Geometry and topology of bubble solutions from gauge theory

2 Departmento
Abstract 1 -BPS type IIB supergravity We study how geometrical and topological aspects of certain 2 solutions are captured by the N = 4 Super Yang-Mills gauge theory in the AdS/CFT context. The type IIB solutions are completely characterized by arbitrary droplets in a plane and we consider, in 1 -BPS operators of the gauge theory with single particular, concentric droplets. We probe the dual 2 traces and extract their one-loop anomalous dimensions. The action of the one-loop dilatation operator can be reformulated as the Hamiltonian of a bosonic lattice. The operators defining the Hamiltonian encode the topology of the droplet. The axial symmetry of the droplets turns out to be essential for obtaining the spectrum of the Hamiltonians. In appropriate BMN limits, the near-BPS spectrum reproduces the spectrum of near-BPS string excitations propagating along each individual edge of the droplet of the dual geometric background. We also study semiclassical regimes for the Hamiltonians. We show that for droplets having disconnected constituents, the Hamiltonian admits different complimentary semiclassical descriptions, each one replicating the semiclassical description for closed strings extending in each of the constituents.
1
Introduction
The notion of emergent geometry in the AdS/CFT correspondence [1] holds certain appeal. The general idea is that judicial choices of BPS operators in the gauge theory are capable of encoding the characteristics of the corresponding dual geometries [2, 3]. 1 -BPS operators of N = 4 SYM A paradigmatic example was provided by [2], focusing on the 2 constructed exclusively from a complex adjoint chiral scalar Z . The dynamics of this sub-sector can be described by gauged harmonic oscillators, which in turn can be mapped into a Fermi problem. The ground state of the system corresponds to a completely occupied Fermi sea forming a disc in the phase space. This distribution, known as the droplet, can then be associated, at the semi-classical level, with the allowed eigenvalues of the scalar field Z [2]. Other 1 2 -BPS ‘excitations’ correspond to disc distortions, further splitting into isolated droplets and creation of holes within these droplets. All these lead to droplets with non-trivial shapes and topologies [3, 4, 5, 6]. This class of 1 2 -BPS states preserve 16 supersymmetries as well as a bosonic R × SO(4) × SO(4) symmetry group. The dual type IIB supergravity backgrounds, known as Lin-Lunin-Maldacena (LLM) geometries, were constructed by fibering two S 3 over a two dimensional plane [3]. The base of the fibration in this construction was proposed to be identified with the droplets describing the eigenvalue distribution in the gauge theory. In particular, the AdS5 × S 5 geometry is shown to correspond to a circular disc droplet, whereas single particle and hole creations were identified with the nucleations of a giant graviton in AdS5 and S 5 respectively [2, 3]. The bottom line of the construction is that when the number of giant gravitons becomes large, the back-reacted non-singular geometry gets parameterized in terms of a single scalar function obeying a Laplace-like differential equation whose boundary conditions on a two dimensional plane can only take the values ± 1 2. Generically, a probe closed string propagating on a LLM geometry should be seen in the dual gauge field theory as a non-BPS operator given by a single trace operator describing the excitations (probe string) on top of a 1 2 -BPS operator (LLM geometry). The study of the one-loop anomalous dimensions for such gauge theory operators turns out to be very rewarding. For the case of simply connected droplets, the gauge theory Hamiltonian describing probe strings was recently derived [7]. This Hamiltonian can be written in terms of certain operators obeying an algebra that encodes, in a simple fashion, the moments describing the geometry of the droplet. Moreover, using the coherent state basis for the operators, a semi-classical action was derived, from which it was possible to reconstruct the region of the LLM droplet plane where the dual string propagates [7]. The approach described in [7, 8, 9] is similar in spirit to the coherent state approach in the context of spinchain/spinning string correspondence [10, 11], where the coherent state action for the integrable spin chain Hamiltonian [12] was identified with the spinning-string action in certain “fast-string” limit. Making the existing picture more precise is the focus of our paper. In particular, we will obtain quantitative information about the dual metric, for droplets of general shapes and topologies, from gauge theory computations. For definiteness we will consider non-BPS excitations, or string probe states, in topologically non-trivial LLM backgrounds, evaluate their scaling dimensions and importantly, distinguish them from their counterparts in AdS5 × S 5 . From the dual gauge theory perspective, the key steps towards these aims amount to defining the appropriate field theory operators and compute the action of the dilatation operator whose eigenvalues give the scaling