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  "body": "\n## Definition\n\nLet $(M, g)$ be a closed oriented [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$.\nLet $\\rho \\colon \\pi_1(M) \\to U(N)$ be a [[unitary representation|74BPYJNZTM]] of the [[fundamental group|S1XK3HXHFX]], and let $E_\\rho$ be the associated flat Hermitian vector bundle.\nAssume that $\\rho$ is acyclic, that is, the cohomology $H^*(M; E_\\rho)$ vanishes.\nFor $0 \\le q \\le n$ let $\\Delta_q$ be the [[Hodge Laplacian|PWG3GKZG80]] on $E_\\rho$-valued $q$-forms.\nBy acyclicity, all eigenvalues of $\\Delta_q$ are positive.\nLet $\\zeta_q(s) = \\sum_\\lambda \\lambda^{-s}$ be its spectral zeta function, summed over its eigenvalues with multiplicity.\nThe *analytic torsion* $T(M, \\rho) \u003e 0$ is defined by\n$$\\log T(M, \\rho) = \\frac12 \\sum_{q = 0}^{n} (-1)^q \\, q \\, \\zeta_q'(0).$$\n\n## Remarks\n\nEquivalently, $T(M, \\rho) = \\prod_q \\big( \\det_\\zeta \\Delta_q \\big)^{(-1)^{q+1} q/2}$ in terms of [[zeta-regularized determinants|YT3ZQ2F7FG]].\nRay 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.\nWithout acyclicity, one defines the torsion as a metric on the determinant line of the cohomology, the Ray–Singer metric.\nConventions differ: some authors use the inverse of $T$, or omit the factor $\\tfrac12$.\n\nExample: let $M = \\R / 2\\pi\\Z$ and let $\\rho$ send the generator of $\\pi_1(M) = \\Z$ to $e^{2\\pi i a}$ with $0 \u003c a \u003c 1$.\nSections of $E_\\rho$ are functions with $f(\\theta + 2\\pi) = e^{2\\pi i a} f(\\theta)$, with eigenfunctions $e^{i(k + a)\\theta}$, $k \\in \\Z$.\nSo $\\Delta_0$ and $\\Delta_1$ both have eigenvalues $(k + a)^2$, and $\\zeta_1(s) = \\zeta(2s, a) + \\zeta(2s, 1 - a)$, with Hurwitz zeta functions.\nFrom $\\zeta'(0, a) = \\log \\Gamma(a) - \\tfrac12 \\log(2\\pi)$ and $\\Gamma(a)\\Gamma(1 - a) = \\pi / \\sin(\\pi a)$ one gets $\\zeta_1'(0) = -2 \\log(2 \\sin \\pi a)$, hence $T(M, \\rho) = 2 \\sin(\\pi a) = \\abs{1 - e^{2\\pi i a}}$.\n\nNon-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.\n",
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