The Ray-Singer torsion for the Lens space $L(p;1,1)$ that is obtained from $S^3\subset \mathbb{C}^2$ by identifying $(z_1,z_2) \sim (e^{2 \pi i/p} z_1,e^{2 \pi i/p} z_2)$ I think can be computed by assuming that all the $n$-forms on $S^3$ satisfy twisted boundary conditions \begin{equation} \phi_n(e^{2 \pi i /p} z_1,e^{2 \pi i /p} z_2) = e^{2\pi i k/p} \phi_n(z_1,z_2) \tag{1} \end{equation} for some $k\in 1,...,p-1$. We can also assume that $k=0$ in which case we have to take zero modes into account. My question is if the Ray-Singer torsion can depend on the choice of twisted boundary conditions throught $k$? If it does depend on $k$, then the torsion is not just a function of the manifold itself, but also depends on what kind of form fields live on it. Is this correct? Also, shall we expect that also $k=0$ would give the same answer for the Ray-Singer torsion as say $k=1$?
My guess is that the torsion is given by $(2 \sin (\pi k/p))^2$ where $k \in 1,...,p-1$. And that one gets yet some other answer for $k=0$. Is my guess correct?
My second question is about Lens space $L(p;\ell,\ell)$. If $p$ is a prime number, then by iterating the identification $(z_1,z_2) \sim (e^{2 \pi i \ell/p} z_1,e^{2 \pi i \ell/p} z_2)$ many times, I seem to reach the conclusion that $L(p;\ell,\ell) = L(p;1,1)$ since there will always be some integer $m$ for which $m \ell =1$ (mod $p$). Is this conclusion correct? If so, then it seems to imply that the Ray-Singer torsion for $L(p;\ell,\ell)$ and $L(p;1,1)$ shall agree? In particular the Ray-Singer torsion of $L(p;\ell,\ell)$ should not depend on $\ell=1,...,p-1$.
By assuming that $\ell k = 1$ (mod $p$) for some $\ell$, we get by iterating (1) above \begin{equation} \phi_n(e^{2 \pi i \ell /p} z_1,e^{2 \pi i \ell /p} z_2) = e^{2\pi i /p} \phi_n(z_1,z_2) \tag{2} \end{equation} which seems to say that torsion on $L(p;\ell,\ell)$ computed with boundary conditions (2) agrees with torsion on $L(p,1,1)$ computed with boundary conditions (1). But if also $L(p;\ell,\ell) = L(p;1,1)$, then this simply says that the torsion of $L(p;1,1)$ does not depend on $k$ in the boundary conditions (1), which is a self-contradiction.
One resolution would be if the torsion is not just a function of the manifold itself, but is a function that depends on the whole package, the manifold $L(p;\ell,\ell)$ plus the twisted boundary conditions (specified by $k$) that are put on the fields on that manifold. Although we can shift $\ell$ to $1$ by iteration, we can not keep $k$ fixed at the same time. So the pair of integers $(\ell,k)$ has an invariant meaning, although they form a conjugacy class $[\ell,k]$ where two elements are identified as $(\ell,k) \sim (1,m)$ where $m$ is a certain integer that depends on $\ell,k,p$ in a certain way.