\section{\bf 1. Introduction.}

This is a further installment of my ongoing numerical evaluations of functional determinants
in various domains, mostly spherical.

In a previous communication, [\putref{dowmassneg}], I computed the determinant for a
scalar field with imaginary mass, $i\alpha$, on a lens space {\it i.e. } a $\mbox{\open\char90}_q$ factor of the
three-sphere without fixed points. In this parallel work, I repeat the calculation for such an
action but now one {\it with} fixed points. It is not anticipated that the results will differ
greatly, ultimately, from those for the whole sphere. Nevertheless there is, I believe, some
technical merit in the intervening calculation and I present it for this reason.

I spend most time on the simplest orbifold, the periodic lune, and then treat, in a later
section, the more general quotient. Some numbers for periodic spherical tetrahedra are
exhibited for potential amusement.

\section{\bf2. Lunes}

The same  eigenstructure holds for lunes as for lens spaces in that the zeta function takes
the same form as in [\putref{dowmassneg}], only the degeneracies change.  In fact one can
proceed indirectly to the generating function  by making this form more precise, as in
[\putref{dowmass}] for the full sphere.  The relevant $\zeta$--function\ is that employed in
[\putref{dowmass}] which I repeat here for the three--lune specifically (see also
[\putref{Dowodd}] for a more extended description of the geometric situation),
  $$
  \zeta(s,a,\alpha,q)\equiv
  \sum_{m_1,m_2,m_3=0}^\infty {1\over\big((a+ q\,m_1+m_2+m_3)^2-\alpha^2\big)^s}\,,
  \eql{zetalune}
  $$

This is actually the $\zeta$--function\ for a lune of {\it any} angle $\pi/q$. When $q$ is an integer one has
a fixed point $\mbox{\open\char90}_q$ action and I restrict to this case from now on. The lune has Neumann
conditions on the (spherical) sides if $a=a_N=1$ and Dirichlet if $a=a_D=1+q$. I will,
shortly, add these two cases to give the eigenstructure of the closed {\it periodic lune},
which is the round (whole) sphere when $q=1$.

The determination of the degeneracies is a combinatorial, lattice problem but one really
needs only their generating function. I define the integer, $l$, by,
  $$
  l\equiv q\,m_1+m_2+m_3\,,
  \eql{ell}
  $$
which now runs from $0$ upwards. The degeneracy, $g(l,q)$, of the {\it level}
$\omega(l)\equiv a+l$ for a given $l$  is the number of (nonnegative) integers
$m_1,m_2,m_3$ such that the equality (\puteqn{ell}) holds. The associated generating
function is classic, going back at least to Euler, and is,
  $$
  h(t,q)=\sum_{l=0}^\infty g(l,q)\,t^l={1\over (1-t)^2\,(1-t^q)}\,.
  \eql{genfunl}
  $$

I now combine the Neumann and Dirichlet problems on the $\pi/q$ lune. This gives the
spectrum on the {\it periodic} $2\pi/q$ lune and I introduce the corresponding total $\zeta$--function,
  $$
  \zeta(s,\alpha,q)\equiv\zeta(s,a_N,\alpha,q)+\zeta(s,a_D,\alpha,q)\,.
  \eql{totzet}
  $$

Both sets of eigenlevels are to be labelled by the same integer $l$ with
$l=0,1,2,\ldots\infty$. Because the constant $a_D=a_N+q$, the Dirichlet generating
function is the Neumann one with $l$ shifted by $q$, and the total generating function is
again standard, being,
  $$
  H(t,q)=\sum_{l=0}^\infty G(l,q)\,t^l={1+t^
  \over (1-t)^2\,(1-t^q)}\,.
  \eql{totgenfun}
  $$
which can be used to compute any individual degeneracy.

