String-like Lagrangians from a generalized geometry

This note will use Hitchin’s generalized geometry and a model of axionic gravity developed by Warren Siegel in the mid-nineties to show that the construction of Lagrangians based on the inner product arising from the pairing of a vector and its dual can lead naturally to the low-energy Lagrangian of the bosonic string. PACS numbers: 02.40.-k, 04.20.-q, 11.25.-w String theory is one of the most popular theories of quantum gravity. It is widely believed to predict gravity, while remaining quantum mechanically finite. Nevertheless, unlike relativity, or gauge theory, it cannot be derived from a simple postulate. This note, inspired by Hitchin’s generalized geometry, and a model of axionic gravity developed by Warren Siegel in the midnineties, will argue that the low energy Lagrangian of the bosonic string follows naturally from the use of an inner product based on the pairing between a vector and its dual, rather than the, arguably less fundamental, standard Riemannian metric. In the generalized geometries developed by Nigel Hitchin and his students [1, 2, 3] geometric objects that are normally defined solely on the tangent bundle T , or cotangent bundle T , are redefined on the vector bundle T ⊕ T . These geometries are endowed with a natural metric arising from the inner product between elements of a vector space and its dual. This inner product has, on a d-dimensional manifold, an O(d, d) symmetry similar to that of the T-dualities found in string theory. The choice of subspaces of T ⊕ T ∗ that are positive, or negative, definite with respect to the natural metric breaks the symmetry to O(d)×O(d) and in turn leads to the generation of a positive definite metric, corresponding to the standard metric on a Riemannian manifold, and a b-field [2, 3]. A model of axionic gravity having similar properties was developed by Warren Siegel in the mid-nineties [4]. It was formulated by adding a second vielbein to an Einstein-Cartan theory of gravity, and has the Lagrangian of the closed, oriented bosonic string at low energies. The extra vielbein combines with the standard one to form an object transforming as a vector of O(d, d). This paper will reappraise Siegel’s model in the light of the recent development of generalized geometries. It will argue that the vielbeins can be understood as a set of d sections of T ⊕ T , coupling in the same way, and having the same transformation properties and relationship to the metric and b-field. In addition to an O(d, d) duality symmetry, the Lagrangian describing Siegel’s model has a GL(d) gauge symmetry that leaves the physical metric and b-field unchanged. The gauge potential for this symmetry is a combination of the vielbeins rather than an independent field. The Lagrangian also contains a scalar field, corresponding to the dilaton, required to make the measure duality invariant, and terms constructed from the vielbeins corresponding to the Ricci scalar and H terms from the low-energy action for the bosonic string (the H term accounts for the behaviour of the b-field via the relation H = db). The Ricci scalar can be regarded as a curvature term for the GL(d) gauge symmetry, and the H term as the analogue of the FμνF μν term arising in conventional theories of vector fields, but with the “vectors” living in T ⊕T ∗ rather than T alone. These are the terms one might expect, given the model’s field content, implying that the low energy Lagrangian of the bosonic string follows naturally from the use of an inner product based on the pairing of elements of the tangent and cotangent spaces in place of an inner product based on a Riemannian metric. This new inner product is arguably more fundamental than the original one; it exists on any differential manifold and does not require the existence of a Riemannian metric. String-like Lagrangians from a generalized geometry 2 1. Axionic Gravity and Generalized Geometry In 1993 Warren Siegel proposed a model of axionic gravity inspired by string theory and based, for a d dimensional manifold, on objects transforming as vectors of O(d, d) [4]. He began with the observation that the action for bosonic string theory can be written as