A tale of two superpotentials: Stability and instability in designer gravity
Abstract
We investigate the stability of asymptotically antide Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the BreitenlohnerFreedman bound. The boundary conditions in these 'designer gravity' theories are defined in terms of an arbitrary function W. Previous work had suggested that the energy in designer gravity is bounded below if (i) W has a global minimum and (ii) the scalar potential admits a superpotential P. More recently, however, certain solutions were found (numerically) to violate the proposed energy bound. We resolve the discrepancy by observing that a given scalar potential can admit two possible branches of the corresponding superpotential, P{sub {+}}. When there is a P{sub } branch, we rigorously prove a lower bound on the energy; the P{sub +} branch alone is not sufficient. Our numerical investigations (i) confirm this picture, (ii) confirm other critical aspects of the (complicated) proofs, and (iii) suggest that the existence of P{sub } may in fact be necessary (as well as sufficient) for the energy of a designer gravity theory to be bounded below.
 Authors:
 Physics Department, UCSB, Santa Barbara, California 93106 (United States)
 Theory Division, CERN, CH1211 Geneva 23 (Switzerland) and APC, 10 rue Alice Domon et Leonie Duquet, 75205 Paris (France)
 Inst. f. Theor. Physik, GeorgAugustUniversitaet, D37077, Goettingen (Germany)
 Publication Date:
 OSTI Identifier:
 21020377
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Physical Review. D, Particles Fields; Journal Volume: 75; Journal Issue: 8; Other Information: DOI: 10.1103/PhysRevD.75.084008; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BOUNDARY CONDITIONS; COSMOLOGY; DE SITTER GROUP; GRAVITATION; INSTABILITY; MASS; MATHEMATICAL SOLUTIONS; POTENTIALS; SCALAR FIELDS; SCALARS; STABILITY; TACHYONS
Citation Formats
Amsel, Aaron J., Marolf, Donald, Hertog, Thomas, and Hollands, Stefan. A tale of two superpotentials: Stability and instability in designer gravity. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVD.75.084008.
Amsel, Aaron J., Marolf, Donald, Hertog, Thomas, & Hollands, Stefan. A tale of two superpotentials: Stability and instability in designer gravity. United States. doi:10.1103/PHYSREVD.75.084008.
Amsel, Aaron J., Marolf, Donald, Hertog, Thomas, and Hollands, Stefan. Sun .
"A tale of two superpotentials: Stability and instability in designer gravity". United States.
doi:10.1103/PHYSREVD.75.084008.
@article{osti_21020377,
title = {A tale of two superpotentials: Stability and instability in designer gravity},
author = {Amsel, Aaron J. and Marolf, Donald and Hertog, Thomas and Hollands, Stefan},
abstractNote = {We investigate the stability of asymptotically antide Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the BreitenlohnerFreedman bound. The boundary conditions in these 'designer gravity' theories are defined in terms of an arbitrary function W. Previous work had suggested that the energy in designer gravity is bounded below if (i) W has a global minimum and (ii) the scalar potential admits a superpotential P. More recently, however, certain solutions were found (numerically) to violate the proposed energy bound. We resolve the discrepancy by observing that a given scalar potential can admit two possible branches of the corresponding superpotential, P{sub {+}}. When there is a P{sub } branch, we rigorously prove a lower bound on the energy; the P{sub +} branch alone is not sufficient. Our numerical investigations (i) confirm this picture, (ii) confirm other critical aspects of the (complicated) proofs, and (iii) suggest that the existence of P{sub } may in fact be necessary (as well as sufficient) for the energy of a designer gravity theory to be bounded below.},
doi = {10.1103/PHYSREVD.75.084008},
journal = {Physical Review. D, Particles Fields},
number = 8,
volume = 75,
place = {United States},
year = {Sun Apr 15 00:00:00 EDT 2007},
month = {Sun Apr 15 00:00:00 EDT 2007}
}

A tale of two superpotentials
We compute the superpotential on the worldvolume theory of Dbranes in the topological LandauGinzburg model associated with the cubic torus. An extended version of mirror symmetry relates this superpotential to the one on the mirror Dbrane. We discuss the equivalence of these two superpotentials by explicitly constructing the openstring mirror map. 
Designer Gravity and Field Theory Effective Potentials
Motivated by the antide Sitter conformal field theory correspondence, we show that there is remarkable agreement between static supergravity solutions and extrema of a field theory potential. For essentially any function V({alpha}) there are boundary conditions in antide Sitter space so that gravitational solitons exist precisely at the extrema of V and have masses given by the value of V at these extrema. Based on this, we propose new positive energy conjectures. On the field theory side, each function V can be interpreted as the effective potential for a certain operator in the dual field theory. 
Energy bounds in designer gravity
We consider asymptotically antide Sitter gravity coupled to tachyonic scalar fields with mass at or slightly above the BreitenlohnerFreedman bound in d{>=}4 spacetime dimensions. The boundary conditions in these ''designer gravity'' theories are defined in terms of an arbitrary function W. We give a general argument that the Hamiltonian generators of asymptotic symmetries for such systems will be finite, and proceed to construct these generators using the covariant phase space method. The direct calculation confirms that the generators are finite and shows that they take the form of the pure gravity result plus additional contributions from the scalar fields. Bymore » 
Erratum: Energy bounds in designer gravity [Phys. Rev. D 74, 064006 (2006)]
Minor errors are corrected in several equations. The corrections do not change the conclusions, but for completeness the conclusions are restated more explicitly.