Testing Strong Gravitational Lensing Effects of Supermassive Black Holes with String-Inspired Metric: Observational Signatures and EHT Constraints

We analyze gravitational lensing in the strong field limit for spherically symmetric string-inspired Euler-Heisenberg black holes, characterized by magnetic charge ($q$) and Einstein-Maxwell-dilaton coupling constants ($\alpha, \beta$) from the low-energy limit of heterotic string theory. Our results show that the string coupling has a weak impact on the positions of relativistic images, deflection angles, photon orbit radii, and shadow sizes, making these black holes indistinguishable from the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black holes with the same mass and charge. Compared to Schwarzschild black holes, the string-inspired Euler-Heisenberg black holes exhibit smaller deflection angles, decreasing with increasing charge. Moreover, the time delay for Sgr A * and M87 * can reach $~11.477$ and $~17349.8$ minutes, respectively, at $q=0.1$ and $\eta=-1$, deviating from Schwarzschild black holes by $~0.0198$ and $~28.9$ minutes, which are not very significant. For Sgr A* and M87*, we determine $\theta_\infty$ range within $(11.52, 26.33)~\mu as$, and $(9.17, 19.78)~\mu as$ respectively, with angular separations $s$ ranging from $(3.29-6.85)~nas$ for Sgr A* and $(2.47-5.15)~nas$ for M87*. EHT bounds on the $\theta_{sh}$ of Sgr A* and M87* within the $1\sigma$ interval bound the $q$ as: for Sgr A* $0.54109\le q \le 0.7796$ and for M87* $0< q \le 0.29107$, while in both the cases, we did not find any bound on the parameter $\eta$. We show that string-inspired Euler-Heisenberg black holes and EHT observations agree in the finite parameter space. A discussion on the effective metric has been included.