A Unique Connection for Born Geometry

Abstract

It has been known for a while that the effective geometrical description of compactified strings on d-dimensional target spaces implies a generalization of geometry with a doubling of the sets of tangent space directions. This generalized geometry involves an O(d, d) pairing η and an O(2d) generalized metric H. More recently it has been shown that in order to include T-duality as an effective symmetry, the generalized geometry also needs to carry a phase space structure or more generally a para-Hermitian structure encoded into a skew-symmetric pairing ω. The consistency of string dynamics requires this geometry to satisfy a set of compatibility relations that form what we call a Born geometry. In this work we prove an analogue of the fundamental theorem of Riemannian geometry for Born geometry. We show that there exists a unique connection which preserves the Born structure (η, ω, H) and which is torsionless in a generalized sense. This resolves a fundamental ambiguity that is present in the double field theory formulation of effective string dynamics.

Vierbein interactions with antisymmetric components

“…Understanding the underlying geometry of any theory which includes gravity is thus crucial. For example, the geometry of string theory and its web of dualities gives rise to interesting new mathematical structures such as the extended space of Double Field Theory [37][38][39] which is related to generalized geometry [40,41] and Born geometry [42,43]. Interestingly, all these setups with intimate relation to quantum gravity contain an antisymmetric structure in addition to the metric.…”

Para-Hermitian geometries for Poisson-Lie symmetric σ-models

“…We summarize the key ideas of para-Hermitian geometry together with the concept of a D-structure and suitably generalized notions of torsion and integrability 2 . This forms the basis to define Born geometry and has been established in [47][48][49], for an executive summary see [65].…”

“…The relation of this mathematical framework to physical systems can be seen in Double Field Theory (DFT) [40][41][42][43][44]. This is a T-duality covariant effective target space theory of closed strings which requires a para-Hermitian structure or the slightly weaker half-integrable structure on the doubled space for consistency [45][46][47][48][49]. A sketchy but short argument why this is the case goes as follows: recall that a complex structure on an even-dimensional real manifold allows us to introduce holomorphic and anti-holomorphic coordinates on this manifold.…”

“…On the other hand Poisson-Lie σ-models provide a rich class of examples for para-Hermitian geometries which can be explicitly constructed on group manifolds. We use them to prove that certain constraints imposed in the recent works on para-Hermitian structures and their connection to generalized geometry [57,58] and DFT [47][48][49]56] can be relaxed. In particular, we find that only the distri butionL associated to the unphysical coordinates has to be integrable to permit the construction of a Courant algebroid over the physical target space.…”

Double Field Theory for the A/B‐Models and Topological S‐Duality in Generalized Geometry

“…A more natural framework for describing fluxes is double field theory (DFT), where the original coordinates conjugate to closed string momentum modes are extended with dual coordinates conjugate to winding modes, and the continuous version of T‐duality becomes a manifest symmetry of the theory (see [] for reviews). The algebroid structure of double field theory was studied in []; in particular, the notion of DFT algebroid was introduced in [] whose derived bracket is the C‐bracket of double field theory, and whose corresponding membrane sigma‐model naturally captures the T‐duality orbit of geometric and non‐geometric flux backgrounds in a single unified description.…”