{"id":"89866ca0-393b-4e9e-95b3-078b6fdd1843","arxiv_id":"2412.15919","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any Coxeter system, the fusion-equivariant Bridgeland stability manifold of a newly constructed 2-Calabi-Yau category is a covering space of a hyperplane complement whose quotient fundamental group is the Artin-Tits group.","lead":"The paper constructs a 2-Calabi-Yau category for every Coxeter system and proves that its fusion-equivariant stability manifold covers a hyperplane complement whose quotient fundamental group is the associated Artin-Tits braid group, up to a Z factor. This yields a new geometric candidate for K(pi,1) spaces for Artin-Tits groups.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 7.21 is the load-bearing transfer: the [BDL23, §4] semistable-object algorithm is asserted to go over verbatim to fusion-enhanced categories, and as stated it is automatic for Stab_C(H) while Proposition 7.23 needs it for the whole component.","rationale":"The reader's weakest-assumption pick, Lemma 7.21, is also the point I find most load-bearing. The covering property of Theorem 7.7 is the core of the paper, and it is exactly the identification of im(Z†_C) with Υ_reg that prevents the central charge from vanishing on root hyperplanes. The paper's appeal to [BDL23, Section 4] is a delegation of a substantial construction to a setting with a different Grothendieck group and different spherical objects, and the text even states the lemma in a form where it is automatic by positivity, which suggests a real gap in the logical chain around Proposition 7.23. I do not think this warrants rejection: the overall strategy follows Bridgeland and Ikeda, the authors are explicit about the delegation, and the rank-two computations in [Hen] may supply the missing details. The honest verdict remains CONDITIONAL pending completion or clarification of Lemma 7.21; hence my recommendation is unchanged from the reader's CONDITIONAL. I also considered Proposition 5.3 (braid relations), but the rank-two reduction and displayed formula (21) make that a less severe gap than the unverified transfer of the semistable-object algorithm.","tokens_in":48772,"tokens_out":16249,"duration_ms":163633,"concrete_test":"For the first non-simply-laced case Γ=I2(5) with C(Γ)=Fib, explicitly execute the [BDL23, §4] algorithm in the fusion-enhanced category D(Γ). For each of the five positive roots α, write the object E_α as an explicit bounded complex of modules Ps⊗E⟨k⟩, compute its class in Λ, and check that E_α is C-spherical (or at least that it is semistable in the stability condition produced by the algorithm). Then feed the result through the five W-translates of a generic chamber point and verify that no translate has Z(α)=0. If all five objects exist, are C-spherical, and have the correct classes, the transfer is credible for the smallest nontrivial case; if any construction fails or produces the wrong class, Lemma 7.21 fails and Proposition 7.23 collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 7.7's covering statement depends on Proposition 7.23, the identification im(Z†_C)=Υ_reg. The proof must exclude central charges vanishing on any positive real root or nonzero imaginary-cone vector. Lemma 7.21 is the only argument ruling out root hyperplanes, and it cites [BDL23, Section 4] with the sentence that the algorithm 'goes over verbatim in our setting', without proof. This transfer is not routine: in [BDL23] root classes live in an integral lattice, while here Λ is a K0(C(Γ))-module and a positive root such as wα_s has coefficients in Z[2cos(π/m)], realized by fusion labels whose Frobenius–Perron dimensions are those numbers. The constructed complexes, their classes, their semistability, and their C-equivariance all need to be re-verified with the fusion-enhanced spherical twists. Moreover, the lemma as written (τ∈Stab_C(H)) is automatic by positivity: for Z(α_s)∈H∪R_{<0}, every positive root has nonzero image. The hard case is an arbitrary τ∈Stab†_C(D), whose central charge could a priori lie on a root hyperplane; that case is not covered by the stated lemma, and Proposition 7.23's reduction to H seems to presuppose the very image identification being proved. If the [BDL23] construction cannot be transferred, root hyperplanes could enter the image and the covering theorem would fail. A second delegated proof, Proposition 5.3, is also load-bearing for the deck-group action, but it is a rank-two reduction with an explicit sketch, so it is less insecure than Lemma 7.21.