\nu \prod_{b=1}^r \prod_{k=1}^\infty
{{\cal P}^{(b)}_k(\nu) + \nu^{(b)}_k \choose \nu^{(b)}_k},&(4\rm b)\cr
%
&{\cal P}^{(b)}_k(\nu) = \hbox{min}(m,k)\delta_{a b} -
2\sum_{j \ge 1} \hbox{min}(k,j)\nu^{(b)}_j
- \sum_{\scriptstyle c = 1 \atop \scriptstyle c \neq b}^r \sum_{j \ge 1}
\hbox{max}(k C_{c b}, \, j C_{b c}) \nu^{(c)}_j, &(4\rm c)\cr
}$$
%
for $1 \le a \le r, \,\, m \ge 0$.
%
Here the symbol ${\quad \choose \quad}$ in (4b) is the
binomial coefficient and the sum extends over all possible
decompositions
%
$\{ \nu^{(b)}_k \mid\,
n_b = \sum_{k=1}^\infty k\nu^{(b)}_k, \,
\nu^{(b)}_k \in {\bf Z}_{\ge 0}, 1 \le b \le r, \, k \ge 1\}$
such that
${\cal P}^{(b)}_k(\nu) \ge 0$ for $1 \le b \le r,\, k \ge 1$.
%
The quantity $\chi_\omega$ in (4a) is the character of the
irreducible $\xr$ module $V_\omega$ with highest weight $\omega$, i.e.,
%
$
\chi_\omega = \chi_\omega(z) =
\hbox{Tr}_{V_\omega}
\hbox{ exp}\bigl(-{2\pi i\over \ell + g}(z + \rho)\bigr),
$
where we have exhibited the dependence on its argument
$z \in {\frak H}^\ast$.
%
We shall simply write $Q^{(a)}_m$ to mean the corresponding
$Q^{(a)}_m(z)$.
%
Then the following recursion relation is known
among the $Q^{(a)}_m$'s [19],
%
$$\eqalignno{
Q^{(a)}_m{}^2 &= Q^{(a)}_{m-1}Q^{(a)}_{m+1} +
Q^{(a)}_m{}^2 \prod_{b=1}^r\prod_{k=0}^{\infty}
Q^{(b)}_k{}^{-2J^{k\, m}_{b\, a}}\,\,\hbox{ for }
1 \le a \le r, \, m \ge 0, &(5\rm a)\cr
%
2J^{n\, k}_{a\, b} &= B_{a b}
\bigl({t_{a b} \over t_a}\delta_{t_b n, \, t_a k} +
{t_{a b} \over t_b}\sum_{j=1}^{t_b-t_a} j
(\delta_{t_b(n+1)-t_aj, t_a k} + \delta_{t_b(n-1)+t_aj, t_a k})
\bigr),&(5\rm b)\cr
&\qquad\hbox{ for } 1 \le a, b \le r, \quad n, k \ge 0,\cr
}$$
%
with the initial condition
$Q^{(a)}_{-1} = 0,\,Q^{(a)}_0 = 1$.
%
In (5b), the sum $\sum_{j=1}^{t_b-t_a}$  is zero unless
$t_b > t_a$ and
$J^{n\, k}_{a\, b}$ is a natural extension of the
${\hat J}^{n\, k}_{a\, b}(0)$ defined in [8]
for the range $0 < n < \ell_a, \, 0 < k < \ell_b$.
%
Now consider the limit $u \rightarrow \infty$ in (2).
%
{}From (5) the resulting algebraic equation admits a constant solution
%
$$
Y^{(a)}_m(\infty) =
{Q^{(a)}_m{}^2 \prod_{b=1}^r\prod_{k=0}^{\infty}
Q^{(b)}_k{}^{-2J^{k\, m}_{b\, a}}
\over
Q^{(a)}_{m+1}Q^{(a)}_{m-1}},\eqno(6)
$$
%
which implies that (2) is a ``$u-$version" of (5).
%
\beginsection 3. Dilogarithm conjecture

Here we formulate our dilogarithm conjecture firstly
then discuss its physical backgorund in the light of
the FR (2).
