Braneworld localisation in hyperbolic spacetime

Abstract

We present a construction employing a type IIA supergravity and 3-form flux background together with an NS5-brane that localises massless gravity near the 5-brane worldvolume. The nonsingular underlying type IIA solution is a lift to 10D of the vacuum solution of the 6D Salam-Sezgin model and has a hyperbolic H (2,2) × S 1 structure in the lifting dimensions. A fully back-reacted solution including the NS5-brane is constructed by recognising the 10D Salam-Sezgin vacuum solution as a "brane resolved through transgression." The background hyperbolic structure plays a key rôle in generating a mass gap in the spectrum of the transverse-space wave operator, which gives rise to the localisation of gravity on the 6D NS5-brane worldvolume, or, equally, in a further compactification to 4D. Also key to the successful localisation of gravity is the specific form of the corresponding transverse wavefunction Schrödinger problem, which asymptotically involves a V = −1/(4ρ 2 ) potential, where ρ is the transverse-space radius, and for which the NS5-brane source gives rise to a specific choice of self-adjoint extension for the transverse wave operator. The corresponding boundary condition as ρ → 0 ensures the masslessness of gravity in the effective braneworld theory. Above the mass gap, there is a continuum of massive states which give rise to small corrections to Newton's law.

Self-tuning at large (distances): 4D description of runaway dilaton capture

“…This could happen at the string scale if the brane UV completes as an object within string theory, or it could happen above or below the scale M if the branes UV complete as vortices in a higher-dimensional field theory. For concreteness we consider the vortex completion, since the extension to string theory of the system used here remains an open question [54][55][56][57]. Since our goal is to explore extra-dimensional approaches to the hierarchy problem, we always take the brane SUSY-breaking scale, M s , much larger than electroweak scales: M s m. If we choose M s M then the vortex sector would be supersymmetric (in that it would preserve at most half of the supersymmetries of the bulk [62][63][64]) with the branes likely arising as BPS solutions.…”

Covert symmetry breaking

“…For the Maxwell vector field, we pick standard Dirichlet boundary conditions at the z = 0 end of the interval I, but Robin boundary conditions (∂ z − 1)A µ = 0 at the z = 1 end. This causes the zero-mode transverse wavefunction to have non-trivial dependence on the transverse coordinate z, similarly to the dependence of the braneworld system of Reference [14] on a transverse radial coordinate. 1 That the key problem starts at fourth order in expansion of the action and is unlikely to be resolved by field redefinitions has recently been highlighted in [16].…”

“…In this paper, we consider a situation without any of the above handholds of full or second-order-in-derivatives consistency. The question we address here is motivated by an observation that one can make in the massless effective theory of supergravity localised on a braneworld submanifold in D = 11 supergravity [14], where the transverse space has an H(2, 2) hyperbolic noncompact structure [15]. This hyperbolic transverse-space structure can be used for dimensional reduction in a standard Kaluza-Klein fashion with fields independent of the transverse coordinates, but, owing to the noncompact transverse structure, the resulting lower dimensional Newton constant vanishes.…”

“…Localisation to the lower dimension in that case arises because there is a mass gap between the zero-level massless fields and the massive fields which, owing to the transverse space's noncompactness, form a continuum in mass starting at the edge of the gap. The transverse-space structure of Reference [14] has the additional advantage that the corresponding Sturm-Liouville problem is integrable when considered as a Schrödinger equation, with a potential of Pöschl-Teller type. This opens the way to analysis of the lower-dimensional effective braneworld theory's field equations beyond linearised order, since integrals over products of the zero-mode transverse wavefunction can be done explicitly.…”

“…1 That the key problem starts at fourth order in expansion of the action and is unlikely to be resolved by field redefinitions has recently been highlighted in [16]. The integrals of general products of the hyperbolic transverse-space wavefunction were given in Reference [14], and the unanticipated values of the resulting effective-theory expansion coefficients starting at fourth order were commented upon in [17].…”