Courant Algebroid Connections and String Effective Actions

Abstract

Courant algebroids are a natural generalization of quadratic Lie algebras, appearing in various contexts in mathematical physics. A connection on a Courant algebroid gives an analogue of a covariant derivative compatible with a given fiber-wise metric. Imposing further conditions resembling standard Levi-Civita connections, one obtains a class of connections whose curvature tensor in certain cases gives a new geometrical description of equations of motion of low energy effective action of string theory. Two examples are given. One is the so called symplectic gravity, the second one is an application to the the so called heterotic reduction. All necessary definitions, propositions and theorems are given in a detailed and self-contained way.

A Unique Connection for Born Geometry

“…We now briefly review these notions before exploring the connection they entertain with the para-Hermitian geometry. We refer the reader to [16] for an excellent review of these geometrical aspects. Let's start by describing the generalized geometry associated with the Courant algebroid given by (E, , E , [ , ] E , π).…”

“…Remarkably, this ambiguity is hidden from us at the one-loop level since it does not affect the construction of the Ricci tensor. The ambiguity disappears in the trace operation [16,43,51] if we demand also compatibility with the dilaton. While quite remarkable, this is still deeply unsettling.…”

Deformed graded Poisson structures, generalized geometry and supergravity

“…In the recent past the generalized differential geometry of T M ⊕ T * M has been exploited to show -with various suitable assumptions -that supergravity, as the supersymmetric theory of gravity in its own right, but also as the effective field theory of superstrings of type IIA and IIB, can be described as some kind of Einstein's General Relativity on this doubled vector bundle. For example, the works [5], [6], [7] and others show this in the framework of Generalized Geometry. In Double Field Theory, where also the coordinates of the base manifold (spacetime) are doubled, similar results -upon suitable projection onto standard target spacetime -were found, see e.g.…”

“…There, the Courant algebroid bracket employed in the definition is the skew-symmetric version. We shall rather stick to a torsion defined in the same fashion but with the Dorfman bracket instead (as in reference [7]), and refer to it as Gualtieri torsion T D G henceforth:…”

“…7) where U, V, W are E-sections and f is a smooth function.With the help of any suitable non-degenerate symmetric bilinear form ·, · , a connection symbol of the first kind defines a generalized connection ∇ (4.6) via∇ W U, V := Γ (W ; U, V ). (4.8)The proof is straightforward, but the important point is that there may be several choices of bilinear forms leading to different connections (some defined on subspaces).…”