{"id":"65871ba7-9288-49fd-80b1-7ee4f4b980dd","arxiv_id":"2501.00394","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"An A_n cluster algebra injects into the equivariant quantum cohomology ring of a partial flag variety, and all-genus Gromov-Witten invariants are mutation-invariant for A-type quiver varieties.","lead":"This paper shows that a type-A cluster algebra can be embedded inside the quantum cohomology ring of a partial flag variety, and proves that all-genus Gromov-Witten invariants are preserved under quiver mutation for spaces that are mutation-equivalent to a type-A quiver. The results connect cluster algebras with enumerative geometry and provide a concrete base case for a broader duality conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The omitted proof of Lemma 2.16 is the load-bearing gap: the entire all-genus Seiberg duality argument inherits an unverified stability-chamber classification.","rationale":"The reader's verdict is CONDITIONAL, with the weakest assumption identified as the omitted proof of Lemma 2.16. My reading of the full text confirms that this lemma is the main structural input to the all-genus Seiberg duality proof: the fixed-point description, the bijection between fixed loci under mutation, and the one-dimensional orbit classification all derive from it. Because the proof is explicitly omitted, the correctness of Section 4 is currently unverified at its foundation. The concern is concrete and testable, and it does not amount to a disagreement with the field or to an ad hominem criticism. I also note a secondary formal issue in Theorem 5.1: the formal variables ξ_i appear in the image of ψ, so the codomain needs to be enlarged or specialized explicitly; the authors' 'let q_i = ...' relation partly addresses this, but the statement still conflates the usual quantum cohomology ring with a ring extension by the ξ_i. This issue is localized and likely repairable, so it does not change the verdict. No reason emerged to reject the paper outright or to upgrade the verdict, so the appropriate recommendation remains CONDITIONAL, expressed here as UNCHANGED relative to the reader's verdict.","tokens_in":48248,"tokens_out":7654,"duration_ms":83032,"concrete_test":"Independently verify Lemma 2.16 for a nontrivial mutation-equivalent quiver, for example the A_4 quiver after mutating at node 2 and then at node 3. Enumerate all GIT chambers for the stability parameter θ by direct King-stability analysis, check that the semistable locus equals the claimed one (nondegenerate matrices, B=0, A_{v1}A_{v2}=0), and confirm that the critical locus is the claimed complete intersection with |Z^S| = ∏_{i=1}^n binom(N_{i+1}, N_i). If the classification holds in several such examples and a proof of Lemma 2.16 is supplied, the concern is settled; if any chamber is missing or the fixed-point count differs, Theorem 4.1 is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central all-genus result, Theorem 4.1, is proved by virtual localization on the varieties Z associated to decorated QPs in Ω_n. The torus fixed locus (Lemma 4.6), the fixed-point bijection under mutation (Lemma 4.7), the classification of one-dimensional torus orbits (Proposition 4.14), and the weight and graph comparisons in Sections 4.4–4.6 all rely on Lemma 2.16. That lemma states the 'only valid stability condition' and describes the critical locus as a complete intersection in a binary-tree quiver variety defined by equations A_{v1}A_{v2}=0. Its proof is explicitly omitted: 'This proof of this Lemma is overly intricate, so we prefer to omit it' (end of Section 2.6). This is not a cosmetic gap: if the stability chamber description is incomplete or incorrect, the torus fixed set can change, the mutation bijection can fail, and the localization comparison of decorated graphs in Section 4.6 has no foundation. I agree with the reader that Lemma 2.16 is the weakest load-bearing assumption. A secondary, more localized well-definedness issue affects Theorem 5.1: the images of cluster variables contain formal variables ξ_i, so the stated codomain QH^*_S(Fl)[t] must be read as an extension or specialization involving the ξ_i, not literally the quantum cohomology ring with formal Kähler parameters; this is repairable but should be stated precisely.