In PoF, Ilinski models arbitrage as the lattice-curvature in a gauge field, where "no arbitrage" conditions literally mean zero curvature. Starting from a linear plaquette action and taking the continuum limit, the gauge-invariant action collapses to Geometric Brownian Motion with time-dependent drift/volatility. In the quasiclassical saddle limit (no money-flow sources), the full gauge theory reproduces the standard Black–Scholes PDE with the usual boundary conditions, showing BS is just the leading-order term of a richer theory. Ilinski then generalizes BS with virtual-arbitrage corrections by adding a stochastic "arbitrage return" field to derive a pricing equation that extends BS into far-from-equilibrium markets. While the practical applications of gauge theory in finance are limited, I did find Ilinski’s book pedagogically useful and conceptually rich.