JT defect amplitudes: reconstruction, a polytope presentation, and the missing gauge-theory input

6 September 2026 · ATTACK · Working research note, not a Publication

Result and scope

In genus zero with two labelled geodesic boundaries and any number of mutually collidable conical defects, the weighted string and dilaton identities, together with the geometric polynomial-degree bound, uniquely determine every amplitude. The answer is a labelled-forest polynomial and is also the lattice volume of an explicit Minkowski sum of two opposite simplices and weighted root segments.

Without that degree bound, the two identities do not determine the answer: an explicit finite contact redefinition preserves both identities at every insertion order while changing amplitudes. This is a renormalization ambiguity, not evidence for a new physical degree of freedom.

These statements isolate a presentation bridge. They do not derive the identities or the degree bound directly from BF/BRST theory. They do not solve quantum gravity, cover every JT chamber, or establish a symplectomorphism with the polytope. The general contact ambiguity and the zero-boundary-length specialization are already present in the literature. A subsequent literature audit also identified the uniqueness proof strategy in Anagnostou–Mullane–Norbury: the argument here adapts their soft-face interpolation mechanism to arbitrary boundary lengths. Novelty of the forest/polytope synthesis has not been established; no independent external review has occurred. See the literature audit for the revised scope and next question.

1. The question being attacked

Jackiw–Teitelboim (JT) gravity has a gauge-theory presentation as BF theory and a geometric presentation in terms of hyperbolic surfaces. In the presence of conical defects, bulk gauge data determine defect holonomy and which collisions are admissible. The geometric presentation additionally supplies weights on collision divisors and identities associated with forgetting marked points.

Wei Gu explicitly leaves open a direct BF or BRST derivation of the weighted string identity, including boundary and collision terms, in §4.7.1 of Conical Defects in JT Gravity from BF Theory: Quantization, Fusion, and Weighted Moduli Spaces, submitted 3 September 2026. This is the selected target: determine exactly which information the gauge-theory presentation must retain to recover those geometric amplitudes.

The first decisive test is not a census of collision patterns. It is whether symmetry and the first two soft-insertion identities determine the amplitudes. They do not by themselves; a single additional structural condition makes them determining in the sector below.

2. Sector, normalization, and established inputs

Let a defect have cone angle $2\pi(1-q_i)$, with fractional deficit $q_i>0$. Work on a genus-zero surface with two labelled geodesic boundaries of lengths $b_1,b_2$, and $k\ge1$ labelled defects. Assume

$$Q=\sum_{i=1}^kq_i<1.$$

Thus every subset of defects can collide. We remain in this one open chamber throughout; no claim is made across walls. Set

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

The complex dimension is $k-1$. We use the standard volume normalization $\int\omega^{k-1}/(k-1)!$. Define

$$(Sf)(x,y)=\int_0^x f(t,y)\,dt+\int_0^y f(x,t)\,dt.$$

The geometric input is:

1. $V_1=1$.
2. String: $V_k(\mathbf q,0)=SV_{k-1}(\mathbf q)$.
3. Dilaton: $\left.\partial_{q_k}V_k\right|_{q_k=0}=cQ_{k-1}V_{k-1}$, where $Q_{k-1}=\sum_{i<k}q_i$.
4. In this chamber $V_k$ is a symmetric polynomial in the deficits of total deficit degree at most $2(k-1)$.

These are inputs, not discoveries of this note. The degree bound follows because the weighted Weil–Petersson cohomology class is quadratic in the deficits and is raised to power $k-1$. The string and dilaton identities are geometric forgetful-map identities. See Eberhardt–Turiaci, 2D dilaton gravity and the Weil-Petersson volumes with conical defects, especially the weighted class and §2.3, and Gu’s normalization in equations (4.120) and (4.137). In this sector the weighted Euler characteristic after removing the new mark is $-Q_{k-1}$; changing cone angle to deficit gives the positive coefficient $4\pi^2Q_{k-1}$.

The values and derivatives at a zero deficit mean polynomial continuation to a chamber face. They concern the normalized marked insertion used in these identities, not an unrenormalized area insertion.

