%Paper: hep-th/9210025
%From: nakanisi@string.harvard.edu (Tomoki Nakanishi)
%Date: Mon, 5 Oct 92 14:38:34 GMT-0400
%Date (revised): Tue, 6 Oct 92 12:22:38 GMT-0400



\input harvmac
\overfullrule=0pt

\Title{HUTP-92/A046}{Rogers Dilogarithm in Integrable Systems*
\footnote{}{\hskip -0.14truein *~ To appear in the proceedings of the XXI
Differential Geometry
 Methods in Theoretical Physics, Tianjin, China 5-9 June 1992.  Preprint
distribution partially supported by NSF Grant PHY~87-14654, and a Packard
Fellowship.}}

\centerline{Atsuo Kuniba}
\bigskip\centerline{Department of Mathematics, Kyushu University}
\centerline{Fukuoka 812 JAPAN}
\vskip0.3cm
\centerline{and}
\vskip0.3cm
\centerline{Tomoki Nakanishi\footnote{$^\dagger$}
{Permanent Address: Department of Mathematics, Nagoya University,
Nagoya 464 Japan}}
\bigskip\centerline{Lyman Laboratory of Physics, Harvard University}
\centerline{Cambridge, MA 02138 USA}

%if too many authors for abstract on same page, say   \vfill\eject\pageno0

\vskip .3in
We discuss some curious aspects of the Rogers dilogarithm appearing
in integrable systems in two dimensions.

\Date{09/92} %replace this line by \draft  for preliminary versions
	     %or specify \draftmode at some point

%if you want double-space, use e.g. \baselineskip=20pt plus 2pt minus 2pt

\newsec{Introduction}
\indent
This note is a brief exposition of the appearance of the
Rogers dilogarithm function in relation to integrable
lattice models and conformal field theory (CFT) in two dimensions.
%
The content is mainly the known facts in refs.${}^{1,2}$ but also includes
a few new informations based on a collaboration with Junji Suzuki.
%
\par
%
The Rogers dilogarithm is a function of a variable $x$ defined by
%
$$L(x) = -{1 \over 2}\int_0^x \bigl(
{log (1-y) \over y} + {log y \over 1-y} \bigr) dy \quad (0 \le x \le 1).
\eqno(1)
$$
%
The following identity is know essentially due to refs.${}^3$
%
$$
{6 \over \pi^2} \sum_{m=1}^\ell L\bigl(
{sin^2 {\pi \over \ell + 2} \over sin^2 {\pi(m+1) \over \ell + 2}}
\bigr) = {3\ell \over \ell + 2},
\quad\hbox{ for } \ell \in {\bf Z}_{\ge 1}.\eqno(2)
$$
%
In the above, the rhs is the well known value of the central charge
for the level $\ell \,\, A^{(1)}_1$ WZW model in conformal field theory.
%
On the other hand, the argument of the dilogarithm
has the form $(Q_m)^{-2}$,
where $Q_m$ is the $m+1$ dimensional
irreducible $A_1$ character specialized to some
``rational point".
%
Thus Eq.\ 2 is connecting the two important quantities in
the $A_1$-related theory, i.e.,
the central charge and the specialized character.
%
\par
%
In fact, there is a conjectural generalization of the identity Eq.\ 2
into arbitrary classical simple Lie algebra $X_r^{4,1}$.
%
%
$$
{6 \over \pi^2} \sum_{a=1}^r \sum_{m=1}^{t_a \ell} L\bigl(
f^{(a)}_m
\bigr) = {\ell dim X_r  \over \ell + g},
\quad\hbox{ for } \ell \in {\bf Z}_{\ge 1}.\eqno(3)
$$
%
Here, $g$ denotes the dual Coxeter number and
$t_a$ is the integer defined as the ratio of the $a-$th Kac
and dual Kac label.
%
The argument $0 \le f^{(a)}_m \le 1$ arises through
thermodynamic Bethe ansatz (TBA) analysis.
%
Here we shall give its definition only for
$X_r = A_r$. (See refs.${}^{1,4,5}$ for the general case.)
%
$$
f^{(a)}_m = f^{(a)}_m(z=0), \qquad
%
f^{(a)}_m(z) = 1 - {Q^{(a)}_{m+1}(z)Q^{(a)}_{m-1}(z)
\over Q^{(a)}_m(z){}^2}, \eqno(4)
$$
%
where $Q^{(a)}_m(z)$ is the irreducible $A_r$ character
with the highest weight $m\Lambda_a$.
%
($z \in $ dual space of the Cartan subalgebra and
$\Lambda_a$ is the $a-$th fundamental weight.
%
Nodes on the Dynkin diagram are enumerated according to ref.${}^1$.)
%
In general, the quantity $Q^{(a)}_m(z)$ is
a Yangian character and we adopt the $z$ dependence as given in
ref.${}^2$.
%
The conjecture (4) was firstly systematically used in the
TBA analysis in refs.${}^{6,1}$.

