This paper explores construction of gauge (diffeomorphism)-invariant observables in anti de Sitter (AdS) space and the related question of how to find a "holographic map" providing a quantum equivalence to a boundary theory. Observables are constructed perturbatively to leading order in the gravitational coupling by gravitationally dressing local field theory operators in order to solve the gravitational constraints. Many such dressings are allowed and two are explicitly examined, corresponding to a gravitational line and to a Coulomb field; these also reveal an apparent role for more general boundary conditions than considered previously. The observables obey a nonlocal algebra, and we derive explicit expressions for the boundary generators of the SO(D-1,2) AdS isometries that act on them. We examine arguments that gravity explains holography through the role of such a boundary Hamiltonian. Our leading-order gravitational construction reveals some questions regarding how these arguments work, and indeed construction of such a holographic map appears to require solution of the non-perturbative generalization of the bulk constraint equations. * giddings@ucsb.edu 0 CONTENTS

By fibering the duality between the E8 × E8 heterotic string on T3 and M-theory on K3, we study heterotic duals of M-theory compactified on G2 orbifolds of the form T7/$$ {\mathbb{Z}}_2^3 $$ ℤ 2 3 . While the heterotic compactification space is straightforward, the description of the gauge bundle is subtle, involving the physics of point-like instantons on orbifold singularities. By comparing the gauge groups of the dual theories, we deduce behavior of a “half-G2” limit, which is the M-theory analog of the stable degeneration limit of F-theory. The heterotic backgrounds exhibit point-like instantons that are localized on pairs of orbifold loci, similar to the “gauge-locking” phenomenon seen in Hořava-Witten compactifications. In this way, the geometry of the G2 orbifold is translated to bundle data in the heterotic background. While the instanton configuration looks surprising from the perspective of the E8 × E8 heterotic string, it may be understood as T-dual Spin(32)/ℤ2 instantons along with winding shifts originating in a dual Type I compactification.

We consider the heterotic string on Calabi-Yau manifolds admitting a Strominger-Yau-Zaslow fibration. Upon reducing the system in the T3-directions, the Hermitian Yang-Mills conditions can then be reinterpreted as a complex flat connection on ℝ3 satisfying a certain co-closure condition. We give a number of abelian and non-abelian examples, and also compute the back-reaction on the geometry through the non-trivial α′-corrected heterotic Bianchi identity, which includes an important correction to the equations for the complex flat connection. These are all new local solutions to the Hull-Strominger system on T3× ℝ3. We also propose a method for computing the spectrum of certain non-abelian models, in close analogy with the Morse-Witten complex of the abelian models.

By fibering the duality between the E 8 × E 8 heterotic string on T 3 and M-theory on K3, we study heterotic duals of M-theory compactified on G 2 orbifolds of the form T 7 /Z 3 2 . While the heterotic compactification space is straightforward, the description of the gauge bundle is subtle, involving the physics of point-like instantons on orbifold singularities. By comparing the gauge groups of the dual theories, we deduce behavior of a "half-G 2 " limit, which is the M-theory analog of the stable degeneration limit of F-theory. The heterotic backgrounds exhibit point-like instantons that are localized on pairs of orbifold loci, similar to the "gauge-locking" phenomenon seen in Hořava-Witten compactifications. In this way, the geometry of the G 2 orbifold is translated to bundle data in the heterotic background. While the instanton configuration looks surprising from the perspective of the E 8 × E 8 heterotic string, it may be understood as T-dual Spin(32)/Z 2 instantons along with winding shifts originating in a dual Type I compactification.

We consider the heterotic string on Calabi-Yau manifolds admitting a Strominger-Yau-Zaslow fibration. Upon reducing the system in the T 3 -directions, the Hermitian Yang-Mills conditions can then be reinterpreted as a complex flat connection on R 3 satisfying a certain co-closure condition. We give a number of abelian and non-abelian examples, and also compute the back-reaction on the geometry through the non-trivial α ′ -corrected heterotic Bianchi identity, which includes an important correction to the equations for the complex flat connection. These are all new local solutions to the Hull-Strominger system on T 3 × R 3 . We also propose a method for computing the spectrum of certain non-abelian models, in close analogy with the Morse-Witten complex of the abelian models.