The $\zeta$--function, (\puteqn{totzet}), can therefore be written in general form exactly as in
[\putref{dowmassneg}],
  $$\eqalign{
   \zeta(s,\alpha,q)&=\sum_{l=0}^\infty {G(l,q)\over\big((1+l)^2-\alpha^2\big)^s}\cr
   &\equiv \sum_{l=0}^{N-1} {G(l,q)\over\big((1+l)^2-\alpha^2\big)^s}+\zeta^{(N)}(s,\alpha,q)\,,
   }
   \eql{lunezet}
  $$
and one easily gets,
  $$\eqalign{
  \zeta'(0,\alpha,q)=\zeta^{(N)}{}'(0,\alpha,q)-\sum_{l=0}^{N-1}G(l,q)\,\log|\big((1+l)^2-\alpha^2\big)|
  +i\pi\sum_{l=0}^{[\alpha]-1}G(l,q)\,.
  }
  \eql{zet2dash}
  $$
(Remember, $a$ is integral.)

Again I have extracted $N$ levels, achieved by removing the first $N$ terms in the Taylor
expansion of the generating function (\puteqn{totgenfun}). I express this subtraction by $
H^{(N)}(t,q)$.
\section{\bf3. Computing the derivative at zero}

The same procedure of back integrating a convergent resolvent, employed previously, will
be followed. Again the the reference point is set to $\alpha=0$ because this $\zeta$--function\  has already
been studied and computational formulae are available, {\it e.g.}\ [\putref{Dowren}]. For
completeness here, I will later repeat these results.

Although the equations closely parallel those detailed elsewhere, I  give the intermediate
steps because I wish to derive the final formula more generally and also to polish the
formalism slightly.

Differentiating (\puteqn{lunezet}) twice with respect $\alpha^2$ then taking the derivative at
$s=0$ and finally back integrating with respect to $\alpha^2$ gives,
  $$
  \zeta^{(N)}{}'(0,\alpha,q)=\int_0^{\alpha^2 }dw\int_0^w dw'\sum_{l=N}^\infty{G(l,q)
  \over\big((1+l)^2-w'\big)^{2}} +C_N(\alpha,q)\,
  \eql{tot0deriv}
  $$
where the last term arises from the constants of integration and involves $ \zeta(s,\alpha,q)$ at
$\alpha=0$. (Any lower limits could be chosen provided the value of $\alpha$ is in the easily
computed range $0\le \alpha\le 1/2$. Another good choice would be the conformal one,
$\alpha=1/2$.)

The first constant of integration requires the computation of
  $$
  {d\over d\alpha^2}\, \zeta^{(N)}{}'(0,\alpha,q)\bigg|_{\alpha=0}
  ={d\over d\alpha^2}\, \zeta'(0,\alpha,q)\bigg|_{\alpha=0}+\sum_{l=0}^{N-1}G(l,q)\,\log(1+l)^2
  $$
using (\puteqn{lunezet}). Now
  $$
  {d\over d\alpha^2}\, \zeta(s,\alpha,q)=s\, \zeta(s+1,\alpha,q)\,.
  \eql{1diff}
  $$
and, noting that $\zeta(s,\alpha,q)$ does not have a pole at $s=1$,then,
  $$
  {d\over d\alpha^2}\, \zeta'(0,\alpha,q)\bigg|_{\alpha=0}=\zeta(1,0,q)\,.
  \eql{1diff0}
  $$
  
The absence of a pole is true for all odd--dimensional, closed manifolds and is, perhaps,
more familiar here as the vanishing of the $C_{1/2}$ heat--kernel coefficient. Even if there
were a pole, the right-hand side of (\puteqn{1diff0}) would still be finite, equal to the pole
remainder. This could be made the basis of a limited computational method that avoids
differentiating twice.
  
  
Therefore
  $$
    C_N(\alpha,q)=\alpha^2\big(\zeta(1,0,q)-A_N(q)\big)+\zeta'(0,0,q)+B_N(q)
  $$
where,
  $$
  B_N(q)=\sum_{l=0}^{N-1}{G(l,q)\log(1+l)^2}\,,
  $$
and
  $$
  A_N(q)=\sum_{l=0}^{N-1}{G(l,q)\over(1+l)^2}\,.
  $$

For the integral in (\puteqn{tot0deriv}), a Bessel Laplace transform gives,
$$\eqalign{
 \sum_{l=N}^\infty{G(l,q)
  \over\big((1+l)^2-\alpha^2\big)^{2}}
  &=\sqrt\pi\int_0^\infty d\tau\,e^{-\tau}\,H^{(N)}\big(e^{-\tau},q\big)\,
  \bigg({\tau\over2\alpha}\bigg)^{3/2}\,I_{3/2}(\alpha\tau)\,.
  }
  \eql{zetdl}
  $$

The quantity $e^{-\tau}\,H^{(N)}$ is the (subtracted) cylinder kernel for the
`square--root' pseudo operator whose eigenvalues are $(1+l)$.

