Infinitesimal Dilogarithm and Algebraic Regulators

• Infinitesimal dilogarithm is a family of additive weight-two regulators defined algebraically using normalized logarithms, coefficient extraction, and Bloch boundaries.
• It satisfies the five-term (pentagon) relation and admits a weight decomposition under scaling, bridging classical dilogarithms with additive K-theory and cyclic homology.
• Recent studies demonstrate its role in cluster identities, regulator maps on curves, and motivic cohomology via explicit formulas over truncated polynomial rings.

The infinitesimal dilogarithm is a family of additive weight-two regulators attached to nilpotent thickenings such as truncated polynomial rings $k_m:=k[t]/(t^m)$ and square-zero extensions. In contrast with the classical dilogarithm, which is multivalued and analytic, infinitesimal dilogarithms are defined algebraically from normalized logarithms, coefficient extraction, Bloch boundaries, and residue constructions. They satisfy five-term identities in Bloch groups, admit weight decompositions under the scaling action $t\mapsto \lambda t$, and, in recent work, have been shown to satisfy the cluster identities attached to periodic mutation sequences; moreover, those cluster identities follow from the infinitesimal pentagon relation (Unver, 19 Sep 2025, Unver, 2019).

1. Classical antecedents and the additive viewpoint

The classical starting point is the dilogarithm

$$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$

defined for $|z|<1$ and analytically continued elsewhere. Two single-valued normalizations that organize its functional equations are the Rogers dilogarithm

$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$

and the Bloch–Wigner dilogarithm

$$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$

These satisfy the five-term relation, which in Bloch-group form is governed by

$$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$

This relation is central to the weight-two Bloch complex

$$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$

and to classical regulator maps (Unver, 19 Sep 2025, Unver, 2019).

The infinitesimal viewpoint replaces analytic continuation and branch choices by algebraic nilpotent thickenings. A basic motivation is the first-order expansion

$$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$

which exhibits the tangent of the classical dilogarithm as a logarithmic differential expression. In the survey literature, additive polylogarithms are therefore treated as tangent objects to the classical theory, closely related to additive $K$-theory, Cathelineau’s complexes, cyclic homology, and infinitesimal motivic cohomology (Unver, 2019).

This viewpoint also explains why the infinitesimal dilogarithm is not a single universally normalized function. The survey literature explicitly treats “several different versions of the weight two regulator in the infinitesimal setting,” including Cathelineau’s four-term theory, Bloch–Esnault and Ünver regulators over truncated polynomial rings, regulators on curves, and square-zero Bloch regulators (Unver, 2019).

2. Definition over truncated polynomial rings

Let $t\mapsto \lambda t$0 be a field of characteristic $t\mapsto \lambda t$1, let $t\mapsto \lambda t$2, and let $t\mapsto \lambda t$3. For $t\mapsto \lambda t$4,

$t\mapsto \lambda t$5

The normalized logarithm is

$t\mapsto \lambda t$6

If $t\mapsto \lambda t$7, write

$t\mapsto \lambda t$8

With

$t\mapsto \lambda t$9

every $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$0 decomposes uniquely as $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$1 with $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$2 and $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$3. For integers $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$4, the infinitesimal dilogarithm is defined by

$$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$5

It depends only on the truncation modulo $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$6, hence factors through $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$7 (Unver, 19 Sep 2025).

For $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$8, there is a single weight $$\operatorname{Li}_2(z):=\sum_{1\le n}\frac{z^n}{n^2},$$9, and one recovers the basic explicit formula

$|z|<1$0

For $|z|<1$1, there are two weights: $|z|<1$2 and

$|z|<1$3

In general there are $|z|<1$4 distinct weights $|z|<1$5 with $|z|<1$6 (Unver, 19 Sep 2025).

Variant	Setting	Formula
$|z|<1$7	$|z|<1$8, $|z|<1$9	$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$0
$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$1	$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$2	$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$3
$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$4	$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$5, $$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$6	$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$7
$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$8	$$\mathrm{L}(z):=\operatorname{Li}_2(z)+\frac12\log(z)\log(1-z)$$9, $$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$0	$$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$1

A structural feature is the scaling action of $$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$2 on $$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$3,

$$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$4

under which the infinitesimal dilogarithm has weight $$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$5: $$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$6 This weight decomposition is fundamental in both the Bloch-group description and the later cluster-theoretic reduction (Unver, 19 Sep 2025).

3. Bloch groups, the pentagon relation, and weight decomposition

For a local ring $$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$7, the Bloch group $$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$8 is the quotient of the free abelian group $$D(z):=\Im(\operatorname{Li}_2(z))+\arg(1-z)\log|z|.$$9 by the subgroup generated by the five-term relations

$$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$0

with $$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$1. The associated Bloch complex is

$$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$2

The infinitesimal dilogarithm is realized on this complex by logarithmic coefficient functionals

$$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$3

For $$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$4,

$$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$5

This depends only on $$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$6, equals $$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$7, and induces

$$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$8

In particular, $$[a]-[b]+[b/a]-\left[\frac{1-a^{-1}}{1-b^{-1}}\right]+\left[\frac{1-a}{1-b}\right].$$9 satisfies the pentagon relation (Unver, 19 Sep 2025).

