[Paper Review] Torsion and nonmetricity in the stringy geometry

B. Sazdović|ArXiv.org|Apr 9, 2003

Geometric and Algebraic Topology2 references4 citations

TL;DR

This paper introduces a geometric framework for bosonic strings propagating in curved spacetime with antisymmetric (B) and dilaton (Φ) background fields, showing that the B-field induces torsion and the dilaton induces nonmetricity in the target space. The key contribution is the formulation of a C-duality relation equating mean extrinsic curvature and its dual, generalizing minimal surfaces to 'stringy C-dual surfaces' in non-Riemannian geometries.

ABSTRACT

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.

Motivation & Objective

• To develop a geometric description of spacetime as perceived by a probe bosonic string in the presence of B-field and dilaton fields.
• To generalize surface geometry to spacetimes with torsion and nonmetricity, particularly for worldsheet dynamics.
• To define new geometric objects—mean extrinsic curvature and dual mean extrinsic curvature—under C-duality.
• To derive a modified integration measure preserving parallel transport in nonmetric spacetimes.
• To unify classical string dynamics with spacetime geometry by identifying the stringy C-dual surface as the solution to field equations.

Proposed method

• Decomposes the general affine connection into Christoffel, contortion (torsion), and nonmetricity parts.
• Defines induced worldsheet metrics and connections, distinguishing two forms due to nonmetricity.
• Introduces mean extrinsic curvature (MEC) and dual mean extrinsic curvature (DMEC) via orthogonal projection of mean torsion.
• Establishes the C-duality relation: MEC = ±DMEC, which reduces to minimal surface condition when torsion and nonmetricity vanish.
• Derives the conformal factor relating intrinsic and induced worldsheet metrics as a function of the dilaton field.
• Constructs a new integration measure for nonmetric spacetimes by requiring invariance under parallel transport and compatibility with integration by parts.

Experimental results

Research questions

• RQ1 How does the presence of an antisymmetric B-field affect the spacetime geometry felt by a bosonic string?
• RQ2 How does the dilaton field induce nonmetricity in the target spacetime geometry?
• RQ3 What is the geometric meaning of the field equations for the string worldsheet in non-Riemannian spacetime?
• RQ4 How can the concept of minimal surfaces be generalized when torsion and nonmetricity are present?
• RQ5 What is the correct integration measure for spacetimes with nonmetricity, ensuring consistency with parallel transport and integration by parts?

Key findings

• The B-field generates spacetime torsion, while the dilaton field induces nonmetricity, leading to a generalized stringy geometry distinct from Riemannian geometry.
• The field equations for the string worldsheet reduce to the C-duality condition: ${}^{ullet}H^{i} = \pm {}^{\ast}H^{i}$, defining a stringy C-dual surface.
• In the absence of torsion, the worldsheet becomes a stringy minimal surface with ${}^{\star}H_{i} = 0$, while in the absence of nonmetricity, it satisfies the C-duality condition.
• In Riemann space-time (vanishing torsion and nonmetricity), the worldsheet is a minimal surface with $H^{i} = 0$.
• The conformal factor relating intrinsic and induced worldsheet metrics is determined by the dilaton field, breaking conformal invariance.
• A new integration measure is derived for nonmetric spacetimes, ensuring invariance under parallel transport and enabling integration by parts, with explicit form $\sqrt{-G_2} \, d^2\sigma$ in the induced metric framework.