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.