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%  *******************  Journal refs **********************

\def\aop#1#2#3{{\it Ann. Phys.} {\bf {#1}} (19{#2}) #3}
\def\cjp#1#2#3{{\it Can. J. Phys.} {\bf {#1}} (19{#2}) #3}
\def\cmp#1#2#3{{\it Comm. Math. Phys.} {\bf {#1}} (19{#2}) #3}
\def\cqg#1#2#3{{\it Class. Quant. Grav.} {\bf {#1}} (19{#2}) #3}
\def\jcp#1#2#3{{\it J. Chem. Phys.} {\bf {#1}} (19{#2}) #3}
\def\ijmp#1#2#3{{\it Int. J. Mod. Phys.} {\bf {#1}} (19{#2}) #3}
\def\jmp#1#2#3{{\it J. Math. Phys.} {\bf {#1}} (19{#2}) #3}
\def\jpa#1#2#3{{\it J. Phys.} {\bf A{#1}} (19{#2}) #3}
\def\mplA#1#2#3{{\it Mod. Phys. Lett.} {\bf A{#1}} (19{#2}) #3}
\def\np#1#2#3{{\it Nucl. Phys.} {\bf B{#1}} (19{#2}) #3}
\def\pl#1#2#3{{\it Phys. Lett.} {\bf {#1}} (19{#2}) #3}
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\def\prp#1#2#3{{\it Phys. Rep.} {\bf {#1}} (19{#2}) #3}
\def\pr#1#2#3{{\it Phys. Rev.} {\bf {#1}} (19{#2}) #3}
\def\prA#1#2#3{{\it Phys. Rev.} {\bf A{#1}} (19{#2}) #3}
\def\prB#1#2#3{{\it Phys. Rev.} {\bf B{#1}} (19{#2}) #3}
\def\prD#1#2#3{{\it Phys. Rev.} {\bf D{#1}} (19{#2}) #3}
\def\prl#1#2#3{{\it Phys. Rev. Lett.} {\bf #1} (19{#2}) #3}
\def\ptp#1#2#3{{\it Prog. Theor. Phys.} {\bf #1} (19{#2}) #3}
\def\tmp#1#2#3{{\it Theor. Mat. Phys.} {\bf #1} (19{#2}) #3}
\def\rmp#1#2#3{{\it Rev. Mod. Phys.} {\bf {#1}} (19{#2}) #3}
\def\zfn#1#2#3{{\it Z. f. Naturf.} {\bf {#1}} (19{#2}) #3}
\def\zfp#1#2#3{{\it Z. f. Phys.} {\bf {#1}} (19{#2}) #3}

\def\asens#1#2#3{{\it Ann. Sci. \'Ecole Norm. Sup. (Paris)} {\bf{#1}} 
(#2) #3} 
\def\aihp#1#2#3{{\it Ann. Inst. H. Poincar\'e (Paris)} {\bf{#1}} (#2) #3} 
\def\cras#1#2#3{{\it Comptes Rend. Acad. Sci. (Paris)} {\bf{#1}} (#2) #3} 
\def\prs#1#2#3{{\it Proc. Roy. Soc.} {\bf A{#1}} (19{#2}) #3}
\def\pcps#1#2#3{{\it Proc. Camb. Phil. Soc.} {\bf{#1}} (19{#2}) #3}
\def\mpcps#1#2#3{{\it Math. Proc. Camb. Phil. Soc.} {\bf{#1}} (19{#2}) #3}

