Gauge theory · research workbench

Matter & global gauge structure

Find the common central kernel of matter multiplets and every compatible central quotient of SU(N).

Scope: One compact SU(N) factor; finite-dimensional continuous complex matter representations. 2≤N≤12; at most 16 multiplets; each Dynkin label 0–30. Kinematic descent only.

Exact resultMatter descends

\begin{aligned}G&=SU(4)/\mathbb Z_{2}\\K&\cong\mathbb Z_{2}\\d_{\rm compatible}&\in\{1,2\}\\\text{chosen quotient admits matter}&=\text{yes}\end{aligned}

Common central kernel: $\mathbb Z_{2}$
Compatible quotient orders: $\{1,2\}$
π₁ of the selected group: $\mathbb Z_{2}$
Center of the selected group: $\mathbb Z_{2}$

Matter multiplets and descent conditions
Matter	Dynkin labels	Dimension	N-ality	Selected quotient
A	$(0,1,0)$	$6$	$2\pmod{4}$	Descends

The common central kernel is generated by $z^{2}$ , with $z=e^{2\pi i/4}\mathbf1$ . Matter permits every quotient listed above; it does not select one. Line operators, discrete theta parameters, and anomaly consistency require additional physical data.

✓ divisor condition✓ common kernel by enumeration✓ descent matches kernel

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Method & interpretation

Intersect central characters

The common kernel has order gcd(N,q₁,…,qₘ), where q=Σ i aᵢ mod N. A quotient of order d admits every field exactly when d divides every charge.

Read the worked guide · Mathematical method

Download script for an independent environment; not executed by this website. Python 3 verifies exact dimensions, the gcd formula, and enumeration of central subgroup elements.

Sources, conventions, and version

Computed locally with exact integer or algebraic arithmetic. Independent human expert review is pending. The export records the conventions and implementation version.

Brian Conrad, Centers of Simply Connected Semisimple Groupspp. 1–2: type A center and the weight-lattice pairing
Thomas D. Cohen, Center symmetry and area lawsProperty (i), equation (3), and the argument using N-ality and the greatest common divisor
Aharony, Seiberg and Tachikawa, Reading between the lines of four-dimensional gauge theories§1.1 and §2: global group, allowed representations, and additional line-operator choices

Convention: su-n-dynkin-center-v1
Algorithm: su-n-matter-kernel-1.0.0

Convention profiles · Reproduce and cite

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