Diffusion models for SU(N) gauge theory

The ordinary Ising model is the right starting point. It separates three ideas that are easy to blur together: what is sampled, where it lives, and which fixed parameter controls its energy.

In the Ising model, each site carries a sampled spin \(s_x=\pm1\). The number \(J\) on a bond is usually a fixed coupling, not a sampled variable: \[ E_{\mathrm{Ising}}[s] =-J\sum_{\langle xy\rangle}s_xs_y. \] The notation \(\langle xy\rangle\) means that \(x\) and \(y\) are nearest neighbors, and the sum counts every bond once. Each local energy term therefore examines two neighboring site spins.

In the standard homogeneous model, we choose the same number \(J\) beforehand for every bond. It is a fixed coupling, not another sampled variable. When \(J>0\), equal neighboring spins lower the energy; when \(J<0\), opposite spins lower it. A more general graph may assign a different fixed value \(J_{xy}\) to each bond: \[ E[s]=-\sum_{\langle xy\rangle}J_{xy}s_xs_y. \] During an ordinary Monte Carlo simulation, the \(J_{xy}\) values remain fixed while the site spins \(s_x\) are sampled.

A lattice gauge theory changes the placement of the dynamical variables. A \(\mathbb Z_2\) gauge theory puts a sampled sign \(u_\ell=\pm1\) on every edge \(\ell\). The smallest gauge-invariant energy term must close into a loop, so it multiplies the four edge signs around a plaquette \(p\): \[ E_{\mathbb Z_2\text{ gauge}}[u] =-\beta\sum_p\prod_{\ell\in p}u_\ell. \] Here \(\beta\), not \(u_\ell\), plays the role of the fixed coupling.

SU(N) gauge theory keeps this edge-and-plaquette architecture but upgrades each sign to a noncommuting matrix \(U_\ell\in\SU{N}\). The product around a square becomes an ordered matrix product, and the trace turns it into a scalar action: \[ S_W[U] =-\frac{\beta}{N}\sum_p \Re\Tr\!\left(U_1U_2U_3^\dagger U_4^\dagger\right) \quad\text{(up to counting conventions).} \]

Start with a complex \(N\times N\) matrix \(U\). Requiring \(U^\dagger U=I\) says it preserves lengths; requiring \(\det U=1\) removes an overall phase. The matrices satisfying both conditions form the special unitary group \(\SU{N}\).

A group is not merely a set: multiplying two allowed matrices gives another allowed matrix, every matrix has an inverse, and multiplication is associative. It is also a smooth curved space. Near the identity, every small motion is generated by a traceless anti-Hermitian matrix \(A\in\su{N}\):

A lattice site is one point in discretized spacetime. Attached to every site \(x\) is an abstract copy of \(\mathbb C^N\): the vector space carrying the fundamental representation of SU(N). In QCD, this is called the local color space.

If the theory contains matter, a field \(\psi(x)\in\mathbb C^N\) picks one vector in that local color space. For SU(3), for example, \(\psi(x)=(\psi_r,\psi_g,\psi_b)^T\) contains the complex amplitudes for the three color basis states. This is one quantum color state with \(N\) components, not \(N\) little colored objects sitting on the node.

Call the neighboring site \(y=x+\hat\mu\). Suppose its field value is the column vector \(\psi(y)\), written using the coordinate convention at \(y\). We cannot compare those numbers directly with \(\psi(x)\), because \(\psi(x)\) uses the convention at \(x\). First translate the neighbor's vector into \(x\)'s coordinates: \[ \underbrace{\psi_{y\to x}}_{\text{neighbor's vector, written at }x} = \underbrace{U_\mu(x)}_{\text{dictionary from }y\text{ to }x} \underbrace{\psi(y)}_{\text{vector written at }y}. \] Now \(\psi_{y\to x}\) and \(\psi(x)\) are written in the same coordinate system and can be compared. To translate the other way, use the inverse dictionary: \[ \psi_{x\to y}=U_\mu^\dagger(x)\psi(x). \] The dagger appears because an SU(N) matrix is unitary, so \(U_\mu^\dagger=U_\mu^{-1}\). Nothing physically travels during this calculation; we are rewriting vectors so both descriptions use the same local coordinates.

What, then, lives on a node? In a theory with matter, a node can carry a dynamical field such as a quark color vector \(\psi(x)\in\mathbb C^N\). But the diffusion papers study pure gauge theory, so no dynamical value is sampled on the nodes. A node supplies only the spacetime label \(x\), an abstract local color space \(\mathbb C^N\), and a freely chosen local basis. The matrix \(\Omega(x)\) describes changing that basis; it is a gauge choice, not an additional physical field. The sampled dynamical data are only the edge matrices \(U_\mu(x)\).

Now change the color coordinates independently at every site: \(\psi(x)\mapsto\Omega(x)\psi(x)\). The link must change with both endpoint bases: \[ U_\mu(x)\longmapsto U_\mu^\Omega(x)= \Omega(x)U_\mu(x)\Omega^\dagger(x+\hat\mu). \] This rule ensures \[ U_\mu^\Omega(x)\psi^\Omega(x+\hat\mu) =\Omega(x)U_\mu(x)\psi(x+\hat\mu), \] so everyone agrees on the transported color state even if they use different coordinates. Individual link matrices depend on those coordinates; physics must be built from combinations in which the basis changes cancel.

