The dilogarithm: a bridge between 3-manifolds and zeta functions

In my undergraduate days I was under the wrong impression that hypergeometric series and things like that were essentially completely understood in the 19th century. In fact, various such transcendental functions and others are at the forefront of modern research. The dilogarithm is one such transcendental function which turns out to be connected to just about everything imaginable. We discuss some of the basics in this post which will hopefully go some way to explaining the attention it has garnered in the past 30 years. We follow this paper of Zagier very closely.

 The dilogarithm function is the function defined by the power series

\[\operatorname{Li}_2(z) = \sum_{n=1}^\infty \frac{z^n}{n^2}, \quad \text{for } |z| < 1.\]

The definition and the name come from the analogy with the Taylor series of the ordinary logarithm around 1:

\[-\log(1 - z) = \sum_{n=1}^\infty \frac{z^n}{n}, \quad \text{for } |z| < 1,\]

which, thinking of this as $\mathrm{Li}_1$, leads similarly to the definition of the polylogarithm:

\[\operatorname{Li}_m(z) = \sum_{n=1}^\infty \frac{z^n}{n^m}, \quad \text{for } |z| < 1, \quad m = 1, 2, \ldots\]

The relation

\[\frac{d}{dz} \operatorname{Li}_m(z) = \frac{1}{z} \operatorname{Li}_{m-1}(z)\]

follows from the power series definition and leads by induction to the extension of the domain of definition of $\operatorname{Li}_m$ to the cut plane $\mathbb{C} \setminus [1, \infty)$; in particular, the analytic continuation of the dilogarithm is given by

\[\operatorname{Li}_2(z) = -\int_0^z \frac{\log(1 - u)}{u} \, du, \quad \text{for } z \in \mathbb{C} \setminus [1, \infty)\]

This obeys a remarkable wealth of functional equations of $\operatorname{Li}_2$:

\[\operatorname{Li}_2\left(\frac{1}{z}\right) = -\operatorname{Li}_2(z) - \frac{\pi^2}{6} - \frac{1}{2} \log^2(-z), \]

\[\operatorname{Li}_2(1 - z) = -\operatorname{Li}_2(z) + \frac{\pi^2}{6} - \log(z) \log(1 - z), \]

\[\operatorname{Li}_2(z^2) = 2 \operatorname{Li}_2(z) + \operatorname{Li}_2(-z), \]

and more generally the “distribution property”:

\[\operatorname{Li}_2(x) = {n} \sum_{z^n = x} \operatorname{Li}_2(z), \quad (n = 1, 2, 3, \ldots).\]

Next, there is the two-variable five-term relation:

\[&\operatorname{Li}_2(x) + \operatorname{Li}_2(y) + \operatorname{Li}_2\left(\frac{1 - x}{1 - x y}\right) + \operatorname{Li}_2(1 - x y) + \operatorname{Li}_2\left(\frac{1 - y}{1 - x y}\right) \\ = \frac{\pi^2}{6} - \log(x) \log(1 - x) - \log(y) \log(1 - y) + \log\left(\frac{1 - x}{1 - x y}\right) \log\left(\frac{1 - y}{1 - x y}\right)\]

The function $\operatorname{Li}_2(z)$, extended as above to $\mathbb{C} \setminus [1, \infty)$, jumps by $2\pi i \log |z|$ as $z$ crosses the cut. Thus, the function

\[\operatorname{Li}_2(z) + i \arg(1 - z) \log |z|\]

(where $\arg$ denotes the branch of the argument lying between $-\pi$ and $\pi$) is continuous.

Surprisingly, its imaginary part

\[D(z) = \Im(\operatorname{Li}_2(z)) + \arg(1 - z) \log |z|\]

is real analytic on $\mathbb{C}$ except at the two points $0$ and $1$, where it is continuous but not differentiable (it has singularities of type $r \log r$ there).

