Kontsevich’s 1½ Logarithm

• Kontsevich’s one-and-a-half logarithm is a family of constructions involving a logarithmic formality morphism in deformation quantization and a truncated logarithmic series in characteristic p.
• It employs logarithmic propagators and a regularized Stokes’ Theorem to define L∞-morphisms that extend from the flat case on ℝᵈ to arbitrary smooth manifolds.
• The construction also appears as a 1-loop correction equating to the square root of the Duflo Jacobian, connecting graph complexes, cluster identities, and regulator formulas.

Kontsevich’s one-and-a-half logarithm is not a single uniformly fixed object across the literature. In deformation quantization, the expression is shorthand for Kontsevich’s conjectural logarithmic version of the formality morphism, obtained by replacing the standard angle-type propagators by logarithmic ones; Alekseev, Rossi, Torossian, and Willwacher proved the existence of this logarithmic formality morphism $U_{\log}$ and its globalization to arbitrary smooth manifolds (Alekseev et al., 2014). In characteristic $p$, often written $1\frac{1}{2}$-logarithm, it denotes the truncated series $\pounds_1(s)=\sum_{1\le i<p}s^i/i$, which appears in infinitesimal dilogarithms, regulators, and cluster identities (Unver, 19 Sep 2025). A further semiclassical interpretation identifies a related logarithmic correction with the $1$-loop factor in Kontsevich’s star product, recovering the square root of the Duflo Jacobian in the linear Poisson case (Cabrera et al., 9 Apr 2026). The shared theme is a logarithmic correction that is intermediate in character: neither a plain logarithm nor an unrelated higher-order term, but a structured refinement with geometric, operadic, and arithmetic consequences.

1. Terminological scope and competing meanings

In the deformation quantization community, “Kontsevich’s one-and-a-half logarithm” is used informally for the logarithmic formality morphism built from logarithmic propagators on compactified configuration spaces of points in the upper half-plane (Alekseev et al., 2014). In arithmetic and cluster-algebra settings, the same expression, or the notation $1\frac12$-logarithm, refers to the truncated logarithmic series

$\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$

defined for a ring of characteristic $p$ (Unver, 19 Sep 2025).

Several later papers use the phrase more loosely. Wang states that the phrase refers to a particular kind of logarithmic singularity or expansion appearing when the Kontsevich–Witten tau-function is rewritten in appropriate variables, especially through Lambert $W$ series and Virasoro-operator transformations (Wang, 2018). Kawazumi and Kuno do not use the phrase explicitly, but their “logarithms of Dehn twists” are presented as a concrete incarnation of the pattern of passing from geometric or group-theoretic data to Lie-theoretic derivations in Kontsevich’s formal symplectic geometry (Kawazumi et al., 2010). Willwacher likewise does not name the phrase explicitly, but describes the graph-complex framework in which a distinguished logarithm-type element lives inside $H^0(GC_2)\cong \mathfrak{grt}_1$ (Willwacher, 2010).

This distribution of meanings shows that the term is context-sensitive. A plausible implication is that “one-and-a-half logarithm” functions less as a single definition than as a recurring label for logarithmic corrections that are intermediate between standard logarithmic objects and more elaborate polylogarithmic, graph-theoretic, or semiclassical structures.

2. The logarithmic formality morphism in deformation quantization

For a smooth manifold $p$0, Kontsevich’s Formality Theorem provides an $p$1-quasi-isomorphism from polyvector fields to multidifferential operators. In the flat case $p$2, the Taylor components are given by sums over admissible graphs, with coefficients obtained from configuration-space integrals over $p$3, where

$p$4

In the standard construction, each edge contributes the angle-type $p$5-form

$p$6

and graph weights arise by integrating the product of these forms over compactified configuration spaces (Alekseev et al., 2014).

Kontsevich’s logarithmic variant replaces the angle propagator by

$p$7

leading to logarithmic weights

$p$8

The phrase “one-and-a-half logarithm” is used there to emphasize that the construction is not simply “everything is logarithmic.” Each edge contributes a $p$9-term, but the argument mixes $1\frac{1}{2}$0 and $1\frac{1}{2}$1, producing a holomorphic/anti-holomorphic hybrid; moreover, near type I boundary strata the logarithmic forms develop $1\frac{1}{2}$2-type singularities. The paper states explicitly that this hybrid behavior is part of what motivates the informal label (Alekseev et al., 2014).

