The algebraic modular functor conjecture in type $A_n$ quantum Teichm\"uller theory

Alexander Shapiro; Gus Schrader

REVIEW 3 major objections 4 minor 48 references

The algebraic modular functor conjecture in type $A_n$ quantum Teichm\"uller theory

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper proves that, for PGL_{n+1}, cutting a marked surface along an essential simple closed curve gives a canonical, mapping-class-equivariant isomorphism between the quantum algebra L_{G,S} and a residue universal Laurent ring built f

desk verdict Resolves a long-open Fock–Goncharov conjecture for PGL_{n+1} with genuinely new tools and unusually honest scope statements; the main soft spot is a stated-but-unproved extension of Theorem 3.10 to quasi-permutations that the mapping-class equivariance rests on. read the letter →

arxiv 2509.03820 v1 pith:G3UVCPTG submitted 2025-09-04 math.QA math.RT

Gus Schrader , Alexander Shapiro This is my paper · ORCID

classification math.QAmath.RT MSC 13F6016T2020G4253D30

keywords algebraicmodularfunctorquantumTeichmüllertheoryclusterPoissonvarietiesalgebrasuniversalLaurentringsPGL_{n+1}openTodachainmappingclassgroup

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves the algebraic modular functor conjecture for G = PGL_{n+1}: the quantum algebra L_{G,S} attached to a marked surface behaves under cutting along an essential simple closed curve as a modular functor should. To get there, the authors introduce two extensions of the quantized Teichmüller framework: enhanced moduli spaces whose 'tacked circles' carry the missing twist-coordinate data, and the residue universal Laurent ring, obtained by localizing the quantum universal Laurent ring and imposing simple-pole residue conditions. In a specially adapted cluster chart, the Dehn twist along the cutting curve becomes n commuting mutations in a rank-2n quantum subtorus, and the gluing problem reduces to the GL_{n+1} open Toda system. The main theorem gives a Γ_{S;c}-equivariant algebra isomorphism eta_c : L_{G,S} → L_{G,S1;ϕ}, and its restriction recovers the centralizer isomorphism (1.1) of the original conjecture. A corner case where the cut surface has a once-punctured torus component is deferred to a companion paper.

What carries the argument

The central object is the residue universal Laurent ring L_{G,S1;ϕ}. The carrying mechanism is a separation of variables in a cluster chart adapted to an isolating cylinder for c: the chart contains a distinguished rank-2n quantum subtorus with vertices s_i, t_i for i = 1,...,n plus two extra frozen directions, and in it the quantum Dehn twist τ_c acts as the product of n commuting mutations followed by n transpositions. The local model is the GL_{n+1} q-difference open Toda chain: the fundamental Hamiltonians H_k(c) are extracted as coefficients of the difference operator Q(q^{-1}z) = H(z)Q(qz) with H(z) = Σ z^k H_k, and an algebraic spectral transform (the Whittaker transform) identifies t

What would settle it

Take G = PGL_2 and S a four-holed sphere with c a separating curve into two pairs of pants. In the isolating cluster chart, the proposed isomorphism must send H_1(c) from formula (5.33) into the residue universal Laurent ring of the cut surface; a direct computation of its image should show it has only simple poles and satisfies condition (6.16). Equivalently, checking the Dehn-twist factorization of Proposition 5.13 at the quantum level—that the relevant conjugate of Q(τ_c) equals the stated product of n commuting mutations—would settle the equivariance claim, since any mismatch in the q-comm

Watch

Extended reading notes

Core claim

Main theorem: for G = PGL_{n+1}, S a marked surface and c an essential simple closed curve, there is a Γ_{S;c}-equivariant algebra isomorphism eta_c : L_{G,S} ≅ L_{G,S1;ϕ}, where S1 is the cut surface with the two new boundary circles promoted to tacked circles and ϕ records the gluing homeomorphism. L_{G,S1;ϕ} is the residue universal Laurent ring: localize the quantum universal Laurent ring of S1 at the canonical mutation-invariant divisors of the tacked circles, take invariants under the product of Weyl groups, and keep elements with only simple poles whose residues satisfy condition (6.16). The isomorphism restricts to the centralizer subalgebra of R_G(c), giving the isomorphism (1.1) of

Load-bearing premise

The load-bearing premise is that two quantum cluster transformations built from the same quiver mutations are equal as soon as their classical or tropical specializations agree, even after the quasi-permutation corrections used here; if that dictionary failed for these specific transformations, the cutting isomorphism would not be well-defined or mapping-class equivariant.

