  The angular correlation of pulsar residuals observed by NANOGrav and other
pulsar timing array (PTA) collaborations show evidence in support of the
Hellings-Downs correlation expected from stochastic gravitational waves (SGW).
In this paper, we offer a non-gravitational wave explanation of the observed
pulsar timing correlations as caused by an ultra-light $L_{\mu} - L_{\tau}$
gauge boson dark matter (ULDM). ULDM can affect the pulsar correlations in two
ways. The gravitational potential of vector ULDM gives rise to a Shapiro
time-delay of the pulsar signals and a non-trivial angular correlation (as
compared to the scalar ULDM case). In addition, if the pulsars have a non-zero
charge of the dark matter gauge group then the electric field of the local dark
matter causes an oscillation of the pulsar and a corresponding Doppler shift of
the pulsar signal. We point out that pulsars carry a significant charge of
muons and thus the $L_{\mu} - L_{\tau}$ vector dark matter contributes to both
the Doppler oscillations and the time-delay of the pulsar signals. Our analysis
shows that the NANOGrav data has a better fit to the $L_{\mu} - L_{\tau}$ ULDM
scenario compared to the SGW or the SGW with Shapiro time-delay hypotheses.
