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The loop-nilpotent CoHA of a tripled quiver is isomorphic to an explicit integral shuffle algebra of polynomials with divisibility conditions.

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2026-07-30 19:29 UTC pith:3C476HZ7

load-bearing objection Solid program paper: explicit shuffle model for loop-nilpotent CoHA, usable BPS/Kac formulas, and a clean Coulomb surjection, under standard genericity.

arxiv 2607.23544 v1 pith:3C476HZ7 submitted 2026-07-26 math.RT hep-thmath-phmath.AGmath.MPmath.QA

The loop-nilpotent cohomological Hall algebra

Shivang Jindal , Andrei Negu\c{t} This is my paper
classification math.RT hep-thmath-phmath.AGmath.MPmath.QA MSC 17B3714F4316G2081R10
keywords cohomological Hall algebraloop-nilpotent CoHAshuffle algebraBPS Lie algebraKac polynomialsCoulomb branchYangianpreprojective algebra
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

Core claim

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.

What carries the argument

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.

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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}.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Circularity Check

2 steps flagged

Sequential multi-paper development with adapted self-citations; geometric CoHA and shuffle S⁺ are independently defined and linked by a non-tautological isomorphism.

specific steps
  1. self citation load bearing [Thm 2.4 proof sketch; Prop. 3.7 sketch]
    "The following result is completely analogous to [JN26a, Theorem 2.6] (also [Neg24, Proposition 3.9]), so we only sketch its proof. ... We claim that Im ι ⊆ S⁺ by adapting to cohomology the K-theoretic argument given in [JN26a, Proposition 2.2] ... opposite inclusion which is due to the fact (Theorem 2.4) that S⁺ is generated by ι(ε_{n,g})."

    The inclusion Im ι = S⁺ that yields the main isomorphism A ≅ S⁺ rests on generation and divisibility arguments whose details are deferred to the authors’ prior K-theoretic paper JN26a (and Neg24). The geometric CoHA side is independent, so this is load-bearing self-citation for the proof, not a definitional loop; it modestly raises circularity only insofar as the cohomological case is not re-proved from scratch.

  2. self citation load bearing [Thm 3.17; Cor. 1.4 / §3.9–3.10]
    "We will now recall the results of [JN26b], which gave an explicit description for the image of g^T_{Q̃,W̃} ↪ V⁺_Q ... Combining this equality with the main result of [JN26b] yields the following formula for the Kac polynomial ... t^{(n,n)′+1} A_{Q,n}(t^{-1}) = (1-t)^r ∑ dim_Q{degree-d polynomials E ... as in Definition 2.3, that satisfy the bounds (16)–(17)} t^{d/2}."

    The explicit polynomial description of the BPS Lie algebra (degree bounds) is imported from the authors’ JN26b and then combined with this paper’s g^{ω-nilp}=ℏg and S⁺ divisibility to rewrite Kac polynomials. The rewrite is a genuine corollary, not a tautology, but the load-bearing characterization of g is a self-citation rather than a new external input.

full rationale

The loop-nilpotent CoHA is defined geometrically via vanishing cycles on the nilpotent locus (Def. 3.2, (89)); the integral shuffle algebra S⁺ is defined by independent I-composition divisibility conditions (Def. 2.3). Theorem 1.1 / Prop. 3.7 identifies them by injectivity of ι (Thm 3.5, geometric localization) plus generation of S⁺ by the explicit classes ε_{n,g} that map to the shuffle generators e_{n,g}. Proofs of generation and of Im ι ⊆ S⁺ are explicitly adapted from the authors’ K-theoretic paper JN26a and from Neg24; the spherical-generation conjecture being proved is their own (JN26b/Neg25). That is ordinary sequential work in a program, not a claim that reduces to its inputs by construction: the geometric object is not defined via the shuffle conditions, and the Kac-polynomial formula (Cor. 1.4) combines the new comparison g^{ω-nilp}=ℏg (Prop. 3.14, geometric support/A¹ argument) with Davison’s external integrality and the prior JN26b degree bounds. No fitted-input-as-prediction pattern exists (pure algebra). No uniqueness theorem is imported to forbid alternatives. Score 2 reflects load-bearing self-citation for proof templates and for the BPS polynomial description used as an ingredient, without circular collapse of the main isomorphism.

