De Sitter vacuum on nucleated D-branes with stringy corrections

We find four-dimensional de Sitter (dS) vacuum solutions on probe D-branes which nucleate in an asymptotically $\text{AdS}_5\times T^{1,1}$ background, including stringy corrections. A sufficiently high chemical potential induced by the wrapped D3-brane charge, breaking supersymmetry in the bulk, is essential to lead to the nucleation of the probe D-brane. We show that stringy corrections can yield a dS vacuum on a spherical D3-brane consistent with the observations without being fine-tuned. Motivated by the fact that the matter fields propagating in the compact extra dimensions can provide solutions for problems in particle physics and cosmology, we also construct a dS vacuum on a probe D5-brane which wraps on a two-torus of the internal manifold $T^{1,1}$. This construction requires turning on a worldvolume U(1) gauge field along the wrapped part of the D5-brane and dissolving some D3-branes in it. The stringy corrections play an important role in yielding a dS vacuum and the field strength of the U(1) gauge field must be sufficiently large to produce a tiny cosmological constant.