%
Let $\log x$ signify the logarithm in the branch
$-\pi < {\rm Im}(\log x) \le \pi$ for $x \neq 0$.
%
Consider a contour on a complex $x-$plane
${\cal C}={\cal C}[f \vert M,N], \,(f \in {\bf C},\, M,N \in {\bf Z})$
from $0$ to $f$ which intersects the branch cut of
$\log x$ for $M$ times and that of $\log(1-x)$ for $N$ times in total.
%
Here the intersection number is counted
as +1 when ${\cal C}$ goes across the cut of $\log x$
(resp. $\log(1-x)$) from the upper (resp. lower) half plane
to the lower (resp. upper) and as $-1$ if opposite.
%
Let $\Log_{\cal C}(f)$ denote the analytic continuation of $\log f$
along the contour ${\cal C}$, namely,
%
$$\eqalignno{
\Log_{\cal C}(f) &= \log f + 2\pi i M, \quad
\Log_{\cal C}(1-f) = \log (1-f) + 2\pi i N.&(7)\cr
}$$
%
Define the multivalued and single-valued Rogers dilogarithms
$L_{\cal C}(f)$ and $L(f)$ by
%
$$\eqalignno{
L_{\cal C}(f) &= -{1 \over 2} \int_{\cal C}
\Bigl( {\Log_{\cal C} (1-x) \over x}
     + {\Log_{\cal C}\, x \over 1 - x }\, \Bigr) dx, &(8{\rm a})\cr
%
L(f) &= -{1 \over 2} \int_{{\cal C}_0}
\Bigl( {\log (1-x) \over x} + { \log \, x \over 1 - x }\, \Bigr) dx,
&(8{\rm b})\cr}
$$
%
where ${\cal C}_0$ is a contour which does not go across the branch cuts
of $\log x$ and $\log (1-x)$.
%
It follows from the definitions that
%
$$\eqalignno{
L(f) &= \cases{
{\pi^2 \over 3}
- {\pi i \over 2} \log f - L(f^{-1}) & if $f \in {\bf R}_{>1}$,\cr
%
-{\pi^2 \over 6}
+ {\pi i \over 2} \log (1 - f) + L({1 \over 1 -f}) & if $f \in {\bf
R}_{<0}$,\cr}
&(9\rm a)\cr
%
L_{\cal C}(f) &= L(f) + \pi i M  \log (1-f) - \pi i N \log f
+ 2\pi^2 M N. &(9\rm b)\cr}
$$
%
%
\par
%
We shall call an element $z \in {\frak H}^\ast$ {\sl regular} if it
satisfies $Q^{(a)}_m(z) \neq
0$ for all $1 \le a \le r,\, 1 \le m \le \ell_a$ and {\sl singular} otherwise.
%
Suppose $z$ is regular and put
$$
f^{(a)}_m(z) =
1 - {Q^{(a)}_{m-1}(z)Q^{(a)}_{m+1}(z) \over Q^{(a)}_m(z){}^2}\quad
(a,m) \in G.\eqno(10)
$$
%
Then $f^{(a)}_m(z) \neq 0, 1, \infty$ for $\forall (a,m) \in G$.
%
Given a set of integers
${\cal S} = \{M^{(a)}_m, N^{(a)}_m \in {\bf Z} \vert (a,m) \in G \}$,
let ${\cal C}_{a,m} = {\cal C}[f^{(a)}_m(z) \vert M^{(a)}_m, N^{(a)}_m]$
be the contour as specified above.