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper associates to an arbitrary Coxeter graph Γ a fusion category C(Γ), a graded Frobenius zigzag algebra zig(Γ) inside C(Γ), and a triangulated 2-Calabi–Yau category D(Γ) = Komb(zig(Γ)-prmod). It constructs an action of the Artin–Tits group B(Γ) on D(Γ) by C-spherical twists, decategorifies this action to a faithful action of the Coxeter group W(Γ) on a K0(C(Γ))-lattice Λ, and studies C(Γ)-equivariant Bridgeland stability conditions. The main result (Theorem 7.7) claims that the connected component Stab†_C(D) containing the linear heart is a covering space of the hyperplane complement Υ_reg(Γ), and that the deck transformation action of π1(Υ_reg/W) is realized by autoequivalences generated by B(Γ) and the shift [2]. Corollary 8.10 and Theorem 8.5 describe the full component Stab†(D) in terms of a Coxeter unfolding ˇΓ and connect the resulting embeddings to LCM-homomorphisms. The paper also formulates faithfulness and K(π,1) conjectures and states a finite-type version (Theorem 7.13).","tokens_in":49110,"tokens_out":8460,"duration_ms":80770,"significance":"If correct, the main theorem substantially extends the Thomas–Bridgeland–Ikeda picture from symmetric Kac–Moody types to arbitrary Coxeter systems, providing a new geometric presentation of the hyperplane complement Υ_reg/W as the deck group action on a stability manifold. The construction of zigzag algebras as distinguished algebra objects in fusion categories, the explicit computation of Hom spaces, and the correction of gaps in Ikeda's argument are valuable contributions in their own right. However, the central image identification and the braid-relation verification are delegated to external sources, so the significance is conditional on those points being supplied.","major_comments":[{"comment":"The only argument ruling out root hyperplanes in the image of Z†_C is Lemma 7.21, whose proof is a one-sentence citation to [BDL23, Section 4] with the assertion that the algorithm 'goes over verbatim'. This transfer is not routine: in [BDL23] the root classes live in a free Z-lattice, whereas here they have coefficients in K0(C(Γ)), and the constructed objects' classes, semistability, and C-equivariance must be re-verified with fusion-enhanced spherical twists. More importantly, the lemma as stated is for τ ∈ Stab_C(H), where it is automatic by positivity: for Z ∈ C, every positive real root α has Z(α) ≠ 0 because α is a nonnegative combination of simple roots. The case needed in Proposition 7.23 is an arbitrary τ ∈ Stab†_C(D) whose central charge could a priori lie on a root hyperplane; Lemma 7.20 cannot reduce to Stab_C(H) for such τ because it presupposes that the central charge lies in Υ_N^reg. Thus the proof of im(Z†_C) = Υ_reg is incomplete as written and appears to rely circularly on the very identification being proved.","section":"§7.6, Lemma 7.21 and Proposition 7.23"},{"comment":"The braid relations among the C-spherical twists are load-bearing: they give the well-defined homomorphism B(Γ) → Br_ST in Theorem 5.6 and hence the deck-group identification in Theorem 7.7. The proof says 'exactly as in the rank two case' and delegates the key computation (21) to [Hen, §2.2], without verifying in this paper that the fusion-enhanced context satisfies all hypotheses of that reference. Please provide a complete proof of (21), or a precise reduction to the cited rank-two computation with every hypothesis checked.","section":"§5.2, Proposition 5.3"},{"comment":"The proof of Lemma 7.22 claims that the support property together with Lemma 7.21 implies that Z′(v) ≠ 0 for every nonzero v on an accumulation ray of positive root rays, and that the statement for all v in the closed imaginary cone follows by convexity. This is not justified as written: the support property constrains only classes of semistable objects, not arbitrary accumulation points of root rays, and convexity of Z(I) does not by itself prevent a positive linear combination of nonzero vectors in a cone from being zero. Since Lemma 7.22 is needed to exclude the imaginary-cone hyperplanes in Proposition 7.23, this step needs a detailed argument.","section":"§7.6, Lemma 7.22"}],"minor_comments":[{"comment":"Theorem 7.13 and Proposition 7.14 are stated as results but their proofs are omitted with a sketch and a reference to similar arguments. If they are not needed for the main theorem, they should be explicitly labeled as conditional or deferred to future work.","section":"§7.3, Theorem 7.13"},{"comment":"The notation H is used both for the upper half-plane in Definition 2.13 and for the linear