Complex symmetric Toeplitz operators on the unit polydisk

In this paper, we study the complex symmetry of Toeplitz operators on the weighted Bergman spaces over the unit polydisk. First, we completely characterize when anti-linear weighted composition operators W(psi,phi)J are conjugations. We then give a sufficient and necessary condition for Toeplitz operators to be complex symmetric with respect to these conjugations. As a consequence, some interesting higher-dimensional complex symmetric phenomena appear on the unit polydisk such as the monomial Toeplitz operators T-zpz over bar q with p = V q for some symmetric permutation matrix V. Surprisingly, these operators T-zVqz over bar q are the only ones that are complex symmetric monomial Toeplitz operators on the unit bidisk.