%
Motivated by the study of the FR (2) which will be discussed later, we define
%
$$\eqalignno{
&{\pi^2 \over 6}c(z,{\cal S}) =
\sum_{(a,m) \in G}\Bigl(
L_{{\cal C}_{a,m}}\bigl(f^{(a)}_m(z)\bigr) -
{\pi i\over 2}D_{{\cal C}_{a,m}}(z)
\Log_{{\cal C}_{a,m}}\bigl(1-f^{(a)}_m(z)\bigr)\Bigr),&(11\rm a)\cr
%
&\pi i D_{{\cal C}_{a,m}}(z) =
\Log_{{\cal C}_{a,m}}\bigl(f^{(a)}_m(z)\bigr) -
\sum_{(b,k) \in G}A^{m\, k}_{a\, b}B_{a b}
\Log_{{\cal C}_{b,k}}\bigl(1-f^{(b)}_k(z)\bigr).&(11\rm b)\cr
}$$
%
By applying (7), (9b) and (11b) to (11a), one can split $c(z,{\cal S})$
into ${\cal S}-$dependent and independent parts as
%
$$\eqalignno{
&c(z,{\cal S}) = c_0(z) - 24T(z,{\cal S}),&(12\rm a)\cr
%
&{\pi^2 \over 6}c_0(z) =
\sum_{(a,m) \in G}\Bigl(
L\bigl(f^{(a)}_m(z)\bigr) -
{\pi i\over 2}d^{(a)}_m(z)
\log\bigl(1-f^{(a)}_m(z)\bigr)\Bigr),&(12\rm b)\cr
%
&\pi i d^{(a)}_m(z) =
\log f^{(a)}_m(z) -
\sum_{(b,k) \in G}A^{m\, k}_{a\, b}B_{a b}
\log\bigl(1-f^{(b)}_k(z)\bigr),&(12\rm c)\cr
%
&T(z,{\cal S}) = {1 \over 2}
\sum_{\scriptstyle (a,m) \in G \atop \scriptstyle (b,k) \in G}
A^{m\, k}_{a\, b}B_{a b} N^{(a)}_m N^{(b)}_k -
\sum_{(a,m) \in G}\bigl({1 \over 2}d^{(a)}_m(z) + M^{(a)}_m\bigr) N^{(a)}_m.
&(12\rm d)\cr}$$
%
Next we introduce an element $\lambda(z) \in {\frak H}^\ast$ by
%
$$
\lambda(z) = {1 \over 2 \pi i}\sum_{a=1}^r\sum_{b=1}^r C_{a b}
\Bigl(\sum_{j=1}^{\ell_b-1} j \log(1-f^{(b)}_j(z)) + \ell_b\bigl(
\log Q^{(b)}_{\ell_b-1}(z) - \log Q^{(b)}_{\ell_b}(z)\bigr)\Bigr)
\Lambda_a.
\eqno(13)
$$
%
{}From now on, we will mainly concern with the specialization
$z = \Lambda \in P_\ell$, which is relevant to our
restricted FR (2) (see (6) and (15) below).
%
\proclaim Conjecture.
%
Let $z \in {\frak H}^\ast$ be regular and put
%
$$\eqalignno{
&c_0(z) = {\ell{\rm dim }X_r \over \ell + g} - r -
24(\Delta^z_{\lambda(z)} + {\cal N}(z)),&(14\rm a)\cr
%
&\Delta^z_y = {(z \vert z + 2\rho)\over 2(\ell + g)} -
{\vert y \vert^2 \over 2\ell}
\quad {\rm for }\,\, y, z \in {\frak H}^\ast.&(14\rm b)\cr
}$$
%
Suppose $\Lambda \in P_\ell$.
%
Then ${\cal N}(\Lambda) \in {\bf Z}$ if $\Lambda$ is regular.
%
In case $\Lambda$ is singular, ${\cal N}(z)$ converges to finitely many
integers depending on the ways $z \in {\frak H}^\ast_{\bf R}$
approaches $\Lambda$.
%

This conjecture has been supported by numerical experiments for
$X_r = A_r, B_r$, $C_r$ and $D_r$ with small values of
the level $\ell$ and rank $r$.
%
It is not difficult to prove ${\cal N}(z) \in {\bf R}$ for
any regular $z \in {\frak H}^\ast_{\bf R}$.
%
Because of the discontinuity of the $\log$ function,
$c_0(z),\, \lambda(z)$ and therefore
${\cal N}(z)$ assume finitely many values when
$z$ approaches a singular $\Lambda$.