Editorial extensions

If this is right

The assignment S ↦ L_{G,S} now satisfies the algebraic modular functor property for PGL_{n+1}: cutting along an essential curve replaces the algebra by the residue universal Laurent ring of the cut surface with gluing datum ϕ.
The isomorphism η_c is Γ_{S;c}-equivariant, so the mapping class group action on L_{G,S} is entirely determined by the action on cut-surface data; in particular, the centralizer of R_G(c) is the Weyl-invariant part of the quotient by the ideal I_c, resolving Conjecture 1.1.
The fundamental Hamiltonians H_k(c) attached to c are independent of the auxiliary isolating triangulation, giving canonical commuting elements in L_{G,S}; the isomorphism η_c restricts to an isomorphism between R_G(c) and the representation ring of PGL_{n+1}.
The construction also covers cut surfaces whose components have a single boundary circle when genus is greater than one, via an auxiliary-curve surgery, so the algebra-level modular functor statement is broader than the original centralizer conjecture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

My inference: the residue universal Laurent ring is a natural target for a true gluing functor on modules: its simple-pole and residue conditions look like the local conditions needed to glue representations, not merely algebras.
My inference: once the announced identification of H_k(c) with q-deformed traces of the monodromy around c is completed, η_c will be an equality between the quantum trace-of-monodromy algebra and the cut-surface character ring; a direct check in the rank-one case using the explicit formula (5.33) would test this.
My inference: the auxiliary-curve surgery used for single-boundary components suggests that arbitrary pants decompositions can be handled by the same mechanism, with Dehn twists along each curve acting as commuting mutations in adapted charts.
My inference: a concrete stress test of the machinery is to verify in the isolating chart that all braid-group relations of the mapping class group lift to quantum cluster transformations; the paper's Corollary 4.33 and Theorem 3.10 predict they do, and failure at any one relation would pinpoint the step that breaks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is the real thing. Schrader and Shapiro prove Conjecture 1.1 for G = PGL_{n+1} and in fact prove the stronger residue-ring isomorphism (1.2). The new constructions—tacked circles, the residue universal Laurent ring, the algebraic Whittaker transform—are original and do real work. The paper is also unusually candid: it says plainly that the genus-one corner case is deferred to [SS25], that the identification of the H_k(c) with q-deformed trace functions is deferred, and that Lemma 6.14 leans on external Coulomb branch results. That honesty makes the paper easier to trust, not harder.

The central argument is long but coherent. Sections 5–8 construct the isolating quiver, localize the Dehn twist, prove cylinder-independence, and get equivariance. I could not machine-check the mutation and dilogarithm computations, and at 143 pages neither will most referees; the proof will need expert verification. Nothing I read contradicts the main theorem.

The soft spot worth naming is the one the stress-test note points at, and it is real but patchable. Theorem 3.10 is stated for composites of mutations. Corollary 4.33 and Proposition 5.13 apply it to quasi-cluster transformations that include quasi-permutations ς_f and ς_c, which act on frozen variables. The classical-implies-quantum statement for mutation composites does not by itself cover conjugating by these isometries. In practice the quantum quasi-permutation is determined by its classical action on A-variables, and the extension probably goes through, but it is not written down, and the well-definedness of the functor Q and the Γ_{S;c}-equivariance of η_c rest on it. This is a missing proof, not a demonstrated contradiction. A referee should ask for a short lemma extending Theorem 3.10 to quasi-cluster transformations, or a revised proof of Corollary 4.33.

Other soft spots are minor: Lemma 6.14(1) is imported from [CW18] rather than proved here, and the two deferred items in [SS25] are part of the full conjecture but not of the advertised theorem. The residue-ring definition is not circular; the right side of (1.2) is not built to match the left by fiat, and the independence results in Section 8 are substantive.

Bottom line: this deserves a serious referee. It is a major claim with a credible proof, and the open questions are specific and honest. I would send it to review and ask for the quasi-permutation lemma before acceptance.

Referee Report

3 major / 4 minor

Summary. The paper proves the algebraic modular functor conjecture for G = PGL_{n+1}: for a marked surface S and an essential simple closed curve c, it constructs a Γ_{S;c}-equivariant algebra isomorphism η_c : L_{G,S} → L_{G,S_1;φ} (Main Theorem, Eq. (1.2)), where L_{G,S_1;φ} is a newly defined residue universal Laurent ring built from the cut surface. This refines Conjecture 1.1 (FG09a, GS19). The proof introduces two extensions of the Fock–Goncharov framework: enhanced moduli spaces with tacked circles and a residue universal Laurent ring obtained by localization, Weyl-group invariants, and simple-pole residue conditions. The main ingredients are special cluster charts isolating the curve c, a realization of the Dehn twist as n commuting mutations, and an algebraic Whittaker transform identifying the relevant local Laurent ring with a variant of the spherical double affine Hecke algebra. The paper is explicit about scope: the genus-1 corner case is deferred to the companion paper [SS25]. The central claim is an independently derived isomorphism theorem, not a circular reformulation.

Significance. If the proof is correct, this resolves a long-standing conjecture of Fock and Goncharov and establishes a modular-functor structure for quantized higher Teichmüller theory in type A_n. The paper is technically rich: it introduces new geometric objects (tacked circles, enhanced moduli spaces), new algebraic objects (residue universal Laurent rings), and gives detailed, largely self-contained arguments in Sections 5–8. It also carefully identifies the exact scope of the result and explicitly defers one boundary case to a companion paper. The main risk is not circularity or the engineering of definitions but a technical gap in an auxiliary comparison theorem: Theorem 3.10 is applied to quasi-cluster transformations that include quasi-permutations, even though the theorem is stated for composites of mutations only. Since this application is load-bearing for the mapping-class-group equivariance of the main isomorphism, the gap must be repaired before the central claim can be regarded as fully established.