3. All-orders uniqueness

Methodological precedent. The proof of Lemma 4.4 in Anagnostou–Mullane–Norbury, §4.2 already combines soft-face data, symmetry, and a degree bound to force uniqueness in the cusp specialization. Their §4.3 uses a related argument. The proof below is a general-boundary adaptation of that strategy; this precedent was identified during the subsequent audit.

Theorem 1. There is at most one symmetric polynomial family satisfying inputs 1–4.

Proof. Let $D_k$ be the difference of two families. The assertion is immediate at $k=1$. Assume it holds at $k-1$. String gives $D_k|_{q_i=0}=0$ for every label $i$, by symmetry. Dilaton gives $\partial_{q_i}D_k|_{q_i=0}=0$. Each identity holds on an open subset of the face, so it holds as a polynomial identity. Therefore $q_i^2$ divides $D_k$ for each $i$. Consequently

$$\prod_{i=1}^kq_i^2\mid D_k.$$

A nonzero difference would have total deficit degree at least $2k$, but its permitted degree is at most $2k-2$. Hence it is zero. This proves the induction. The argument can be applied coefficientwise in $x,y$, or at each fixed pair of boundary lengths. ∎

This proves uniqueness against all polynomial alternatives in this class, not merely a chosen family of counterterms. In fact the same argument only needs a degree bound strictly below $2k$.

4. A forest formula at every order

Put

$$P_m(x,y)=S^m1 =\sum_{j=0}^m\binom mj\frac{x^jy^{m-j}}{j!(m-j)!}.$$

The two integration operators commute, which proves the second equality. Let $\mathcal F_k$ be the set of all simple labelled forests on $[k]$, including isolated vertices. For a forest $F$, write $e(F)$ for its edge count and $d_i(F)$ for the degree of vertex $i$.

Theorem 2. In the specified chamber,

$$\boxed{ V_k(\mathbf q;x,y)= \sum_{F\in\mathcal F_k} c^{e(F)}P_{k-1-e(F)}(x,y) \prod_{i=1}^kq_i^{d_i(F)}. }$$

Proof. At $k=1$ the sole forest gives 1. Setting $q_k=0$ leaves only forests in which $k$ is isolated. Removing that vertex replaces $P_{k-1-e}$ by $P_{k-2-e}$; applying $S$ reverses this replacement, giving string.

Taking the derivative at $q_k=0$ instead leaves exactly the forests in which $k$ has degree one. Delete this leaf. Any forest on $[k-1]$ can be obtained, and attaching the leaf to vertex $i$ contributes $cq_i$. Summing the attachments contributes $cQ_{k-1}$, giving dilaton. A forest has at most $k-1$ edges, so the deficit degree is at most $2k-2$. Symmetry is immediate. Theorem 1 now identifies this family with the geometric volume. ∎

For example,

$$V_2=x+y+cq_1q_2,$$

$$V_3=\frac{x^2}{2}+2xy+\frac{y^2}{2} +c(x+y)(q_1q_2+q_1q_3+q_2q_3) +c^2q_1q_2q_3(q_1+q_2+q_3).$$

At $x=y=0$, only trees survive. The weighted Cayley identity, proved directly by the Prüfer code, gives for $k\ge2$

$$V_k(\mathbf q;0,0)=c^{k-1}\Bigl(\prod_iq_i\Bigr)Q^{k-2}.$$

This specialization is already known: it is Lemma 4.4 in Anagnostou–Mullane–Norbury, Weil-Petersson volumes, stability conditions and wall-crossing, in their Losev–Manin chamber. It is a control and an attribution boundary, not a new result claimed here.

5. An exact convex-polytope presentation

Let $\Delta=\operatorname{conv}(e_1,\ldots,e_k)$, and use volume normalized by the difference lattice

$$A_{k-1}=\{z\in\mathbb Z^k:\sum_i z_i=0\}.$$

A lattice fundamental parallelepiped has volume 1; a unimodular $(k-1)$-simplex has volume $1/(k-1)!$. Define the Minkowski sum

$$\mathcal P_k=x\Delta+y(-\Delta) +c\sum_{i<j}q_iq_j[e_i,e_j].$$

All summands lie in parallel affine hyperplanes. Translations do not affect this volume.

