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Commutativizing the target erases the A₂ wall-crossing identity

Statement refuted

The A₂ quantum-dilogarithm wall identity is not preserved if the skew quantum torus product is replaced by ordinary commutative multiplication while its factor normalization is kept. In the commutative formal power-series ring in $x_{1}, x_{2}$ over $Q (t)$ , the resulting claimed equality is $E_{t^{2}} (x_{2}) E_{t^{2}} (x_{1}) = E_{t^{2}} (x_{1}) E_{t^{2}} (t x_{1} x_{2}) E_{t^{2}} (x_{2}), E_{t^{2}} (x) = \sum_{n \geq 0} \frac{t^{n} x^{n}}{\prod_{r = 1}^{n} (t^{2 r} - 1)} .$

Facts & Assumptions

Given: A formal variable $t$ and the series displayed in the refuted claim.

[F1]

The degree-one coefficient of $E_{t^{2}} (x)$ is $c_{1} = t / (t^{2} - 1)$ , directly from its defining series.

[F2]

The A₂ chamber products arise from the completed skew Hall integration and the subobject-first normalization (Euler and skew integration are multiplicative for finite acyclic quivers, Completed Hall wall crossing for finite acyclic quivers).

Counterexample

Given: The commutative power-series ring and the displayed candidate identity.

1.1F1algebra

In the left side, the coefficient of bidegree $(1, 1)$ can only come from the linear term of each of the two factors. It is therefore $c_{1}^{2}$ .

1.2F1algebra

In the right side, that bidegree receives the contribution $c_{1}^{2}$ from the linear terms of the first and third factors, and the contribution $t c_{1}$ from the linear term of the middle factor. All remaining terms have a coordinate degree exceeding $(1, 1)$ in at least one variable. Thus its coefficient is $c_{1}^{2} + t c_{1}$ .

2.1F1F2step 1.1step 1.2algebra∎

The difference is $t c_{1} = t^{2} / (t^{2} - 1)$ , a nonzero rational function (and nonzero at every finite-field specialization $t = \sqrt{q} > 1$ ). The two formal series are unequal, so ordinary commutative multiplication cannot retain the A₂ wall identity. This is exactly why the target skew twist is part of the theorem's normalization.

Depends on

Used by

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Dependency tree · two levels

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Sources

Tom Bridgeland, Hall algebras and Donaldson–Thomas invariants, §4.6 (standard reference, not scraped)
Olivier Schiffmann, Lectures on Hall Algebras, Example 3.15 (standard reference, not scraped)