Analytic torsion

Remarks

Equivalently, $T(M, \rho) = \prod_q \big( \det_\zeta \Delta_q \big)^{(-1)^{q+1} q/2}$ in terms of zeta-regularized determinants. Ray and Singer showed that $T(M, \rho)$ does not depend on the metric $g$ when $n$ is odd, and that $T(M, \rho) = 1$ when $n$ is even. Without acyclicity, one defines the torsion as a metric on the determinant line of the cohomology, the Ray–Singer metric. Conventions differ: some authors use the inverse of $T$, or omit the factor $\tfrac12$.

Example: let $M = \R / 2\pi\Z$ and let $\rho$ send the generator of $\pi_1(M) = \Z$ to $e^{2\pi i a}$ with $0 < a < 1$. Sections of $E_\rho$ are functions with $f(\theta + 2\pi) = e^{2\pi i a} f(\theta)$, with eigenfunctions $e^{i(k + a)\theta}$, $k \in \Z$. So $\Delta_0$ and $\Delta_1$ both have eigenvalues $(k + a)^2$, and $\zeta_1(s) = \zeta(2s, a) + \zeta(2s, 1 - a)$, with Hurwitz zeta functions. From $\zeta'(0, a) = \log \Gamma(a) - \tfrac12 \log(2\pi)$ and $\Gamma(a)\Gamma(1 - a) = \pi / \sin(\pi a)$ one gets $\zeta_1'(0) = -2 \log(2 \sin \pi a)$, hence $T(M, \rho) = 2 \sin(\pi a) = \abs{1 - e^{2\pi i a}}$.

Non-example: for the trivial representation of $\pi_1(S^1)$, $H^0(S^1; \C) = \C$ is nonzero, so the representation is not acyclic, $\Delta_0$ has the eigenvalue $0$, and the formula above does not apply as stated.