PT-Symmetric Quaternionic Spacetime from String Theory: Bridging D3-Brane Dynamics with Cosmological Observations

We present a $\mathcal{PT}$-symmetric quaternionic extension of spacetime, derived non-perturbatively from the Dirac--Born--Infeld (DBI) action of D3-branes in Type IIB string theory. Unlike traditional compactification, our approach embeds rotational degrees of freedom directly into four-dimensional spacetime via a T-dualized NS--NS $B$-field. The resulting quaternionic metric is given by: \[ G_{\mu\nu} = g_{\mu\nu}^{(R)} + \mathbf{i}\, g_{\mu\nu}^{(i)} + \mathbf{j}\, g_{\mu\nu}^{(j)} + \mathbf{k}\, g_{\mu\nu}^{(k)}, \] where $\mathbf{i}, \mathbf{j}, \mathbf{k}$ are rotational generators induced by the $B$-field topology. These structures emerge naturally from flux quantization ($b = 5.834 \times 10^{-16}~\text{m}^{-1}$) and string coupling ($\epsilon$, related to $g_s$), and they encode dark energy and dark matter as geometric effects.Solving the Einstein equations with this quaternionic metric yields modified Friedmann equations. For $\epsilon \approx 2$, the model predicts a dark energy density of \[ \rho_{\text{imag}} \approx \frac{\epsilon^2}{1 + \epsilon^2} M_{\rm pl}^2 H_0^2 \sim 2.8 \times 10^{-47}~\text{GeV}^4, \] which aligns with the $\Lambda$CDM value from Planck 2018 observations.Furthermore, the galactic gravitational potential becomes: \[ \Phi_{\text{total}} = -\frac{GM}{r} + \frac{1}{2} b r, \] predicting flattened rotation curves at 200--300 km/s, pending validation against real galactic data.$\mathcal{PT}$-symmetry ensures all physical observables remain real despite the quaternionic structure. We propose Bayesian tests using Planck and DESI data to constrain the parameters $(\epsilon, b)$, offering a bridge between non-perturbative string theory and observable cosmology.