The function $D(z)$, which is real-valued on $\mathbb{C}$, can be expressed in terms of a function of a single real variable, namely:

\[D(z) = \frac{1}{2} D\left( \frac{z}{\bar{z}} \right) + D\left( \frac{1 - \frac{1}{z}}{1 - \frac{1}{\bar{z}}} \right) + D\left( \frac{1}{1 - z} \cdot \frac{1}{1 - \bar{z}} \right)\]

which expresses $D(z)$ for arbitrary complex $z$ in terms of the function

\[D(e^{i\theta}) = \Im\left(\operatorname{Li}_2(e^{i\theta})\right) = \sum_{n=1}^\infty \frac{\sin(n\theta)}{n^2}\]

(Note that the real part of $\operatorname{Li}_2$ on the unit circle is elementary:

\[\Re(\operatorname{Li}_2(e^{i\theta})) = \frac{\pi^2}{6} - \frac{\theta(2\pi - \theta)}{4} = \sum_{n=1}^\infty \frac{\cos(n\theta)}{n^2})\]

All of the functional equations satisfied by $\operatorname{Li}_2(z)$ lose the elementary correction terms (constants and products of logarithms) when expressed in terms of $D(z)$. In particular, one has the 6-fold symmetry:

\[D(z) = D\left(1 - \frac{1}{z}\right) = D\left(\frac{1}{1 - z}\right) = -D\left(\frac{1}{z}\right) = -D(1 - z) = -D(-z)\]

and the five-term relation:

\[D(x) + D(y) + D\left( \frac{1 - x}{1 - x y} \right) + D(1 - x y) + D\left( \frac{1 - y}{1 - x y} \right) = 0\]

The functional equations become even cleaner if we think of $D$ as being a function not of a single complex number but of the cross-ratio of four such numbers, i.e., if we define:

\[D(z_0, z_1, z_2, z_3) = D\left( \frac{(z_0 - z_2)(z_1 - z_3)}{(z_0 - z_3)(z_1 - z_2)} \right), \quad (z_0, z_1, z_2, z_3 \in \mathbb{C})\]

Then the symmetry properties say that $D$ is invariant under even and anti-invariant under odd permutations of its four variables. The five-term relation takes on the attractive form:

\[\sum_{i=0}^4 (-1)^i D(z_0, \dots, \hat{z}_i, \dots, z_4) = 0, \quad (z_0, \dots, z_4 \in \mathbb{P}^1(\mathbb{C}))\]

and the multi-variable formula generalizes to the following beautiful formula:

\[D(z_0, z_1, z_2, z_3) = n \cdot D(a_0, a_1, a_2, a_3)\]

for $z_0, a_1, a_2, a_3 \in \mathbb{P}^1(\mathbb{C})$, where $f: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$ is a function of degree $n$ and $a_0 = f(z_0)$. (Equation (1) is the special case when $f$ is a polynomial, so that $f^{-1}(\infty) = \infty$ with multiplicity $n$.)

We now classify all of the relations that the dilogarithm satisfies! 

By a ``functional equation of the dilogarithm'' we mean any collection of integers $n_i$ and rational or algebraic functions $x_i(t)$ of one or several variables such that

\[\sum n_i \operatorname{Li}_2(x_i(t))\]

is a finite combination of products of two logarithms, or such that

\[\sum n_i D(x_i(t))\quad \text{(resp. } \sum n_i L(x_i(t)) \text{ if all the } x_i(t) \text{ are real)}\]

is constant (resp. locally constant).

Lemma:  (1) Any rational function of one variable is equivalent modulo the five-term relation to a linear combination of linear functions.

(2)  Any functional equation of the dilogarithm with rational functions of one variable as arguments is a consequence of the five-term relation.

(Part (2) is to be interpreted up to constants, i.e., the five-term relation suffices to give all relations $\sum_i D(x_i(t)) = C$, but not necessarily to determine $C$.)

Proof: (1) Let $f(t)$ be an element of the field $\mathbb{C}(t)$ of rational functions in one variable. We want to show that the element $[f] \in \mathbb{Z}[\mathbb{C}(t)]$ is equivalent modulo five-term relations to a $\mathbb{Z}$-linear combination of elements of the form $[a_i t + b_i]$. We do this by induction on the degree.