The double integration with respect to $\alpha^2$ is best effected using Bessel properties, {\it e.g.}\
Petiau, [\putref{Petiau}], chap.III \S\S12--15. Set $z=\alpha\tau$ and use the easily derived
relation,
    $$\eqalign{
  \int_0^zz_1 dz_1\ldots\int_0^{z_{p-1}}z_p dz_p\,{z_p}^{-\lambda}\,I_{\lambda}(z_p)
  =z^{p-\lambda}\,
  I^{(p)}_{\lambda-p}(z)\,,
   }
   $$
where $F^{(p)}$ stands for the removal of the first $p$ terms in the power series
expansion of $F$. In $d$ dimensions, $p=d-1$ and $\lambda=d/2$.


\begin{ignore}
   $$\eqalign{
  \int_0^\alpha\,2\alpha'\, d\alpha' &\int_0^{\alpha'}\,2\alpha''\, d\alpha'' \bigg({\tau\over2\alpha''}
  \bigg)^{\lambda}\,I_{\lambda}(\alpha''\tau)\cr
   &=\bigg({\tau^2\over2}\bigg)^{\lambda-2}\,\int_0^zz' dz'\int_0^{z'}z'' dz''\,
   z''^{-\lambda}\,I_{\lambda}(z'')\cr
   &=\bigg({\tau^2\over2}\bigg)^{\lambda-2}\,
  \int_0^z z'dz' \bigg[z''^{-\lambda+1}\,I_{\lambda-1}(z'')\bigg|^{z'}_0\bigg]\cr
  &=\bigg({\tau^2\over2}\bigg)^{\lambda-2}\,
  \bigg[z^{-\lambda+2}\,I_{\lambda-2}(z)-{2^{2-\lambda}\over\Gamma(\lambda-1)}
  -z^2{2^{-\lambda}\over\Gamma(\lambda)}\bigg]\cr
  &=\bigg({\tau^2\over2z}\bigg)^{\lambda-2}\,
  \bigg[I_{\lambda-2}(z)-{(z/2)^{\lambda-2}\over\Gamma(\lambda-1)}
  -{(z/2)^\lambda\over2\Gamma(\lambda)}\bigg]
   }
   $$
\end{ignore}
The value of $\lambda$ relevant here is $3/2$  for which, if needed ($p=2$),
  $$
  I_{-1/2}(z)=\sqrt{2\over\pi z}\,\cosh z\,.
  $$

Hence putting everything together, the final, computable formula from (\puteqn{zet2dash})
and (\puteqn{tot0deriv}) is (I give the real part only),
  $$\eqalign{
  \zeta'(0,\alpha,q)&=\sqrt{2\pi\alpha}\int_0^\infty d\tau\,{1\over\sqrt\tau}\,
  e^{-\tau}\,H^{(N)}\big(e^{-\tau},q\big)\,
   I^{(2)}_{-1/2}(\alpha\tau)\cr
  &+\sum_{l=0}^{N-1}G(l,q)\,\bigg(\log{(l+1)^2\over|\alpha^2-(l+1)^2|}
  -\alpha^2 {1\over (l+1)^2}\bigg)\cr
  &+\alpha^2\zeta(1,0,q)+\zeta'(0,0,q)\,.
  }
  \eql{compf}
  $$

The last two terms have to be found independently. I give the computable expressions
derived from those in [\putref{Dowren}] by setting $\alpha$ equal to zero,
$$\eqalign{
  \zeta(1,0,q)&={1\over4}\int_0^\infty dx \,
  {\rm Re\,}{\tau\,\coth q\tau/2\over\sinh^2\tau/2} \cr
  \zeta'(0,0,q)&={1\over2}\int_0^\infty dx \,
  {\rm Re\,}{\coth q\tau/2\over\tau\sinh^2\tau/2}\,,\cr
  }
  \eql{alpha0}
  $$
with $\tau=x+iy$, $y$ having to lie between $0$ and the first singularity of the integrand
above the real axis.

