%Paper: hep-th/9206093
%From: GROVE@pierre.mit.edu (Heather Grove, 6-304A, 617-253-4852)
%Date: Wed, 24 Jun 1992 13:15:35 -0400 (EDT)


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\centerline{{\bf GAUGE THEORIES FOR GRAVITY ON A LINE}\footnote{*}{This
work is supported in part by funds
provided by the U. S. Department of Energy (D.O.E.) under contract
\#DE-AC02-76ER03069.}}
\vskip 24pt
\centerline{R. Jackiw}
\vskip 12pt
\centerline{\it Center for Theoretical Physics}
\centerline{\it Laboratory for Nuclear Science}
\centerline{\it and Department of Physics}
\centerline{\it Massachusetts Institute of Technology}
\centerline{\it Cambridge, Massachusetts\ \ 02139\ \ \ U.S.A.}
\vskip 1in
\centerline{Submitted to: {\it Theoretical and Mathematical Physics}}
\vskip 1in
\centerline{\it In Memoriam}
\medskip
\centerline{\bf M. C. Polivanov}
\vfill
%\noindent Recent Problems in Mathematical Physics, Salamanca, Spain,
%June 1992, XIX International Colloquium on Group Theoretical Methods in
%Physics, Salamanca, Spain, July, 1992.
%\bigskip
\centerline{ Typeset in $\TeX$ by Roger L. Gilson}
\vskip -12pt
\noindent CTP\#2105\hfill June 1992
\eject
\baselineskip 24pt plus 2pt minus 2pt
\noindent{\bf I.\quad INTRODUCTION}
\medskip
\nobreak
Professor M. C.
Polivanov and I met only a few times, during my infrequent visits to
the-then Soviet Union in the 1970's and 1980's.  His hospitality  at the
Moscow Steclov Institute made the trips a pleasure, while the scientific
environment that he provided made them professionally valuable.
But it is the human contact that I remember most vividly and shall now miss
after his death.  At a time when issues of conscience were both pressing for
attention and difficult/dangerous to confront, Professor Polivanov made a deep
impression with his quiet but adamant commitment to justice.  I can only
guess at the satisfaction he must have felt when his goal of gaining freedom
for Yuri Orlov was attained, and even more so these days when human rights
became defensible in his country; it is regrettable that he cannot now enjoy
the future that he strived to attain.

One of our joint interests was the Liouville theory,$^{1,\,2}$
which in turn can be
viewed as a model for gravity in two-dimensional space-time. Some recent
developments in this field are here summarized and dedicated to Polivanov's
memory, with the hope that he would have enjoyed knowing about them.

We study lower-dimensional gravity both for pedagogical reasons --- one
expects that the dimensional reduction effects sufficient simplification to
permit thorough analysis, while still retaining useful content to inform the
physical $(3+1)$-dimensional problem --- and also, if one is lucky, there are
practical applications --- {\it e.g.\/}
idealized cosmic strings are described by
$(2+1)$-dimensional gravity, while the still lower-dimensional models are used
in statistical mechanics.

The drastic dimensional reduction to $(1+1)$ dimensions --- gravity on a line,
{\it i.e.\/}, {\it lineal\/} gravity --- is not devoid of interest, provided
dynamical equations are not based on the Einstein tensor $G_{\mu\nu} =
R_{\mu\nu} - {1\over 2} g_{\mu\nu} R$, which vanishes identically in two
dimensions.

In a proposal of several years ago,$^3$
it was suggested that gravity equations be
based on the Riemann scalar $R$, the simplest entity that encodes
in two dimensions all local geometric information about space-time.
Moreover, in an action formulation it is necessary to introduce an additional
scalar field, which acts as a Lagrange multiplier that enforces the equation of
motion for $R$.  Thus we are dealing with scalar-tensor theories, or --- to
use the contemporary string nomenclature --- ``dilaton'' gravities.