%
Hereafter we shall arbitrarily fix one way of letting
$z \rightarrow \Lambda\, (z: \hbox{regular},\, z \in {\frak H}^\ast_{\bf R})$
and all the formulas involving $\Lambda \in P_\ell$ should
be understood as defined by
this limit whenever $\Lambda$ is singular.
%
{}From numerical tests,
$Q^{(a)}_m(\Lambda) = Q^{(a)}_{\ell_a}(\Lambda)
Q^{(a)}_{\ell_a-m}(\Lambda)^\ast$ seems valid for any
$-1 \le m \le \ell_a+1$ and $\Lambda \in P_\ell$,
where $\ast$ denotes complex conjugation.
%
Note in particular that it implies
%
$$Q^{(a)}_{\ell_a+1}(\Lambda) = 0. \eqno(15)$$
%
Assuming this and (A.10,11) in [8], one can show that
$\lambda(\Lambda)$ is finite and satisfies
%
$$\eqalignno{
\Lambda &\equiv \lambda(\Lambda) \quad \hbox{ mod } Q,&(16\rm a)\cr
%
T(\Lambda,{\cal S})
&= -{1 \over 2\ell}\vert \lambda(\Lambda) + \beta({\cal S}) \vert^2 +
{1 \over 2\ell}\vert \lambda(\Lambda)\vert^2
+ \hbox{integer},&(16\rm b)\cr
%
\beta({\cal S}) &= \sum_{a=1}^r\sum_{m=1}^{\ell_a-1} m N^{(a)}_m \alpha_a
\in Q,&(16\rm c)\cr}
$$
%
where the integer part is dependent on $\Lambda$ and ${\cal S}$.
%
Combining (16b) with (12a) and admitting the conjecture, we have
%
$$
c(\Lambda,{\cal S}) =
{\ell{\rm dim }X_r \over \ell + g} - r -
24(\Delta^\Lambda_{\lambda(\Lambda) + \beta({\cal S})}
+ \hbox{integer}) \quad \hbox{for any } \, \Lambda \in P_\ell.
\eqno(17)
$$
%
The RHS of (17) with the congruence
properties (16a,c) is well known as the
central charge $-$ 24(scaling dimension mod ${\bf Z}$) of the
level $\ell \, \axr$ parafermion CFT [22].
%
As the integer set ${\cal S}$ is chosen variously, $\beta({\cal S})$ (16c)
ranges over the root lattice $Q$.
%
Thus all the spectra in the parafermion CFTs seem to come out
from the Rogers dilogarithm function through the quantity
$c(\Lambda,{\cal S})$ (11a).
%
This is our main observation in this Letter.
%
We note that in the case $\Lambda = 0 \in P_\ell$, $\lambda(0) = 0$
holds and the earlier conjecture in [20,8] corresponds to
${\cal N}(0) = 0$.
%
\par
%
There are actually two sources for considering the
quantity $c(\Lambda,{\cal S})$ (11a) in connection to the
parafermion CFTs, which we shall now explain.
%
The physical meaning of the integer set ${\cal S}$ is yet to be clarified
in the light of these connections.
%
%
The first source is to calculate finite-size corrections
to the ground state energies for $\axr$ fusion RSOS models.
%
In [3] Kl\"umper and Pearce evaluated them for $\axr = A^{(1)}_1$
via the Rogers dilogarithm and showed that
the resulting exponents in regime I/II become the parafermion values
in agreement with the earlier result in [21].
%
They started from the FR
essentially identical to the
restricted $U_q(A^{(1)}_1)$ FR (2), where
$Y^{(1)}_m(u)$ indeed is the finite-size correction
part of the row-to-row transfer matrices.
%
Such an interpretation is not known so far for the
$\uqaxr$ FR (2) in general.
%
However we have found that the $c(\Lambda,{\cal S})$ naturally arises
as the ``$c$ for excitations" in their sense
through a heuristic calculation extending the $A^{(1)}_1$ case.
%
Although the treatment in [3] is not necessarily identical to our
formulation here, the regime I/II result therein effectively
confirms our conjecture for the simplest case $\axr = A^{(1)}_1$.
