  In this paper, a family of arbitrarily high-order structure-preserving
exponential Runge-Kutta methods are developed for the nonlinear Schr\"odinger
equation by combining the scalar auxiliary variable approach with the
exponential Runge-Kutta method. By introducing an auxiliary variable, we first
transform the original model into an equivalent system which admits both mass
and modified energy conservation laws. Then applying the Lawson method and the
symplectic Runge-Kutta method in time, we derive a class of mass- and
energy-preserving time-discrete schemes which are arbitrarily high-order in
time. Numerical experiments are addressed to demonstrate the accuracy and
effectiveness of the newly proposed schemes.