 As $N$ tends to infinity, the integral term in (\puteqn{zet2dash}) vanishes leaving a standard regularised
expression equivalent to that used in, {\it e.g.}, [\putref{ADH}] for the round sphere. In fact, for
largish $N$, one could drop the integral, with a reduced accuracy penalty.
  
Incidentally, an equation similar to (\puteqn{compf}) will be valid for a more general spectrum
because the Bessel transform separates $\lambda$ from the mass parameter, $\alpha^2$, in the
eigenvalues $\lambda-\alpha^2$, [\putref{DoandKi}]. Whether all quantities in (\puteqn{compf}) are
available depends on the system.

\section{\bf4. The results}

Figure 1 shows the variation of the logdet for the three lunes, $q=1$ (the round sphere),
$q=2$ and $q=3$ only, the round sphere being typical for these low values of $q$. Figure 2
enlarges the region $0<\alpha<1$ and demonstrates that, as for lens spaces, any deformation
from roundness eliminates the maximum at $\alpha=1/2$, the conformal point. Figure 3 plots
the Hartle--Hawking probability, $|\Psi_{HH}|^2$, and Figure 4 displays the same numbers
plotted against the variable $\sigma=1/4-\alpha^2$, used in [\putref{ADH,ADKS}]. Like these
references  I have, without loss of generality, set $N=2$ in the free Sp$(N)$ CFT$_3$ of
anticommuting scalars according to the dS/CFT scheme.

\epsfxsize=5truein \epsfbox{lunedetsnegfig1.eps}


If one wishes to plot the wavefunction(al) itself, then the imaginary part in (\puteqn{zet2dash})
comes into play. Since the degeneracy, $G(l,q)$, alternates between even and odd as $l$
increases, the zeros of the wavefunction will alternate between points of inflection and
extrema, which are maxima and minima, in turn.


Figure 5 displays the results for three higher values of $q$ which have a more complicated
initial behaviour.

\epsfxsize=5truein \epsfbox{lunedetsnegfig2.eps}

\epsfxsize=5truein \epsfbox{lunedetsnegfig3.eps}

\epsfxsize=5truein \epsfbox{lunedetsnegfig4.eps}

\epsfxsize=5truein \epsfbox{lunedetsnegfig5.eps}


\section{\bf5. Other orbifold quotients}

The lune is the simplest orbifold quotient of the three--sphere and its analysis serves as an
introduction to the general case. In fact the (allowed) factors of the three--sphere can all be
treated in a unified way, as I now bring forward, if only for the sake of completing a list. The
quotients are classified by the regular 4--polytope reflection groups, of which there are only
a finite number.

The eigenvalues all have the same general form, (\puteqn{ell}) being replaced by,
  $$
  l=\omega_1\,n_1+\omega_2\,n_2+\omega_3\,n_3\,,
  $$
with the parameters, $\omega_i$, taking particular integer values. For convenience I list these
as vectors$\,$ : $(3,4,5)$ for the polytope group $[3,3,3]$, $(4,6,8)$ for $[3,3,4]$,
$(6,8,12)$ for $[3,4,3]$ and $(12,20,30)$ for $[3,3,5]$. I have labelled the groups by their
Coxeter symbol. (The Schl\"afli symbol of the corresponding 4--polytope is
$\{\omega_1,\omega_2,\omega_3\}$.)

The quotients are fundamental domains for the action and tile the three--sphere. In the
periodic case they can be pictured as (doubled) spherical tetrahedra, or limiting forms
thereof.

The degeneracy generating functions are classic and follow from Molien's theorem and
invariant theory. I give them in the guise of the cylinder kernel, $K^{1/2}$,
  $$\eqalign{
  K^{1/2}(\tau,\bom)&=2\cosh(\omega_0\,\tau/2)\prod_{i=1}^3{1\over2\sinh(\omega_i\tau/2}\cr
  &=t{1+t^{\omega_0}\over(1-t^{\omega_1})(1-t^{\omega_2})(1-t^{\omega_3})}\,,
  }
  \eql{CK}
  $$
where $\omega_0$ is the number of reflections and equals $1+\sum_{i=1}^3(\omega_i-1)$. Note
that $K^{1/2}(\tau,\bom)$ is antisymmetric in $\tau$. (See [\putref{ChandD}].)