Write $f(t) = A(t)/B(t)$, where $A(t)$ and $B(t)$ are polynomials of degree $\leq n$, not both constant. Since we are working modulo the five-term relation, we can replace $f$ by $1/f$ or $1/(1 - f)$ if necessary to ensure that both $A(t)$ and $C(t) := B(t) - A(t)$ are non-constant. Choose a root $a$ of $A(t)$ and a root $c$ of $C(t)$, and set

\[g(t) = \frac{c - a}{t - a}, \quad A^*(t) = g(t) A(t), \quad D(t) = B(t) - A^*(t),\]

so that $\deg(A^*) \leq n - 1$, $\deg(D) \leq n$, and $D(c) = 0$.

Then modulo the five-term relation, we have

\[[f] \equiv -[g] + [fg] - \left[\frac{1 - f}{1 - fg}\right] + \left[\frac{1 - (1 - g)}{1 - fg}\right].\]

Each rational function appearing on the right has numerator of degree $\leq n - 1$ and denominator of degree $\leq n$. Moreover, if $B$ has degree $\leq n - 1$, then all terms on the right have both numerator and denominator of degree $\leq n - 1$. Therefore, we have reduced any rational function with numerator and denominator of degrees $\leq (n, n)$ to a combination of rational functions with degrees $\leq (n - 1, n)$, and so on. 

Iterating this procedure, we reduce to degree $(0, 1)$, i.e., to linear functions (possibly after inversion). Note that the number of applications of the five-term relation needed grows exponentially with the degree $n$; more precisely, it is at most $(1 + \sqrt{2})^{2n}/4$.

(2) By what we have just proved, any element $\xi(t) \in \mathbb{Z}[\mathbb{C}(t)]$ can be written modulo the five-term relation as $\xi_0 + \sum n_i [\varphi_i(t)]$, where $\xi_0 \in \mathbb{Z}[\mathbb{C}]$, $n_i \in \mathbb{Z}$ and the $\varphi_i$ are non-constant linear functions of $t$. We can write each $\varphi_i(t)$ as

\[\frac{t - c_i}{c_i' - c_i}\]

with $c_i, c_i' \in \mathbb{C}$ distinct and (since we may replace $[\varphi_i(t)]$ by $-[1 - \varphi_i(t)]$ modulo the five-term relation) $0 \leq \arg(c_i' - c_i) < \pi$.

The derivative of $D(\varphi_i(t))$ is proportional to

\[(t - c_i)^{-1} \log|t - c_i'| - (t - c_i')^{-1} \log|t - c_i|,\]

and since these functions are linearly independent for different $i$ (as one sees by examining their singularities), we deduce that $D(\xi(t))$ is constant if and only if $n_i = 0$ for all $i$, i.e., if $\xi(t) \equiv \xi_0$ modulo the five-term relation, as claimed.

This proof also shows that every element of $\mathbb{Z}[\mathbb{C}(t)] / (\text{5-term relation})$ has a unique representative of the form $\xi_0 + \sum n_i [a_i t + b_i]$ with $0 \leq \arg(a_i) < \pi \blacksquare$.

Volumes of 3-manifolds

We work in the upper half-space model of hyperbolic 3-space. An ideal tetrahedron is a tetrahedron whose vertices are all in

\[\partial \mathbb{H}^3 = \mathbb{C} \cup \{\infty\} = \mathbb{P}^1(\mathbb{C}).\]

Let $\Delta$ be such a tetrahedron. Although the vertices are at infinity, the (hyperbolic) volume is finite. It is given by

\[\operatorname{Vol}(\Delta) = D(z_0, z_1, z_2, z_3) \]

where $z_0, \dots, z_3 \in \mathbb{C}$ are the vertices of $\Delta$ and $D$ is the function defined earlier. 

In the special case that three of the vertices of $\Delta$ are $\infty$, $0$, and $1$, the previous equation reduces to the formula (due essentially to Lobachevsky) $\operatorname{Vol}(\Delta) = D(z)$.

