{"id":"55985de3-b4a0-4c9f-bd85-006163e82604","arxiv_id":"2607.23544","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Loop-nilpotent CoHAs of tripled quivers are isomorphic to an explicit integral shuffle algebra, yielding generators, Coulomb-branch surjections, BPS characterizations, and a new Kac-polynomial formula.","lead":"The paper gives an explicit shuffle-algebra description of the loop-nilpotent cohomological Hall algebra of a tripled quiver, and uses it to link CoHAs, Coulomb branches, BPS Lie algebras, and Kac polynomials. It supplies generators, a supercommutativity statement, a new polynomial formula for Kac polynomials, and identifies the ADE case with a dual of the Yangian.","discovery_kind":"extension","skeptic_critique":null,"referee_report":null,"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is the cohomological counterpart of their K-theoretic shuffle work, and the payoff is concrete. The main new object is the integral shuffle algebra S^{+} of color-symmetric polynomials with the I-composition divisibility conditions (Def 2.3). Under Assumption Ь they get A^{T,ω-nilp} ≅ S^{+} (Thm 1.1 / Prop 3.7), then generators e_{n,g}, supercommutativity at ℏ=0 (Prop 2.7, full combinatorial argument), a surjection from the shifted double onto the quantized Coulomb branch H_{d|k,ℓ}, spherical generation of the localized algebra (proving their earlier conjectures), and—most useful for computation—an explicit degree-plus-divisibility description of the BPS Lie algebra that yields a new formula for Kac polynomials as dimensions of spaces of polynomials (Cor 1.4). For ADE they identify the loop-nilpotent CoHA with the positive half of the Drinfeld–Gavarini dual of the Yangian.\n\nWhat works: the reduction of the geometric isomorphism to generation of S^{+} by the e_{n,g} plus injectivity of ι is clean once you accept the analogy with JN26a; the specialization argument for supercommutativity is written out carefully; the BPS comparison g^{ω-nilp} = ℏ g (Prop 3.14) via support and the A^{1}-action is the right geometric move; and the Coulomb surjection is the natural cohomological twin of the Finkelberg–Frassek–Tsymbaliuk picture. Citations to DM20, BFN, FT, and their own concurrent papers are appropriate sequential development, not circularity.\n\nSoft spots are real but proportionate. Several proofs are sketched “as in JN26a” or reduced to residue identities and base-change for non-proper JH maps; a referee will want those filled or pointed more precisely. Everything load-bearing sits under the usual torus genericity (Ъ, Ь, geometric (29)); if those fail the statements need rephrasing, which is standard for the area but should be flagged. No machine-checked proofs or code, just ordinary dense math.\n\nThis is for people already working on CoHAs, Coulomb branches, or Kac polynomials who want an effective model rather than another existence statement. It deserves a serious referee. I would engage with it and expect to cite the Kac formula and the generators.","headline":"Solid program paper: explicit shuffle model for loop-nilpotent CoHA, usable BPS/Kac formulas, and a clean Coulomb surjection, under standard genericity.","tokens_in":35461,"tokens_out":597,"would_cite":true,"duration_ms":13147,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["17B37","14F43","16G20","81R10"],"pacs":[],"model":"grok-4.5","headline":"The loop-nilpotent CoHA of a tripled quiver is isomorphic to an explicit integral shuffle algebra of polynomials with divisibility conditions.","keywords":["cohomological Hall algebra","loop-nilpotent CoHA","shuffle algebra","BPS Lie algebra","Kac polynomials","Coulomb branch","Yangian","preprojective algebra"],"falsifier":"For a concrete quiver and a torus that violates one of the genericity assumptions, compute both the loop-nilpotent CoHA (via Borel–Moore homology of the nilpotent locus) and the candidate shuffle algebra S⁺; if they are not isomorphic as R-algebras, the main theorem is false.","tokens_in":35594,"feed_emoji":"🔀","tokens_out":870,"duration_ms":18156,"temperature":0.7,"pith_summary":"The paper constructs an explicit algebraic model for the loop-nilpotent cohomological Hall algebra of a tripled quiver with its canonical cubic potential: it is isomorphic to a concrete subalgebra of color-symmetric polynomials that obey a family of divisibility conditions indexed by compositions of the dimension vector. This identification turns geometric questions about the CoHA into calculations with polynomials. From it the authors obtain generators for both the loop-nilpotent and the full preprojective CoHAs, prove that the loop-nilpotent algebra