Since the initial proposal, various models have been studied.  Here I shall
describe two that are selected by their group theoretical properties:
they can be formulated as gauge theories based on groups relevant to
space-time: de~Sitter or anti-de~Sitter (in $(1+1)$-dimensions
both groups are $SO(2,1)$, although the geometries are different)
and Poincar\'e.  The first of these is
the one proposed originally;$^3$  it is governed by the action
$$I_1 = \int d^2x\,\sqrt{-g}\, \eta (R-\Lambda)\eqno(1)$$
The second is ``string-inspired'' and has been recently studied for purposes
of modeling (on a line!) black hole physics;$^4$ its action is
$$\bar{I}_2 = \int d^2x\,\sqrt{-\bar{g}}\, e^{-2\varphi}\left( \bar{R} - 4
\bar{g}^{\mu\nu} \partial_\mu\varphi\partial_\nu \varphi - \Lambda\right)
\eqno(\overline{2})$$
(Notation: time and space carry the metric tensor
$g_{\mu\nu}$ with signature $(1,-1)$.  The two-vector $x^\mu = (t,x)$ will be
frequently presented in light-cone components $x^\pm \equiv {1\over
\sqrt{2}}{(t\pm x)}$.  Tangent space components are labeled by Latin letters
$a,b,\ldots$, and the Minkowski metric tensor $h_{ab}=\diag (1,-1)$
raises/lowers these indices.  Also we use the anti-symmetric tensor
$\epsilon^{ab}$, $\epsilon^{01} = 1$.)

In (1), $R$ is the scalar curvature built from $g_{\mu\nu}$, $\eta$ is a
world scalar Lagrange multiplier related to the dilaton, while $\Lambda$ is a
cosmological constant.  In ($\overline{2}$)
we temporarily use an over-bar to denote a
differently scaled metric tensor $\bar{g}_{\mu\nu}$ from which $\bar{R}$ is
constructed, while $\varphi$ is the dilaton.  Formula ($\overline{2}$)
arises naturally
from string theory, restricted to a two-dimensional target space, with the
anti-symmetric tensor field identically vanishing.
In the string context, matter is
taken to couple to $\bar{g}_{\mu\nu}$; for our purposes in the absence of
matter it is convenient to redefine variables by $\bar{g}_{\mu\nu}=
e^{2\varphi} g_{\mu\nu}$, $\eta = e^{-2\varphi}$.  Then ($\overline{2}$)
becomes
$$I_2 = \int d^2x\, \sqrt{-g}\, \left( \eta R - \Lambda\right) \eqno(2)$$
but it is to be remembered that because of the redefinition, the ``physical''
metric tensor is $g_{\mu\nu}/(-2\eta)$.  Note that (2) is invariant against
shifting $\eta$ by a constant, because $\sqrt{-g}\,R$ is a total derivative.

It is seen that the two models (1) and (2) differ in the placement of the
Lagrange multiplier with the cosmological term:  in (1) $\eta$ multiplies
$\Lambda$, in (2) the $\eta$ factor is absent from $\Lambda$.
Of course in the limit $\Lambda=0$, the difference disappears.

We now describe the interesting gauge group structure of (1) and (2) which we
name {\it (anti)~de~Sitter gravity\/} and {\it extended Poincar\'e gravity\/},
respectively.
\goodbreak
\bigskip
\noindent{\bf II.\quad (ANTI)~DE~SITTER GRAVITY}
\medskip
\nobreak
The equations of motion that follow from varying $\eta$ and $g_{\mu\nu}$ in (1)
are
$$R=\Lambda\eqno(3)$$
$$\left( {\cal D}_\mu {\cal D}_\nu - g_{\mu\nu} {\cal D}^2\right) \eta +
{\Lambda\over 2} g_{\mu\nu}\eta = 0 \eqno(\hbox{4a})$$
The second equation, with ${\cal D}_\mu$ the space-time covariant derivative,
can be decomposed into traceless and trace parts.
$$\eqalignno{\left( {\cal D}_\mu {\cal D}_\nu - {1\over 2} g_{\mu\nu} {\cal
D}^2\right) \eta &= 0 & (\hbox{4b}) \cr
\left( {\cal D}^2 - \Lambda\right) \eta &= 0 &(\hbox{4c}) \cr}$$

