  This paper focuses on indefinite stochastic mean-field linear-quadratic
(MF-LQ, for short) optimal control problems, which allow the weighting matrices
for state and control in the cost functional to be indefinite. The solvability
of stochastic Hamiltonian system and Riccati equations is presented under both
positive definite case and indefinite case. The optimal controls in open-loop
form and closed-loop form are obtained, respectively. Moreover, the dynamic
mean-variance problem can be solved within the framework of the indefinite
MF-LQ problem. Other two examples shed light on the theoretical results
established.