becomes supercommutative after setting the equivariant parameter ħ to zero, and produce a surjection onto the quantized Coulomb branch algebra of the corresponding quiver gauge theory. They also characterize the BPS Lie algebra by degree bounds and the same divisibility conditions, yielding a new expression for the Kac polynomials of the quiver as dimensions of spaces of polynomials. For ADE quivers the construction recovers the positive half of the Drinfeld–Gavarini dual of the Yangian, and a spherical-generation conjecture for the localized shuffle algebra is settled.","feed_headline":"Loop-nilpotent CoHA equals an explicit shuffle algebra","feed_subtitle":"Divisibility conditions on polynomials give generators, Coulomb maps, and a new Kac formula","key_machinery":"The integral shuffle algebra S⁺: the R-subalgebra of color-symmetric polynomials whose specializations along every I-composition are divisible by an explicit product of linear factors involving ħ and the arrow weights. It is generated by the elements e_{n,g} and is the image of the loop-nilpotent CoHA.","core_discovery":"Under a genericity assumption on the torus parameters, the natural map from the loop-nilpotent CoHA into the zero-potential CoHA, composed with the Feigin–Odesskii shuffle isomorphism, is an R-algebra isomorphism onto the integral shuffle algebra S⁺ of color-symmetric polynomials that satisfy the I-composition divisibility conditions of Definition 2.3.","pith_inferences":["The same divisibility conditions should give effective algorithms for computing low-rank Kac polynomials and BPS dimensions that avoid finite-field point counting.","The shifted double construction supplies a uniform algebraic home for both CoHA actions and Coulomb-branch difference operators, suggesting a single presentation that interpolates Higgs and Coulomb sides.","For quivers with loops the failure of triangular decomposition of the integral double may encode new relations among monopole operators that are invisible after localization."],"forward_implications":["The loop-nilpotent CoHA surjects onto every quantized Coulomb branch algebra of the corresponding framed quiver gauge theory.","After setting ħ=0 the loop-nilpotent CoHA is supercommutative.","The BPS Lie algebra of the full preprojective CoHA is cut out by explicit degree bounds plus the same divisibility conditions, giving a polynomial-space formula for Kac polynomials.","For ADE quivers the loop-nilpotent CoHA is the positive half of the Drinfeld–Gavarini dual of the Yangian.","The localized shuffle algebra is generated by the single-variable elements e_{i,k}."],"fun_headline_variants":["Loop-nilpotent CoHA is the integral shuffle algebra of I-divisible polynomials","Genericity makes loop-nilpotent CoHA equal Feigin–Odesskii shuffle S⁺","Loop-nilpotent CoHA ≅ color-symmetric polynomials with I-composition divisibility","Shuffle model identifies loop-nilpotent CoHA with integral S⁺ under genericity","Divisibility conditions cut out loop-nilpotent CoHA inside the shuffle algebra"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The torus parameters must satisfy several genericity conditions (ħ nonzero and certain linear combinations of the arrow weights never vanish or collide with integer multiples of ħ); without them the injectivity and generation statements fail.","fun_headline_variants_meta":{"raw":{"variants":["Loop-nilpotent CoHA is the integral shuffle algebra of I-divisible polynomials","Genericity makes loop-nilpotent CoHA equal Feigin–Odesskii shuffle S⁺","Loop-nilpotent CoHA ≅ color-symmetric polynomials with I-composition divisibility","Shuffle model identifies loop-nilpotent CoHA with integral S⁺ under genericity","Divisibility conditions cut out loop-nilpotent CoHA inside the shuffle algebra"]},"model":"grok-4.5","effort":"low","cost_usd":0.004502,"raw_usage":{"total_tokens":1270,"prompt_tokens":725,"num_sources_used":0,"completion_tokens":96,"cost_in_usd_ticks":45024000,"prompt_tokens_details":{"text_tokens":725,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":449,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":725,"tokens_out":96,"duration_ms":8782,"temperature":1.0,"reasoning_tokens":449,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T19:29:46.255958+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"For a concrete quiver and a torus that violates one of the genericity assumptions, compute both the loop-nilpotent CoHA (via Borel–Moore homology of the nilpotent locus) and the candidate shuffle algebra S⁺; if they are not isomorphic as R-algebras, the main theorem is false.","supporting_citations":[],"review_version":1}