%
We shall emphasize that for higher rank algebras,
the present $c(\Lambda,{\cal S})$ only corresponds to
``regime I/II-like region" with a special
``fusion type $\forall s_a > 1$" in the sense of
section 4.2 in [8].
%
Thus we expect further generalizations should be possible related to
various coset CFTs.
%
See for example the regime III/IV result in [3] for $A^{(1)}_1$.
%
\par
%
The second source is to study integrable perturbations [14] of parafermion
CFTs.
%
In [9-13], ultraviolet Casimir energies in various
perturbed CFTs have been obtained by using
integral equations of the form
$$
Rm^{(a)}_j \hbox{ch } u = \epsilon^{(a)}_j(u)
+ \sum_{(b,k) \in G} \int_{- \infty}^\infty
dv \, \Psi^{j k}_{a b}(u - v)\, \log
\bigl(1 + \hbox{exp}(-\epsilon^{(b)}_k(v)) \bigr),\eqno(18)
$$
for $(a,m) \in G$, which originates in the TBA.
%
Here $m^{(a)}_j$ denotes mass,
$\epsilon^{(a)}_j(u)$ is the energy of the physical
excitation with rapidity $u$,
$\Psi^{j k}_{a b}(u)$ is some kernel and
$R$ is the system size corresponding to the inverse temperature in the TBA.
%
In addition, there are some observations that
(18) also yields low-lying excitation energies
by introducing ``imaginary chemical potentials" [23,27,28].
%
Our $c(\Lambda,{\cal S})$ stems from a modification of (18) so as to
match the logarithm of (2) under the identification
$Y^{(a)}_j(u) = e^{-\epsilon^{(a)}_j(u)}$
up to the LHS which is tending to zero
in the ultraviolet (or high temperature) limit.
%
In particular, we choose $\forall m^{(a)}_j>0$ as $j-$independent, add
the imaginary term $\pi i D_{{\cal C}_a,j}(\Lambda)$ (11b) on the RHS and take
$\Psi^{j k}_{a b}(u)$ to
be the universal kernel occurring in the $\uqaxr$ Bethe equations, i.e.,
$2\pi\Psi^{j k}_{a b}(u) = \int dx e^{iux} (\delta_{a b}\delta_{j k} -
{\hat M}_{a b}(\pi x/2){\hat A}^{(\ell)\, j k}_{\quad\, a b}(\pi x/2))$
using (2.10) of [8].
%
Then an analogous calculation to [11] amounts to considering
the $c(\Lambda,{\cal S})$ as the ``$c$ for excitations".
%
\par
%
These observations
have led to the quantity
$c(\Lambda,{\cal S})$ which has been conjectured to possess
the remarkable property (17).
%
Our $\uqaxr$ FR (2) lies as their common background.
%
It is to be analyzed further
to actually yield the physical
consequences in the $\uqaxr$ Bethe ansatz systems.
%
The details omitted here will appear elsewhere.
%
\par \vskip0.1cm
%
The authors thank P.A.Pearce for supporting
their visit to University of Melbourne, kind hospitality and
explaining the work [3].
%
They also thank M.T.Batchelor, R.J.Baxter, V.V.Bazhanov, J.Suzuki,
M.Wakimoto and R.B.Zhang
for valuable comments.
%
This work is supported by the Australian Research Council.
