Falk Hassler

A bstract: Due to the exponential growth in the number of terms, computing α ^′ -corrections to string theory’s low-energy effective actions is a challenging matter. In order to fix all the couplings, one has usually to deal with a large number of string scattering amplitudes. This difficulty can be overcome by exploiting T-duality, which severely constrains the allowed structure of the effective action. It is then convenient to work in a formulation where T-duality is a manifest symmetry. Building on a series of previous works by some of the authors, [1–3], we present a refined version of an elegant and effective procedure that allows to obtain all the higher-derivative corrections of the NS-NS sector of string theories at order α ^′ and α ^{′ 2} , up to an overall coefficient. We dub this approach α ^′ -bootstrap , since it is based only on consistency conditions and avoids the direct computation of scattering amplitudes. The procedure relies on an infinite dimensional algebraic structure that we present in full detail, and it is conjectured to work at all orders. Although, at the moment, it still misses the ζ -like corrections starting at order α ^{′ 3} , the ease with which it can be generalized is promising for future developments in this direction.

Cartan geometry provides a unifying algebraic construction of curvature and torsion, based on an underlying model Lie algebra; a viewpoint that can be extended naturally to the higher algebraic structures underlying supergravity. We present a Cartan-geometric framework for generalized geometries governed by a differential graded Lie algebra, extending previous results. The extended tangent bundle admits the action of both a global duality group G and a local gauge group H . This algebraic structure is implemented via a brane current algebra; the phase space Poisson structure of p -branes. Within this Cartan-inspired framework, we define a hierarchy of generalized connections and compute their linearized torsion and curvature tensors, including the higher curvatures required by the tensor hierarchy. This provides a systematic construction of curvature and torsion tensors in generic generalized geometries.

Compatibility with T-duality severely constrains higher-derivative corrections to the low-energy supergravity limits of string theory. For example, it suggests that Lorentz transformations for heterotic strings are modified in precisely the way required for the Green-Schwarz anomaly cancellation mechanism. A systematic procedure to construct the resulting generalized Green-Schwarz transformations is the generalized Bergshoeff-de Roo identification (gBdRi) ; . Although it in principle allows computing α ′ -corrections to higher and higher orders, technically it becomes unfeasible beyond α ′ 2 . We revisit this problem with an alternative approach to the gBdRi, which we have recently developed. It gives rise to a very simple all-order transformation law whose closure we verify by explicitly computing the resulting gauge algebra.

Duality covariant curvature and torsion tensors in double field theory/generalized geometry are central in analyzing consistent truncations, generalized dualities, and related integrable σ-models. They are constructed systematically with the help of a larger, auxiliary space in a procedure inspired by Cartan geometry originally proposed by Poláček and Siegel for bosonic strings. It pivots around a maximally isotropic group that captures the generalized structure group of the physical space. We show how dropping the isotropy condition on this group allows us to describe heterotic/type I strings. As an immediate application, we construct a new family of heterotic backgrounds that interpolates between the two-dimensional cigar and trumpet backgrounds.

We revisit duality-covariant higher-derivative corrections which arise from the generalized Bergshoeff-de Roo (gBdR) identification, a prescription that gives rise to a two parameter family of α ^′ -corrections to the low-energy effective action of the bosonic and the heterotic string. Although it is able to reproduce all corrections at the leading and sub-leading (α ^′2) order purely from symmetry considerations, a geometric interpretation, like for the two-derivative action and its gauge transformation is lacking. To address this issue and to pave the way for the future exploration of higher-derivative (=higher-loop for the β-functions of the underlying σ-model) corrections to generalized dualities, consistent truncations and integrable σ-models, we recover the gBdR identification’s results from the Poláček-Siegel construction that provides a natural notion of torsion and curvature in generalized geometry.

We revisit duality-covariant higher-derivative corrections which arise from the generalized Bergshoeff-de Roo (gBdR) identification, a prescription that gives rise to a two parameter family of $α'$-corrections to the low-energy effective action of the bosonic and the heterotic string. Although it is able to reproduce all corrections at the leading and sub-leading ($α'^2$) order purely from symmetry considerations, a geometric interpretation, like for the two-derivative action and its gauge transformation is lacking. To address this issue and to pave the way for the future exploration of higher-derivative (=higher-loop for the $β$-functions of the underlying $σ$-model) corrections to generalized dualities, consistent truncations and integrable $σ$-models, we recover the gBdR identification's results from the \PS{} construction that provides a natural notion of torsion and curvature in generalized geometry.

Duality covariant curvature and torsion tensors in double field theory/generalized geometry are central in analyzing consistent truncations, generalized dualities, and related integrable $σ$-models. They are constructed systematically with the help of a larger, auxiliary space in a procedure inspired by Cartan geometry originally proposed by Poláček and Siegel for bosonic strings. It pivots around a maximally isotropic group that captures the generalized structure group of the physical space. We show how dropping the isotropy condition on this group allows us to describe heterotic/type I strings. As an immediate application, we construct a new family of heterotic backgrounds that interpolates between the two-dimensional cigar and trumpet backgrounds.