Double Field Theory as the Double Copy of Yang-Mills

We show that double field theory naturally arises from the color-kinematic double copy of Yang-Mills theory. A precise double copy prescription for the Yang-Mills action at quadratic and cubic order is provided that yields the double field theory action in which the duality invariant dilaton has been integrated out. More precisely, at quadratic order this yields the gauge invariant double field theory, while at cubic order it yields the cubic double field theory action subject to a gauge condition that originates from Siegel gauge in string field theory.

General String Cosmologies at Order $alpha{}^{,3}$

We compute the cosmological reduction of general string theories, including bosonic, heterotic and type II string theory to order $alpha^{3}$, i.e., with up to eight derivatives. To this end we refine recently introduced methods that allow one to bring the reduced theory in one dimension to a canonical form with only first-order time derivatives. The resulting theories are compatible with a continuous $O(d,d,mathbb{R})$ invariance, which in turn fixes the B-field couplings.

Homotopy Transfer and Effective Field Theory II: Strings and Double Field Theory

We continue our study of effective field theory via homotopy transfer of $L_infty$-algebras, and apply it to tree-level non-Wilsonian effective actions of the kind discussed by Sen in which the modes integrated out are comparable in mass to the modes that are kept. We focus on the construction of effective actions for string states at fixed levels and in particular on the construction of weakly constrained double field theory. With these examples in mind, we discuss closed string theory on toroidal backgrounds and resolve some subtle issues involving vertex operators, including the proper form of cocycle factors and of the reflector state. This resolves outstanding issues concerning the construction of covariant closed string field theory on toroidal backgrounds. The weakly constrained double field theory is formally obtained from closed string field theory on a toroidal background by integrating out all but the doubly massless states and homotopy transfer then gives a prescription for determining the theorys vertices and symmetries. We also discuss consistent truncation in the context of homotopy transfer.

Beta functions for the duality-invariant sigma model

The $O(d,d)$ invariant worldsheet theory for bosonic string theory with $d$ abelian isometries is employed to compute the beta functions and Weyl anomaly at one-loop. We show that vanishing of the Weyl anomaly coefficients implies the equations of motion of the Maharana-Schwarz action. We give a self-contained introduction into the required techniques, including beta functions, the Weyl anomaly for two-dimensional sigma models and the background field method. This sets the stage for a sequel to this paper on generalizations to higher loops and $alpha$ corrections.

String Dualities at Order $alpha^{,3}$

We compute the cosmological reduction of the fourth powers of the Riemann tensor claimed to arise in string theory at order $alpha^{,3}$, with overall coefficient proportional to $zeta(3)$, and show that it is compatible with an $O(9,9)$ symmetry. This confirms the general result in string theory, due to Sen, that classical string theory with $d$-dimensional translation invariance admits an $O(d,d)$ symmetry to all orders in $alpha$.

Gauge Invariant Perturbation Theory via Homotopy Transfer

We show that the perturbative expansion of general gauge theories can be expressed in terms of gauge invariant variables to all orders in perturbations. In this we generalize techniques developed in gauge invariant cosmological perturbation theory, using Bardeen variables, by interpreting the passing over to gauge invariant fields as a homotopy transfer of the strongly homotopy Lie algebras encoding the gauge theory. This is illustrated for Yang-Mills theory, gravity on flat and cosmological backgrounds and for the massless sector of closed string theory. The perturbation lemma yields an algorithmic procedure to determine the higher corrections of the gauge invariant variables and the action in terms of these.

Old Dualities and New Anomalies

We revisit the question whether the worldsheet theory of a string admits a global O(d,d) symmetry. We consider the truncation of the target space theory in which fields are independent of d coordinates, which is O(d,d,R) invariant. The worldsheet theory is not O(d,d,R) invariant, unless it is truncated by setting winding and center-of-mass momenta to zero. We prove consistency of this truncation and give a manifestly O(d,d,R) invariant action, generalizing a formulation due to Tseytlin by including all external and internal target space fields. It is shown that, due to chiral bosons, this symmetry is anomalous. The anomaly is cancelled by a Green-Schwarz mechanism that utilizes the external B-field.

Toward Exotic 6D Supergravities

We investigate exotic supergravity theories in 6D with maximal (4,0) and (3,1) supersymmetry, which were conjectured by C. Hull to exist and to describe strong coupling limits of ${cal N}=8$ theories in 5D. These theories involve exotic gauge fields with non-standard Young tableaux representations, subject to (self-)duality constraints. We give novel actions in a 5+1 split of coordinates whose field equations reproduce those of the free bosonic (4,0) and (3,1) theory, respectively, including the (self-)duality relations. Evidence is presented for a master exceptional field theory formulation with an extended section constraint that, depending on the solution, produces the (4,0), (3,1) or the conventional (2,2) theory. We comment on the possible construction of a fully non-linear master exceptional field theory.

Homotopy Transfer and Effective Field Theory I: Tree-level

We use the dictionary between general field theories and strongly homotopy algebras to provide an algebraic formulation of the procedure of integrating out of degrees of freedom in terms of homotopy transfer. This includes more general effective theories in which some massive modes are kept while other modes of a comparable mass scale are integrated out, as first explored by Sen in the context of closed string field theory. We treat $L_infty$-algebras both in terms of a nilpotent coderivation and, on the dual space, in terms of a nilpotent derivation (corresponding to the BRST charge of the field theory) and provide explicit formulas for homotopy transfer. These are then shown to govern the integrating out of degrees of freedom at tree level, while the generalization to loop level will be explored in a sequel to this paper.

Duality Invariance and Higher Derivatives

We dimensionally reduce the spacetime action of bosonic string theory, and that of the bosonic sector of heterotic string theory after truncating the Yang-Mills gauge fields, on a $d$-dimensional torus including all higher-derivative corrections to first order in $alpha$. A systematic procedure is developed that brings this action into a minimal form in which all fields except the metric carry only first order derivatives. This action is shown to be invariant under ${rm O}(d,d,mathbb{R})$ transformations that acquire $alpha$-corrections through a Green-Schwarz type mechanism. We prove that, up to a global pre-factor, the first order $alpha$-corrections are uniquely determined by ${rm O}(d,d,mathbb{R})$ invariance.