  This paper is concerned with an indefinite linear-quadratic mean field games
of stochastic large-population system, where the individual diffusion
coefficients can depend on both the state and the control of the agents.
Moreover, the control weights in the cost functionals could be indefinite. A
direct approach is used to derive the $\epsilon$-Nash equilibrium strategy.
First, we formally solving an $N$-player game problem within a vast and finite
population setting. Subsequently, decoupling or reducing high-dimensional
systems by introducing two Riccati equations explicitly yields centralized
strategies, contingent on the state of a specific player and the average state
of the population. As the population size $N$ goes infinity, the construction
of decentralized strategies becomes feasible. Then, we demonstrated they are an
$\epsilon$-Nash equilibrium. Numerical examples are provided to demonstrate the
effectiveness of the proposed strategies.