The main analytic obstacle was open since 1999: the forms $1\frac{1}{2}$3 do not automatically extend smoothly to the compactification, and near a collapsing cluster $1\frac{1}{2}$4 one obtains a local expansion

$1\frac{1}{2}$5

with $1\frac{1}{2}$6 basic for a circle action rotating the cluster. Alekseev, Rossi, Torossian, and Willwacher resolve this by introducing local torus actions on charts of the compactified configuration spaces and proving a Regularized Stokes’ Theorem for top-minus-one degree forms with such boundary singularities. In particular, if $1\frac{1}{2}$7 is regularizable on a compact manifold with corners $1\frac{1}{2}$8, then

$1\frac{1}{2}$9

This regularized Stokes formula restores the standard graphical proof of the $\pounds_1(s)=\sum_{1\le i<p}s^i/i$0-relations. The paper proves that top-degree logarithmic forms are regular, so the weights $\pounds_1(s)=\sum_{1\le i<p}s^i/i$1 exist; that only the expected type I two-point collapses and type II operadic boundary terms survive; and that higher interior collapses vanish, yielding the logarithmic analogue of Kontsevich’s Vanishing Lemma. The resulting structure maps define an $\pounds_1(s)=\sum_{1\le i<p}s^i/i$2-morphism

$\pounds_1(s)=\sum_{1\le i<p}s^i/i$3

whose first Taylor component is the Hochschild–Kostant–Rosenberg map. The same paper proves the requisite globalization conditions, including vanishing on vector fields and on linear vector fields, and concludes that $\pounds_1(s)=\sum_{1\le i<p}s^i/i$4 globalizes from $\pounds_1(s)=\sum_{1\le i<p}s^i/i$5 to any smooth manifold (Alekseev et al., 2014).

3. The characteristic-$\pounds_1(s)=\sum_{1\le i<p}s^i/i$6 $\pounds_1(s)=\sum_{1\le i<p}s^i/i$7-logarithm

In arithmetic usage, Kontsevich’s one-and-a-half logarithm is the truncated series

$\pounds_1(s)=\sum_{1\le i<p}s^i/i$8

defined for $\pounds_1(s)=\sum_{1\le i<p}s^i/i$9 an odd prime and $1$0 a ring of characteristic $1$1 (Unver, 19 Sep 2025). It is the ordinary logarithm power series truncated at degree $1$2, with the sign convention determined in the cited papers. The characteristic-$1$3 setting is essential: the coefficient $1$4 is unavailable, so the series beyond degree $1$5 is no longer meaningful as a formal series in $1$6. The paper therefore presents $1$7 as the largest part of the logarithmic series that survives modulo $1$8 (Unver, 19 Sep 2025).

This function enters the characteristic-$1$9 infinitesimal dilogarithm. For $1\frac12$0 and $1\frac12$1, with

$1\frac12$2

the infinitesimal dilogarithm is

$1\frac12$3

Thus $1\frac12$4 is the logarithmic factor inside a characteristic-$1\frac12$5 dilogarithmic object (Unver, 19 Sep 2025).

Ünver’s “The Chow-Kontsevich dilogarithm” presents a closely related variant. There the starting point is again Kontsevich’s function

$1\frac12$6

which the paper says Kontsevich called the $1\frac12$7-logarithm because it satisfies the four-term functional equation. Ünver uses it to construct a characteristic-$1\frac12$8 additive dilogarithm

$1\frac12$9

and then a curve-level regulator, the Chow-Kontsevich dilogarithm, from triples of functions on a curve to the ground field (Ünver, 2023).

A common misconception is to identify this characteristic-$\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$0 object with the logarithmic propagator construction of deformation quantization. The two are distinct. One is a truncated scalar-valued series in characteristic $\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$1; the other is a family of configuration-space differential forms defining a logarithmic $\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$2-morphism. The coincidence lies in the label and in the logarithmic-correction motif, not in a direct equality of definitions.

4. Functional equations, cluster identities, and regulators

The characteristic-$\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$3 $\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$4-logarithm is governed by strong functional identities. In “Infinitesimal Dilogarithm Satisfies Cluster Identities,” the infinitesimal dilogarithm $\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$5, hence $\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$6, is shown to satisfy cluster identities attached to $\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$7-periodic mutation sequences in cluster patterns. For such a period, if $\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$8, the paper derives

$\pounds_1(s)=\sum_{1\le i<p}\frac{s^i}{i},$9

and proves that these cluster identities are consequences of the pentagon relation in the infinitesimal setting (Unver, 19 Sep 2025).

A particularly prominent consequence is Kontsevich’s four-term functional equation. From the $p$0 cluster period, the paper obtains

$p$1

The same paper also records basic involutive identities such as

$p$2

again reflecting logarithmic symmetry in characteristic $p$3 (Unver, 19 Sep 2025).

Ünver’s regulator theory places these identities in a motivic and curve-theoretic setting. The Chow-Kontsevich dilogarithm is a map

$p$4

for a smooth projective curve $p$5, built from local residues and the characteristic-$p$6 additive dilogarithm. On $p$7, the regulator specializes to the Kontsevich function itself: $p$8 The paper also defines an infinitesimal invariant of cycles

$p$9

and proves that if two irreducible cycles are equivalent modulo $W$0, then their values under both the classical infinitesimal regulator $W$1 and the characteristic-$W$2 regulator $W$3 agree (Ünver, 2023).

These results identify the characteristic-$W$4 one-and-a-half logarithm as a regulator kernel rather than merely a formal power series. It is the part of the infinitesimal dilogarithm that survives in residue formulas, Bloch-group expressions, and cluster-theoretic functional equations.