The above geometric dynamics may be presented in a gauge theoretical
fashion.$^5$  To this end one
uses the (anti) de~Sitter group with Lorentz generator $J$ and
translation generators $P_a$ satisfying the $SO(2,1)$ algebra (for
$\Lambda\not=0$).
$$\left[ P_a,J\right] = e_a{}^b P_b\ \ ,\qquad \left[ P_a, P_b\right] =
-{\Lambda\over 2} \epsilon_{ab} J \eqno(5)$$
The gauge connection one-form is introduced $A=A_\mu\,dx^\mu$ and expanded
in terms of the generators,
$$A = e^a P_a+\omega J\eqno(6)$$
where $e^a_\mu$ is the {\it Zweibein\/} and $\omega_\mu$ is the
spin-connection.  The curvature two-form
$$F = dA + A^2 \eqno(7)$$
becomes
$$F = f^a P_a + fJ = \left( De\right)^a P_a + \left( d\omega- {\Lambda\over 4}
e^a \epsilon_{ab} e^b\right) J \eqno(8)$$
$$\left( De\right)^a \equiv de^a + \epsilon^a{}_b \omega e^b\eqno(9)$$
It is seen that $d\omega$ is the scalar curvature density and $f^a$ is the
torsion density, each expressed in terms of $e^a$ and $\omega$, which at this
stage are independent variables.

The Lagrange density
$$\eqalign{{\cal L}'_1 &= \sum\limits^2_{A=0} \eta_A F^A
= \eta_a \left( De\right)^a +
\eta_2 \left( d\omega - {\Lambda\over 4}e^a \epsilon_{ab} e^b\right) \cr
F^A &= \left( f^a, f\right)\ \ ,\qquad \eta_A = \left( \eta_a,
\eta_2\right)}\eqno(10)$$
is gauge invariant: the three field strengths $F^A$ transform covariantly
according to the three-dimensional adjoint representation, while the
Lagrangian multiplet triplet $\eta_A$ transforms by the coadjoint
representation.

The equation obtained from (10) by varying $\eta_a$ gives the
condition of vanishing torsion, and allows evaluating the spin connection in
terms of the {\it Zweibein\/}.
$$\omega = e^a \left(h_{ab} \epsilon^{\mu\nu} \partial_\mu e^b_\nu\right)\big/
\det e\eqno(11)$$
The equation which follows upon variation of $\eta_2$ regains (3) once (11) is
used.  Variation
of $e^a$ and $\omega$ produces equations for the Lagrange multipliers
$\eta_a$ and $\eta_2$, respectively, the latter of course coinciding with
$\eta$ in the geometric formulations (1), (3) and (4).
$$\eqalignno{d\eta_a + \epsilon_a{}^b \omega \eta_b - {\Lambda\over 2}
\epsilon_{ab} \eta_2 e^b &= 0 &(\hbox{12a}) \cr
d\eta_2 + \eta_a \epsilon^a{}_b e^b &= 0 &(\hbox{12b})\cr}$$
Upon taking a space-time
covariant derivative of (12b) and using (12a) to eliminate
$\eta_a$, we recover (4).  Finally we see that when $\omega$ is eliminated from
${\cal L}'_1$ with the help of (11), so that the torsion (9) vanishes,
what remains is the Lagrange density of
(1), expressed in terms of {\it Zweibeine\/}.

Thus the geometric formulation of this gravity theory is contained within the
(anti)~de~Sitter
group theoretical framework for solutions with $\det e\not=0$, but
see below.

Explicit classical solutions to the
equations are easy to find.  Working within the
geometric framework, we use coordinate invariance to choose a conformally
flat metric tensor.
$$g_{\mu\nu} = h_{\mu\nu} \exp 2\sigma \eqno(13)$$
Then (3) becomes the Liouville equation,
$$\square \sigma = {\Lambda\over 2} \exp 2\sigma \eqno(14)$$
studied by Polivanov.$^1$
Its general solution depends on two arbitrary functions
of the two light-cone variables, $F(x^+)$, $G(x^-)$,
$$\exp 2\sigma = {F'(x^+) G'(x^-)\over \left( 1 - {\displaystyle{\Lambda\over
4}} FG\right)^2 } \eqno(15)$$
whose derivatives fulfill the consistency condition $F'G'>0$.  But the residual
coordinate invariance within the conformal gauge allows choosing $F(x^+) =
x^+, G(x^-) = x^-$, hence
$$\exp 2\sigma = {1\over \left( 1 - {\displaystyle{\Lambda\over 8}}
x^2\right)^2} \eqno(16)$$
In conformal gauge, (4b) reduces to
$$\partial_\mu V_\nu + \partial_\nu V_\mu - h_{\mu\nu} h^{\alpha\beta}
\partial_\alpha V_\beta =0\eqno(17)$$
where $V_\mu$ is defined by
$$V_\mu \exp 2\sigma=\partial_\mu\eta \eqno(18)$$
Equation (17) is just the (flat-space) conformal Killing equation with
solutions in terms of arbitrary functions of a single light-cone variable.
$$V_- = V_-(x^+)\ \ ,\qquad V_+ = V_+(x^-)\eqno(19)$$
Finally the remaining equation (4c) together with (18) restricts these
functions, so that the solution for $\eta$ takes the form
$$\eta = {\alpha_a x^a + \alpha_2 \left( 1 +
{\displaystyle{\Lambda\over 8}}x^2\right) \over 1 -
{\displaystyle{\Lambda\over 8}} x^2} \eqno(20)$$
where $\alpha_a$ is a constant two-vector and $\alpha_2$ is a constant scalar.