\def\amsh#1#2#3{{\it Abh. Math. Sem. Ham.} {\bf {#1}} (19{#2}) #3}
\def\am#1#2#3{{\it Acta Mathematica} {\bf {#1}} (19{#2}) #3}
\def\aim#1#2#3{{\it Adv. in Math.} {\bf {#1}} (19{#2}) #3}
\def\ajm#1#2#3{{\it Am. J. Math.} {\bf {#1}} ({#2}) #3}
\def\amm#1#2#3{{\it Am. Math. Mon.} {\bf {#1}} (19{#2}) #3}
\def\adm#1#2#3{{\it Ann. der  Math.} {\bf {#1}} ({#2}) #3}
\def\aom#1#2#3{{\it Ann. of Math.} {\bf {#1}} (19{#2}) #3}
\def\cjm#1#2#3{{\it Can. J. Math.} {\bf {#1}} (19{#2}) #3}
\def\cpde#1#2#3{{\it Comm. Partial Diff. Equns.} {\bf {#1}} (19{#2}) #3}
\def\cm#1#2#3{{\it Compos. Math.} {\bf {#1}} (19{#2}) #3}
\def\dmj#1#2#3{{\it Duke Math. J.} {\bf {#1}} (19{#2}) #3}
\def\invm#1#2#3{{\it Invent. Math.} {\bf {#1}} (19{#2}) #3}
\def\ijpam#1#2#3{{\it Ind. J. Pure and Appl. Math.} {\bf {#1}} (19{#2}) #3}
\def\jdg#1#2#3{{\it J. Diff. Geom.} {\bf {#1}} (19{#2}) #3}
\def\jfa#1#2#3{{\it J. Func. Anal.} {\bf {#1}} (19{#2}) #3}
\def\jlms#1#2#3{{\it J. Lond. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\jmpa#1#2#3{{\it J. Math. Pures. Appl.} {\bf {#1}} ({#2}) #3}
\def\ma#1#2#3{{\it Math. Ann.} {\bf {#1}} ({#2}) #3}
\def\mz#1#2#3{{\it Math. Zeit.} {\bf {#1}} ({#2}) #3}
\def\ojm#1#2#3{{\it Osaka J.Math.} {\bf {#1}} ({#2}) #3}
\def\pams#1#2#3{{\it Proc. Am. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\pems#1#2#3{{\it Proc. Edin. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\pja#1#2#3{{\it Proc. Jap. Acad.} {\bf {A#1}} (19{#2}) #3}
\def\plms#1#2#3{{\it Proc. Lond. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\pgma#1#2#3{{\it Proc. Glasgow Math. Ass.} {\bf {#1}} (19{#2}) #3}
\def\qjm#1#2#3{{\it Quart. J. Math.} {\bf {#1}} (19{#2}) #3}
\def\qjpam#1#2#3{{\it Quart. J. Pure and Appl. Math.} {\bf {#1}} ({#2}) #3}
\def\rcmp#1#2#3{{\it Rend. Circ. Mat. Palermo} {\bf {#1}} (19{#2}) #3}
\def\rms#1#2#3{{\it Russ. Math. Surveys} {\bf {#1}} (19{#2}) #3}
\def\top#1#2#3{{\it Topology} {\bf {#1}} (19{#2}) #3}
\def\tams#1#2#3{{\it Trans. Am. Math. Soc.} {\bf {#1}} (19{#2}) #3}
\def\zfm#1#2#3{{\it Z.f.Math.} {\bf {#1}} ({#2}) #3}
% *******************   Main text *********************
%\begin{ignore}
\vglue 1.5truein
\begin{title}  
%\vglue 20truept
\righttext {MUTP/98/2}
\righttext{hep-th/9802029}
%\leftline{\today}
%\vskip 30truept
\centertext {\Bigfonts \bf Conformal anomaly in 2d}
\centertext{\Bigfonts \bf dilaton--scalar theory}
\vskip 20truept 
\centertext{J.S.Dowker\footnote{dowker@a3.ph.man.ac.uk}}
\vskip 7truept
\centertext{\it Department of Theoretical Physics,\\
The University of Manchester, Manchester, England}
\vskip10truept
\vskip 20truept
\centertext {Abstract}
\vskip10truept
\begin{narrow}
The discrepancy between the anomaly found by Bousso and Hawking and that of
other workers is explained by the omission of a zero mode contribution to
the effective action.
\end{narrow}
\vskip 5truept
\righttext {February 1998}
\vskip 60truept
%\righttext{Typeset in \jyTeX}
\vfil
\end{title}
\pagenum=0
%\end{ignore}
\section {\bf 1. Introduction}
A number of recent, and not so recent, papers have been concerned with the
conformal anomaly in the dilaton--scalar system in two-dimensional
gravity. This anomaly takes the general, local form 
$$
{T^\mu}_\mu=T(x)={1\over24\pi}(R-6\nabla^\mu\phi\nabla_\mu\phi
+\al\square\phi)
\eql{anom}$$
where $\phi$ is the dilaton field. The treatments agree on the first two
terms but the coefficient, $\al$, of the total divergence is subject to
some discussion. Firstly there is the question of whether the correct
two-dimensional reduction of the spherical four-dimensional theory has 
been taken. The reduction adopted by Elizalde, Naftulin and Odintsov, 
[\pref{ENO}], Mukhanov, Wipf and Zelnikov, 
[\pref{MWZ}], and by Kummer, Liebl and Vassilevich (KLV),
[\pref{KLV,KLV2}], produces $\al=6$. (This follows from the choice 
$\vphi=\psi=\phi$ in [\pref{KLV}]. See also Chiba and Siino, 
[\pref{CandS}]). Bousso and Hawking (BH), [\pref{BoandH}], effectively
choose a measure such that, in the notation of KLV, 
$\psi=0$ and $\vphi=\phi$. BH obtain the value $\al=-2$ while KLV's
formula gives $\al=4$. This latter value is also obtained by Ichinose,
[\pref{Ichinose}]. In this brief note we consider only this 
discrepancy since it seems to be a clear mathematical contradiction. 
The existence of a discrepancy, actually between the $\al=-2$ and $\al=6$ 
values, was early noted by Nojiri and Odintsov, [\pref{NandO}], who ascribed 
it entirely to total divergence ambiguities. For completeness, by repeating 
some standard material, we will here confirm the value $\al=4$ and then 
indicate where we think the calculation of BH breaks down. 