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims two main results. First, Theorem 4.1 asserts all-genus Seiberg duality for quiver varieties associated to decorated quivers with potential that are mutation-equivalent to the A_n-type quiver; the proof uses virtual localization and a comparison of decorated graphs after a canonical bijection of torus fixed points. Second, Theorem 5.1 asserts an injective Q-algebra homomorphism from the A_n cluster algebra into the equivariant quantum cohomology ring of the partial flag variety, with cluster variables mapped to equivariant Chern polynomials; the proof reduces the homomorphism statement to quantum cohomological exchange relations, which are established through the abelian/nonabelian correspondence.","tokens_in":48555,"tokens_out":9177,"duration_ms":97435,"significance":"If both theorems are correct, the paper would connect cluster algebras and quantum cohomology in a new and explicit way, and would extend Seiberg duality for A-type quivers from genus zero to all genera. The computational strategy is coherent and uses well-established tools: the localization formulae of Graber–Pandharipande and Liu, and the abelian/nonabelian correspondence in the form proved by Ciocan-Fontanine–Kim–Sabbah and Webb. The explicit quantum exchange relations in Sections 5.3–5.5 are checkable and are a strength of the manuscript. However, the current text does not establish one of the central structural inputs, and the definition and injectivity of the cluster-to-quantum map have gaps; for these reasons the main theorems are not yet fully supported.","major_comments":[{"comment":"The classification of valid stability conditions for every decorated QP in Ω_n, together with the description of the critical locus Z as a complete intersection in a binary-tree quiver variety, is stated in Lemma 2.16, but its proof is explicitly omitted: \"This proof of this Lemma is overly intricate, so we prefer to omit it.\" This lemma is not a peripheral technicality: Lemma 4.6 describes the torus fixed locus using this description, Lemma 4.7 constructs the fixed-point bijection under mutation from it, Proposition 4.14 classifies one-dimensional torus orbits by an induction that presupposes it, and the graph and weight comparisons in Sections 4.4–4.6 rely on Proposition 4.14. If the stability-chamber classification is incomplete or incorrect, the fixed locus, the orbit structure, and therefore the localization comparison underlying Theorem 4.1 are not established. This is a load-bearing gap and must be fixed by a complete proof or a precise reference.","section":"§2.6, Lemma 2.16"},{"comment":"The stated codomain of ψ is QH_S^*(Fl)[t], but the images ψ(x_i)=(-1)^{N_i}ξ_i c_t^S(S_i) contain formal variables ξ_i. The relation qi = (-1)^{N_i+N_{i+1}} ξ_{i-1}ξ_{i+1}^{-1} expresses Kähler variables in terms of ξ_i, so the target ring must be described precisely—for example as an extension or localization in which the ξ_i are adjoined and the q_i are subsequently identified—before ψ can be regarded as a map into QH_S^*(Fl)[t]. In addition, part (2) of Theorem 5.1 defines ψ on a non-initial cluster variable by choosing an arbitrary non-initial seed containing that variable; the paper gives no argument that this image is independent of the choice of seed and of the mutation path. The injectivity statement is also stated in Theorem 5.1 but its proof is addressed only later; see the next comment. The codomain issue and the well-definedness proof must be supplied.","section":"§5, Theorem 5.1"},{"comment":"The proof of injectivity of ψ is not rigorous as written. After invoking the Laurent phenomenon, the argument asserts that if ψ(α)=0 then a minimal exponent i0 has coefficient a_{i0}=0, because \"this term will not be affected by the quantum reduction.\" But ψ(α) is an element of a quantum cohomology ring, where the product is deformed by quantum corrections and the ring has nontrivial relations; a coefficient comparison in ξ^i and t-power is not automatically valid unless a filtration or basis making the claimed leading term well-defined is established. The partial order on exponent vectors does not, by itself, control quantum corrections involving sums of products of initial variables. Since \"injective\" is part of the central claim of Theorem 5.1, this argument needs to be replaced by a genuine proof, for instance by specializing q to zero and using the classical equivariant cohomology basis.","section":"§5.5, Theorem 5.2(2)"},{"comment":"Equations (70) and (71), which are the exchange relations for the three-adjacent-node cases, are not proved in the manuscript: the text states, \"The proof is similar with that of (69). One can prove by induction on m... We leave the detail to readers.