The {\it Zweibein\/} and spin connection of the gauge theoretical formulation
are given by related formulas.  The former, the ``square root'' of the
metric tensor, becomes (apart from an arbitrary Lorentz transformation on the
tangent-space indices)
$$e^a_\mu =  \delta^a_\mu\exp\sigma = {1\over 1 -
{\displaystyle{\Lambda\over 8}} x^2} \delta^a_\mu \eqno(21)$$
while the latter is
$$\omega_\mu = - h_{\mu\alpha}\epsilon^{\alpha\beta} \partial_\beta
\sigma\eqno(22)$$
The Lagrange multiplier $\eta_2$
coincides with $\eta$, while Eq.~(12) for $\eta_a$ is solved by
$$\eta_a \exp \sigma = \epsilon_a{}^\mu \partial_\mu \eta \eqno(23)$$
Of course the general solution is an arbitrary coordinate transformation of
the above.

Finally we observe that the gauge theoretical formulation allows an
alternative group theoretical
presentation of solutions.  The field equations following from
(10), upon respective variation of $\eta_A$ and $A$, are
$$F=0\eqno(24)$$
$$dH + [A,H] = 0 \eqno(25)$$
$A$, $F$ and $H= \eta_a h^{ab} P_b +{2\over\Lambda} \eta_2 J$
belong to the $SO(2,1)$ algebra (the factor $2/\Lambda$
is a consequence of the group
metric).  Equation (24) implies that $A$ is a pure gauge given
by an arbitrary element $U$ of the $SO(2,1)$ group,
$$A = U^{-1}dU \eqno(26)$$
while the Lagrange multiplier is then determined by (25) to be
$$H = U^{-1} \Phi\,U\eqno(27)$$
where $\Phi$ is a constant element in the algebra.  The explicit group and
algebra elements that correspond to the above solution, Eqs.~(20) -- (23), are
$$\eqalignno{U &= \exp \left( {i\pi\over \sqrt{ 1 -
{\displaystyle{\Lambda\over 8}} x^2}}\left( - {1\over 2}x^a\epsilon_a{}^b  P_b
+ J\right)\right) &(28) \cr\noalign{\hbox{and}}
\Phi &={2\over\Lambda} \alpha_a \epsilon^{ab} P_b - \alpha_2 J   &(29) \cr}$$
$U$ is unique up to a constant gauge transformation.

Within the gauge theoretical framework, an even simpler solution to (24) and
(25) is available: $A=0$, $H=\Phi$, which makes no sense geometrically: not
only $\det e$, but both the connections $e^a$ and $\omega$ vanish!  But in
fact use can be made of such solutions:
when presented with a geometrically singular configuration,
perform any gauge transformation producing non-singular connections,
for example with the group element $U$ above. So we see that the group
theoretical framework, even in its $\det e=0$ sector, contains adequate
information for encoding the gravity theory.
\goodbreak
\bigskip
\noindent{\bf III.\quad EXTENDED POINCAR\'E GRAVITY}
\medskip
\nobreak
Equations of motion of the string-inspired gravitational theory (2) are,
from varying~$\eta$
$$R = 0 \eqno(30)$$
and from varying $g_{\mu\nu}$
$$\left( {\cal D}_\mu {\cal D}_\nu - g_{\mu\nu} {\cal D}^2\right) \eta +
{\Lambda\over 2}  g_{\mu\nu} = 0 \eqno(\hbox{31a})$$
which is equivalent to
$${\cal D}_\mu {\cal D}_\nu \eta = {\Lambda\over 2} g_{\mu\nu}
\eqno(\hbox{31b})$$
Note that (31a) differs from (4a) by the absence of $\eta$ in the last term.

