Torsion and zeta functions
2019/09/10\[ \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\range}{range} \newcommand{\even}{\mathrm{even}} \newcommand{\odd}{\mathrm{odd}} \newcommand{\c}{\mathfrak{c}} \newcommand{\Lie}{\mathcal{L}} \]
I want to understand the naive idea behind the relationships between torsions and zeta functions. We'll finish by having a finite dimensional appreciation of the analytic torsion.
Cohomological definition of torsion
We start with an overly trivial setting of a very short complex of vector spaces:
\[ (C^\bullet, d) := 0 \to C^0 \xrightarrow{d} C^1 \to 0 \]
with \(C^\even=C^0\), \(C^\odd = C^1\). Choosing a basis \(\c\) for \(C\) gives a matrix representation of \(d\) from which we may calculate the determinant. The torsion is then \( \tau_{(C^\bullet, \c)} = \det \left( d : C^\even \to C^\odd \right). \)
More generally an acyclic complex \((C^\bullet, d)\) has a contraction \(\eta\) satisfying \(d\eta + \eta d = 1\). The torsion, with respect to a basis \(\c\), is
\[ \tau_{(C^\bullet, \c)} = \det \left( d + \eta : C^\even \to C^\odd \right). \]
We can push the calculations much further if we have a metric floating around:
Naively we should think of \(\eta\) being built from the adjoint of \(d\) in the presence of some metric on \(C^\bullet\). This would give \(d^*\) and \(\Delta = (d+d^*)^2\) from which \(\eta = \Delta^{-1} d^*\).
We see \(C^j\) decomposed as \(K^j\oplus B^j = \ker d \oplus \range d^*\). And \(\Delta = d d^* |_K + d^*d |_B\) so we introduce notation for the determinants: \(\delta_j = k_j \cdot b_j\). Note that \(b_j = k_{j+1}\).
We write \(\tau^2 = \det ((d+\eta)(d+\eta)^*:C^\odd\to C^\odd)\) and on \(K^j\oplus B^j\) the operator is diagonal: \((d d^*) \oplus (\Delta^{-2}d^* d)\) whose determinant is \(k_j b_j^{-1}\). Therefore
\[ \tau^2 = \frac{k_1 k_3 \cdots k_N}{b_1 b_3\cdots } = \frac{k_1 k_3 \cdots k_N}{k_2 k_4\cdots } = \frac{\delta_1 \delta_3^3 \cdots \delta_N^N}{\delta_2^2 \delta_4^4\cdots } = \prod_{k=1}^N (\det \Delta)^{(-1)^{k+1} k} \]
In the above \(N\) is the largest non-zero degree in \(C^\bullet\). (It must be odd for the torsion to be interesting to me.)
Zeta function as determinant of operator
Consider a symmetric positive operator \(A\) on \(\C^n\). Writing it's eigenvalues as \(0\le\lambda_1\le\dots\lambda_n\), one usually defines the associated zeta function as
\[ \zeta_A(s) = \sum_{\lambda_j > 0} \lambda_j^{-s}. \]
We will define the zeta function dependent on two parameters:
\[ \zeta_A(s,\lambda) = \sum_{\lambda_j \neq -\lambda} (\lambda_j + \lambda)^{-s} \]
which is effectively the zeta function for \(A+\lambda\). Under a change of variables calculation using the definition of the Gamma function, we may write
\[ \zeta_A(s, \lambda) = \frac{1}{\Gamma(s)} \int_0^\infty t^{s-1} \tr (e^{-t(A+\lambda)} - \Pi(\lambda)) \, dt \]
where \(\Pi(\lambda)\) is the projection operator onto the null space of \(A+\lambda\). From either formula for \(\zeta\), we see that \( \log \det (A+\lambda) = -\left.\partial_s \zeta_A(s,\lambda) \right|_{s=0} . \)
Analytic torsion
Given a manifold \(M\) of dimension \(N\), the Laplacian \(\Delta\) acting on differential forms provides a zeta function \(\zeta_\Delta(s,\lambda)\). This provides the regularised determinant
\[ \log \det \Delta := -\left.\partial_s \zeta_\Delta(s,0) \right|_{s=0} \]
This provides the definition of analytic torsion \(\tau\) introduced by Ray and Singer:
\[ 2 \log \tau = \left.\sum_{k=1}^N k (-1)^k \partial_s \zeta_\Delta(s,\lambda) \right|_{s=0, \lambda=0} \]
which is equal to the Reidemeister torsion. This has been proved by Cheeger and Müller.