Group theory for gauge theories

A gauge theory needs a group that acts on every field, a local action, and the global data appropriate to its spacetime and observables. The Lie algebra determines the infinitesimal gauge couplings. The global group restricts the permitted matter and Wilson representations. Neither choice alone specifies the interactions or a complete quantum theory.

This guide concerns ordinary four-dimensional gauge theories on a spin spacetime, with a compact connected internal gauge group and finite-dimensional complex matter representations in characteristic zero. Lorentz representations and particle statistics are additional data. Identifying a gauge-center element with spacetime fermion parity requires a different combined structure and is outside the descent tests used here.

Choose matter and a global form · Check a product group · Declare the fields

The algebra enters the covariant derivative

For one simple factor, let $T_R^a$ be Hermitian generators in the matter representation $R$, with $[T_R^a,T_R^b]=if^{abc}T_R^c$. Choose the convention

$$D_\mu=\partial_\mu+igA_\mu, \qquad A_\mu=A_\mu^aT_R^a, \qquad [D_\mu,D_\nu]=igF_{\mu\nu}.$$

Expanding the commutator gives

$$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu +ig[A_\mu,A_\nu].$$

With $\psi\mapsto U_R(x)\psi$, the transformation $D_\mu\mapsto U_R D_\mu U_R^{-1}$ makes $D_\mu\psi$ transform like $\psi$ and $F_{\mu\nu}$ transform by conjugation. The gauge potential itself has an inhomogeneous derivative term; it is a connection, not an ordinary adjoint matter field. Constant gauge transformations act on it by the adjoint representation.

For a product group, sum the terms $ig_s A_{s\mu}^aT_{R_s}^a$ over factors, acting on their separate tensor factors. An abelian term is $ig_1qB_\mu$. Both the charge normalization and the circle period must be fixed. Grzadkowski et al., §2, equations (2.2) and (2.4), use this plus-sign convention for the Standard Model covariant derivative and curvature.

The quadratic-invariant reference fixes $\operatorname{tr}_{\mathbf N}(T^aT^b)=\delta^{ab}/2$ for $SU(N)$. Its trace indices are representation factors that can enter loop calculations; the particle spins, number of species, and other physical factors still have to be supplied.

Choose a global group that admits the fields

For a simply connected compact simple group $\widetilde G$, a candidate global form is $G=\widetilde G/\Gamma$ with $\Gamma\subset Z(\widetilde G)$. A representation $R_s$ descends exactly when

$$R_s(\gamma)=I\quad\text{for every }\gamma\in\Gamma.$$

For several matter multiplets, compute the common central kernel $K=\bigcap_s\ker(R_s|_{Z(\widetilde G)})$. Any $\Gamma\subset K$ is compatible with these local representations. There is no requirement to divide out all of $K$. The lattice description underlying this test is in Conrad, Centers of Simply Connected Semisimple Groups, pp. 1–2.

For $SU(N)$, take $z=e^{2\pi i/N}I$ and Dynkin labels $a_i$ in consecutive $A_{N-1}$ order. The center charge is

$$t(R)=\sum_{i=1}^{N-1}i a_i\pmod N, \qquad |K|=\gcd(N,t(R_1),\ldots,t(R_m)).$$

The matter/global-form derivation proves this criterion and treats empty inventories. It is a test of central action, not an anomaly calculation.

For a concrete inventory, take an $SU(6)$ theory with complex scalar fields in $\Lambda^2\mathbf6$ and $\Lambda^4\mathbf6$, plus adjoint gauge fields. The two scalar dimensions are $15$, but their labels and center charges differ:

Field representation	Dynkin labels	Center charge
$\Lambda^2\mathbf6$	$(0,1,0,0,0)$	$2$
$\Lambda^4\mathbf6$	$(0,0,0,1,0)$	$4$
Adjoint	$(1,0,0,0,1)$	$0$

Here $h=\gcd(6,2,4)=2$, so the compatible groups are $SU(6)$ and $SU(6)/\langle z^3\rangle$. Directly, $z^3=-I$ acts on an exterior power $\Lambda^k\mathbf6$ as $(-1)^k$, hence trivially on both scalars. Adding $\Lambda^3\mathbf6$, of dimension $20$ and charge $3$, changes the gcd to one and rules out that quotient. The local algebra and its gauge-boson count remain unchanged.

Enter this inventory in the quotient workbench. Its answer establishes the allowed representations; it does not choose scalar masses, a potential, or a vacuum.

Even a global group leaves quantum choices

The distinction is visible already in pure gauge theories with algebra $\mathfrak{su}(2)$. Adjoint local fields admit either $SU(2)$ or $SO(3)=SU(2)/\mathbb Z_2$. The doublet is an allowed Wilson representation of $SU(2)$, while an ordinary Wilson representation of $SO(3)$ must have integer spin.

There are also two line-spectrum choices for the same $SO(3)$ group. Write $(z_e,z_m)\in\mathbb Z_2\times\mathbb Z_2$ for the electric and magnetic center-charge classes. The possibilities are

Theory	Allowed center-charge classes of genuine lines
$SU(2)$	$(0,0),(1,0)$
$SO(3)_+$	$(0,0),(0,1)$
$SO(3)_-$	$(0,0),(1,1)$

These are classes, not a listing of every line weight. Mutual locality requires $z_ez'_m-z_mz'_e=0\pmod2$ between allowed lines. The two $SO(3)$ choices share their purely electric representations but differ in whether a minimal magnetic line is purely magnetic or dyonic. This is the example in Aharony–Seiberg–Tachikawa, §1.2, pp. 4–5; equation (1.4).

