Topological Flux on a Context Manifold Generates Nonreciprocal Collective Dynamics

TL;DR

This paper shows that nonreciprocal collective dynamics can arise from internal topology when active agents interact through a Chern-Simons gauge field defined on a compact context manifold. The gauge field is fast-relaxing and, via a Gauss-law constraint $\frac{\kappa}{2\pi} B(c,t)=\rho(c,t)$, induces an antisymmetric interaction kernel; eliminating the field yields nonlinear, nonreciprocal corrections $v = v^{(0)} + v^{(1)} + \cdots$ that drive chiral flows. In continuum form, density evolves as $\partial_t\rho = -\nabla\cdot(\rho v) + D_c \Delta\rho$, with Fourier components showing how $\hat{v}^{(0)}$ and higher-order terms generate nonreciprocity. Numerical simulations on a toroidal context manifold reveal long-lived vortices, finite circulation, and pronounced hysteresis under parameter sweeps, confirming that internal topology alone can control directional influence and irreversible dynamics in active matter.

Abstract

Non-reciprocal interactions, where the influence of agent $i$ on $j$ differs from that of $j$ on $i$, are fundamental in active and living matter. Yet, most models implement such asymmetry phenomenologically. Here we show that non-reciprocity can emerge from internal topology alone. Agents evolve on an internal ``context manifold'' coupled to a Chern-Simons gauge field. Because the gauge field is first order in time, it relaxes rapidly; eliminating it yields an effective transverse, antisymmetric interaction kernel that generically produces chiral waves, persistent vorticity, and irreversible state transitions. Numerical simulations reveal clear signatures of broken reciprocity: long-lived vortex cores, finite circulation, asymmetric information flow, and a nonzero reciprocity residual. The dynamics further exhibit pronounced hysteresis under parameter sweeps, demonstrating memory effects that cannot occur in reciprocal or potential-driven systems. These results identify Chern-Simons gauge fields as a minimal and universal source of directional influence and robust non-reciprocal collective behavior.

Figures (4)

• Figure 1: Final velocity distributions in physical space for two sample runs with $\gamma=3.0$.
• Figure 2: Final vorticity distributions in physical space for two sample runs with $\gamma=3.0$.
• Figure 3: Final reciprocity distributions in physical space for two sample runs with $\gamma=3.0$. Reciprocity values greater than zero indicates non-reciprocal interaction.
• Figure 4: The variation of Mean Alignment $\mathrm{P}$, Signed Circulation Dominance (SCD), Enstrophy $\mathrm{E}$, and Maximum Vorticity $\omega$ with the forward and backward sweeps of $\gamma$. Forward and backward sweeps do not yield same trajectories and give rise hysteresis effect.