\newsec{Scaling Dimensions from Dilogarithm}
\indent
There is a generalization${}^2$ of Eq.\ 3 so as to include
the parafermion scaling dimensions (modulo  integer) in CFT
%
$\Delta^{\Lambda}_{\lambda} = {(\Lambda \vert \Lambda + 2\rho) \over
2(\ell + g)} - {\vert \lambda \vert^2 \over 2g}$,
where $\Lambda$ is a level $\ell$ dominant integral weight of
$X^{(1)}_r$ and $\lambda \in \Lambda + $ root lattice.
%
This is achieved by considering the specialization
$ f^{(a)}_m(z=\Lambda)$ instead of the principal one in Eq.\ 4.
%
Now that the $ f^{(a)}_m(\Lambda)$ is a
complex number in general, one can consider
various analytic continuations $L_{a,m}(x)$ of $L(x)$.
%
Leaving all the technical points${}^2$, we have a conjecture
%
$$
{6 \over \pi^2} \sum_{a=1}^r \sum_{m=1}^{t_a \ell-1}
L_{a,m}\bigl(f^{(a)}_m(\Lambda)\bigr)
+ \hbox{ Logarithmic terms} = {\ell dim X_r  \over \ell + g} - r
 -24 (\Delta^{\Lambda}_{\lambda} + \hbox{ integer}). \eqno(5)
$$
%
Here, $\lambda$
depends on the integration contour along which
the $L(x)$ is analytically continued.
%
Its explicit form and the logarithmic terms can be found in ref.${}^2$.
%
See also refs.\ ${}^{7,8,9,10}$ for some physical aspects.
%
Here we shall only present a conjecture
on the value $Q^{(a)}_{t_a\ell}(\Lambda)$,
under which the congruence $\lambda \equiv \Lambda$ (modulo  root lattice)
can be verified directly.
%
Put $k =$ number of the Kac labels $a_i \, (0 \le i \le r)$ equal to 1.
%
Then we conjecture that $Q^{(a)}_{t_a\ell}(\Lambda)$ is a
$k-$th root of unity as
%
$$\eqalign{
Q^{(a)}_{t_a\ell}(\Lambda) &=
\hbox{exp}(-2\pi i c(\Lambda){\overline \gamma}_a/k)
\quad \hbox { for } X_r \neq D_r,\cr
%
&= \hbox{exp}(2\pi i c_2(\Lambda)\gamma^{(2)}_a/k)
\quad \hbox { for } X_r =  D_r, \, r = \hbox{odd},\cr
%
&= \hbox{exp}(2\pi i ((c_2(\Lambda)-rc_1(\Lambda))\gamma^{(1)}_a
                    + c_1(\Lambda)\gamma^{(2)}_a)/k)
\quad \hbox { for } X_r = D_r,\, r = \hbox{even}.\cr
}\eqno(6)$$
%
Here, for $\Lambda = \sum_{a=1}^r \mu_a \Lambda_a$,  we have set
%
$c(\Lambda)
= \sum_a \gamma_a \mu_a \,  \hbox{ mod }\, k \,
\hbox{ for } X_r \neq D_r$ and
$c_i(\Lambda)
= \sum_a \gamma_a^{(i)} \mu_a \,\, \hbox{ mod }\,  k/(3-i)
\, \hbox{ for } X_r = D_r, \,
i = 1,2$
and
$\gamma$ is the rank-dimensional integer vector given by
%
$$
\eqalign{
&A_r, B_r, C_r, E_6: \gamma_a = a,\cr
&D_r: \gamma^{(1)}=(0,\ldots,0,1,1), \quad
\gamma^{(2)} = (2,4,6,\ldots,2(r-2),r-2,r),\cr
&E_7: \gamma = (0,0,0,1,0,1,1),\cr
&E_8,F_4,G_2: \gamma = (0,\ldots,0).\cr
}\eqno(7)$$
%
Finally, ${\overline \gamma} = -\gamma$ if $X_r = A_r$ and
${\overline \gamma} = \gamma$ of the dual algebra of  $X_r$ if
$X_r \neq A_r, D_r$.
%
Eq.\ 6 is a special solution of
$\prod_b Q^{(b)}_{t_b \ell}(\Lambda)^{C_{a b}}=1$
where $C$ is the Cartan matrix of $X_r$.
%
{}From numerical tests it also seems valid that
$Q^{(a)}_m(\Lambda) = Q^{(a)}_{t_a\ell}(\Lambda)
Q^{(a)}_{t_a\ell - m}(\Lambda){}^{\ast}$ for
$-1 \le m \le t_a\ell + 1$ and
$Q^{(a)}_{t_a\ell + 1}(\Lambda) = 0$
for any level $\ell$ dominant
integral weight $\Lambda$.
%
These are interesting arithmetic properties of
the specialized Yangian characters.
%
We note especially that $Q^{(a)}_m(0)$ appears${}^1$
as the high temperature limit
of $log ( entropy )$ per site in the TBA system connected to $X_r$.
%
This implies that $Q^{(a)}_m(0)$ yields the largest eigenvalue
of the incidence matrix for a fusion $X^{(1)}_r$ RSOS model.