major comments (3)

[§4.8, Corollary 4.33] The proof of Corollary 4.33 invokes Theorem 3.10 to conclude equality of quantum quasi-cluster transformations from equality of their classical specializations. However, Theorem 3.10 is explicitly stated only for quantum cluster transformations obtained as composites of cluster mutations (3.10). The transformations in Corollary 4.33 arise from square moves and shifts, which by (4.19), (4.23), and (4.30) are compositions of mutations with the quasi-permutations ς_f and ς_c. These quasi-permutations act nontrivially on frozen variables, e.g. ς_f(ξ_ℓ) = ξ_{φ(f)} + ξ_{c,i} in (4.20). The theorem as stated does not control such relabelings. This is not a cosmetic issue: Corollary 4.33 is used to define the functor Q : xPt(S) → Cl_Q in (4.32), and hence to prove the well-definedness of the mapping class group action. Please provide a proof of the needed extension — for instance, a criterion fo
[§5.5, Proposition 5.13] Proposition 5.13 realizes the Dehn twist τ_c in the isolating cluster as the composite µ_c = ∏(s_j,t_j) ∘ ∏ µ_{s_j} in (5.27). The proof states that this follows from Theorem 3.10 or from Corollary 4.33. Both routes depend on the same unproved extension of Theorem 3.10 to transformations containing the quasi-permutations ς_f and ς_c. Since the localization of the Dehn twist to the n-mutation composite is the mechanism by which the Γ_{S;c}-equivariance of η_c is obtained, this gap propagates to Theorem 8.36 and to the main theorem (1.2). The statement may well be true, but it needs a proof that does not simply cite a theorem whose hypotheses exclude the transformations being used.
[§8.34–8.36 (via §5.13 and §4.33)] Theorems 8.34 and 8.36 establish cylinder-independence and Γ_{S;c}-equivariance of the cutting isomorphism. These theorems are justified through the functor Q and Proposition 5.13, and hence inherit the gap described above. I am not asking the authors to prove a general classical-implies-quantum theorem for all quasi-cluster transformations; it is enough to prove it for the specific quasi-permutations ς_f and ς_c defined in (4.19)–(4.23), which have a very special form. But the current text does not supply this lemma, and without it the central equivariance claim is not established.

minor comments (4)

[§3.6–3.8] The paper defines quasi-permutations and quasi-cluster transformations, but does not state any analogue of Theorem 3.10 for them. Adding such a statement — even as a conjecture or a remark — would make the later usage transparent.
[§4.30] Remark 4.34 warns that the conclusion of Corollary 4.33 is false under a weaker hypothesis on the equivalence of graphs. This is helpful, but the remark would be even more useful if it sketched a counterexample or explained precisely where the hypothesis on bijections φ_1, φ_2 enters.
[Notation] The notation Γ_{S;c} is used both for the centralizer of the Dehn twist and for the enhanced Ptolemy groupoid in Definition 2.28 and surrounding text. The ambiguity is manageable but could confuse readers; a distinction such as Γ_S vs. xΓ_S would help.
[§6.4] The algebraic Whittaker transform is a substantial technical component, but the discussion of the Whittaker kernel W^{(n+1)}(λ) is compressed. In particular, the assertion that it equals Macdonald's P-polynomial at t=0 is stated without the standard normalization details. A pointer to the exact normalization in Macdonald's book would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main cutting isomorphism is proven by explicit cluster/chart and Toda–Whittaker constructions against external benchmarks, not by defining the target ring to match the source.

full rationale

The Main Theorem (1.2) is a substantive isomorphism, not a reformulation. Its target L_{G,S1;phi} is not introduced as a renaming or image of L_{G,S}; it is obtained from L_{G,S1} by localization, W(c±)-invariants, and residue conditions (Definition 8.6), with the residue characterization modeled on the external GKV framework [GKV97]. The proof constructs eta_c through isolating cluster charts, the Baxter operator / algebraic Whittaker transform, and Theorems 8.31, 8.34, and 8.36; no fitted parameter is renamed as a prediction. The only near-circularity-shaped concern is Corollary 4.33, which applies Theorem 3.10 (cited to external works FG09a, Kel11, KN11) to quasi-cluster transformations that include quasi-permutations. That is a possible proof gap or an extension of the stated theorem, but it is not circular: it does not assume the main isomorphism, and the equality of classical quasi-cluster transformations is derived from the geometry of the bicolored graphs rather than from the target algebra. Self-citations to GS19 and SS18 provide auxiliary lemmas and analogous computations, but none assumes Conjecture 1.1 or the Main Theorem, and the central derivation is self-contained enough that no load-bearing step reduces by definition to its own inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 4 invented entities

This is pure mathematics with no fitted numerical parameters. The ledger is dominated by standard cluster-algebra and representation-theory results imported from the literature, plus four new objects introduced by the paper (tacked circles, the enhanced moduli space, the residue universal Laurent ring, and the fundamental Hamiltonians). The load-bearing external imports are Theorem 3.10, BZ05's upper bound theorem, Goodearl-Yakimov's identification for the Toda cluster algebra, and the Coulomb-branch localization results of BFN/FT/CW. Two disclosed gaps sit in the ledger: the expected but unproven isomorphism of Remark 2.19(a), and the general-G version of Lemma 6.14(1) promised by Klyuev.