\" Since these identities are part of Theorem 5.2 and hence of the proof that ψ is a ring homomorphism, the paper should include the complete induction or give a precise reduction to the already proved cases. Leaving the details to the reader is not sufficient for a central claim of this kind.","section":"§5.5.3, Lemma 5.16"}],"minor_comments":[{"comment":"In item 2 of Conjecture 3.8, the two displayed cases both state the condition N_f(k)>N_a(k); the second case should presumably be N_f(k)<N_a(k). Please correct this typo, which currently makes the formula ambiguous.","section":"§3.3, Conjecture 3.8"},{"comment":"The sentence preceding Example 2.17 reads \"This proof of this Lemma is overly intricate, so we prefer to omit it\", which contains a grammatical error; more importantly, it should be removed once a proof is supplied, since an explicit admission of an omitted proof is incompatible with a research article's claims.","section":"§2.6, Lemma 2.16"},{"comment":"The proof of Lemma 4.7 states \"I′_v2 ⊂ I′_v since I′_v2 ∩ I_v = ∅\"; the implication is not immediate from the displayed definitions and should be expanded or corrected, since Lemma 4.7 is used to construct the fixed-point bijection in the Seiberg duality theorem.","section":"§4.2, Lemma 4.7"},{"comment":"The Kähler parameters q_i are treated both as formal variables and as elements of the quantum cohomology ring. Please clarify the ring in which the relation qi = (-1)^{N_i+N_{i+1}} ξ_{i-1}ξ_{i+1}^{-1} is imposed, and distinguish these parameters from the equivariant parameters λ_1,...,λ_{N_{n+1}}.","section":"§5, Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper's main ideas are promising and the technical apparatus is standard, but the omitted proof of Lemma 2.16 is the kind of gap that prevents acceptance in its current form. The referee report has identified the same issue as the main concern. If the authors can supply a complete proof of Lemma 2.16 and repair the well-definedness and injectivity arguments for ψ, the revised manuscript could be appropriate for publication; I would not recommend acceptance before those points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper proves two things: an injective Q-algebra homomorphism from the A_n cluster algebra into the equivariant quantum cohomology ring of a partial flag variety, and all-genus Seiberg duality for every quiver mutation-equivalent to the A_n quiver. The second is the heavier result and the main event. On reading, the cluster-variable image is a clean construction: Chern polynomials of quotient bundles match the exchange relations after introducing xi variables to absorb signs and Kaehler factors. The proof of the quantum product identities in Section 5 is a genuine piece of work, using abelian/nonabelian correspondence and explicit Schur polynomial computations, and the computations are detailed. The localization comparison for the Seiberg duality theorem is also carefully set up: fixed points, one-dimensional orbits, weights, and decorated graph contributions are all compared explicitly. The authors also honestly note in Remark 5.3(4) that a similar exchange-relation equation appears in Fomin-Williams-Zelevinsky, which is the right thing to do.\n\nThe soft spot is exactly where the reader and stress-test put it: Lemma 2.16. The entire Section 4, including fixed point classification, orbit description, mutation bijection, and graph comparison, depends on the stability phase description and the critical locus being a complete intersection defined by A_{v1}A_{v2}=0 in a binary-tree quiver variety. The proof is omitted with the comment that it is overly intricate. For a lemma this load-bearing, that is not acceptable in a self-contained paper. If the chamber description is wrong or incomplete, Theorem 4.1 falls. This is repairable if the authors can supply the proof or a detailed sketch, but it is not cosmetic.\n\nThere is also a smaller well-definedness issue in Theorem 5.1: the image of psi contains formal variables xi_i, so the codomain QH^*_S(Fl)[t] must be read as an extension ring or the xi_i must be specialized. The statement as written is not literally correct. The injectivity proof using the partial order on xi-exponents is plausible but terse.