Theorem 3. For nonnegative boundary variables $x,y$,

$$\operatorname{Vol}_{A_{k-1}}(\mathcal P_k)=V_k(\mathbf q;x,y).$$

Proof. Adding a segment $t[0,v]$ in a primitive lattice direction increases the volume of a convex body by $t$ times the quotient-lattice volume of its projection along $v$. This follows by integrating line fibers. Iterating gives a multiaffine expansion in the segment parameters.

The vectors $e_i-e_j$ indexed by a graph are linearly independent exactly when the graph is a forest. Thus cyclic edge sets have zero coefficient. A forest extends to a spanning tree, whose edge vectors are a lattice basis of $A_{k-1}$. Its independent directions are therefore unimodular, with no extra index factor.

Quotient by an $e$-edge forest. The quotient contracts its components; $\Delta$ projects to the standard simplex of dimension $d=k-1-e$, and $-\Delta$ to its negative. Their Minkowski sum has volume $P_d(x,y)$. To check this last statement without a mixed-volume formula, identify the simplex with $\operatorname{conv}(0,e_1,\ldots,e_d)$. Its difference body $x\Delta_d-y\Delta_d$ consists precisely of vectors whose positive coordinates sum to at most $x$ and whose negative magnitudes sum to at most $y$. An orthant with $j$ positive coordinates contributes $x^jy^{d-j}/(j!(d-j)!)$, and there are $\binom dj$ such orthants. Summation gives $P_d$.

Substitution of the segment lengths $cq_iq_j$ now gives exactly the forest formula of Theorem 2. ∎

This uses the classical forest/unimodularity mechanism of graphical zonotopes, with an elementary proof provided here. It gives an equality of scalar volumes. It does not yet identify the physical reduced BF phase space, its measure, or the canonical hyperbolic symplectic form with this convex body. The Losev–Manin space is known to be toric; the further identification needed for a direct physical derivation is not inferred solely from that fact.

6. Sewing to asymptotic JT boundaries

Use the standard trumpet and measure

$$Z_{\mathrm{tr}}(\beta,b)=\frac{e^{-b^2/(4\beta)}}{2\sqrt{\pi\beta}}, \qquad b\,db.$$

Writing $x=b^2/2$, the Laplace transform of $\int_0^x f(t)dt$ is $2\beta$ times the transform of $f$. The two-boundary transform of 1 is $\sqrt{\beta_1\beta_2}/\pi$. Thus, with $B=2(\beta_1+\beta_2)$,

$$Z_{0,2;k}(\boldsymbol\beta;\mathbf q) =\frac{\sqrt{\beta_1\beta_2}}{\pi} \sum_{F\in\mathcal F_k} c^{e(F)}B^{k-1-e(F)} \prod_iq_i^{d_i(F)}.$$

This is for labelled insertions; no $1/k!$ is included. Fugacity expansions use their usual factorials. It is a consequence of the geometric reconstruction, not a separate direct BF derivation.

7. Why the soft identities alone are insufficient

The general issue is established, not new: Witten discusses finite renormalization and contact ambiguities in products of JT defect operators in the introduction to Matrix Models and Deformations of JT Gravity. The following construction isolates an ambiguity invisible to both of the specific soft identities.

For $q,p>0$ and $q+p<1$, let

$$C_\eta(q,p)=\eta q^2p^2(1-q-p)^2.$$

A covariant, nonderivative local contact prescription can change a product by

$$[O_q(X)O_p(Y)]_\eta-[O_q(X)O_p(Y)]_0 =C_\eta(q,p)\delta_g(X,Y)O_{q+p}(X).$$

Here the covariant delta is normalized with the same integration measure as the integrated operators. The fused operator preserves total deficit. This term is compatible with local gauge covariance; this assertion is not a derivation of every BF/BRST boundary Ward identity.

There is a consistent formal all-orders completion. Reparametrize the defect fugacity source by

$$\lambda'_Q=\lambda_Q+ \frac12\int_{q+p=Q}C_\eta(q,p)\lambda_q\lambda_p.$$

For a finite set of labelled deficits this means summing, in each correlator, over disjoint pair matchings. Replace each paired pair by its fused charge and multiply by $C_\eta$; leave the other labels unchanged. This is simply the chain rule for the generating function. Since the source map has identity linear part, it is invertible as a formal power series. It preserves the charge Euler grading.