heart in Section 7.1. This is a recurring source of potential confusion; consider renaming one of them.","section":"§2.2 and §7.1"},{"comment":"The notation Stab_C(D) uses C both for a fusion category and for a complexified chamber. The authors do note the convention, but a different symbol for the chamber (for example, C_ch) would improve readability.","section":"§1 and §3"},{"comment":"The proof of Proposition 7.11 invokes Proposition 7.25, but Proposition 7.25 only concerns Ψ(H) ⊂ H, not Ψ(P_s) ≅ P_s for all s. The intended implication can be obtained from Proposition 7.26, whose proof is stated to be 'essentially identical' and omitted. Please clarify the logical dependence.","section":"§7.3, Proposition 7.11"},{"comment":"The notation Komb appears alongside Kom^b; please standardize.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the paper is clearly written, but the two principal delegations — Lemma 7.21 and Proposition 5.3 — are exactly the points that distinguish this work from prior constructions. The referee report's conditional verdict is appropriate: the manuscript should not be accepted until the image identification is proved without circularity and the braid-relation transfer is supplied. There is also substantial overlap with [Hen], [DHL23], and [QZ23]; the introduction acknowledges this, but the dependence on [Hen] for a load-bearing step should be made more precise."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this is a plausible and genuinely new generalization of the Bridgeland–Ikeda picture to arbitrary Coxeter systems. The construction of zigzag algebras as algebra objects in fusion categories built from Temperley–Lieb–Jones categories, with Frobenius–Perron dimensions matching 2cos(pi/m), is a real achievement and likely of independent interest. The paper is honest, well-structured, and it explicitly corrects a gap in Ikeda's work (Remark 7.28). If the main theorem holds, it gives candidate K(pi,1) spaces for all Artin–Tits groups and ties the K(pi,1) conjecture to faithfulness of the B-action on D(Γ). That is a worthwhile target.\n\nThe soft spots are concentrated exactly where the reader's report says. Lemma 7.21 is load-bearing: it is the only step ruling out root hyperplanes in the image of Z^†_C, and the proof is one sentence saying the algorithm from [BDL23, Section 4] goes over verbatim. That transfer is not obviously routine. In [BDL23] root classes live in an integral lattice; here they live in a K0(C)-module with coefficients in Z[2cos(pi/m)], and you need to re-verify that the constructed complexes are semistable and C-equivariant. The stress-test note is also right that the lemma as stated is automatic for τ in Stab_C(H), while Proposition 7.23 needs it for the whole component. The reduction to H in Lemma 7.20 only works for points already in (Z^†_C)^{-1}(Υ_reg), which is what the proof is trying to establish, so as written there is a real circularity risk. This is the main gap. Proposition 5.3 is also delegated to the first author's thesis, but it is a rank-two reduction with an explicit sketch, so I am less worried. Proposition 8.9's 'best checked independently by the reader' is a hand-wave, but it is a minor point in the unfolding section.\n\nI want to give credit where it is due: the paper does not hide these gaps, the categorical setup is coherent, and the claimed theorem is not fitted to a target answer. No red flags on data or circular fitting. The unfolding story and the connection to LCM-homomorphisms are genuinely interesting and should be preserved.\n\nWho is this for? Representation theorists working on stability conditions, braid groups, and categorical actions. A serious referee should engage with it, but the referee should demand a complete proof of Lemma 7.21 and a clarification of the reduction in Proposition 7.23. My recommendation: send it to peer review, and treat the current version as conditionally acceptable pending those additions.","headline":"A genuinely new generalization of the Bridgeland–Ikeda stability picture to arbitrary Coxeter systems, with a strong construction and an honest write-up, but the main covering theorem currently rests on one delegated proof (Lemma 7.21) that is not routine.","tokens_in":49669,"tokens_out":1971,"would_cite":true,"duration_ms":22169,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16G20","18G80","20F36","20F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any Coxeter graph, the paper constructs a triangulated category whose fusion-equivariant stability component is a covering of a hyperplane