Less obvious is that (possibly after removing from $M$ a finite number of closed geodesics) there is always a triangulation into ideal tetrahedra. The part of such a tetrahedron going out towards a vertex at infinity will then either tend to a cusp of $M$ or else spiral in around one of the deleted curves.

Let these tetrahedra be numbered $\Delta_1, \dots, \Delta_n$ and assume (after an isometry of $\mathbb{H}^3$ if necessary) that the vertices of $\Delta_\nu$ are at $\infty$, $0$, $1$, and $z_\nu$. Then

\[\operatorname{Vol}(M) = \sum_{\nu=1}^n \operatorname{Vol}(\Delta_\nu) = \sum_{\nu=1}^n D(z_\nu).\]

From the work of Jørgensen and Thurston, one knows that the set of volumes of hyperbolic 3-manifolds $\operatorname{Vol}$ is a countable and well-ordered subset of $\mathbb{R}^+$ (i.e., every subset has a smallest element) of order type $\omega^{\omega}$.

The formula for the volume as it stands says nothing about this set since any real number can be written as a finite sum of values $D(z)$ with $z \in \mathbb{C}$. However, the parameters $z_\nu$ of the tetrahedra triangulating a complete hyperbolic 3-manifold satisfy an extra relation, namely

\[\sum_{\nu=1}^n z_\nu \wedge (1 - z_\nu) = 0.\]

This is a consequence of work of Neumann and Zagier, but the proof of the main theorem is a rather involved combinatorial argument that I won't be able to do justice to. The set of numbers $\sum_{\nu=1}^n D(z_\nu)$ with $z_{\nu}$ satisfying the above equation is countable, essentially because writing out the relations forces all the numbers involved to be algebraic.

Values of zeta functions

In this section we will see how to use the above techniques to compute special values of zeta functions of number fields, a huge area of interest in number theory.

Let $\mathbb{F}:=\mathbb{Q}(\alpha)$ be a number field. The minimal polynomial $f$ of $\alpha$, which is irreducible over $\mathbb{Q}$, in general becomes reducible over $\mathbb{R}$, where it splits into $r_1$ linear and $r_2$ quadratic factors (thus $r_1 \geq 0$, $r_2 \geq 0$, $r_1 + 2r_2 = N$). It also in general becomes reducible when it is reduced modulo a prime $p$, but if $p \nmid d_f$ then its irreducible factors modulo $p$ are all distinct, say $r_{1,p}$ linear factors, $r_{2,p}$ quadratic ones, etc.\ (so $r_{1,p} + 2r_{2,p} + \cdots = N$). Then $\zeta_F(s)$ is the Dirichlet series given by an Euler product

\[\prod_p Z_p(p^{-s})^{-1}\]

where $Z_p(t)$ for $p \nmid d_f$ is the monic polynomial

\[(1 - t)^{r_{1,p}} (1 - t^2)^{r_{2,p}} \cdots\]

of degree $N$, and $Z_p(t)$ for $p \mid d_f$ is a certain monic polynomial of degree $\leq N$. Thus $(r_1, r_2)$ and $\zeta_F(s)$ encode the information about the behavior of $f$ (and hence $F$) over the real and $p$-adic numbers, respectively.

Let $\mathcal{O}$ denote the ring of integers of $F$ (this is the $\mathbb{Z}$-lattice in $\mathbb{C}$ spanned by $1$ and $\sqrt{-a}$ or $(1+\sqrt{-a})/2$, depending whether $d = -4a$ or $d = -a$). Then the group $\Gamma = \mathrm{SL}_2(\mathcal{O})$ is a discrete subgroup of $\mathrm{SL}_2(\mathbb{C})$ and therefore acts on hyperbolic space $\mathbb{H}^3$ by isometries. A classical result of Humbert gives the volume of the quotient space $\mathbb{H}^3/\Gamma$ as

\[\frac{|d|^{3/2} \zeta_F(2)}{4\pi^2}.\]

