Torsion and nonmetricity in the stringy geometry

In the present article, we study the space-time geometry felt by probe bosonic string moving in antisymmetric and dilaton background fields. This space-time geometry we shall call the stringy geometry. In particular, the presence of the antisymmetric field leads to the space-time torsion, and the presence of the dilaton field leads to the space-time nonmetricity. We generalize the geometry of surfaces embedded in space-time to the case when torsion and nonmetricity are present. We define the mean extrinsic curvature for Minkowski signature and introduce the concept of mean torsion. Its orthogonal projection defines the dual mean extrinsic curvature. In this language, one field equation is just the equality of mean extrinsic curvature and dual mean extrinsic curvature, which we call self-duality relation. In the torsion and nonmetricity free case, the world-sheet is a minimal surface, specified by the requirement that mean extrinsic curvature vanishes. Generally, it is stringy self-dual (anti self-dual) surface. In the presence of the dilaton field, which breaks conformal invariance, the conformal factor which connects intrinsic and induced metrics, is determined as a function of the dilaton field itself. We also derive the integration measure for the space-time with stringy nonmetricity.