\n\nOverall, this is a serious paper with substantial new results, but it is not ready to be taken as proved. It deserves peer review, not desk rejection. A referee should ask for a complete proof or detailed outline of Lemma 2.16 and a precise statement of the codomain in Theorem 5.1. If those are supplied, the paper would be a strong contribution; as it stands, treat the all-genus theorem as conditional.","headline":"Strong, plausible results, but the omitted proof of Lemma 2.16 makes the all-genus Seiberg duality theorem conditional.","tokens_in":49085,"tokens_out":3910,"would_cite":false,"duration_ms":39124,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13F60","14N35","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The $A_n$ cluster algebra embeds injectively into the equivariant quantum cohomology ring of a partial flag variety, and all-genus Seiberg duality holds for A-type quiver mutations.","keywords":["cluster algebras","quantum cohomology","quiver varieties","Gromov-Witten invariants","Seiberg duality","flag varieties","quiver mutations","equivariant localization"],"falsifier":"Compute, for the $A_3$-mutation-equivalent quiver of Example 2.17 with decorations $N_1<N_2<N_3<N_4$, the torus-fixed locus from the equations $A_1A_2=0$, $A_2B=0$, $BA_1=0$ under the stability condition $\\sigma_1>0$, $\\sigma_2<0$, $\\sigma_3+\\sigma_2>0$, and verify Lemma 4.6's description of fixed points and Lemma 4.7's bijection under the mutation $\\mu_2$; if any fixed set fails to match the predicted subset rules, or if a genus-one invariant computed on both sides differs, the claimed all-genus Seiberg duality fails.","tokens_in":48025,"feed_emoji":"📐","tokens_out":11709,"duration_ms":109414,"temperature":0.7,"pith_summary":"This paper proves two things about A-type quivers. The first is that the $A_n$ cluster algebra embeds into the equivariant quantum cohomology ring of the partial flag variety $Fl(N_1,\\dots,N_{n+1})$, extended by a formal variable $t$: each initial cluster variable is sent to a Chern polynomial of a tautological bundle, and each non-initial cluster variable to a Chern polynomial of a quotient bundle $S_j/S_i$, with signs and $\\xi$-factors chosen so that the cluster exchange relations become quantum product relations. The second is the all-genus Seiberg duality conjecture for every quiver with potential mutation-equivalent to an $A_n$ quiver: the Gromov–Witten invariants of the critical locus of the mutated potential coincide with those of the flag variety once insertions and curve classes are matched. A sympathetic reading is that cluster variables are not formal symbols but actual cohomology classes, and that quiver mutation is an invariance symmetry of all genera, not just genus zero. The paper also proposes a general conjecture extending this cluster-algebra-in-quantum-cohomology construction to arbitrary quivers with potential.","feed_headline":"Cluster algebra found inside a quantum cohomology ring","feed_subtitle":"Every cluster variable is a Chern polynomial, and A-type quiver mutations preserve all-genus invariants.","key_machinery":"Two mechanisms carry the paper. The first is the comparison of Gromov–Witten invariants by torus localization: after Lemma 2.16 identifies the critical locus $Z$ of the potential as a complete intersection defined by equations $A_{v_1}A_{v_2}=0$ inside a binary-tree quiver variety, Lemma 4.7 gives a canonical bijection $\\varphi: Z^S \\to (Z')^S$ between finite torus-fixed sets, Proposition 4.15 extends it to one-dimensional orbit closures, and Proposition 4.16 shows flag weights are exactly preserved; the decorated-graph localization formula then forces the all-genus invariants to agree term by term. The second is the abelian/nonabelian correspondence for quantum cohomology, which reduces the needed identities on the flag variety to explicit relations in the abelianized toric variety; the output is the quantum cohomological cluster exchange relation above, which matches the cluster exchange relation after the sign and $\\xi$-factors are absorbed.","core_discovery":"The central discovery is that the cluster algebra and the enumerative geometry of A-type quivers are the same structure viewed from two sides. On the algebraic side, Theorem 5.1 constructs a map $\\psi: A_n \\to QH_S^*(Fl)[t]$ with $\\psi(x_i)=(-1)^{N_i}\\xi_i c_t^S(S_i)$ on initial variables and $\\psi(x'_v)=(-1)^{N_j-N_i}\\xi_j\\xi_i^{-1}c_t^S(S_j/S_i)$ on a non-initial variable whose associated decoration is $N_j-N_i$; with Kähler variables set to $q_i=(-1)^{N_i+N_{i+1}}\\xi_{i-1}\\xi_{i+1}^{-1}$, the map is a well-defined injective $\\mathbb{Q}$-algebra homomorphism. The proof is reduced to the quantum cohomological cluster exchange relation $c_t^S(S_m/S_{p+1}) * c_t^S(S_k/S_l) = c_t^S(S_m/S_l) * c_t^S(S_k/S_{p+1}) + \\prod_{a=p+1}^k (-1)^{N_a+N_{a-1}}q_a\\, c_t^S(S_m/S_{k+1}) * c_t^S(S_p/S_l)$. On the geometric side, Theorem 4.1 proves all-genus Seiberg duality for any two quivers with potential in $\\Omega_n$ related by a quiver mutation: torus localization gives a bijection of fixed loci and of one-dimensional orbits, the weights of every flag and the combinatorics of every decorated graph are preserved, so the genus-$g$, $m$-marked, curve-class-$\\beta$ invariants agree after applying the canonical cohomology isomorphism and curve class transformation.","pith_inferences":["One consequence the paper leaves implicit is that the injectivity of $\\psi$ gives a new faithful representation of the cluster algebra: distinct Laurent expressions in the initial variables cannot coincide after passing to quantum cohomology, so quantum products detect cluster-algebraic nontriviality.","If a stability classification analogous to Lemma 2.16 can be supplied for other mutation classes, the same localization mechanism should yield all-genus Seiberg duality for D-type and other quivers, with the cluster exchange relation playing the role of the quantum product relation.","Because $\\psi$ is not surjective, the cluster algebra is a proper subalgebra; one could ask whether its saturation or localization in $QH_S^*(Fl)[t]$ recovers the full quantum cohomology ring, linking cluster exchange relations to known presentations of flag-variety quantum cohomology.","A testable extension is to specialize the formal variable $t$ and compare the induced map on ordinary cohomology; this would produce ordinary-cohomology cluster representations and might expose the role of the $q_i$ factors more directly."],"forward_implications":["For every $A_n$-mutation-equivalent quiver with potential, the all-genus Gromov–Witten invariants of the critical locus are equal to those of the partial flag variety under an explicit matching of insertions and curve classes; in particular, the equivariant quantum cohomology rings are isomorphic after the Kähler variable change of Corollary 4.2.","The cluster algebra $A_n$ sits inside $QH_S^*(Fl)[t]$ as a subalgebra, so every cluster variable has a geometric meaning as a signed, $\\xi$-scaled Chern polynomial of a quotient bundle of tautological bundles.","Because $\\psi$ is injective, relations among cluster variables are governed by relations among these Chern polynomials, and any relation in the cluster algebra must already be a consequence of the quantum cohomological cluster exchange relations.","The general conjecture of the paper, if proved, would extend the same dictionary to arbitrary quivers with potential, making cluster exchange relations into quantum product relations for quiver varieties.","The localization comparison gives a canonical correspondence between torus-fixed points and one-dimensional orbits across mutation, so the full Gromov–Witten invariant, not just the genus-zero part, is invariant under A-type quiver mutation."],"supporting_citations":[{"why":"Supplies the localization theorem converting equivariant Gromov–Witten invariants to sums over torus-fixed loci, used in the proof of Theorem 4.1.","marker":"[23]"},{"why":"Provides the decorated-graph localization formula that organizes the contributions compared under quiver mutation.","marker":"[34]"},{"why":"Establishes the abelian/nonabelian correspondence for Frobenius manifolds, the main tool for proving the quantum cohomological cluster exchange relations.","marker":"[10]"},{"why":"Gives the abelian/nonabelian correspondence for I-functions, used to restrict the correspondence to small quantum cohomology without parameter change.","marker":"[43]"},{"why":"Extends abelianization and quantum Lefschetz to orbifold quasimap I-functions, supporting the computations in Section 5.","marker":"[42]"},{"why":"Provides the specialization rule for Novikov variables of the abelianized flag variety used in deriving the quantum relations.","marker":"[24]"},{"why":"Foundational cluster algebra paper supplying the exchange relations and the Laurent phenomenon used to prove injectivity of $\\psi$.","marker":"[19]"},{"why":"Defines quiver mutations with potentials and reduced quivers with potentials, the formal framework for all decorated quivers in $\\Omega_n$.","marker":"[13]"},{"why":"Classifies quivers mutation-equivalent to an $A_n$ quiver, giving the recursive description of $\\Omega_n$ used throughout.","marker":"[7]"},{"why":"Formulates the Seiberg