Axiom & Free-Parameter Ledger

0 free parameters · 8 axioms · 2 invented entities

The paper works in standard derived algebraic geometry / equivariant cohomology of quiver stacks, importing Kontsevich–Soibelman CoHAs, Davison–Meinhardt integrality, Braverman–Finkelberg–Nakajima Coulomb branches, and Finkelberg–Frassek–Tsymbaliuk shuffle-to-difference-operator maps. Load-bearing extra structure is the list of genericity assumptions on (ℏ, u_α) and the definition of the integral shuffle algebra S⁺ by divisibility. No numerical free parameters are fitted.

axioms (8)
  • domain assumption Kontsevich–Soibelman CoHA multiplication via vanishing cycles on quiver representation stacks is associative and well-defined for the tripled quiver with cubic potential (and for the loop-nilpotent Serre subcategory).
    Used as the geometric source of A^{T,ω-nilp} throughout §3; cited from KS11, Dav23, DM20.
  • domain assumption Dimensional reduction identifies the cubic-potential CoHA (resp. loop-nilpotent version) with Borel–Moore homology of the preprojective/commuting locus Z_{Q⁺} (resp. ω-nilpotent locus).
    Dav17 Thm A.11 / YZ20; used for π₁, π₂ and for comparing ι with closed embeddings (§3.4–3.5).
  • domain assumption Relative and absolute cohomological integrality: Sym of BPS sheaves recovers the CoHA (DM20 Thms A,C and equivariant upgrades).
    Used to define g^T and g^{T,ω-nilp} and the PBW isomorphism Sym(g[u]) ≅ A_rat (§3.7–3.8).
  • ad hoc to paper Assumption Ъ: ℏ ≠ 0 and no nontrivial nonnegative integer relation Σ(x_α u_α + y_α u_ᾱ) = 0.
    Required for spherical generation of S⁺_loc (Thm 1.2 / 2.2).
  • ad hoc to paper Assumption Ь: ℏ ≠ 0 and the listed non-overlap / freeness conditions on arithmetic progressions u_α + ℤℏ for arrows and loops.
    Required for A^{T,ω-nilp} ≅ S⁺ and for the Coulomb surjection (Thm 1.1, 1.3, Prop 3.7).
  • ad hoc to paper Geometric assumption: ℏ ≠ 0 and u_α ∉ ℤℏ for every loop α of Q.
    Used for freeness, injectivity of ι/ι', and BPS support comparison (Thm 3.5, Prop 3.14, Cor 1.4).
  • domain assumption Finkelberg–Frassek–Tsymbaliuk homomorphism Φ: shifted double shuffle → difference operators, with image of e_{n,g}, f_{n,g} equal to dressed monopole operators when n ≤ d.
    Imported as Thm 2.9 / Prop 2.10 to obtain the surjection onto H_{d|k,ℓ}.
  • standard math Standard facts on perverse t-structures, vanishing-cycle functors, and equivariant localization for torus actions on stacks.
    Background for §3.2–3.5; KS90, Ach21, etc.
invented entities (2)
  • Integral loop-nilpotent shuffle algebra S⁺ (Definition 2.3) independent evidence
    purpose: Provide the explicit polynomial model isomorphic to A^{T,ω-nilp} and the source of generators and BPS conditions.
    Defined by I-composition specializations and χ-weighted divisibility by linear factors in ℏ, u_α; cohomological counterpart of the K-theoretic S⁺ from JN26a/Neg24.
  • Shifted double loop-nilpotent CoHA S^r ⊂ S^r_loc no independent evidence
    purpose: Algebraic stand-in for a doubled CoHA that surjects onto quantized Coulomb branches H_{d|k,ℓ}.
    Defined via shuffle generators plus Cartan elements with r-dependent commutation (65)–(66); not given a direct geometric construction in the paper.

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read the original abstract

We give an explicit shuffle algebra model for the loop-nilpotent cohomological Hall algebra (CoHA) of a tripled quiver with canonical cubic potential. As consequences, we (1) relate the loop-nilpotent CoHA to the quantized Coulomb branch algebra of the corresponding quiver gauge theory, (2) show that the loop-nilpotent CoHA is supercommutative after specialization at $\hbar=0$, (3) give generators for both the loop-nilpotent CoHA and the full preprojective CoHA, and (4) obtain an explicit characterization of the BPS Lie algebra of the full preprojective CoHA via certain degree and divisibility conditions. This gives a new formula for the Kac polynomials of the quiver in terms of the dimensions of certain vector spaces of polynomials. We also prove a conjecture on the spherical generation of the localized shuffle algebra and show that for ADE quivers, the loop-nilpotent CoHA is the positive half of Drinfeld-Gavarini dual of the Yangian.

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