To give a gauge theoretical formulation,$^6$ we make use of the {\it centrally
extended\/} Poincar\'e group, whose algebra is
$$\left[ P_a, J\right] = \epsilon_a{}^bP_b\ \ ,\qquad \left[ P_a, P_b\right] =
\epsilon_{ab} I \eqno(32)$$
where the central element $I$ commutes with $P_a$ and $J$.  Consequently the
connection $A$ and curvature $F$ now become
$$\eqalignno{A &= e^a P_a + \omega J + a I &(33) \cr\noalign{\vskip 0.2cm}
F &= dA + A^2 = f^a P_a+ f J+ g I \cr
&=\left( De\right)^a P_a + d\omega J + \left( da + {1\over 2}e^a \epsilon_{ab}
e^b\right) I &(34) \cr}$$
Here $a$ and $g$ are the additional connection and curvature associated with
the central element in the algebra.

This magnetic-like extension of the Poincar\'e group may be viewed as an
unconventional contraction of the de~Sitter group: The ordinary Poincar\'e
algebra (Eq.~(32) without the central element) is the $\Lambda \to 0$
contraction of the $SO(2,1)$ algebra (5).  However, owing to the well-known
ambiguity of two-dimensional angular momentum, in (5) one may replace $J$ by
$J - 2I/\Lambda$ before taking the $\Lambda\to 0$ limit, which then leaves
(32).

The extension reflects a 2-cocycle in the composition law for representatives
of
the Poincar\'e group.  If the group acts on coordinates $x^a$ by
$$x^a\longrightarrow \bar{x}^a = {\cal M}^a{}_b x^b + q^a \eqno\hbox{(35a)}$$
where ${\cal M}$ is a finite Lorentz transformation
$${\cal M}^a{}_b = \delta^a{}_b \cosh \alpha + \epsilon^a{}_b \sinh \alpha
\eqno(\hbox{35b})$$
and $q^a$ is a finite translation, the composition law for these is
$$\eqalignno{{\cal M}_{(12)} &= {\cal M}_{1} {\cal M}_{2} &(\hbox{36a})
\cr
q_{(12)} &= q_{1} + {\cal M}_1 q_{2} &(\hbox{36b})\cr}$$
However, the composition law for a representation $G({\cal M},q)$ containing
the extension (32) in its algebra acquires a 2-cocycle.
$$G\left( {\cal M}_1, q_1\right) G\left( {\cal M}_2, q_2\right) =\exp\left\{
{i\over 2} q^a_1 \epsilon_{ab} \left( {\cal M}_1 q_2\right)^b \right\} G
\left( {\cal M}_1 {\cal M}_2, q_1 + {\cal M}_1 q_2\right) \eqno(37)$$
($I$ is represented by $i=\sqrt{-1}$.)

A finite gauge transformation, generated by the gauge function $\Theta$,
$$\Theta = \theta^a P_a + \alpha J + \beta I \eqno(38)$$
produces the following transformations on the connections.
$$\eqalign{e^a \to \bar{e}^a &= \left( {\cal M}^{-1}\right)^a_{\ b} \left( e^b
+ \epsilon^b{}_c \theta^c \omega + d\theta^b\right) \cr
\omega \to \bar{\omega} &= \omega + d\alpha \cr
a\to \bar{a} &= a - \theta^a \epsilon_{ab} e^b - {1\over 2} \theta^2\omega +
d\beta + {1\over 2} d\theta^A \epsilon_{ab} \theta^b \cr}\eqno(39)$$
The multiplet of curvatures $F^A = \left( f^a, f,g\right)$ transforms by the
adjoint $4\times4$ representation of the extended group,
$$\eqalign{f^a \to \bar{f}^a
&= \left( {\cal M}^{-1}\right)^a_{\ b} \left( f^b +
\epsilon^b{}_c \theta^c f\right) \cr
f \to \bar{f} &= f \cr
g\to \bar{g} &= g -\theta^a \epsilon_{ab} f^b - {1\over 2} \theta^2 f
\cr}\eqno(40)$$
or
$$\eqalign{ F^A \to \bar{F}^A &= \sum\limits^3_{B=0} \left(
U^{-1}\right)^A_{\ B} F^B\cr
U&= \left( \matrix{
{\cal M}^a{}_b & - \epsilon^a{}_c \theta^c & 0 \cr
0 & 1 & 0 \cr
\theta^c \epsilon_{cd} {\cal M}^d{}_b & - \theta^2/2 & 1 \cr}\right)
\cr}\eqno(41)$$
The upper left $3\times3$ block in $U$ comprises the adjoint representation
of the conventional Poincar\'e group with $q^a$ of (35) identified with
$-\epsilon^a{}_c\theta^c$, while the fourth row and column arise from the
extension.  Note that in the above realization of the gauge action on $F$, the
extension is not visible: $I$ is represented by ${\bf O}$.
 On the other hand, an
additional connection and curvature ($a,g$) are present.