On the other hand, $\mathbb{H}^3/\Gamma$ (or, more precisely, a certain covering of it of low degree) can be triangulated into ideal tetrahedra with vertices belonging to $\mathbb{P}^1(F) \subset \mathbb{P}^1(\mathbb{C})$, and this leads to a representation

\[\zeta_F(2) = \frac{\pi^2}{3|d|^{3/2}} \sum_\nu n_\nu D(z_\nu)\]

with $n_\nu \in \mathbb{Z}$ and $z_\nu \in F$ itself rather than in the much larger field $\mathbb{Q}(e^{2\pi i n/|d|})$.

If $r_2 = 1$ but $N > 2$, then one can again associate to $F$ (in many different ways) a discrete subgroup $\Gamma \subset \mathrm{SL}_2(\mathbb{C})$ such that $\mathrm{Vol}(\mathbb{H}^3/\Gamma)$ is a rational multiple of

\[\frac{|d|^{1/2} \zeta_F(2)}{\pi^{2N - 2}}.\]

This manifold $\mathbb{H}^3/\Gamma$ is now compact, so the decomposition into ideal tetrahedra is a little less obvious than in the case of imaginary quadratic $F$, but by decomposing into non-ideal tetrahedra (tetrahedra with vertices in the interior of $\mathbb{H}^3$) and writing these as differences of ideal ones, it was shown that the volume is an integral linear combination of values of $D(z)$ with $z$ of degree at most 4 over $F$.

For $F$ completely arbitrary, there is still a similar statement, except that now one gets discrete groups $\Gamma$ acting on $\mathbb{H}_3^{r_2}$. The final result is that

\[\frac{|d|^{1/2} \cdot \zeta_F(2)}{\pi^{2(r_1 + r_2)}}\]

is a rational linear combination of $r_2$-fold products

\[D(z^{(1)}) \cdots D(z^{(r_2)})\]

with each $z^{(i)}$ of degree $\leq 4$ over $F$ (more precisely, over the $i$th complex embedding $F^{(i)}$ of $F$, i.e., over the subfield $\mathbb{Q}(\alpha^{(i)})$ of $\mathbb{C}$, where $\alpha^{(i)}$ is one of the two roots of the $i$th quadratic factor of $f(x)$ over $\mathbb{R}$).

We now come full circle and use the machinery we developed to produce more dilogarithm identities. For this purpose, it is convenient to think of \( L \) or \( D \) as functions of real or complex oriented 3-simplices. By an oriented \( n \)-simplex in \( \mathbb{P}^1(\mathbb{C}) \), we mean an \( (n+1) \)-tuple of points in \( \mathbb{P}^1(\mathbb{C}) \) together with an ordering up to even permutations; more precisely, such a simplex has the form \( [x_0,\dots,x_n] \) with \( x_j \in \mathbb{P}^1(\mathbb{C}) \) and with the convention that

\[[x_{\pi(0)},\dots,x_{\pi(n)}] = \operatorname{sgn}(\pi)[x_0,\dots,x_n] \quad \text{for } \pi \in S_{n+1}.\]

Let \( C_n \) denote the free abelian group on oriented \( n \)-simplices. There are boundary maps \( \partial : C_n \to C_{n-1} \) defined by the usual formula

\[\partial([x_0,\dots,x_n]) = \sum_{i=0}^n (-1)^i [x_0,\dots,\hat{x}_i,\dots,x_n],\]

and the sequence (with \( \varepsilon([x]) = 1 \))

\[\cdots \longrightarrow C_4 \xrightarrow{\partial} C_3 \xrightarrow{\partial} C_2 \xrightarrow{\partial} C_1 \xrightarrow{\partial} C_0 \xrightarrow{\varepsilon} \mathbb{Z} \longrightarrow 0\]

is exact.