assumptions (7)

standard math Theorem 3.10: composites of quantum cluster transformations coincide iff their classical and tropical specializations coincide.
Imported from [FG09a, Kel11, KN11]; used in Corollary 4.33 and Proposition 5.13 to prove well-definedness of the Ptolemy to cluster functor and localization of the Dehn twist.
standard math Quantum Laurent phenomenon: the upper bound U^A_Q equals the quantum upper cluster algebra L^A_Q, and universal Laurentness can be checked by 1-step mutations.
BZ05 Theorem 5.1; used in Section 3.5 (Corollary 3.19) and throughout for membership in universal Laurent rings.
domain assumption A_Toda(GL_{n+1}) = L_Toda(GL_{n+1}) and the explicit presentation of L_Toda as a localization of a quantum coordinate ring of a unipotent cell.
Goodearl-Yakimov [GY21] Theorem B; used in Section 6.1; load-bearing for the spectral side of the Whittaker transform.
domain assumption The equivariant K-theory localization theorem identifies the Coulomb branch convolution algebra for GL_{n+1} with the algebra D_res of residue-constrained q-difference operators.
Based on [BFN16], [FT17], [CW18]; Lemma 6.14(1) depends on it. Section 6.3 defers a conceptual proof for general G to a forthcoming paper by Klyuev, a disclosed gap.
domain assumption For adjoint G, the coordinate ring of the cluster Poisson variety P_Q(G,S) is isomorphic to that of the moduli space P_{G,S}.
Invoked in Remark 2.19(b) via [She22]; the analogous statement for A-varieties (Remark 2.19(a)) is stated as expected with no known reference.
standard math Zig-zag and web combinatorics of ideal bicolored graphs of rank n, including the codistance identity of Proposition 4.28.
From [Gon17] and [FG06b]; used in Section 4 to construct cluster K2 coordinates on the enhanced moduli spaces.
domain assumption The residue characterization of Hecke algebras of Ginzburg-Kapranov-Vasserot transfers to the settings of Definitions 6.4 and 8.6.
The sufficiency of the simple-pole plus residue conditions for identifying D_res, and later the residue universal Laurent ring, is the crux of Sections 6 and 8; the paper proves the needed statements (Lemmas 6.6, 6.12, 6.14) in the GL_{n+1} case.

invented entities (4)

Tacked circles (boundary circles with distinguished tacks) on marked surfaces
purpose: Carry additional boundary data whose coordinate is Poisson-conjugate to the central length coordinate, providing algebro-geometric analogs of Fenchel-Nielsen twist coordinates for gluing.
Introduced in Definitions 2.12 and 2.20 and justified internally: the external check is that the gluing theorem (1.2) holds and that the resulting isomorphism is canonical.
Enhanced moduli space P^circle_{G,S} of decorated local systems on surfaces with tacked circles
purpose: Host for the classical limit of the cutting construction; carries a cluster Poisson structure and an enhanced compatible pair.
Internal construction of Section 4; its correctness is supported by Propositions 4.31 and 4.32 and by the main theorem.
Residue universal Laurent ring L_{G,S1;phi}
purpose: A localization-plus-invariants-plus-residue-conditions refinement of the quantum universal Laurent ring of the cut surface; the target of the cutting isomorphism that reconstructs L_{G,S}.
Defined in Definition 8.6; the justification is the main theorem (1.2) and the independence results of Theorems 8.34 and 8.36. No external falsifiable handle outside this paper is given.
Fundamental quantum Hamiltonians H_k(c)
purpose: Generate the commutative subalgebra R_G(c) attached to a simple closed curve c; candidates for the q-deformed traces chi^q_V(c) of the conjecture.
Canonicity is proven within this paper (Theorem 8.34), and the spectral interpretation as character-ring generators is proven via Pieri rules (6.41). The identification with traces of monodromy is deferred to the companion paper [SS25] (Remark 5.18), so no external falsifiable handle appears here.

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Pith. "Pith review of The algebraic modular functor conjecture in type $A_n$ quantum Teichm\"uller theory." pith.science (2026). https://pith.science/paper/G3UVCPTG

@misc{pith2026250903820,
  author       = {Pith},
  title        = {Pith review of: The algebraic modular functor conjecture in type $A_n$ quantum Teichm\"uller theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G3UVCPTG}},
  note         = {Machine review of arXiv:2509.03820}
}

abstract

Fock and Goncharov introduced a quantization of higher Teichm\"uller theory using cluster Poisson varieties and their noncommutative deformations, associating to a complex semisimple Lie group $G$ and a marked surface $S$ a quantum algebra $\mathbb{L}_{G,S}$ equipped with an action of the surface mapping class group. They conjectured that these quantizations form an algebraic analog of a modular functor: cutting a surface along a simple closed curve should correspond to a canonical gluing isomorphism for the associated algebras. In this paper we prove this conjecture for $G = \mathrm{PGL}_{n+1}$. Our approach requires two extensions of the Fock-Goncharov framework: (1) enhanced moduli spaces incorporating additional boundary data, providing algebro-geometric analogs of Fenchel-Nielsen twist coordinates; and (2) the residue universal Laurent ring, a refinement of the quantum universal Laurent ring obtained by localizing and imposing residue conditions. Using these tools, we construct canonical cutting isomorphisms that are equivariant under mapping class group actions and suffice to reconstruct the entire algebra $\mathbb{L}_{G,S}$ from data associated to the cut surface.