In this representation, the extended algebra possesses a non-singular Killing
metric, which is unavailable without the extension.
$$h_{AB} = \left( \matrix{ h_{ab} & \phantom{-}0 & \phantom{-}0 \cr
0 & \phantom{-}0 & -1 \cr
0 & -1 & \phantom{-}0 \cr}\right) \eqno(42)$$
It is true that ${}^TUh U = h$; this allows raising and lowering the indices
$(A,B)$.

An invariant Lagrange density is now constructed with an extended multiplet of
Lagrange multipliers $\eta_A$,
$$\eqalign{
{\cal L}'_2 &= \sum\limits^3_{A=0} \eta_A F^A = \eta_a \left( De\right)^a +
\eta_2 d\omega + \eta_3 \left( da + {1\over 2} e^a \epsilon_{ab} e^c\right)\cr
F^A &= \left( f^a, f,g\right)\ \ ,\qquad \eta_a = \left( \eta_a,
\eta_2,\eta_3\right) \cr}\eqno(43)$$
which obey the coadjoint transformation law,
$$\eta_A \to \bar{\eta}_A = \sum^3_{B=0} \eta_B U^B{}_A
\eqno(44)$$
or in components
$$\eqalign{\eta_a \to \bar{\eta}_a &= \left( \eta_b - \eta_3 \epsilon_{bc}
\theta^c\right) {\cal M}^b{}_a \cr
\eta_2 \to \bar{\eta}_2 &= \eta_2 - \eta_a \epsilon^a{}_b \theta^b - {1\over
2}\eta_3 \theta^2 \cr
\eta_3\to\bar{\eta}_3 &= \eta_3 \cr}\eqno(45)$$

Using the invariant metric (42), other group invariants may be constructed.
$$\eqalignno{{\cal F}^2 &= \sum^3_{A,B=0} {}^*F^A h_{AB} F^B &(46) \cr
M &=-{2\over\Lambda} \sum^3_{A,B=0} \eta_A h^{AB} \eta_B &(47) \cr}$$
where ${}^*F^A$ is the 0-form ${1\over 2} \epsilon^{\mu\nu} F^A_{\mu\nu}$, dual
to the 2-form $F^A$.

We recognize in (43) the torsion $\left( De\right)^a$ and curvature $d\omega$
densities, which vanish as a consequence of varying $\eta_a$ and $\eta_2$,
respectively.  Thus Eq.~(30) is regained.  The Lagrange multiplier $\eta$ in
(2) corresponds to $\eta_2$ in the present formulas
and the equation for it, obtained by varying
$\omega$, is as in the (anti)~de~Sitter model, (12b),
$$d\eta_2 + \eta_a \epsilon^a{}_b e^b = 0 \eqno(\hbox{48a})$$
while the equation for $\eta_a$, obtained by varying $e^a$, differs from
(12a),
$$d\eta_a + \epsilon_a{}^b \omega \eta_b + \eta_3\epsilon_{ab} e^b=0
\eqno(\hbox{48b})$$
We need a value for $\eta_3$ to close the system (48).  The equation for that
multiplier is obtained by varying $a$,
$$d\eta_3 = 0 \eqno(\hbox{48c})$$
and a constant, cosmological solution
$$\eta_3 = - {\Lambda\over 2}\eqno(\hbox{48d})$$
renders (48b) similar to (12a),
$$d\eta_a + \epsilon_a{}^b \omega \eta_b - {\Lambda\over 2}\epsilon_{ab}
e^b = 0 \eqno(\hbox{48e})$$
except that there is no factor of $\eta_2$ in the last, cosmological term of
(48e).  This of course has the consequence that when (48a) and (48e) are
combined as before, the second order equation that emerges for $\eta = \eta_2$
reproduces (31).