\section{\bf 2. The dynamics and the anomaly}
In its simplest form the (matter) action adopted is, in 2d,
$$
S_m=-{1\over2}\int_\man \,e^{-2\phi}
\nabla^\mu f\,\nabla_\mu f\,\sqrt{g}d^2x
$$
where $f$ is the scalar matter field, with the corresponding field
operator
$$
A=e^{-2\phi}(-\square+2\nabla^\mu\phi\nabla_\mu).
\eql{op}$$

The most rapid method of finding the anomaly relies on its standard 
expression, $\ze(0,x)$, in terms of the local \zf\
associated with $A$, or, entirely equivalently, of the heat-kernel 
coefficient, $C^{(2)}_1(x)$, [\pref{DandC}]. To this end the operator 
$A$ is rewritten
$$
A=-e^{-2\phi}\big((\nabla^\mu-\nabla^\mu\phi)
(\nabla_\mu-\nabla_\mu\phi)+V\big)
$$
where 
$$V=\square\phi-\nabla^\mu\phi\nabla_\mu\phi.
$$

Introducing the auxiliary metric 
$$g'_{\mu\nu}=e^{2\phi}g_{\mu\nu}
$$
$A$ can also be written as
$$
A=-\big(({\nabla'}^\mu-{\nabla'}^\mu\phi)
({\nabla'}_\mu-{\nabla'}_\mu\phi)+V'\big)
$$
with
$$
V'=\square'\phi-{\nabla'}^\mu\phi{\nabla'}_\mu\phi=e^{-2\phi}\,V,
$$
where ${\nabla'}^\mu$ is ${\nabla'}_\mu,\,=\nabla_\mu$, raised by 
$g'^{\mu\nu}$.

When computing the eigenvalues of $A$, the scalar product of the $f$'s is
defined using the covariant measure of the $g$ metric. However, the trivial
Weyl potential, ${\nabla'}_\mu\phi$, can be removed by the gauge 
transformation, $f\to f'=\exp(-\phi)\,f$. The $f'$'s are 
normalized using the auxiliary metric, $g'$, and have the field operator $A'$
where
$$
A'=e^{-\phi}Ae^{\phi}=-\big(\square'+V'\big).
$$
The formal computation of $\ze(0)$ can thus proceed as for the standard
Laplacian by treating, temporarily, $g'$ as the metric. We will therefore 
find the integrated anomaly
$$
T=\ze(0)={1\over4\pi}\int\,C_1^{(2)}(g',x)\, \sqrt{g'}\,d^2x
$$
and can use the expression for $C_1$ derived many years ago,
$$
C^{(n)}_1(g',x)={R'\over6}+V',
\eql{cee1n}$$
where the coordinate system has been extended artificially to an 
$n$-dimensional one for later use.

The local trace anomaly, expressed as a density in the auxiliary metric is,
[\pref{DandC}], 
$$
T'(x)={1\over4\pi}C^{(2)}_1(g',x)={1\over4\pi}\bigg({R'\over6}+V'\bigg).
\eql{auxanom}$$
In order to obtain a density in the original metric, $g$, one simply
rewrites the $g'$ in (\peq{auxanom}) in terms of $g$ and removes the
resulting overall factor of $e^{-2\phi}$ to allow for the change in
the $\sqrt{g}$'s. As advertised we find,
$$
T(x)={1\over24\pi}(R-6\nabla^\mu\phi\nabla_\mu\phi+4\square\phi).
$$

Being based on standard techniques, this discussion adds nothing
material to the earlier treatments. However, as an amusing novelty, one 
can check the total derivative term, $\square'\phi$, in (\peq{cee1n}) in
the following way.

Instead of introducing the gauge potential we treat the operator $A$ as
it stands in (\peq{op}) and, further, work in $n$ dimensions. The idea here
is to use our previous technique, [\pref{Dowk}], of deriving the total 
derivative term in the {\it local} coefficient from the {\it integrated} 
coefficient (from which of course this term is absent).  

To save writing a lot of primes we replace $g'$ in $A$ by $g$ 
which should be thought of simply as a generic metric. 
(This is only a notational convenience for the purposes of this check.) 