The survey literature places this in a broader additive-regulator picture. Over truncated rings, Bloch–Esnault and Ünver construct maps $$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$0 with the same five-term behavior and the same scaling decomposition, and for $$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$1 one obtains the explicit regulator

$$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$2

Collectively, these maps induce an isomorphism

$$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$3

and the recent cluster-identity paper states the corresponding weight-two regulator consequence as

$$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$4

being an isomorphism on the infinitesimal part of $$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$5 (Unver, 2019, Unver, 19 Sep 2025).

A key structural theorem in the cluster setting is the infinitesimal reduction theorem. If $$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$6 vanishes on constants and satisfies the pentagon relation, then $$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$7 automatically satisfies all cluster period identities. The mechanism uses the infinitesimal part $$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$8, idempotents $$B_2(k)\xrightarrow{\delta}\Lambda^2 k^\times,\qquad \delta([a])=(1-a)\wedge a,$$9 for the scaling-weight decomposition, and the fact that

$$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$0

is induced by $$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$1. This gives a precise sense in which cluster identities reduce to the five-term relation in the infinitesimal setting (Unver, 19 Sep 2025).

4. Cluster identities and periodic mutation sequences

The cluster-theoretic framework fixes a rank $$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$2 cluster pattern

$$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$3

and its associated $$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$4-pattern

$$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$5

With an initial vertex $$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$6, a free $$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$7-pattern, a skew-symmetrizer

$$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$8

and a $$\operatorname{Li}_2(x+\varepsilon y)=\operatorname{Li}_2(x)-\varepsilon\,y\,\frac{\log(1-x)}{x},$$9-periodic mutation sequence

$K$0

one obtains a period identity in wedge form. There exists a proper algebraic set $K$1 such that for admissible $K$2,

$K$3

Here $K$4 denotes evaluation of the rational function $K$5 at $K$6 (Unver, 19 Sep 2025).

From this wedge identity, the main additive cluster identity in characteristic $K$7 is

$K$8

for $K$9, provided $t\mapsto \lambda t$00 for all $t\mapsto \lambda t$01. The flatness condition ensures that the Bloch-complex expressions are defined in units of $t\mapsto \lambda t$02 (Unver, 19 Sep 2025).

The significance of the result is twofold. First, it gives additive analogues of the Rogers dilogarithm identities attached to cluster periods, in the sense of Nakanishi and collaborators. Second, the reduction theorem shows that these identities do not depend on special analytic properties of the chosen regulator: any function on $t\mapsto \lambda t$03 that vanishes on constants and satisfies the infinitesimal pentagon automatically satisfies the same cluster period identities. The paper states this explicitly as a reduction from cluster identities to the pentagon relation, implemented through the weight decomposition of $t\mapsto \lambda t$04 and the isomorphisms furnished by the maps $t\mapsto \lambda t$05 (Unver, 19 Sep 2025).

This also clarifies the algebraic role of normalization. In the classical setting, Rogers’ normalization is used to remove multivaluedness. In the infinitesimal setting, the normalized logarithm $t\mapsto \lambda t$06 and truncation operators replace branch choices, so the identities become algebraic identities in $t\mapsto \lambda t$07 and $t\mapsto \lambda t$08 rather than single-valued analytic continuations (Unver, 19 Sep 2025).

5. Characteristic $t\mapsto \lambda t$09 avatars and Kontsevich-type logarithms

The characteristic $t\mapsto \lambda t$10 theory has two closely related, but notationally distinct, strands in the cited literature. In the cluster-identity paper, for an odd prime $t\mapsto \lambda t$11 and a ring $t\mapsto \lambda t$12 of characteristic $t\mapsto \lambda t$13, Kontsevich’s one-and-a-half logarithm is written

$t\mapsto \lambda t$14

For $t\mapsto \lambda t$15, one sets

$t\mapsto \lambda t$16

and defines the characteristic-$t\mapsto \lambda t$17 infinitesimal dilogarithm

$t\mapsto \lambda t$18

It also admits the Bloch-complex expression

$t\mapsto \lambda t$19

which is the exact analogue of the characteristic $t\mapsto \lambda t$20 Bloch-complex formula (Unver, 19 Sep 2025).

In this setting, cluster period identities persist in characteristic $t\mapsto \lambda t$21: $t\mapsto \lambda t$22 The paper also records explicit consequences. Mutation involutivity yields

$t\mapsto \lambda t$23

For the $t\mapsto \lambda t$24 cluster algebra one obtains a pentagon-type identity, and after the substitutions $t\mapsto \lambda t$25, $t\mapsto \lambda t$26, together with $t\mapsto \lambda t$27, this becomes the Kontsevich four-term identity; at first order $t\mapsto \lambda t$28, it linearizes to the classical four-term identity for $t\mapsto \lambda t$29. A further explicit periodic identity is computed for the $t\mapsto \lambda t$30 cluster algebra (Unver, 19 Sep 2025).