The remaining equation of the gauge theoretical formulation, obtained by
varying $\eta_3$
$$da = - {1\over 2} e^a \epsilon_{ab} e^b\eqno(49)$$
and allowing evaluation of $a$, has no counterpart in the geometric
formulation.
Equation (49) can always be locally integrated because the right side is
a two-form, hence closed in two dimensions.  However in general, there will be
singularities in $a$, since upon integrating (49) over a two-space, the
right side gives the total ``volume,'' which could be a well-defined
non-vanishing quantity, while the left side always integrates to zero if
the manifold is closed and bounded, and $a$ is non-singular.

Note that upon eliminating $\omega$ in ${\cal L}'_2$ with the zero-torsion
equation $\left( De\right)^a=0$ and evaluating $\eta_3$ at $-\Lambda/2$, ${\cal
L}'_2$ coincides with the Lagrange density in (2), now expressed in terms of
{\it Zweibeine\/}, apart from the total derivative $-\Lambda/2\,da$, which
does not contribute to equations of motion.

Thus here again, the group theoretical formulation reproduces the geometric
one, for solutions with $\det e\not=0$, but again see below.
However, the former is more flexible:
Eq.~(48c) is satisfied with vanishing $\eta_3$; this corresponds to a
vanishing cosmological constant.  Thus the gauge theory built on the {\it
extended\/} Poincar\'e group possesses as a solution a {\it non-extended\/}
system.  It is interesting therefore that here the cosmological term is an
integration constant, and not inserted {\it a priori\/} into the theory.

Finding explicit
solutions is straightforward.  In the geometric formulation, (3) is
solved by a flat metric tensor.
$$g_{\mu\nu} = h_{\mu\nu}\eqno(50)$$
Then (31) immediately gives
$$- 2\eta= M- {\Lambda\over 2}\left( x - x_0\right)^2  \eqno(51)$$
with $M$ and $x_0$ being integration constants,
the former reflecting the $\eta$-translation invariance mentioned earlier.

Interest in the model$^4$
derives precisely from the above ``black-hole'' solution with mass $M$ [in
terms of the ``physical'' metric $g_{\mu\nu}/(-2\eta)$], located at $x_0$.
An arbitrary coordinate transformation of this configuration produces the
general solution.

The gauge theoretical counterparts of the above are a flat {\it Zweibein\/}
(apart
from a constant tangent-space Lorentz transformation)
$$e^a_\mu = \delta^a_\mu\eqno(52)$$
and a vanishing spin connection.
$$\omega = 0 \eqno(53)$$
Taking in (48c) the cosmological solution for $\eta_3$, allows solving (48e)
for $\eta_a$
$$\eta_a = {\Lambda\over 2}\epsilon_{a\mu} \left( x^\mu - x^\mu_0\right)
\eqno(54)$$
and from (48a) $\eta_2=\eta$ is recovered to be as in (51).  Finally (49) is
solved for $a$.
$$a_\mu = {1\over 2} \epsilon_{\mu\nu}  x^\nu \eqno(55)$$
with a pure gauge contribution $\partial_\mu \chi$ left arbitrary.
The potential in (55)
corresponds to a constant ``magnetic field,'' as is appropriate with our
``magnetic-like'' extension of translations.

Note the two invariants defined in (46) and (47): ${\cal F}^2$ vanishes since
$F^A$ does, while $M$ is recognized as the ``black hole'' mass.