Figures

Figures reproduced from arXiv: 2509.03820 by the authors.

**Figure 1.** Examples of decorated surfaces. Definition 2.6. Let S1 and S2 be decorated surfaces. A decorated embedding ι: S1 ãÑ S2 is an oriented embedding of surfaces which respects walls, regions, colors, and marked points. A decorated isotopy is a continuous path through the space of decorated embeddings. In what follows, we shall only consider decorated surfaces up to decorated isotopies. Recall that a data of a (Betti) G-l… view at source ↗

**Figure 2.** Decorated monogons. Example 2.10. Let π : G{N Ñ G{B denote the natural projection. Then the decorated character stacks corresponding to surfaces shown on [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Decorated cylinders and disks [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (62 more)

**Figure 4.** Figure 4: Decorated triangle. We say that a decorated surface is simple2 if each of its T-regions is either a boundary Tdisk with a single marked point, an interior T-disk with a single marked point, or a boundary T-annulus with at most one marked point, the unique marked point…

**Figure 5.** Figure 5: Decorated surface and marked surface. Definition 2.13. Given a marked surface S, we define its A-lift SA to be the decorated surface where all punctures are replaced by internal T-disks with a single marked point, and its P-lift SP to be the one where all punctures are…

**Figure 6.** Figure 6: Pentagon relation. Proposition 2.25 ([CF99]). The Ptolemy complex is connected and simply connected: (1) for any pair of objects, ∆1 , ∆2 , there exists a 1-morphism F : ∆1 Ñ ∆2 ; 3Let us point out that our definition of the Ptolemy complex differs from the one given i…

**Figure 7.** Figure 7: , where the homeomorphism ϕ is indicated by horizontal yellow segments connecting tacked circles c˘. This procedure yields the same ideal triangulation ∆ if the initial one ∆1 is replaced by its image under the diagonal product of Dehn twists τc` τc´ . Hence it descend…

**Figure 8.** Figure 8: The umbral move UC;e0 . Lemma 2.35. Any morphism in Pt|c| pSq can be factored into a sequence of umbral moves and flips preserving the isolating cylinder of c. Proof. Since the cutting functor intertwines umbral moves on S with those on the cut surface S 1 , it suffice…

**Figure 9.** Figure 9: Note that possibly vi “ v 1 i , in which case the e 1 i , e2 i are loops. Then the flip Fbi replaces bi with d 1 i , and flip Fdi replaces di with b 1 i . Thus, performing moves U˘ we change ∆2 into a triangulation containing triangles ∆1 ˘. The latter can be turned in…

**Figure 10.** Figure 10: Bi-colored graphs of rank 3 in ideal triangles. A bi-colored graph Γ Ă S is trivalent if each of its non-boundary vertices is such. Evidently, every equivalence class of bi-colored graphs contains a bipartite one and a trivalent one, and we will often use those in the…

**Figure 11.** Figure 11: Graph Γ∆ on an annulus, one boundary of which is a tacked circle while the other contains 2 marked points. Let Γ Ă S be a trivalent bicolored graph on a marked surface S. Assume further that the graph Γ in the vicinity of a tacked circle c can be described as follows.…

**Figure 12.** Figure 12: Positive shift at a tacked circle. Let Γ “ Γ∆ for some ideal triangulation ∆ of S, as in Example 4.3, c P CpSq be a tacked circle, and ∆ the special triangle of ∆ containing c. Then the positive shift σc corresponds to the flip of ∆ at the edge of ∆, which follows c a…

**Figure 13.** Figure 13: Square move at a face f P FpΓq. Remark 4.9. As explained in [FG06b], alternatively see Section 4.8 here, for any pair of ideal triangulations ∆, ∆1 of S, the graphs Γ∆ and Γ∆1 of the same rank are related to each other by a sequence of square moves and shifts at tacke…

**Figure 14.** Figure 14: Left pane: zig-zags on an ideal bicolored graph Γ∆ of rank 3. Right pane: the corresponding ideal A3-web. The zig-zags on a bicolored graph Γ Ă Sr are almost identical to the zig-zags on Γ˝ Ă Sr˝. Indeed, the only ones that can possibly differ are those that contain e…

**Figure 15.** Figure 15: Faces in Z ` i pt˜q for a bipartite graph. Dashed edges indicate a possibly non-zero number of edges of Γ. r 4.3. Quivers from ideal bicolored graphs. To each ideal bipartite graph Γ Ă S of rank n we now associate a quiver QΓ with seed ΘΓ “ pIΓ, I˚ Γ ,ΛΓ,p¨, ¨qΓ, teℓu…

