JT gravity: a bridge between presentations

The missing translation

Jackiw–Teitelboim (JT) gravity is a two-dimensional model of gravity. Its BF gauge-theory presentation fixes defect holonomy and admissible fusion. The hyperbolic-surface presentation supplies additional collision-divisor weights and identities for forgetting a marked defect.

Wei Gu, §4.7.1, explicitly leaves a direct BF or BRST derivation of the weighted string identity open in the preprint submitted on 3 September 2026. The target here is that presentation bridge. The amplitudes to be recovered are already known geometrically; a direct derivation would not by itself establish new dynamics.

A chamber with two boundaries

Take a genus-zero surface with two labelled geodesic boundaries and $k\ge1$ labelled defects of cone angles $2\pi(1-q_i)$ . Assume $q_i>0$ and $Q=\sum_iq_i<1$ , so every cluster can collide. Write

$x=b_1^2/2,\quad y=b_2^2/2,\quad c=4\pi^2, \qquad V_k=V^{\mathrm{wt}}_{0,2;k}(b_1,b_2;\mathbf q).$

The geometric inputs are the unit volume $V_1=1$ , the two soft identities

$V_k(\mathbf q,0)=SV_{k-1},\qquad \left.\partial_{q_k}V_k\right|_{q_k=0} =cQ_{k-1}V_{k-1},$ $Sf=\int_0^x f(t,y)\,dt+\int_0^y f(x,t)\,dt,$

and total deficit degree at most $2k-2$ . These come from the quadratic weighted Weil–Petersson class and geometric forgetful-map identities; see Eberhardt–Turiaciand Gu's equations (4.120), (4.137). They are established inputs, not identities derived from BF theory in this note. The soft limit concerns the normalized marked insertion and polynomial continuation to a chamber face.

One uniqueness argument at every order

The proof of Anagnostou–Mullane–Norbury, Lemma 4.4, already uses soft-face data, symmetry, and a degree bound to determine the cusp polynomial. The argument here adapts that interpolation strategy to arbitrary boundary lengths. This methodological precedent is distinct from the known cusp formula quoted below.

Suppose two symmetric polynomial amplitude families satisfy those inputs. Inductively, their difference $D_k$ and its first normal derivative vanish on every face $q_i=0$ . Hence

$\prod_{i=1}^k q_i^2\mid D_k, \qquad \deg_{\mathbf q}D_k\le2k-2.$

A nonzero multiple has degree at least $2k$ . Therefore the difference is zero. This proves uniqueness uniformly in the number of defects, against every polynomial alternative in the class.

The unique family has an explicit form. Let $\mathcal F_k$ denote all labelled forests on the defect labels, including isolated vertices, and put

$P_m(x,y)=S^m1= \sum_{j=0}^m\binom mj\frac{x^jy^{m-j}}{j!(m-j)!}.$ $V_k=\sum_{F\in\mathcal F_k} c^{|E(F)|}P_{k-1-|E(F)|}(x,y) \prod_iq_i^{\deg_F(i)}.$

To prove the formula, setting a charge to zero leaves exactly the forests where that vertex is isolated, giving string. Taking its first derivative leaves exactly the forests where it is a leaf. Deleting the leaf and summing its possible attachment vertices gives the dilaton factor $cQ_{k-1}$ . A forest has at most $k-1$ edges, giving the degree bound. Uniqueness completes the proof.

$V_2=x+y+cq_1q_2,$ $\begin{aligned} V_3={}&\tfrac12x^2+2xy+\tfrac12y^2\\ &+c(x+y)(q_1q_2+q_1q_3+q_2q_3)\\ &+c^2q_1q_2q_3(q_1+q_2+q_3). \end{aligned}$

At zero boundary lengths, only trees survive. The resulting weighted Cayley formula $c^{k-1}(\prod_iq_i)Q^{k-2}$ is already known: it is Lemma 4.4 of Anagnostou–Mullane–Norbury. It is a control, not a new result claimed here.

The same amplitudes as convex volumes

Let $\Delta=\operatorname{conv}(e_1,\ldots,e_k)$ . The forest polynomial is exactly the volume of the Minkowski sum

$\mathcal P_k=x\Delta+y(-\Delta) +c\sum_{i<j}q_iq_j[e_i,e_j], \qquad \operatorname{Vol}_{A_{k-1}}(\mathcal P_k)=V_k.$

Volume is normalized so a fundamental cell of the difference lattice $A_{k-1}=\{z\in\mathbb Z^k:\sum_i z_i=0\}$ has volume one. Adding a root segment contributes the volume of the projection along that direction. Independent root segments are precisely forests, and their lattice index is one. Quotienting by a forest contracts its components and projects the two simplices to opposite simplices of dimension $k-1-|E(F)|$ .

The volume of two such opposite simplices is $P_m$ : splitting their difference body into orthants gives one product of simplex volumes per orthant. The segment expansion is therefore exactly the forest formula. This uses the classical graphical-zonotope mechanism; the complete elementary argument is in the proof packet.

The subsequent attack proves that every interior member of this family has the full braid normal fan, so it is a genuine moment polytope for the Losev–Manin toric variety. That strengthens the convex statement, but it does not identify the physical symplectic form: the same fan supports many measures.