complement, with the Artin–Tits group and the Serre shift acting as deck transformations.","keywords":["Bridgeland stability conditions","Artin-Tits groups","Coxeter systems","zigzag algebras","fusion categories","spherical twists","hyperplane complements","K(pi,1) conjecture"],"falsifier":"Compute the image of the central-charge map for a Coxeter system where the transferred algorithm is not verified, and look for a positive real root alpha whose hyperplane is hit by a central charge in the image; such a point would contradict Proposition 7.23 and Theorem 7.7. More directly, search for a C-equivariant stability condition whose central charge vanishes on a positive real root or on a nonzero element of the closed imaginary cone, which Lemmas 7.21 and 7.22 assert never happens.","tokens_in":48479,"feed_emoji":"🗺️","tokens_out":7200,"duration_ms":62231,"temperature":0.7,"pith_summary":"This paper constructs, for every Coxeter graph, a triangulated category that carries a natural action of the associated Artin–Tits group, and then studies the space of Bridgeland stability conditions on it that are equivariant for a fusion-category action. The central result is that a distinguished connected component of this equivariant stability space is a covering space of the hyperplane complement Upsilon_reg, and that the deck transformations of the quotient covering to Upsilon_reg modulo the Coxeter group are exactly the Artin–Tits braid group together with the Serre shift [2]. If correct, this realizes the fundamental group of the quotient hyperplane complement as autoequivalences of a triangulated category, giving a geometric K(pi,1)-style model for Artin–Tits groups. The construction works for arbitrary Coxeter systems, going beyond the symmetric Kac-Moody cases studied earlier.","feed_headline":"Stability spaces turn Artin-Tits groups into deck transformations","feed_subtitle":"A covering-space theorem makes the fundamental group of a Coxeter hyperplane complement act by autoequivalences.","key_machinery":"The central object is the zigzag algebra zig(Gamma), built as a graded Frobenius algebra object in a fusion category C(Gamma) associated to Gamma; each edge labeled m contributes an object of Frobenius-Perron dimension 2cos(pi/m). Its category of graded projective modules, after passing to bounded homotopy categories, gives the triangulated 2-Calabi-Yau category D(Gamma). The action of the Artin-Tits group is generated by C-spherical twists, two-term complexes of bimodules whose braiding relations mirror the Coxeter relations. The covering theorem is proved by showing that the central-charge map restricted to the C-equivariant component is a local homeomorphism onto Upsilon_reg, that the group actions are free and properly discontinuous, and that the quotient is Upsilon_reg/W; the identification of the image rests on a semistable-object algorithm for every positive real root.","core_discovery":"Theorem 7.7 states that for any Coxeter graph with Artin-Tits group B and Coxeter group W, the C-equivariant stability manifold of the category D contains a connected component Stab^dagger_C(D) for which the central-charge map is a covering onto the hyperplane complement Upsilon_reg. The composite covering to Upsilon_reg/W has deck-transformation group generated by the image of B, acting through C-spherical twists, and by the degree-two Serre shift [2]; when W is infinite, the fundamental group of Upsilon_reg/W is Z times B, and the Z factor is realized by [2]. For symmetric Kac-Moody Coxeter graphs, the construction recovers the earlier stability-space covering theorems as a special case. The paper also proves that the larger non-equivariant component Stab^dagger(D) covers the hyperplane complement of an unfolded Coxeter graph, with the equivariant component embedded inside it, yielding a geometric realization of LCM-homomorphism embeddings between Artin-Tits groups.","pith_inferences":["A testable check that goes beyond the paper is to verify, in a small infinite Coxeter example, that the transferred algorithm really produces a semistable object for every positive real root; failure would localize the main gap in the image identification.","If the covering theorem holds, the equivariant stability component gives a candidate intrinsic Teichmuller space for arbitrary Artin-Tits groups, and proving its contractibility would settle the K(pi,1) conjecture for all Coxeter types, not just finite ones.","Because the equivalence between D(Gamma) and the unfolded simply-laced category identifies the non-equivariant stability spaces, any