\newsec{Functional Relations}
\indent
The Yangian character  $Q^{(a)}_m(z)$ is known to satisfy
interesting recursion relations${}^{4,5}$.
%
For example in $X_r = A_r$ case,
%
$$
Q^{(a)}_m(z){}^2 = Q^{(a)}_{m+1}(z)Q^{(a)}_{m-1}(z) +
                   Q^{(a+1)}_m(z)Q^{(a-1)}_m(z)\eqno(8)
$$
%
and similar relations are known for all the other algebras.
%
Furthermore there is a ``spectral parameter dependent version
(or Yang-Baxterization)" of these relations.
%
Below we shall describe it briefly for $X_r = A_r$.
%
In ref.${}^6$ Bazhanov and Reshetikhin wrote down a system of
functional relations among the row to row transfer matrices
for the fusion $A^{(1)}_r$ model${}^{11}$.
%
$$\eqalignno{
T^{\xi, \eta}(u) &= \hbox{det}\, \bigl(
T^{\xi, (\eta_i-i+j)\Lambda_1}(u+\eta_1+i-\eta_i-1)
\bigr)_{1 \le i,j \le \eta^{\prime}_1} &(9\hbox{a})\cr
%
&= \hbox{det}\, \bigl(
T^{\xi, \Lambda_{\eta^{\prime}_i-i+j}}(u+\eta_1-j)
\bigr)_{1 \le i,j \le \eta_1}\, , &(9\hbox{b})\cr
%
}$$
%
where $\xi$ (resp. $\eta$) is the Young diagram representing
the fusion type in the horizontal (resp. vertical)
direction and the $T^{\xi, \eta}(u)$ is
transferring the states into the vertical direction.
%
$\eta^{\prime} = [\eta^{\prime}_1, \ldots , \eta^{\prime}_{\eta_1}]$
is the transpose of
$\eta = [\eta_1, \ldots , \eta_{\eta^{\prime}_1}]$
and $u$ is the spectral parameter entering the solution of the
Yang-Baxter equation that underlies the model${}^{11}$.
%
Eq.\ 9 is a ``quantum analogue" of the
2nd Weyl character formula (Jacobi-Trudi's formula)${}^6$.
%
(In fact Eq.\ 9 is different from that in ref.${}^6$ and the
alternation is based on a private communication with V.V. Bazhanov.
%
Note that $T^{\xi, \eta}(u)$ may be regarded as representing an eigenvalue
thanks to the commutatibity.)
%
Now let $\eta$ be the $a$ by $m$ rectangular Young diagram
corresponding to $m\Lambda_a$ and write the $T^{\xi, \eta}(u)$
as $T^{(a)}_m(u)$.
%
Then from Eq.\ 9 one can prove the functional relation
%
$$
T^{(a)}_m(u)T^{(a)}_m(u+1) = T^{(a)}_{m+1}(u) T^{(a)}_{m-1}(u+1) +
                   T^{(a+1)}_m(u) T^{(a-1)}_m(u+1),\eqno(10)
$$
%
which is a ``Yang-Baxterization" of Eq.\ 8.
%
{}From this we find that the combination
%
$$y^{(a)}_m(u+{a+m \over 2}) =
{T^{(a)}_{m+1}(u)T^{(a)}_{m-1}(u+1) \over
T^{(a+1)}_m(u)T^{(a-1)}_m(u+1)} \eqno(11)
$$
solves essentially the following
$U_q(A^{(1)}_r)$ functional relation proposed in ref.${}^2$
%
$$
y^{(a)}_m(u+{1 \over 2})y^{(a)}_m(u-{1 \over 2})
= {(1 + y^{(a)}_{m+1}(u))(1 + y^{(a)}_{m-1}(u)) \over
   (1 + y^{(a+1)}_m(u)^{-1}) (1 + y^{(a-1)}_m(u)^{-1})}. \eqno(12)
$$
%
As noted in ref.${}^2$, the above equation is also satisfied by
$e^{\epsilon^{(a)}_m(u)}$ ($\epsilon^{(a)}_m(u)$: pseudo energy )
in the TBA at $\infty$ temperature.
%
In this way, one can roughly say that the TBA equation
has a solution in terms of
``Yang-Baxterized" Yangian characters.
%
It is interesting to note that Eq.\ 10 can be viewed as a simplest
example of the Pl\"ucker relation under Eq.\ 9.