Thus a matter-kernel calculator cannot specify the genuine line spectrum or a discrete theta choice. A theory also needs its couplings and action, allowed bundle sectors and topological terms, spacetime assumptions, and any relevant boundary conditions. Quantum anomaly cancellation is another test; it is not implied by successful representation descent.

A product-group example: the Standard Model

Fix the compact product $\widetilde G=SU(3)_c\times SU(2)_L\times U(1)$ with circle parameter of period $2\pi$ and integer charge $q=6Y$. Electric charge is $Q_{\mathrm{em}}=T_3+Y$. In all-left notation, one generation and the Higgs carry

$$\begin{aligned} Q&:(\mathbf3,\mathbf2)_1,& u^c&:(\overline{\mathbf3},\mathbf1)_{-4},& d^c&:(\overline{\mathbf3},\mathbf1)_2,\\ L&:(\mathbf1,\mathbf2)_{-3},& e^c&:(\mathbf1,\mathbf1)_6,& H&:(\mathbf1,\mathbf2)_3. \end{aligned}$$

Here $u^c,d^c,e^c$ are left-handed conjugates of right-handed fields; $H$ is a scalar. Their common central kernel is generated by

$$\gamma=(e^{2\pi i/3}I_3,-I_2,e^{2\pi i/6}), \qquad K=\langle\gamma\rangle\cong\mathbb Z_6.$$

If $t_3$ is color triality and $t_2$ is weak parity, its phase is $\exp[2\pi i(2t_3+3t_2+q)/6]$. For example, $Q$ gives $2+3+1=6$, while $u^c$ gives $4-4=0$. All six multiplets have trivial phase. The Standard Model global-group article proves that this is the entire common kernel and obtains the four allowed orders $d=1,2,3,6$ for $\Gamma_d=\langle\gamma^{6/d}\rangle$. Compare Tong, Line Operators in the Standard Model, §3, pp. 7–9 and footnote 2.

The four groups give identical local invariant spaces for these fields. To see why, let $\pi:\widetilde G\to G_d$ be the quotient map. Each field action is $\rho=\bar\rho\circ\pi$, and surjectivity of $\pi$ gives $V^{\widetilde G}=V^{G_d}$ for any tensor or polynomial space built from them. This statement concerns the local algebraic invariant problem; the global theories can still differ in their probes and bundles.

Carry the same conventions into branching and interactions

A proposed unification uses an actual homomorphism. With the same $q=6Y$ convention, the block map is

$$\Phi(A,B,z)=\operatorname{diag}(z^{-2}A,z^3B)\in SU(5).$$

Its kernel is $\langle\gamma\rangle$. Taking the exterior square of $\mathbf5=(\mathbf3,\mathbf1)_{-2}\oplus(\mathbf1,\mathbf2)_3$ yields

$$\mathbf{10}\downarrow\Phi =(\overline{\mathbf3},\mathbf1)_{-4} \oplus(\mathbf3,\mathbf2)_1 \oplus(\mathbf1,\mathbf1)_6.$$

Thus $u^c,Q,e^c$ fit into the ten for this map. The SU(5) guide tracks the remaining multiplets and the charge sign used by the calculator. The block-map construction is described in Baez–Huerta, §3.1, pp. 34 and 38–39. This embedding is additional structure; the observed local field assignments alone do not establish an $SU(5)$ theory.

Next ask for an interaction. In all-left notation the up-type Yukawa monomial is $Qu^cH$. Its integer charge is $1-4+3=0$, and an explicit contraction is

$$\mathcal O_{pr}= \epsilon_{\alpha\beta}\epsilon_{ij} Q_p^{A i\alpha}\,u^c_{r,A}{}^{\beta}\,H^j.$$

Here $A$ is a color index, $i,j$ are weak indices, $\alpha,\beta$ are undotted Lorentz indices, and $p,r$ label flavors. Color contracts through $\mathbf3\otimes\overline{\mathbf3}\to\mathbf1$, and the two weak and two spin doublets each use an antisymmetric invariant. The fermions are distinct species, so this contraction is nonzero. Its canonical dimension is $3/2+3/2+1=4$.

There is one contraction per ordered flavor pair: $n_g^2$, hence nine for three generations. This is a count of complex Yukawa tensor entries before flavor-basis changes; it is not a count of physical masses and mixing parameters. The right-handed form and the other Yukawas are in Grzadkowski et al., equation (2.1); the matter guide translates their conjugations.

Use the fixed-monomial workbench to check the declared fields and this contraction count. The Standard Model enumeration extends the same inventory through canonical dimension six for derivative-free scalar/Weyl polynomials. It omits derivatives and field strengths and is not a complete SMEFT basis.

A reproducible gauge-theory calculation therefore retains the group and quotient kernel, each field’s Lorentz and gauge representations, abelian periods and charges, flavors, statistics, and the requested operation. Group theory supplies precise constraints on these choices; the action, quantum consistency, and dynamics require their own analysis.