The quantity multiplying $t$ in (\puteqn{CK}), is the generating function, $H(t,\bom)$, {\it cf }\
(\puteqn{totgenfun}). The two terms in the numerator correspond to Neumann and Dirichlet
conditions on the boundary of the fundamental domain. Adding these yields the spectrum on
a {\it periodic} domain, obtained by sticking two fundamental domains together and
identifying boundaries. This doubled domain has singularities of codimensions 2 and 3.

The analysis is virtually identical to that in the preceding sections and leads to the final
formula,
  $$\eqalign{
  \zeta'(0,\alpha,\bom)&=\sqrt{2\pi\alpha}\int_0^\infty d\tau\,{1\over\sqrt\tau}\,
  K^{1/2(N)}(\tau,\bom)\,
   I^{(2)}_{-1/2}(\alpha\tau)\cr
  &+\sum_{l=0}^{N-1}G(l,\bom)\,\bigg(\log{(l+1)^2\over|\alpha^2-(l+1)^2|}
  -\alpha^2 {1\over (l+1)^2}\bigg)\cr
  &+\alpha^2\zeta(1,0,\bom)+\zeta'(0,0,\bom)\,.
  }
  \eql{compf2}
  $$

The equations which generalise (\puteqn{alpha0}) can be obtained, as above, from
[\putref{Dowren}],
$$\eqalign{
  \zeta(1,0,\bom)&={1\over4}\int_0^\infty dx \,
  {\rm Re\,}{\tau\,\cosh\big((\omega_1+\omega_2+\omega_3-2)
  \tau/2\big)\over\sinh(\omega_1\tau/2) \sinh(\omega_2\tau/2)\sinh(\omega_3\tau/2)}\cr
  \zeta'(0,0,\bom)&={1\over2}\int_0^\infty dx \,
  {\rm Re\,}{\cosh\big((\omega_1+\omega_2+\omega_3-2)
  \tau/2\big)\over\tau\,\sinh(\omega_1\tau/2) \sinh(\omega_2\tau/2)\sinh(\omega_3\tau/2)}
  \,.\cr
  }
  \eql{alphaomega0}
  $$
  
Figure 6 gives some numerical results.

\epsfxsize=5truein \epsfbox{lunedetsnegfig6.eps}

To complete the list, Figures 7 and 8 give the results for the (double) dihedral group for
which $\bom=(q,2,1)$.

\epsfxsize=5truein \epsfbox{lunedetsnegfig7.eps}

\epsfxsize=5truein \epsfbox{lunedetsnegfig8.eps}

Finally, to illustrate the sort of behaviour that occurs, in Figure 9 I display the
Hartle--Hawking probability for the $[3,3,5]$ case, over a fair range of $\alpha$. There is
actually a shallow zero at $\alpha=1$, (see Figure 6). I should also note that there is no
apparent pattern to the parity of the degeneracies.

\epsfxsize=5truein \epsfbox{lunedetsnegfig9.eps}

\section{\bf6. Discussion}

I have presented more applications of a simple numerical method of computing functional
determinants (for negative mass--squared) on spherical quotients, and have shown that one
does not have to work too hard to take the more complicated degeneracies into account.
The extension to higher dimensions, is straightforward, if desired.

Concerning the numerical results, I draw no conclusions above and beyond the remark that
the, possibly unwelcome [\putref{ADH,ADKS}], large--$\alpha$ divergence still seems to be
present.

Calculationally, the $\bom$ in (\puteqn{CK}) do not have to take the specific polytope values.
They can be any integers, but would then seem to lose geometric significance.

On a purely technical note, I have differentiated twice to get a convergent resolvent. In fact,
when calculating the determinant it is possible to get away with differentiating once. This
works well for the round sphere, where the degeneracies are simple, and standard
summations can be employed. However things are more involved for the higher factors and
I could not get this procedure to work very easily, although I have not tried hard in this
regard.

It is possible to discuss the case of lunes of {\it arbitrary} angle but then the generating
function (\puteqn{totgenfun}) cannot be used, at least not to extract the degeneracies..
  
 \vglue 20truept

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