The K2 avatar of the higher Weil-Petersson form

Definitions and mathematical statement

Let $S$ be a closed oriented genus-$g\geq2$ surface and $H_n$ an $SL(n,{\mathbb{R}})$ Hitchin component. The source writes $A={\mathbb{Q}}(H_n)$ and a central extension $1\to K_2(A)\to E(n,A)\to SL(n,A)\to1$, and asks whether the logarithmic differential of the resulting surface class is a Weil--Petersson form. A precise corrected version first chooses a smooth irreducible algebraic moduli model $Y$ over a characteristic-zero field $k$, containing the intended Hitchin locus in its real points, and a generically descended tautological representation over $A=k(Y)$. An arbitrary constant $A$-valued representation would not represent the varying moduli symplectic form. A coarse character variety need not itself carry such a universal matrix representation without a framing, stack, or field extension.

For a specified central extension $1\to K_2(A)\to\widetilde G_A\to SL(n,A)\to1$, choose generators $a_i,b_i$ with $\prod_i[a_i,b_i]=1$ and lift their images under $\rho$. The product of lifted commutators gives \[ q_\rho=\prod_{i=1}^g[\widetilde\rho(a_i),\widetilde\rho(b_i)]\in K_2(A), \] independent of the central choices of lifts, with orientation/sign conventions fixed. Over a field in the stable range $n\geq3$, a Steinberg extension realizes the usual $K_2$ kernel. Rank two needs an unstable extension or a specified compatible stabilization; the displayed exact sequence cannot silently be assumed to have the same stable kernel. The source's symbol $E(n,A)$ is treated as its extension notation, not automatically the customary elementary matrix group.

For Milnor symbols the logarithmic differential is \[ d\log\{f,g\}=\frac{{\,\mathrm{d}} f}{f}\wedge\frac{{\,\mathrm{d}} g}{g}\in\Omega^2_{A/k}. \] Initially this is a rational algebraic form, not automatically a regular differential form on all of $Y$ or $H_n$. Trivial codimension-one tame symbols of the $K_2$ class supply the unramified regularity input in the smooth model. The aid reports that Fock--Goncharov's 2006 motivic surface class gives the generalized Weil--Petersson/Goldman form after matching the algebraic model, central extension, orientation, and invariant quadratic-form normalization. The three stated checks are generic tautological descent, codimension-one regularity, and normalization compatibility; their verification for a literal reading of the AIM notation is not supplied.

With trace pairings $B_n(U,V)=\operatorname{Tr}_n(UV)$, the Goldman form is \[ \omega_n([\alpha],[\beta])=\langle B_n(\alpha\smile\beta),[S]\rangle \] on the smooth representation-moduli locus. For the principal Fuchsian embedding $F_n$ induced by the irreducible $n$-dimensional representation of $SL(2)$, the aid gives the exact test \[ F_n^*\omega_n=\frac{n(n^2-1)}6\,\omega_2. \] Identifying $\omega_2$ with the conventionally normalized Weil--Petersson form requires its scalar normalization as well. Thus the literal question is not a normalization-free equality, and a higher-rank form cannot simply restrict with factor one under all trace conventions. The corrected motivic comparison is a supplied literature assertion with missing full theorem details, not an independently verified identity for every possible model or extension.

Short English statement

The supplied corrected answer identifies the surface $K_2$ class with the Goldman/generalized Weil--Petersson form only after specifying a tautological algebraic model, regularity, and matching extension normalization. Rank two needs an additional extension caveat. The principal Fuchsian restriction has trace-normalization factor $n(n^2-1)/6$, so an unqualified scalar-free equality would be misleading.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-topology-notes.json, notes[276]; canonical tag "question".
• AIM workshop Representations of surface groups, Question 4.7 attributed to Olivier Guichard, https://aimath.org/WWN/surfacegroups/surfacegroups.pdf. It refers to an omitted preceding question. No complete original literature citation, proposer year, or remarks are supplied.
• Statement status is corrected_verified. Verification reports separating the central-extension display from "SL(n)" and removing the next heading "5 Other questions"; the mathematical reconstruction follows that inherited check.
• Supplied reference: Fock--Goncharov's 2006 motivic surface-class construction, without title, theorem number, or complete authoritative publication details.
• Supplied research attempt 1, registry updated 14 August 2026: partial result, low novelty confidence, machine-generated and not peer reviewed. It supplies the descent/regularity/normalization audit and principal trace-index test.