Impact of Kalb–Ramond fields and perfect fluid dark matter on black hole shadows and gravitational lensing

We study the shadow and gravitational lensing features of static, spherically symmetric black holes in the presence of a Kalb–Ramond (KR) field and perfect fluid dark matter (PFDM). The KR field, derived from string theory, has a Lorentz-violating parameter $a$, whereas PFDM is defined by the density parameter $\beta$. The resulting KR-PFDM metric modifies null geodesics, photon spheres, and related observables. We analyze photon motion in vacuum and in dispersive plasma environments, considering homogeneous, singular isothermal sphere (SIS), and non-singular isothermal sphere (NSIS) plasma profiles. The numerical results show that, in vacuum, increasing $a$ from $-0.2$ to 0.2 reduces the photon sphere radius $r_{\mathrm{ph}}$ from $\sim$3.00 $M$ to $\sim$2.85 $M$ for fixed $\beta =0.2$, while raising $\beta$ from 0.1 to 0.3 in $a=0$ reduces $r_{\mathrm{ph}}$ by $\approx 5\text{\%}$. Consequently, the shadow radius $r_{\mathrm{sh}}$ decreases by up to 10% for the same parameter variations. In a homogeneous plasma with ${\omega }_p^2/{\omega }_0^2=0.5$, the shadow radius is smaller by $\sim$15% compared to vacuum. Weak lensing analysis shows that for an impact parameter $b=5M$, the deflection angle $\hat{\alpha }$ decreases from $\sim$1.2 rad to $\sim$0.9 rad as $a$ increases from $-0.2$ to 0.2 in a uniform plasma, with SIS and NSIS profiles producing progressively smaller deflections. The magnifications of the images drop by $\sim$8% as $\beta$ increases from 0.1 to 0.3, while the larger $a$ slightly enhances the magnification. Our results demonstrate that both Lorentz symmetry breaking and PFDM act to compactify the photon sphere, shrink the shadow, and weaken gravitational lensing signatures. The magnitude of these effects, comparable to or exceeding 5%–15% in key observables, suggests that future high-resolution VLBI facilities such as ngEHT could place meaningful constraints on $a$ and $\beta$.