**Figure 16.** Figure 16: Quiver Q9 Γ from the bi-colored graph on a triangle. Let us now describe the form p¨, ¨qΓ˝ in terms of the graph Γr “ π ´1 pΓq on the universal cover Sr of S. The graph Γ gives rise to a pair of abelian groups r ΛΓr “ à f˜PFpΓrq Zxef˜y and Λp Γr “ ź f˜PFpΓrq Zxef˜y to…

**Figure 17.** Figure 17: Quiver for a triangle with notches. Inside each non-special triangle ∆ we draw the quiver Q described in the beginning of this section, see [PITH_FULL_IMAGE:figures/full_fig_p036_17.png]

**Figure 18.** Figure 18: Quiver Q∆ from a triangulation on [PITH_FULL_IMAGE:figures/full_fig_p036_18.png]

**Figure 19.** Figure 19: Quivers Q9 ∆ (on the left) and Q∆ (on the right) for an annulus with one special and one non-special triangle. Note that unless we have a non-special triangle sharing two edges with a special one, the bases teiu and te9iu coincide, and hence Q∆ “ Q9 ∆. Thus, it remain…

**Figure 20.** Figure 20: Square move and quiver mutation. Now we consider the case ℓ1 “ ec,i P ICpSq and ℓ2 “ ℓ P IΓ˝ . First, assume that there is no lift ˜f P π ´1 pfq which is simultaneously incident to one of the zig-zags zi´1pc˜q, zipc˜q, and ˇzpt˜q for a lift ˜c of c and a tack t˜ on ˜c…

**Figure 21.** Figure 21: Zig-zags and on an ideal bicolored graph Γr∆ of rank 3. Example 4.29 [PITH_FULL_IMAGE:figures/full_fig_p042_21.png]

**Figure 22.** Figure 22: Triangulations and quiver for G “ SL2. F¯ b´1 F¯ t0 F¯ b0 F¯ t1 F¯ b1 F¯ c´1 F¯ c0 F¯ c1 2 4 2 4 2 1 3 1 3 F¯ b´1 F¯ t0 F¯ b0 F¯ t1 F¯ b1 F¯ c´1 F¯ c0 F¯ c1 2 4 2 4 2 1 3 1 3 [PITH_FULL_IMAGE:figures/full_fig_p043_22.png]

**Figure 23.** Figure 23: Unrollings of the triangulations from [PITH_FULL_IMAGE:figures/full_fig_p043_23.png]

**Figure 24.** Figure 24: Mutation at vertex 3 in a self-folded triangle for G “ SL3. triangulation we have A9 1 “ A1 “ rν1pb1q ^ ν2pc1qs, A9 2 “ rν1pb1q ^ ν1pb0q ^ ν1pc1qs, A2 “ A9 2{A1, and A3 “ A9 3 “ rν1pc1q ^ ν2pb1qs. The square move at face 3 corresponds to a Pl¨ucker relation A3A9 1 3 “…

**Figure 25.** Figure 25: A flip. On the other hand, suppose c is a tacked circle contained in the special triangle ∆c, and d˘ be the remaining two sides of ∆c, so that d` follows d´ as we go around c in the positive direction. To the flips Fd˘ we associate the quasi-permutations µd˘ “ ς ˘1 c …

**Figure 26.** Figure 26: Triangulation ∆ of a torus with a tacked circle. Now given a general pair ∆1 , ∆2 , choose any sequence d “ pd1, . . . dkq of flips such that ∆1 “ Fdk ¨ ¨ ¨ Fd1 p∆2 q. Given any two such sequences d, d 1 , both cluster transformations QpFdq “ QpFdk q ˝ ¨ ¨ ¨ ˝ QpFd1 q…

**Figure 27.** Figure 27: Graphical representation of skew-pairing for Q∆. 1 2 1 2 3 [PITH_FULL_IMAGE:figures/full_fig_p049_27.png]

**Figure 28.** Figure 28: Black-white graph Γ∆1. classical cluster transformation reads ς ` c : Ypc,1q ÞÑ Y 1 pc,1q pY 1 2 q ´1 , Yi ÞÑ Y 1 i , i “ 1, 2, 3. But now observe that the element ρ of the mapping class group of S induced by the 120 degree counterclockwise rotation satisfies ρp∆1 q “…

**Figure 29.** Figure 29: Black-white graph Γ∆2 . This time we observe that the element τ of the mapping class group given by the Dehn twist along the p0, 1q-curve on the torus satisfies τ p∆2 q “ ∆, and so induces a quasi-permutation morphism ςτ : Y 1 2 ÞÑ Y3, Y 1 3 ÞÑ Y2, Y 1 1 ÞÑ Y1, Y 1 pc…

**Figure 30.** Figure 30: (omitting all vertices on lines labelled 0 and n ` 1 along with all adjacent arrows, so that the quivers Q˘1 and Q˘n have 3 vertices each). Given a word i, we read it left to right and draw quivers Qa for each a P A. We then amalgamate adjacent vertices in each of the…

**Figure 31.** Figure 31: Braid move pi, i ` 1, iq Ø pi ` 1, i, i ` 1q. i ´ 1 i i ` 1 i ´ 1 i i ` 1 [PITH_FULL_IMAGE:figures/full_fig_p052_31.png]