\beginsection References

\item{1.}{A.Kuniba, ``Thermodynamics of the $U_q(X^{(1)}_r)$
Bethe ansatz system with $q$ a root of unity" , to appear in
Nucl.Phys.B.}
%
\item{2.}{A.Kuniba and T.Nakanishi,
``Spectra in conformal field theories from the Rogers dilogarithm",
ANU preprint (1992) SMS-042-92.}
%
\item{3.}{A.N.Kirillov and N.Yu.Reshetikhin,
J.Phys.{\bf A20} (1987) 1587.}
%
\item{4.}{A.N.Kirillov, Zap.Nauch.Semin.LOMI {\bf 164} (1987) 121.}
%
\item{5.}{A.N.Kirillov and N.Yu.Reshetikhin,
Zap.Nauch.Semin.LOMI {\bf 160} (1987) 211.}
%
\item{6.}{V.V.Bazhanov and N.Yu.Reshetikhin, J.Phys.{\bf A23} (1990)
1477.}
%
\item{7.}{M.J.Martins, Phys.Rev.Lett. {\bf 67} (1991) 419.}
%
\item{8.}{T.R.Klassen and E.Melzer, Nucl.Phys.{\bf B370} (1992) 511.}
%
\item{9.}{P.Fendley, Nucl.Phys. {\bf B374} (1992) 667.}
%
\item{10.}{A.Kl\"umper and P.A.Pearce, Physica {\bf A183} (1992) 304.}
%
\item{11.}{M.Jimbo, A.Kuniba, T.Miwa and M.Okado, Commun.Math.Phys.
{\bf 119} (1988) 543.}
%
\bye