The smallest closed loop is a square called a plaquette, hence the letter \(P\). A square needs two directions: \(\mu\) names its first axis and \(\nu\) its second. Thus \(P_{\mu\nu}(x)\) means “the plaquette based at site \(x\), lying in the \(\mu\)-\(\nu\) plane, with the orientation fixed by the ordered pair \((\mu,\nu)\).” Its four factors represent the edges in the \(+\mu,+\nu,-\mu,-\nu\) directions. For example, \(P_{12}(x)\) is the square in lattice directions 1 and 2 that starts at \(x\): \[ P_{\mu\nu}(x)=U_\mu(x)U_\nu(x+\hat\mu) U_\mu^\dagger(x+\hat\nu)U_\nu^\dagger(x). \] The ordered subscripts matter because SU(N) matrices generally do not commute. Reversing the orientation produces the inverse loop matrix (up to the corresponding shift of base point). It transforms as \(P\mapsto\Omega(x)P\Omega^\dagger(x)\), so its trace is unchanged. The trace measures the curvature trapped in that square.

The symbol \(U_p\) means the plaquette matrix: the ordered product of the four link matrices around one square \(p\). For the plaquette based at \(x\) in the \(\mu\)-\(\nu\) plane, \[ U_p\equiv P_{\mu\nu}(x) = U_\mu(x)\, U_\nu(x+\hat\mu)\, U_\mu^\dagger(x+\hat\nu)\, U_\nu^\dagger(x). \] The daggered matrices traverse the final two edges backward. Thus \(U_p\) is one SU(N) matrix answering: “after transporting around this closed square, what net transformation remains?” If every internal vector returns unchanged, then \(U_p=I\): the plaquette is locally flat. The action needs to measure how far every \(U_p\) is from that ideal.

First compress the matrix to one real, gauge-invariant number: \[ w_p=\frac{1}{N}\Re\Tr U_p. \] The normalized trace reaches its maximum \(w_p=1\) when \(U_p=I\). Therefore \(1-w_p\) is zero for a flat plaquette and positive when going around the square produces a nontrivial rotation.

Add that local cost over every plaquette: \[ \boxed{ S_W[U] =\beta\sum_p\left(1-\frac{1}{N}\Re\Tr U_p\right) =\frac{\beta}{N}\sum_p\left(N-\Re\Tr U_p\right). } \] This is the Wilson gauge action. The lattice and all link matrices \(U_\mu(x)\) are the variables; \(\beta\) is one fixed coupling. In a common four-dimensional SU(N) convention, \(\beta=2N/g_0^2\), where \(g_0\) is the bare gauge coupling.

After Wick rotating time, quantum expectation values become averages over classical-looking Euclidean field configurations: \[ \langle\mathcal O\rangle= \frac{1}{Z}\int\!\mathcal DU\;\mathcal O[U]e^{-S_W[U]}, \qquad Z=\int\!\mathcal DU\;e^{-S_W[U]}. \]

Low-action fields are more probable, but they are not the only fields that matter. The enormous number of less-than-minimal configurations contributes entropy. Standard lattice calculations use Markov-chain Monte Carlo, especially HMC, to draw correlated samples from this Boltzmann weight. At fine lattice spacing, long autocorrelation times and frozen topological sectors can make those samples expensive.

A diffusion model creates a bridge between hard samples \(U_0\sim p_{\rm data}\) and an easy reference distribution. On a compact group, the natural easy distribution is Haar measure: uniform over the group.

The intrinsic construction in Komijani, Marinkovic, and Turgut multiplies each link by many tiny random group elements: \[ U_{t+\Delta t} = \exp\!\left[\sigma(t)\sqrt{\Delta t}\,\eta_t\right]U_t, \qquad \eta_t\in\su{N}. \] Because the exponential of a traceless anti-Hermitian matrix is in SU(N), every intermediate state remains a valid gauge field. As accumulated variance grows, memory of \(U_0\) disappears and \(U_t\) approaches Haar.

At time \(t\), noisy fields have density \(\rho_t(U)\). Its score is the tangent vector \[ s(U,t)=T^a\partial^a\log\rho_t(U)\in\su{N}. \] It answers a local question: “which infinitesimal group motion most increases log probability?”

A neural network \(s_\theta(U,t)\) learns this vector from noised training fields. Reversing the diffusion uses the score to oppose the entropy-producing random walk. The deterministic member of the reverse family is the probability-flow ODE \[ \frac{dU_t}{dt} = -\frac12\sigma^2(t)s_\theta(U_t,t)U_t, \] integrated from noisy \(t=1\) toward data \(t=0\).

Both papers learn to generate finite-lattice equilibrium gauge configurations, but only one treats SU(N) as the curved group it is throughout diffusion. Use the staged comparison below; each click adds one design decision.

Set stochastic reverse noise to zero and freeze the trained score model. Integrating its probability-flow ODE gives a deterministic map \(g:\mathcal Z\to\mathcal C\) from easy base fields \(Z\) to generated gauge fields \(U=g(Z)\).

Given any differentiable gauge-invariant loss \(\mathcal L(U)\), optimize its pullback: \[ \widetilde{\mathcal L}(Z)=\mathcal L(g(Z)). \] Backpropagation through the full ODE gives the gradient at the base field. Every outer update is made intrinsically on each SU(N) base link: \[ Z_{\ell}^{(k+1)} = \exp\!\left[-\eta A_{\ell}^{(k)}\right]Z_{\ell}^{(k)}, \qquad A_{\ell}^{(k)} = \operatorname{grad}_{Z_\ell} \mathcal L(g(Z^{(k)}))\in\su{N}. \]

To first order, mapping this step through \(g\) is Riemannian gradient descent on the generated manifold. In Euclidean notation, \[ \Delta U\approx-\eta\,J_gJ_g^\top\nabla_U\mathcal L. \] The learned generator supplies a geometry-aware preconditioner \(J_gJ_g^\top\): base-space motion is pushed into directions the model knows how to generate.