**Figure 32.** Figure 32: Shuffle pi, ´iq Ø p´i, iq. 1 2 1 2 [PITH_FULL_IMAGE:figures/full_fig_p052_32.png]

**Figure 33.** Figure 33: Merge p1, 1q Ñ p1q. 5.2. Reduction of cyclic Weyl words. In this subsection we describe certain reductions of cyclic Weyl words, which were explained to us by M. Gekhtman. In what follows these reductions yield sequences of quiver mutations which we use over and over …

**Figure 34.** Figure 34: Quiver Q4 cyl (on the left) and its image under Φp1,2q 1 Φ p1,1q 1 (on the right). 5.3.1. Quiver transformation Φ1. Consider the quiver Qn cyl » Q∆ for a triangulation ∆ of a cylinder with a single marked point on each boundary component, see the left side of [PITH_F…

**Figure 35.** Figure 35: shows Q4 cone, where positions of vertices have been changed without relabelling. Following the quiver through the steps of the transformation Φ1, one can see that the bottom n mutable rows of Qn cone form a subquiver Qn w0,c, while the top n ´ 2 mutable rows of Qn co…

**Figure 36.** Figure 36: Quiver Q4 frust (on the left) and its image under Φp1,2q 2 Φ p1,1q 2 (on the right). Example 5.4. The step Φp1,3q 2 consists of consecutive mutations at the shaded nodes on the right pane of [PITH_FULL_IMAGE:figures/full_fig_p058_36.png]

**Figure 37.** Figure 37: Quiver Q4 ∆;c . where A is the Cartan matrix of type An. We write ΛSě0 for the rank 2pn ` 1q-sublattice of Λ∆;c spanned by the eh˘ together with the esi , eti , and ΛSă0 for the sublattice spanned by all other e-basis vectors for ΛS. Hence we have a non-orthogonal dir…

**Figure 38.** Figure 38: Quiver Q4 Sě0 . 5.3.3. Quiver transformation Φc` . Now let S 1 be a surface with a pair of tacked circles c˘. Suppose that ∆1 is an ideal triangulation of S 1 containing isolating cylinders for both c˘, so that ∆1 is locally of the form shown in the right pane of [PI…

**Figure 39.** Figure 39: Quiver Q4 sf (on the left) and its image under Φp0q c` (on the right). We now define a quiver transformation Φc` “ Φ3 ˝ Φ p0q c` (5.22) of Q∆1 which modifies only the subquiver Qn sfp`q. The preliminary wave Φp0q c` consists of ‚ apply transformation (4.16) at nodes p…

**Figure 40.** Figure 40: Graphical representation of pairings between basis vectors for the basis teℓu associated to the subquiver of Q4 ∆1 ;c˘ in the vicinity of tacked circle c` (on the left), and for the corresponding dotted basis te9ℓu (on the right). Notation 5.11. Given a pair of tacked…

**Figure 41.** Figure 41: Bicolored graph Γ∆`;c` . and so an easy computation shows that σipAvj q “ $ ’’& ’’% Avi`1Ac,i`1 Ac,i j “ i AviAc,i Ac,i`1 j “ i ` 1 Avj else. (5.24) Hence in a c-isolating cluster the braid group acts by a quasicluster automorphism, which proves the action factors thr…

**Figure 42.** Figure 42: Bicolored graph Γ∆;c. The cluster transformation for the shift σc˘ is expressed via the quasi-permutation ςc˘ described in Lemma 4.22, which in Q∆1 ;c˘ simplifies to τc˘ “ ςc˘ peℓq “ # e 1 ιpℓq ´ e9 1 ιpv ˘ i q , if ℓ “ pc˘, iq, e 1 ιpℓq , otherwise, (5.28) where ι i…

**Figure 43.** Figure 43: Subquiver of Qp∆;c (on the left) and that of Qp1 ∆;c (on the right) for n “ 4. Now consider the cluster transformation µBaxter defined as the composition of the following 2n ` 1 mutations starting from the quiver Qp∆;c: µBaxter “ µsn µtn . . . µs2 µt2 µs1 µt1 µa. (5.3…

**Figure 44.** Figure 44: Quiver Q3,2 (on the left) and its image under µ3,2 (on the right). Using the dictionary between Weyl words and quivers described in Section 5,we can now present the transformation Φn,k as a sequence of mutations µn,k. Let Qn,k be the quiver with 2n ` 2k ` 1 vertices, …

**Figure 45.** Figure 45: Shifts of vertices during µ3,2 . commute. Within each even column, highlighted by dashed rectangles, we read circled vertices bottom to top, and the non-circled ones top to bottom. Sequences of mutations at circled and non-circled vertices at a given even column commu…

**Figure 46.** Figure 46: Mutation sequence µ3,2 . and µ p2n`2k`2q,‚ n,k “ 1. Let us also notice that by reflecting the parallelogram on [PITH_FULL_IMAGE:figures/full_fig_p088_46.png]

**Figure 47.** Figure 47: Mutation sequence µ3,2 . In the remainder of this section, we make extensive use of tropical cluster transformations. We start with the following useful result. Lemma 7.5. Let yi denote the tropical cluster variables of the quiver Qn,k, and y 1 i be the image of yi un…