faithfulness statement for D(Gamma) would embed B(Gamma) into a simply-laced Artin-Tits group, an open question the paper leaves."],"forward_implications":["If the Artin-Tits group acts faithfully on D, then Stab^dagger_C(D) is simply connected and is the universal cover of Upsilon_reg; faithfulness is known for finite Coxeter groups, so in finite type Stab^dagger_C(D) is contractible and Upsilon_reg/W is a K(pi,1) space.","The deck-transformation description realizes pi_1(Upsilon_reg/W), which is B or Z times B, as autoequivalences of D, with spherical twists matching Coxeter reflections at the level of central charges.","The closed embedding of covering spaces from the equivariant component to the unfolded component realizes LCM-homomorphism embeddings geometrically, and faithfulness of the B-action on D would imply injectivity of those embeddings.","The construction extends the known stability-space covering descriptions from symmetric Kac-Moody types to arbitrary Coxeter systems, with the fusion-category action playing an essential role outside that class."],"supporting_citations":[{"why":"Supplies the classical hyperplane complement Omega_reg whose fundamental group modulo W is the Artin-Tits group, the target of the covering theorem.","marker":"[Van83]"},{"why":"Provides the proof strategy for the covering property in the preprojective-algebra/Kac-Moody setting, which the paper adapts and corrects.","marker":"[Ike14]"},{"why":"Provides chamber and wall-crossing arguments used to identify the image of the central-charge map and to prove the group actions are free.","marker":"[Bri09]"},{"why":"Supplies the algorithm that constructs a semistable object of class alpha for each positive real root alpha, transferred verbatim in Lemma 7.21.","marker":"[BDL23]"},{"why":"Establishes that fusion-equivariant stability conditions form a closed complex submanifold, a technical foundation for Stab_C(D).","marker":"[DHL23]"},{"why":"Develops the rank-two case of C-spherical twists and fusion-equivariant stability conditions that the paper generalizes to arbitrary Coxeter systems.","marker":"[Hen]"},{"why":"Provides the imaginary cone for arbitrary Coxeter systems, used to define the hyperplane complement Upsilon_reg and to prove its topology.","marker":"[Dye19]"},{"why":"Introduces LCM-homomorphisms between Artin-Tits groups, whose embeddings are geometrically realized by the unfolding construction.","marker":"[Cri99]"},{"why":"Gives related finite-type fusion-equivariant stability results, showing the new construction agrees with known ones in finite types.","marker":"[QZ23]"}],"fun_headline_variants":["Stability covering theorem ties Artin-Tits groups to deck transformations","Artin-Tits groups act as deck transformations on stability spaces","Stability manifolds reveal Artin-Tits groups as deck transformations","Artin-Tits group acts by deck transformations in stability theory"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the image of the central-charge map is exactly Upsilon_reg assumes that for every positive real root alpha the fusion-enhanced category has a semistable object of class alpha; the paper transfers an algorithm from the symmetric Kac-Moody setting without giving a full proof of that transfer.","fun_headline_variants_meta":{"raw":{"variants":["Stability covering theorem ties Artin-Tits groups to deck transformations","Artin-Tits groups act as deck transformations on stability spaces","Stability manifolds reveal Artin-Tits groups as deck transformations","Artin-Tits group acts by deck transformations in stability theory"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000811,"raw_usage":{"total_tokens":3486,"prompt_tokens":805,"completion_tokens":2681,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":421,"completion_tokens_details":{"reasoning_tokens":2608}},"tokens_in":421,"tokens_out":2681,"duration_ms":18410,"temperature":1.0,"reasoning_tokens":2608,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:57:09.453856+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the image of the central-charge map for a Coxeter system where the transferred algorithm is not verified, and look for a positive real root alpha whose hyperplane is hit by a central charge in the image; such a point would contradict Proposition 7.23 and Theorem 7.7. More directly, search for a C-equivariant stability condition whose central charge vanishes on a positive real root or on a nonzero element of the closed imaginary cone, which Lemmas 7.21 and 7.22 assert never happens.","supporting_citations":[],"review_version":1}