The gauge theoretical solution may of course also be presented in a group
theoretical fashion, since the equations are of the same form as in (24) and
(25), with all quantities belonging to the {\it extended\/} algebra and group.
The explicit formulas,
corresponding to the ``black hole'' solution, Eqs.~(50) -- (55),
are as follows.  The group element $U$
that leads to the pure gauge connection $A=U^{-1} dU$ is
$$U = \exp x^aP_a\eqno(56)$$
up to a constant gauge transformation.
The constant algebra element $\Phi$ that gives
$H=\eta_a h^{ab} P_b - \eta_3 J - \eta_2 I = U^{-1} \Phi U$
is (placement of $\eta_2$ and $\eta_3$ dictated by the group metric
(42), {\it viz.\/} $\eta^A = h^{AB} \eta_B)$
$$\Phi = {\Lambda\over 2} x^a_0 \epsilon_a{}^b P_b
 + {\Lambda\over 2} J + \left( {M\over 2} - {\Lambda\over 4}
x_0^2 \right) I \eqno(57)$$

As in the (anti)~de~Sitter model, we see that after a further
gauge transformation we pass
to the geometrically singular configuration $A=0$, $H=\Phi$.  This gives an
especially succinct account of the relevant geometric information : $\Phi$
encodes the integration constants, which characterize the intrinsic geometry:
the cosmological constant $\Lambda$, the ``black hole'' mass $M$ and
location $x_0$. A geometry is
built with these characteristics once a gauge transformation is performed, say
with the above $U$, to obtain non-singular connections.
\goodbreak
\bigskip
\noindent{\bf IV.\quad CONCLUSION}
\medskip
\nobreak
The two models here considered are special: their geometric dynamics possess a
gauge theoretical formulation.  The extended Poincar\'e model exhibits
the intriguing possibility of a cosmological term that is an integration
constant, as are the ``black hole'' mass $M$ and location $x_0$; all three
are encoded in the Lagrange multipliers of the theory.

Both models can also be obtained by dimensional reduction from
$(2+1)$ dimensions:  To obtain (anti)~de~Sitter
gravity in its geometric formulation
one begins$^3$ with the Einstein theory/Hilbert action (with cosmological
term),
suppresses dependence on the third dimension, sets $g_{\mu 2}$ to zero for
$\mu=0,1$ and $g_{22}$ to $\eta^2$; for the gauge theoretical formulation
one starts with the {\it Dreibein\/}-spin connection form of the theory, which
also is equivalent to a Chern--Simons, ${\cal O}(2,2)$ or ${\cal O}(3,1)$
model.$^7$  Extended Poincar\'e gravity can be similarly constructed, but the
higher-dimensional theory has to be suitably extended by an Abelian ideal.
Indeed it is found that {\it both\/} the (anti)~de~Sitter
and extended Poincar\'e
$(1+1)$ dimensional theories arise as {\it different\/} dimensional
reductions of a {\it single\/},
extended $(2+1)$-dimensional gravity.$^8$  This and another interesting topic
--- the coupling of matter consistently with the gauge principle$^9$ --- are
beyond the scope of our review.  In yet a further investigation one could
study non-topological theories in which invariants (46) and/or (47) are added
to the Lagrange density (43).

In conclusion, we note that dynamics determined by a group has been familiar
in physics since the invention of Yang--Mills theory.  However, the examples
described here offer a new possibility: in the Lie algebra that determines a
gauge theory one can allow an extension.  This gives rise to richer
dynamics within the same group theoretical structure, and in the gravity model
studied above produces the cosmological constant.
\goodbreak
\bigskip
\centerline{\bf ACKNOWLEDGEMENTS}
\medskip
\nobreak
The review was prepared with the assistance of D.~Cangemi, particularly in
finding the explicit solutions; this I gratefully acknowledge.
\vfill
\eject
\centerline{\bf REFERENCES}
\medskip
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of the Cauchy Problem for the Liouville Equation $\varphi_{tt}(t,x) -
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\medskip
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\medskip
\item{3.}C. Teitelboim, ``Gravitation and Hamiltonian Structure in Two
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\medskip
\item{4.}H. Verlinde, ``Black Holes and Strings in Two Dimensions,'' in {\it
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\medskip
\item{5.}T. Fukuyama and K. Kamimura, ``Gauge Theory of Two-Dimensional
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discussed in these papers.
\medskip
\item{6.}D. Cangemi and R. Jackiw, ``Gauge Invariant Formulations of Lineal
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\medskip
\item{7.}A. Ach\'ucarro and P. Townsend, ``A Chern--Simons Action for
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\medskip
\item{8.}D. Cangemi, in preparation.
\medskip
\item{9.}G. Grignani and G. Nardelli, in preparation.
\par
\vfill
\end