duality conjecture for Gromov–Witten theory that Theorem 4.1 proves in the A-type case.","marker":"[39]"}],"fun_headline_variants":["A_n cluster algebra embeds in equivariant quantum cohomology","All-genus Seiberg duality proved for A-type quiver varieties","Cluster variables map to Chern polynomials in quantum cohomology","Quantum cohomology ring contains cluster algebra for A-type","A-type quiver mutations preserve all-genus GW invariants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 2.16's classification of the valid stability conditions and of the critical loci as complete intersections defined by $A_{v_1}A_{v_2}=0$ in a binary-tree quiver variety, and the paper explicitly says it omits the proof of that lemma; if this description is wrong or incomplete, the fixed-point bijections, orbit classifications, and the all-genus Seiberg duality comparison built on it do not follow.","fun_headline_variants_meta":{"raw":{"variants":["A_n cluster algebra embeds in equivariant quantum cohomology","All-genus Seiberg duality proved for A-type quiver varieties","Cluster variables map to Chern polynomials in quantum cohomology","Quantum cohomology ring contains cluster algebra for A-type","A-type quiver mutations preserve all-genus GW invariants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000778,"raw_usage":{"total_tokens":3546,"prompt_tokens":1156,"completion_tokens":2390,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":772,"completion_tokens_details":{"reasoning_tokens":2303}},"tokens_in":772,"tokens_out":2390,"duration_ms":15295,"temperature":1.0,"reasoning_tokens":2303,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:52:52.927848+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for the $A_3$-mutation-equivalent quiver of Example 2.17 with decorations $N_1<N_2<N_3<N_4$, the torus-fixed locus from the equations $A_1A_2=0$, $A_2B=0$, $BA_1=0$ under the stability condition $\\sigma_1>0$, $\\sigma_2<0$, $\\sigma_3+\\sigma_2>0$, and verify Lemma 4.6's description of fixed points and Lemma 4.7's bijection under the mutation $\\mu_2$; if any fixed set fails to match the predicted subset rules, or if a genus-one invariant computed on both sides differs, the claimed all-genus Seiberg duality fails.","supporting_citations":[{"cited_title":"Graber and R","cited_arxiv_id":null,"evidence_quote":"Supplies the localization theorem converting equivariant Gromov–Witten invariants to sums over torus-fixed loci, used in the proof of Theorem 4.1."},{"cited_title":"Localization in Gromov-Witten Theory and Orbifold Gromov-Witten Theory","cited_arxiv_id":"1107.4712","evidence_quote":"Provides the decorated-graph localization formula that organizes the contributions compared under quiver mutation."},{"cited_title":"Ciocan-Fontanine, B","cited_arxiv_id":null,"evidence_quote":"Establishes the abelian/nonabelian correspondence for Frobenius manifolds, the main tool for proving the quantum cohomological cluster exchange relations."},{"cited_title":"Webb, The abelian-nonabelian correspondence for I-functions, Int","cited_arxiv_id":null,"evidence_quote":"Gives the abelian/nonabelian correspondence for I-functions, used to restrict the correspondence to small quantum cohomology without parameter change."},{"cited_title":"Webb, Abelianization and quantum Lefschetz for orbifold quasimap I-functions, Adv","cited_arxiv_id":null,"evidence_quote":"Extends abelianization and quantum Lefschetz to orbifold quasimap I-functions, supporting the computations in Section 5."},{"cited_title":"Gu and E","cited_arxiv_id":null,"evidence_quote":"Provides the specialization rule for Novikov variables of the abelianized flag variety used in deriving the quantum relations."},{"cited_title":"Fomin and A","cited_arxiv_id":null,"evidence_quote":"Foundational cluster algebra paper supplying the exchange relations and the Laurent phenomenon used to prove injectivity of $\\psi$."},{"cited_title":"Derksen, J","cited_arxiv_id":null,"evidence_quote":"Defines quiver mutations with potentials and reduced quivers with potentials, the formal framework for all decorated quivers in $\\Omega_n$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies quivers mutation-equivalent to an $A_n$ quiver, giving the recursive description of $\\Omega_n$ used throughout."},{"cited_title":"Ruan, Nonabelian gauged linear sigma model, Chin","cited_arxiv_id":null,"evidence_quote":"Formulates the Seiberg duality conjecture for Gromov–Witten theory that Theorem 4.1 proves in the A-type case."}],"review_version":1}