The function \( D : \mathbb{C} \to \mathbb{R} \) defines a function \( D : C_3 \to \mathbb{R} \) which associates to a 3-simplex in \( \mathbb{P}^1(\mathbb{C}) \) the value of \( D \) on the cross-ratio of its vertices,

\[D([a,b,c,d]) = D\left( \frac{(a-d)(b-c)}{(a-c)(b-d)} \right),\]

as in Chapter I. This is well-defined (i.e., transforms under the action of \( \pi \in S_4 \) by \( \operatorname{sgn}(\pi) \)) and is 0 on \( \partial(C_4) \) by the five-term relation, since the element \( V(x,y) \) in (17) is simply the boundary of the 4-simplex \( [\infty,0,x,1,y^{-1}] \) and any 4-simplex is equivalent to such a one under the action of \( \mathrm{PGL}_2(\mathbb{C}) \) on \( C_3 \). (Notice that \( D \) is invariant under the action of \( \mathrm{PGL}_2(\mathbb{C}) \) on \( C_3 \), since the cross-ratio is.) Because of the exactness of (22), we can say equivalently that \( D \) vanishes on \( \ker(C_3 \xrightarrow{\partial} C_2) \), or that it factors through \( \partial \): if we define a map

\[\tilde{D} : C_2 \to \mathbb{R} \quad \text{by} \quad \tilde{D}([a,b,c]) = -D([a,b,c,\infty]),\]

(here ``\( \infty \)'' could be replaced by any other fixed base-point \( x_0 \in \mathbb{P}^1(\mathbb{C}) \)), then for every oriented 3-simplex \( [a,b,c,d] \in C_3 \) we have

\[\tilde{D}(\partial([a,b,c,d])) = \tilde{D}(-[a,b,c] + [a,b,d] - [a,c,d] + [b,c,d])\]

\[= D([a,b,c,\infty]) - D([a,b,d,\infty]) + D([a,c,d,\infty]) - D([b,c,d,\infty])\]

\[= D([a,b,c,d]) - \partial([a,b,c,d,\infty]) = D([a,b,c,d])\]

and hence \( D = \tilde{D} \circ \partial \).

We can think of an element \( \xi \in \ker(C_3 \xrightarrow{\partial} C_2) \) as a closed, triangulated, oriented near-3-manifold \( M \), smooth except possibly at its vertices (it is a union of oriented tetrahedra glued to each other along their faces, and is hence automatically smooth on the interior of its 3-, 2- or 1-simplices, while at a vertex its topology is that of a cone on some compact oriented surface), together with a map \( \varphi \) from the vertices of \( M \) to \( \mathbb{P}^1(\mathbb{C}) \). Any such element \( \xi = [\sigma_i] \) gives an identity

\[D(M, \varphi) := \sum_i D(\sigma_i) = 0\]

among values of the Bloch-Wigner dilogarithm. This identity can be written explicitly as a combination of five-term equations by the calculation above (just replace each simplex \( \sigma = [a,b,c,d] \) of \( M \) by \( [\sigma,\infty] := [a,b,c,d,\infty] \)).

We can also perform the same construction over \( \mathbb{R} \), starting from a triangulated near-3-manifold \( M \) and a map \( \varphi \) from its 0-skeleton to \( \mathbb{P}^1(\mathbb{R}) \); then the element \( L(M,\varphi) \) defined as the sum over the 3-simplices \( \sigma \) of \( M \) of the value of \( L \) at the cross-ratio of the images of the vertices of \( \sigma \) under \( \varphi \) will be an integral multiple of \( \pi^2/2 \) by virtue of the functional equation of the Rogers dilogarithm.

This is only the beginning of a what promises to be a deep and beautiful story that runs through many other areas of mathematics, most notably that one can define something called a Bloch group by imposing certain 5-term identities analogous to those satisfied by the dilogarithm on the group of formal sums of elements of $\mathbb{F}^{\times}$. This has, conjecturally, the same order as some important things in algebraic $K-$theory, and the various generalisations of the dilogarithm also feature in the tower of conjectures that follow from this one. Much has been conjectured, and progress seems to have been rather slow in the area, so perhaps the interested reader will have something to explore in their free time. For more details on this tantalising mystery, see the various papers by Zagier.