Now insert an additional charge $s$. At $s=0$, terms where it is paired vanish because $C(q,0)=0$. Unpaired terms obey the original string identity. At first derivative, paired terms still vanish because $\partial_sC(q,s)|_{s=0}=0$; unpaired terms obey dilaton with the same total charge. This proves preservation of both soft identities at every order. No enumeration at increasing order is used in the proof.

The coefficient and its first derivatives also vanish on the collision wall $q+p=1$. It may be extended by zero beyond that wall as a $C^1$ function. No higher wall smoothness or unique physical ultraviolet prescription is asserted.

The redefinition is not trivial at fixed source coordinates: $C(1/4,1/4)=\eta/1024$. In the two-boundary sector it gives

$$\widetilde V_2=V_2+C_\eta(q,p),$$

because the fused one-defect volume is 1. The second soft derivative detects it:

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

The polynomial degree condition excludes this redefinition: even its lowest nonzero total degree is four, whereas $V_2$ has degree at most two. Theorem 1 gives the stronger conclusion for arbitrary polynomial alternatives in the sector.

For an optional disk control, the one-defect kernel is

$$Z_1(\beta;Q)=\frac{e^{\pi^2(1-Q)^2/\beta}}{2\sqrt{\pi\beta}}.$$

At $q=p=1/4$, the change in the two-defect disk kernel is $\eta Z_1(\beta;1/2)/1024$. In the blunt disk chamber the established baseline is $Z_2=2\beta Z_1(\beta;Q)$, so $\partial_\beta(Z_2/Z_1)=2$ is invariant under this ultralocal nonderivative counterterm class. It is not claimed invariant against arbitrary derivative counterterms.

Changing amplitudes at fixed uncalibrated coupling labels is not enough to distinguish physical theories. This construction exhibits missing normalization data, not hidden propagating modes or new black-hole dynamics.

8. Verification and evidence limits

The companion jt-contact-ward-check.py is a self-contained Python standard-library verifier using exact rational polynomial arithmetic. It checks the contact jets, fusion-wall jets, coefficient $1/1024$, charge grading, a four-label source-expansion/matching comparison, and the forest identities and degree bound through $k=4$.

Two controls do not use the soft-identity reconstruction:

• A direct $\overline{\mathcal M}_{0,5}$ intersection calculation checks the $k=3$ polynomial using the quadratic weighted Weil–Petersson class, including the triple-collision divisor. The identities used are $\kappa_1^2=5$, $\kappa_1\psi_i=3$, $\psi_i^2=1$, $\psi_i\psi_j=2$ for distinct labels, and the standard boundary-divisor intersections.
• A rational convex-hull calculation checks the two-dimensional polytope at $(q_1,q_2,q_3)=(1/10,1/5,3/10)$, $c=4,x=2,y=3$. Its exact lattice area is $25947/1250$, agreeing with the forest polynomial.

All checks passed. The all-orders claims follow from the analytic arguments above, not from these finite tests. The two reconstructions use different calculations but were prepared in this same research run; they are not independent external review.

9. Dependency and influence ledger

The user’s presentation-loss idea motivates asking whether quotienting removes measurable dynamics or merely forgets a contact prescription. The conclusion here separates those alternatives instead of assuming the former.

10. Next decisive question

Revised after the literature audit: First specify the gravitational integration cycle, boundary state, normalized defect operator, and contact-renormalization prescription. Then determine whether BF/BRST theory fixes the measure at a two-defect collision and its soft boundary terms without importing the weighted geometric class. The known $V_2=x+y+4\pi^2qp$ is a control, not a new result.

Only a Ward or measure-reduction mechanism with a credible extension to arbitrary insertion number justifies resuming the uniform degree question. The larger goal remains to derive both soft identities and the bound $\deg_q V_k<2k$ directly. Matching small volumes, choosing a counterterm convention, or finding a toric presentation of already imported geometry does not complete the BF bridge. See the literature audit for established edge-mode, localization, and toric precedents.

If the contact freedom cannot be removed from physical input, formulate the result as a family of renormalization prescriptions and determine which observables survive their equivalence. Do not turn the next run into an enumeration of more collision strata or assert that a coordinate redefinition creates physical dynamics.