5. The $W$5-loop correction, symplectic groupoids, and the Duflo factor

A different but closely related interpretation appears in semiclassical analysis. Cabrera and Ledesma describe Kontsevich’s one-and-a-half logarithm as the subtle extra factor in the $W$6-loop part of the star product formula. In the linear case $W$7, they state that it is exactly the square root of the Duflo Jacobian

$W$8

and more generally it is built from configuration-space integrals of logarithmic kernels, namely wheels (Cabrera et al., 9 Apr 2026).

Their framework is a symplectic groupoid $W$9 equipped with a half-density along multiplication. Given a non-vanishing half-density $H^0(GC_2)\cong \mathfrak{grt}_1$0 on $H^0(GC_2)\cong \mathfrak{grt}_1$1, the canonical enhancement is

$H^0(GC_2)\cong \mathfrak{grt}_1$2

where $H^0(GC_2)\cong \mathfrak{grt}_1$3 is the Liouville half-density. The main existence theorem states that $H^0(GC_2)\cong \mathfrak{grt}_1$4 is associative and that every other associative enhancement has the form

$H^0(GC_2)\cong \mathfrak{grt}_1$5

with $H^0(GC_2)\cong \mathfrak{grt}_1$6 satisfying a multiplicative $H^0(GC_2)\cong \mathfrak{grt}_1$7-cocycle condition. Equivalence classes of nonvanishing enhancements are in bijection with $H^0(GC_2)\cong \mathfrak{grt}_1$8 (Cabrera et al., 9 Apr 2026).

Applied to Kontsevich’s star product, the leading loop factor in the Fourier-integral-operator form is

$H^0(GC_2)\cong \mathfrak{grt}_1$9

and the corresponding enhancement satisfies

$p$00

The term $p$01 is the logarithmic defect after subtracting the canonical Jacobian contribution $p$02. The paper interprets this as the one-and-a-half logarithm: an additive $p$03-cocycle measuring the deviation of the analytic $p$04-loop factor from the canonical Liouville enhancement (Cabrera et al., 9 Apr 2026).

For coordinate Poisson manifolds, Cabrera and Ledesma prove that this formal $p$05-cocycle is exact, so the Kontsevich enhancement is equivalent to the canonical one. In the linear Poisson case $p$06, the statement is sharper: with

$p$07

they show

$p$08

Hence the Duflo square-root factor is not an ad hoc correction but the canonical Jacobian attached to symplectic groupoid multiplication (Cabrera et al., 9 Apr 2026).

A frequent misunderstanding is to treat this $p$09-loop logarithm as unrelated to the earlier propagator-based logarithmic formality morphism. The papers do not identify them outright, but they do place them in the same configuration-space and wheel-graph environment. A plausible implication is that the two viewpoints describe complementary aspects of the same logarithmic sector of deformation quantization: one at the level of $p$10-morphisms, the other at the level of semiclassical amplitudes.

6. Graph complexes, formal symplectic geometry, and other later extensions

Willwacher’s identification

$p$11

places distinguished logarithm-type constructions inside the zeroth cohomology of Kontsevich’s graph complex. The paper states that a distinguished element of $p$12, described informally in later expositions as Kontsevich’s one-and-a-half logarithm, is represented by a graph cocycle whose representatives have a nonzero coefficient in front of the wheel graph with $p$13 spokes, matching the leading Lie word $p$14 of the corresponding $p$15 (Willwacher, 2010). This provides a graph-complex location for logarithmic phenomena already visible in deformation quantization and Duflo theory.

Kawazumi and Kuno give a topological analogue. For a symplectic expansion $p$16 of the completed group ring of a surface group, they define

$p$17

where $p$18 and $p$19 is cyclic symmetrization. For a simple closed curve $p$20, the Dehn twist satisfies

$p$21

Their paper presents this as a logarithm-type passage from loop data to derivations in Kontsevich’s formal symplectic Lie algebra, although it explicitly notes that the phrase “one-and-a-half logarithm” is not itself used in the text (Kawazumi et al., 2010).

Wang’s work on the Kontsevich–Witten and Hodge tau-functions uses the phrase for a different analytic pattern: a logarithmic singularity structure tied to Lambert $p$22, branch differences $p$23, and Virasoro-operator transformations. The paper proves that the multiplicative constant in Alexandrov’s conjectural formula is

$p$24

by identifying the Lambert $p$25-based series with the Virasoro-only series. There the “one-and-a-half logarithm” is associated with the branch-difference structure of Lambert $p$26 rather than with characteristic-$p$27 truncations or logarithmic propagators (Wang, 2018).

These later usages should be read cautiously. They do not erase the two principal definitions; rather, they extend the phrase to settings where logarithmic corrections, infinitesimal generators, and wheel-type graph structures play analogous roles. This suggests a broader encyclopedia-level conclusion: Kontsevich’s one-and-a-half logarithm is best understood as a family of related logarithmic constructions centered on three technically precise cores—the logarithmic formality morphism, the characteristic-$p$28 truncated logarithm $p$29, and the $p$30-loop/Duflo correction—each with its own domain, but all shaped by Kontsevich’s configuration-space, graph-theoretic, and formal-geometric methods.