Quantization of the topological $\sigma$-model and the master equation of the BF formalism

Abstract

We quantize the topological $\sigma$-model. The quantum master equation of the Batalin-Vilkovisky formalism $\Delta_\rho \Psi = 0$ appears as a condition which eliminates the exact states from the BRST invariant states $\Psi$ defined by $Q \Psi = 0$. The phase space of the BV formalism is a supermanifold with a specific symplectic structure, called the fermionic K\"ahler manifold.