**Figure 48.** Figure 48: Quivers Q5 c;k for k “ 7, 8, 9, 13 (on the left) and k “ 10, 11, 12 (on the right). the vertex of ∆ opposite the side e, that is Φ∆;A starts and ends with a mutation at the node c1 in the notations of [PITH_FULL_IMAGE:figures/full_fig_p093_48.png]

**Figure 49.** Figure 49: Quiver Q5 1 [PITH_FULL_IMAGE:figures/full_fig_p096_49.png]

**Figure 50.** Figure 50: Quiver Q5 2 [PITH_FULL_IMAGE:figures/full_fig_p097_50.png]

**Figure 51.** Figure 51: Quiver Q5 3 [PITH_FULL_IMAGE:figures/full_fig_p098_51.png]

**Figure 52.** Figure 52: Quiver Q5 4 [PITH_FULL_IMAGE:figures/full_fig_p099_52.png]

**Figure 53.** Figure 53: Quiver Q5 5 [PITH_FULL_IMAGE:figures/full_fig_p100_53.png]

**Figure 54.** Figure 54: Quiver Q5 6 [PITH_FULL_IMAGE:figures/full_fig_p101_54.png]

**Figure 55.** Figure 55: Quiver Q5 7 [PITH_FULL_IMAGE:figures/full_fig_p102_55.png]

**Figure 56.** Figure 56: Quiver Q5 8 [PITH_FULL_IMAGE:figures/full_fig_p103_56.png]

**Figure 57.** Figure 57: Quiver Q5 9 [PITH_FULL_IMAGE:figures/full_fig_p104_57.png]

**Figure 58.** Figure 58: Quiver Q5 10 [PITH_FULL_IMAGE:figures/full_fig_p105_58.png]

**Figure 59.** Figure 59: Quiver Q5 11 [PITH_FULL_IMAGE:figures/full_fig_p106_59.png]

**Figure 60.** Figure 60: Quiver Q5 12 [PITH_FULL_IMAGE:figures/full_fig_p107_60.png]

**Figure 61.** Figure 61: Quiver Q5 13 [PITH_FULL_IMAGE:figures/full_fig_p108_61.png]

**Figure 62.** Figure 62: Possible local quivers I ´´ Toda, I`´ Toda, I`` Toda in the definition of a locally glueable pair for P GL5. Definition 8.14. A locally glueable pair of handle signature ε “ pεa, εbq P t``, ´`, ´´u consists of a pair of quivers pQ, Qcutq together with the following ad…

**Figure 63.** Figure 63: Possible local quivers I ´´ hat , I`´ hat , I`` hat in the definition of a locally glueable pair for P GL3. (2) A decomposition of index sets Icut “ I ă0 cut \ I ě0 cut and a bijection I ě0 cut » I ε hat such that the sublattice in ΛQcut corresponding to directions te…

**Figure 64.** Figure 64: The quivers Qˆ 2,1 and Qˆ 1,2 [PITH_FULL_IMAGE:figures/full_fig_p135_64.png]

**Figure 65.** Figure 65: The quivers Qˆcut 2,1 and Qˆcut 1,2 . Observe that the quiver Qˆcut m,n in the corresponding locally glueable pair pQˆm,n, Qˆcut m,nq can be obtained from Qˆcut n,m by applying the cluster transformation µ cut n,m “ ςn,m ˝ mź `1 j“1 µm,0 pvj,aq, where µm,0pvj,aq is th…

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Cluster structure on genus 2 spherical DAHA: seven-colored flower

S.\,Arthamonov, L.\,Chekhov, P.\,Di Francesco, R.\,Kedem, G.\,Schrader, A.\,Shapiro, M.\,Shapiro. ``Cluster structure on genus 2 spherical DAHA: seven-colored flower.'' arXiv:2402.16074 (2024)

work page Pith review arXiv 2024

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11Symplectic geometry of Teichm\"uller spaces for surfaces with ideal boundary.'' Communications in Mathematical Physics 405, no

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work page 2018

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``Quantum character varieties and braided module categories.'' Selecta Mathematica 24, no

D.\,Ben-Zvi, A.\,Brochier, D.\,Jordan. ``Quantum character varieties and braided module categories.'' Selecta Mathematica 24, no. 5 (2018): 4711-4748

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A.\,Braverman, M.\,Finkelberg, H.\,Nakajima. ``Towards a mathematical definition of Coulomb branches of 3-dimensional N= 4 gauge theories, II.'' arXiv:1601.03586 (2016)

arXiv 2016

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A.\,Braverman, M.\,Finkelberg, H.\,Nakajima. ``Coulomb branches of 3d N= 4 quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes).'' arXiv:1604.03625 (2016)

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A.\,Berenstein, S.\,Fomin, A.\,Zelevinsky. ``Cluster algebras III: Upper bounds and double Bruhat cells.'' Duke Mathematical Journal 126, no. 1 (2005): 1-52

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``Extended Goldman symplectic structure in Fock--Goncharov coordinates.'' Journal of Differential Geometry 124, no

M.\,Bertola, D.\,Korotkin. ``Extended Goldman symplectic structure in Fock--Goncharov coordinates.'' Journal of Differential Geometry 124, no. 3 (2023): 397-442

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