%
%
\beginsection References

\noindent
\item{[1]}{R.J.Baxter, Physica{\bf 18D} (1986) 321}
%
%
\item{[2]}{A.Kl\"umper, M.T.Batchelor and P.A.Pearce, J.Phys.A{\bf 24} (1991)
3111}
%
\item{[3]}{A.Kl\"umper and P.A.Pearce, ``Conformal weights of RSOS lattice
models
and their fusion hierarchies", preprint No.23-1991 (1991)}
%
\item{[4]}{I.Affleck, Phys.Rev.Lett.{\bf 56} (1986) 746}
%
\item{[5]}{H.M.Babujan, Nucl.Phys.{\bf B215}[FS7] (1983) 317;}
\item{}{H.J.de Vega and M.Karowski, Nucl.Phys.{\bf B285}[FS19] (1987) 619;}
\item{}{A.N.Kirillov and N.Yu.Reshetikhin, J.Phys.A{\bf 20} (1987) 1587}
%
\item{[6]}{V.V.Bazhanov and Yu.N.Reshetikhin, Int.J.Mod.Phys.A{\bf 4}
(1989) 115}
%
\item{[7]}{V.V.Bazhanov and Yu.N.Reshetikhin, J.Phys.A{\bf 23} (1990) 1477}
%
%
\item{[8]}{A.Kuniba, ``Thermodynamics of the $\uqaxr$ Bethe Ansatz System
With $q$ A Root of Unity", ANU preprint (1991)}
%
\item{[9]}{Al.B.Zamolodchikov, Nucl.Phys.{\bf B342} (1990) 695;
{\bf B358} (1991) 497}
%
\item{[10]}
{V.V.Bazhanov and N.Yu.Reshetikhin, Prog.Theor.Phys.Suppl.{\bf 102} (1990) 301}
%
\item{[11]}{M.J.Martins, Phys.Rev.Lett.{\bf 65} (1990) 2091}
%
\item{[12]}{T.R.Klassen and E.Melzer, Nucl.Phys.{\bf B338} (1990) 485;
{\bf B350} (1991) 635}
%
\item{[13]}{V.A.Fateev and Al.B.Zamolodchikov, Phys.Lett.{\bf B271} (1991) 91}
%
\item{[14]}{A.B.Zamolodchikov, Int.J.Mod.Phys.{\bf A4} (1989) 4235}
%
\item{[15]}{H.W.J.Bl\"ote, J.L.Cardy
and M.P.Nightingale,  Phys.Rev.Lett.{\bf 56} (1986) 742}
%
\item{[16]}{L.Lewin, {\sl Polylogarithms and associated functions},
(North-Holland, 1981)}
%
\item{[17]}{C.N.Yang and C.P.Yang, J.Math.Phys.{\bf 10} (1969) 1115}
%
\item{[18]}
{N.Yu.Reshetikhin and P. B. Wiegmann, Phys.Lett{\bf B189} (1987) 125}
%
\item{[19]}{A.N.Kirillov and N.Yu.Reshetikhin,
Zap.Nauch.Semin.LOMI{\bf 160} (1987) 211}
%
\item{[20]}{A.N.Kirillov, Zap.Nauch.Semin.LOMI{\bf 164} (1987) 121 and private
communications}
%
\item{[21]}
{G.E.Andrews, R.J.Baxter and P.J.Forrester,
J.Stat.Phys.{\bf 35} (1984) 193;}
\item{}{E.Date, M.Jimbo, A.Kuniba, T.Miwa and M.Okado,
Nucl.Phys.{\bf B290}}
\item{}{[FS20] (1987) 231; Adv.Stud.in Pure Math.{\bf 16}
(1988) 17}
%
\item{[22]}{V.A.Fateev and A.B.Zamolodchikov,
Sov.Phys.JETP {\bf 62} (1985) 215;}
\item{}{D.Gepner, Nucl.Phys.{\bf B290}[FS20] (1987) 10}
%
\item{[23]}{T.R.Klassen and E.Melzer, Nucl.Phys.{\bf B370} (1992) 511}
%
\item{[24]}{V.G.Kac, {\sl Infinite dimensional Lie algebras},
(Cambridge University Press, 1990)}
%
%
\item{[25]}{M.Jimbo, Lett.Math.Phys.{\bf 10} (1985) 63;}
\item{}{V.G.Drinfel'd, ICM proceedings, Berkeley (1987) 798}
%
\item{[26]}{Al.B.Zamolodchikov, Phys.Lett.{\bf B253} (1991) 391}
%
\item{[27]}{M.J.Martins, Phys.Rev.Lett.{\bf 67} (1991) 419}
\item{[28]}{P.Fendley, Nucl.Phys.{\bf B372} (1992) 533}
%
%
\bye