The toric presentation does not fix the measure

For any $\eta\ge0$ , replace every root-segment length by

$w_{ij}^{(\eta)}=4\pi^2q_iq_j +\eta q_i^2q_j^2(1-q_i-q_j)^2.$

These lengths stay positive. If two adjacent labels $a,b$ are swapped at rank $r$ , the corresponding exposed vertices differ by

$\bigl(w_{ab}^{(\eta)}+x\mathbf1_{r=1} +y\mathbf1_{r=k-1}\bigr)(e_a-e_b).$

The coefficient is strictly positive, so no wall of the braid fan collapses. All deformations are moment polytopes on the same smooth toric variety. Their volumes are the same forest formula with $w_{ij}^{(\eta)}$ in place of $4\pi^2q_iq_j$ .

The added term has a double zero when either charge becomes soft. Deleting an isolated forest vertex and then a leaf proves, uniformly in the number of defects, that string and dilaton are unchanged. Nevertheless,

$W_2^{(\eta)}=x+y+4\pi^2qp +\eta q^2p^2(1-q-p)^2,$

so the amplitude varies. The deformation and its first normal derivative also vanish at the two-defect wall $q+p=1$ . Thus the full fan, both all-orders soft identities, and the first wall jet still do not select the physical coefficient.

For more than two defects this is a test of the toric and soft data; it does not assert pointwise equality with the Goldman form on the whole separated locus. The combined obstruction including the restricted BF cohomology class is already complete in the two-defect case below.

The missing datum is a relative collision class

Let $X$ be the weighted compactification, $U$ its separated-defect locus, and $D$ a collision divisor. The canonical section of $\mathcal O_X(D)$ is nowhere zero on $U$ . Hence

$[D]\in\ker\!\left(H^2(X)\longrightarrow H^2(U)\right).$

The separated Goldman/Weil–Petersson form cannot determine this extension coefficient. Small changes in it remain Kähler, and a Thom representative can place the change entirely near the collision. With two defects, $X\cong \mathbb P^1$ and the collision is a point: the freedom is literally an undetermined area, or moment-interval length.

Gu's BF calculation derives the reduced measure on the smooth separated locus but explicitly imports weighted global geometry for this extension. The fixed-fan theorem shows that permutohedral toricity cannot remove that import. A direct BF derivation would have to calculate the relative class or a scheme-independent observable sensitive to it.

What the two soft identities fail to see

The degree bound is essential. A local fused-defect contact term with coefficient

$C_\eta(q,p)=\eta q^2p^2(1-q-p)^2,\qquad q+p<1,$

preserves charge and local gauge covariance. Both its value and first derivative vanish when either defect becomes soft. It nevertheless changes the two-defect volume by $C_\eta$ .

This is not just a two-insertion construction. Make the formal source substitution

$\lambda'_Q=\lambda_Q+\frac{1}{2} \int_{q+p=Q}C_\eta(q,p)\lambda_q\lambda_p.$

The chain rule sums over disjoint fused pairs at every order. Terms pairing the soft label vanish through its first derivative; all other terms retain the same total charge and obey the original identities. Thus string and dilaton survive at all orders. A second soft derivative detects the change:

$\left.\partial_p^2(\widetilde V_2-V_2)\right|_{p=0} =2\eta q^2(1-q)^2.$

Witten's Matrix Models and Deformations of JT Gravity already explains the general contact-renormalization ambiguity. The example here isolates freedom invisible to these two identities. It violates the geometric degree bound, which removes the ambiguity in the stated polynomial class. It represents a contact prescription or coupling reparametrization, not automatically a physically distinct theory.

Why this program is paused

The attack gate was a canonical BF-to-moment-polytope map or an obstruction. The obstruction is now explicit. Selecting a contact convention and then reconstructing more amplitudes would import the datum the bridge was meant to derive.

The program should reopen only if a genuinely BF-native calculation fixes the relative collision class, or if a new observable survives consistent source and field redefinitions. No such observable has been found here. The next research mode is therefore a new hunt, not another JT amplitude layer.

What a change of presentation must preserve

Blommaert–Mertens–Verschelde already study JT horizon edge states and the distinction between local BF equivalence and a gravitational integration prescription. Boundary degrees of freedom are established parts of this subject. The question here requires a specific measure or observable whose translation is missing.

A useful comparison is the tree bijection of Budd–Meeusen–Zonneveld, submitted 10 December 2025. In a different cusp/geodesic-boundary sector, it transports the Weil–Petersson measure and recovers a geodesic-distance statistic. Our fixed-fan theorem upgrades the convex bodies to moment polytopes, but simultaneously proves that this still does not select the physical measure or carry an observable. The Losev–Manin space is already a permutohedral toric variety; toricity alone is not the missing BF derivation.

Read the source comparison and revised research question for the methodological precedents, physical scope, and remaining obligations.

Proof, checks, and attribution

The all-orders statements have analytic proofs. Exact rational checks verify the identities through four defects, the three-defect volume by a separate moduli-space intersection calculation, and a polytope area by a convex-hull calculation. These different calculations were prepared in the same run; they are not external independent review.