A second characteristic-$t\mapsto \lambda t$31 line appears in the Chow–Kontsevich dilogarithm paper. There, a variant of the one-and-a-half logarithm is denoted

$t\mapsto \lambda t$32

and it is used to define

$t\mapsto \lambda t$33

The paper emphasizes that over truncated polynomial rings in characteristic $t\mapsto \lambda t$34, dilogarithmic phenomena split into two independent pieces, one “classical additive” and one “purely characteristic $t\mapsto \lambda t$35.” It further proves, for $t\mapsto \lambda t$36 an algebraic closure of $t\mapsto \lambda t$37, that

$t\mapsto \lambda t$38

induces an isomorphism on the infinitesimal part of $t\mapsto \lambda t$39, explaining why both regulators are needed in characteristic $t\mapsto \lambda t$40 (Ünver, 2023).

The same paper constructs the Chow–Kontsevich dilogarithm

$t\mapsto \lambda t$41

for a smooth proper curve $t\mapsto \lambda t$42, defined by

$t\mapsto \lambda t$43

It is well-defined, functorial, vanishes on boundaries, and on $t\mapsto \lambda t$44 satisfies

$t\mapsto \lambda t$45

whenever $t\mapsto \lambda t$46 (Ünver, 2023).

6. Curves, infinitesimal regulators, and motivic significance

The infinitesimal dilogarithm is not confined to the Bloch group of a truncated ring; it also appears as the local input for regulators on curves and for infinitesimal motivic cohomology. For smooth projective curves $t\mapsto \lambda t$47 over $t\mapsto \lambda t$48, one can define regulators

$t\mapsto \lambda t$49

using a sheafified weight-three Bloch complex, the additive dilogarithm maps $t\mapsto \lambda t$50, and a correction 1-form $t\mapsto \lambda t$51 whose residues compensate for choices of liftings. The regulator is given by an explicit sum over closed points,

$t\mapsto \lambda t$52

and is independent of all choices, vanishes on coboundaries, and has $t\mapsto \lambda t$53-weight $t\mapsto \lambda t$54 (Unver, 2020).

The same work proves a generalized reciprocity theorem for cycles. If $t\mapsto \lambda t$55 satisfy the smoothness and normal-crossings hypotheses $t\mapsto \lambda t$56 and are congruent modulo $t\mapsto \lambda t$57, then

$t\mapsto \lambda t$58

for every $t\mapsto \lambda t$59. In this form, infinitesimal dilogarithmic regulators function as deformation invariants of cycles over truncated polynomial rings (Unver, 2020).

A parallel square-zero theory is developed in the infinitesimal Bloch regulator paper. Let $t\mapsto \lambda t$60 be a finite-type $t\mapsto \lambda t$61-scheme, with a smooth reduced subscheme $t\mapsto \lambda t$62 defined by a square-zero ideal sheaf $t\mapsto \lambda t$63 that is locally free on $t\mapsto \lambda t$64. Then there are functorial regulators

$t\mapsto \lambda t$65

and

$t\mapsto \lambda t$66

where $t\mapsto \lambda t$67 is the first André–Quillen homology sheaf. The local input is a generalized additive dilogarithm

$t\mapsto \lambda t$68

depending on a local splitting $t\mapsto \lambda t$69, satisfying the five-term relation and glued globally through explicit homotopies $t\mapsto \lambda t$70. Under the local-freeness hypothesis on $t\mapsto \lambda t$71, $t\mapsto \lambda t$72 is an isomorphism (Unver, 2019).

These constructions establish the motivic role of the infinitesimal dilogarithm. The survey literature describes the infinitesimal complexes $t\mapsto \lambda t$73, $t\mapsto \lambda t$74, and $t\mapsto \lambda t$75 as tangent versions of the Bloch motivic complex, and presents the additive dilogarithms $t\mapsto \lambda t$76, $t\mapsto \lambda t$77, curve regulators, and Bloch regulators as the weight-two infinitesimal counterparts of classical polylogarithmic regulator maps. In this sense, the infinitesimal dilogarithm occupies the interface of Bloch groups, cyclic homology, André–Quillen homology, additive $t\mapsto \lambda t$78-theory, and cluster algebra periodicity (Unver, 2019, Unver, 2019, Unver, 2020).

A persistent organizing principle across these developments is reduction to the five-term relation. In the classical setting, the dilogarithm’s functional equations are expected to be controlled by the pentagon. In the infinitesimal setting, this principle becomes explicit: the regulators are built so that the Bloch boundary, logarithmic coefficient maps, and weight decomposition force their identities—on Bloch groups, on cluster periods, and on curves—to be consequences of the infinitesimal pentagon relation (Unver, 19 Sep 2025).