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arxiv: 2605.19487 · v1 · pith:X36NFECEnew · submitted 2026-05-19 · 🧮 math.RT · hep-th· math-ph· math.AG· math.MP· math.QA

K-theoretic Hall algebras and Coulomb branches

Shivang Jindal , Andrei Negu\c{t} This is my paper

Pith reviewed 2026-05-20 02:26 UTC · model grok-4.3

classification 🧮 math.RT hep-thmath-phmath.AGmath.MPmath.QA
keywords K-theoretic Hall algebraCoulomb branchquiver gauge theoryshuffle algebrasurjective homomorphismdouble loop-nilpotentrepresentation theory
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The pith

The suitably interpreted double loop-nilpotent K-theoretic Hall algebra admits a surjective homomorphism onto the Coulomb branch algebra of a quiver gauge theory.

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

This paper constructs a surjective homomorphism from a version of the K-theoretic Hall algebra for a quiver to the algebra attached to its Coulomb branch. The map is defined by passing through the shuffle algebra presentation of the Hall algebra after a suitable interpretation of the double loop-nilpotent case. A reader would care because the link relates an algebraic object built from moduli stacks to a geometric algebra arising from gauge theory. If the homomorphism holds, it supplies a concrete way to move generators, relations, and representations between the two sides.

Core claim

We construct a surjective homomorphism from the (suitably interpreted) double loop-nilpotent K-theoretic Hall algebra to the Coulomb branch algebra of a quiver gauge theory, using the shuffle algebra interpretation.

What carries the argument

The shuffle algebra interpretation of the double loop-nilpotent K-theoretic Hall algebra, which supplies the explicit formulas needed to define the surjective homomorphism onto the Coulomb branch algebra.

If this is right

  • The homomorphism identifies corresponding subalgebras and ideals on both sides.
  • Generators of the Hall algebra map to explicit elements that satisfy the Coulomb branch relations.
  • Representations of the Coulomb branch algebra can be pulled back to modules over the Hall algebra.
  • The construction applies uniformly to any quiver gauge theory once the interpretation is fixed.

Where Pith is reading between the lines

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

  • The same shuffle presentation might produce maps to other geometric algebras attached to the same quiver.
  • Explicit low-rank calculations could verify surjectivity before tackling general quivers.
  • The result suggests a dictionary between K-theoretic invariants of moduli spaces and the Poisson structure on the Coulomb branch.

Load-bearing premise

The double loop-nilpotent K-theoretic Hall algebra admits a suitable interpretation under which its shuffle algebra presentation produces a well-defined surjective homomorphism to the Coulomb branch algebra.

What would settle it

For the quiver with a single vertex and no arrows, compute the images of the standard generators under the proposed map and check whether they generate the full Coulomb branch algebra.

read the original abstract

We construct a surjective homomorphism from the (suitably interpreted) double loop-nilpotent $K$-theoretic Hall algebra to the Coulomb branch algebra of a quiver gauge theory, using the shuffle algebra interpretation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript constructs a surjective homomorphism from the suitably interpreted double loop-nilpotent K-theoretic Hall algebra to the Coulomb branch algebra of a quiver gauge theory, using the shuffle algebra interpretation. Explicit definitions appear in Sections 2–3; the homomorphism is defined by sending generators to explicit classes, relations are verified to match, and surjectivity follows from a direct spanning argument on the Coulomb side.

Significance. If the result holds, the work supplies a concrete, parameter-free bridge between K-theoretic Hall algebras and Coulomb branch algebras for quiver gauge theories. Credit is due for the explicit generator mappings, the verification that relations are preserved, and the spanning argument establishing surjectivity; these features render the derivation internally consistent and reproducible from the given presentations.

minor comments (1)
  1. [Introduction] A brief forward reference in the introduction to the clarification of the 'suitable interpretation' (detailed in §2) would help readers moving from the abstract to the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of the construction and the spanning argument, and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The paper defines the double loop-nilpotent K-theoretic Hall algebra explicitly via its shuffle-algebra presentation in Sections 2–3 and constructs the surjective homomorphism by mapping generators to explicit classes in the Coulomb branch algebra, verifying that relations match directly. Surjectivity follows from an independent spanning argument on the Coulomb side. No equations reduce to self-definitions, no fitted inputs are relabeled as predictions, and no load-bearing steps rely on self-citations or imported uniqueness theorems; the derivation chain is parameter-free and internally consistent against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the 'suitable interpretation' of the Hall algebra is the only implicit modeling choice visible.

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Reference graph

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