Because $A$ is not conformally covariant in $n$
dimensions the calculation is not quite straightforward, but the necessary
formalism is available in [\pref{Dowk}]. The behaviour of $A$ under scale
changes $g_{\mu\nu}\to\overline g_{\mu\nu}=\la^2 g_{\mu\nu}$ is easily
determined to be
$$
(\overline A+\overline U)\overline f=\la^{-(n+2)/2}A f,\quad \overline f=
\la^{(2-n/2}f,
$$
where $\overline U$ measures the loss of conformal covariance
and equals, (\cf [\pref{Dowk}]),
$$
\overline U=(n-2)\overline\nabla^\mu\om\overline\nabla_\mu\phi+
\xi(n)(n-1)\big(2\la^{-1}\overline\square\la-(n-2)\overline\nabla^\mu
\om\,\overline\nabla_\mu\om\big)
$$
with $\xi(n)=(n-2)/4(n-1)$ and $\om=-\ln\la$. The second part of 
$\overline U$ is connected with the noncovariance of the Laplacian. 

Working around $\om=0$ (when $\overline U$ vanishes) and applying 
perturbation theory in $\overline U$ allows one to relate the 
relevant \zfs\ and thence the heat-kernel coefficients to obtain, 
[\pref{Dowk}],
$$\eqalign{
{1\over\sqrt{g}}{\de C_k^{(n)}[e^{-2\om}g]\over\de\om(x)}\bigg|_{\om=0}=&
-(n-2k)\,C_k^{(n)}(g,x)\cr
&+\bigg((n-2)\big(\square\phi+\nabla_\mu\phi\nabla^\mu\big)
+2(n-1)\xi(n)\,\square\bigg)C_{k-1}^{(n)}(g,x).\cr}
\eql{coeffvar}$$
We use this equation to find the local coefficient on the right from the
variation of the integrated one on the left.

As our application we set $k=1$. Using the fact that $C_0$ is the 
Weyl volume term, $C_0^{(n)}(g,x)=1$, we quickly find
$$
C^{(2)}_1(g,x)=\square\phi+
\lim_{n\to2}{1\over n-2}{1\over\sqrt{g}}{\de C_1^{(n)}
[e^{-2\om}g]\over\de\om(x)}\bigg|_{\om=0}.
\eql{cee1}$$

The point of this little exercise is simply to say that, assuming we know 
only the integrated $C_1^{(n)}$, 
$$
C_1^{(n)}[g]=\int\,\bigg({R\over6}-(\nabla\phi)^2\bigg)\,\sqrt{g}\,d^nx,
$$
then the variation and limit in (\peq{cee1}) easily yield
$$
C^{(2)}_1(g,x)=\square\phi+{R\over6}-(\nabla\phi)^2
$$
showing the resurrection of the total derivative contribution.

\section{\bf 3. Discussion}
We now turn to the question raised earlier concerning the origin of the 
discrepancy with the result of BH, [\pref{BoandH}]. 
The problem arises when BH assume, after their equn. (3.4), that the manifold
has the topology of the two-sphere, for then there is a zero mode of the
Laplacian and one cannot use the quoted Polyakov form for the 
effective action (equn. (3.3)). It is better to use the antisymmetrical 
cocycle function, $W[\overline g,g]\sim W[\overline g]-W[g]$, for the 
conformal change $g\to\overline g$.  

As we have shown, [\pref{Dowk2}], because of the zero 
mode, apart from the standard contribution, there is an additional term of 
the form
$$
\De W[\overline g,g]=
{1\over48\pi |\man|}\int\ln\bigg({g\over\overline g}\bigg)\sqrt{g}\, d^2x\,
\int R\sqrt{g}\, d^2x
$$
where $|\man|=|\man(g)|$ is the two-surface area.

Computing this for the uniform rescaling, $\overline g_{\mu\nu}=
\exp(2\phi_c)g_{\mu\nu}$, yields an extra contribution which cancels the
change in the effective action used by BH -- the last term in equn. (3.5).
(This must be so for consistency and is the whole point of [\pref{Dowk2}].) 
If we carried on with the analysis as in BH, then we would conclude that 
$q_3,\,=\al/24\pi$, were zero. 

One way of partially retrieving the situation is to use the cocycle 
function, $W[\overline g,g]$, (as we 
should). Then the last term in equn. (3.3) is replaced by
$$
{1\over2}q_3\int \big(\overline R\sqrt{\overline g}-R\sqrt{g}\,
\big)\phi\,d^2x
$$
which vanishes when $\phi$ is uniform by topological invariance, but which 
still has the required variation and everything is consistent. However it 
is not then possible to deduce 
the value of $q_3$ in a simple way, at least not by conformal transformations
in two dimensions alone.

If we wish to avoid a zero mode, then it is necessary to include 
boundary terms in the effective action, 
{\it when this last is being evaluated}. In any case, zero mode or boundary,
the additional contributions remove any discrepancies and also, 
incidentally, render nugatory the specific criticisms by Nojiri and 
Odintsov, [\pref{NandO}], of Bousso and Hawking's choice of term in the 
effective action. The problem is not so much the ambiguity in this term, 
rather it is its incorrect behaviour for uniform $\phi$.



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\end{putreferences}
%\end{ignore}

\bye



