#### How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

2. Complete and sign the license agreement.

3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2213  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

# memo_has_moved_text();Poincaré–Einstein holography for forms via conformal geometry in the bulk

A. Rod Gover, Emanuele Latini and Andrew Waldron

Publication: Memoirs of the American Mathematical Society
Publication Year: 2015; Volume 235, Number 1106
ISBNs: 978-1-4704-1092-6 (print); 978-1-4704-2224-0 (online)
DOI: http://dx.doi.org/10.1090/memo/1106
Published electronically: September 23, 2014
Keywords:Differential forms, conformally compact, Q-curvature, conformal harmonics, Poincaré–Einstein, AdS/CFT, holography, scattering

View full volume PDF

View other years and numbers:

Chapters

• Chapter 1. Introduction
• Chapter 2. Bulk conformal geometry and extension problems
• Chapter 3. Tractor exterior calculus
• Chapter 4. The exterior calculus of scale
• Chapter 5. Higher form Proca equations
• Chapter 6. Obstructions, detours, gauge operators and $Q$-curvature
• Appendix A. The ambient manifold
• Appendix B. List of common symbols

### Abstract

We study higher form Proca equations on Einstein manifolds with boundary data along conformal infinity. We solve these Laplace-type boundary problems formally, and to all orders, by constructing an operator which projects arbitrary forms to solutions. We also develop a product formula for solving these asymptotic problems in general. The central tools of our approach are (i) the conformal geometry of differential forms and the associated exterior tractor calculus, and (ii) a generalised notion of scale which encodes the connection between the underlying geometry and its boundary. The latter also controls the breaking of conformal invariance in a very strict way by coupling conformally invariant equations to the scale tractor associated with the generalised scale. From this, we obtain a map from existing solutions to new ones that exchanges Dirichlet and Neumann boundary conditions. Together, the scale tractor and exterior structure extend the solution generating algebra of Gover and Waldron to a conformally invariant, Poincaré–Einstein calculus on (tractor) differential forms. This calculus leads to explicit holographic formulæfor all the higher order conformal operators on weighted differential forms, differential complexes, and Q-operators of Branson and Gover (2005). This complements the results of Aubry and Guillarmou where associated conformal harmonic spaces parametrise smooth solutions.

### References [Enhancements On Off] (What's this?)

• [1] O. Aharony, S.S. Gubser, J.M. Maldacena et al., Large $N$ field theories, string theory and gravity, Phys. Rept. 323, 183-386 (2000), [[href]]http://arxiv.org/abs/hep-th/9905111arXiv:hep-th/9905111.
• [2] M.T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Diff. Geom. 18, 701 (1983).
• [3] E. Aubry, C. Guillarmou, Conformal harmonic forms, Branson-Gover operators and Dirichlet problem at infinity, arXiv:0808.0552.
• [4] T.N. Bailey, M.G. Eastwood, and A.R. Gover, Thomas's structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), 1191–1217.
• [5] T.P. Branson, Conformally covariant equations on differential forms, Comm. Partial Differential Equations, 4 (1983), 393.
• [6] T.P. Branson, Sharp inequalities, the functional determinant, and the complementary series, Trans. Amer. Math. Soc. 347 (1995), 3671–3742.
• [7] T. Branson and A.R. Gover, Electromagnetism, metric deformations, ellipticity and gauge operators on conformal 4-manifolds, Differential Geom. Appl. 17 (2002), 229–249, [[href]]http://arxiv.org/abs/hep-th/0111003arXiv:hep-th/0111003.
• [8] T. Branson, A.R. Gover, Conformally invariant non-local operators, Pacific J. Math. 201 (2001), 19–60.
• [9] T. Branson, A.R. Gover, Conformally invariant operators, differential forms, cohomology and a generalisation of Q-curvature, Comm. Partial Differential Equations 30 (2005), 1611–1669.
• [10] A. Čap, and A.R. Gover, Tractor bundles for irreducible parabolic geometries. Global analysis and harmonic analysis, Sémin. Congr. 4, 129, Soc. Math. France 2000.
• [11] A. Čap, and A.R. Gover, Standard tractors and the conformal ambient metric construction, Ann. Global Anal. Geom. 24 (2003), 231–295, [[href]]http://arxiv.org/abs/math/0207016arXiv:math/0207016.
• [12] P. Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal. 57 (1984), 154–206.
• [13] D. Cherney, E. Latini, A. Waldron, $(p,q)$-form Kähler Electromagnetism, Phys. Lett. B674, 316-318 (2009), [[href]]http://lanl.arxiv.org/abs/0901.3788arXiv:0901.3788.
• [14] S. Deser, R. I. Nepomechie, Anomalous Propagation Of Gauge Fields In Conformally Flat Spaces, Phys. Lett. B132, 321 (1983); “Gauge Invariance Versus Masslessness In De Sitter Space”, Annals Phys. 154, 396 (1984).
• [15] S. Deser and A. Waldron, Partial masslessness of higher spins in (A)dS, Nucl. Phys. B 607 (2001), 577–604, arXiv:hep-th/0103198; Gauge invariances and phases of massive higher spins in (A)dS, Phys. Rev. Lett. 87 (2001), 031601, arXiv:hep-th/0102166; Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations, Phys. Lett. B 513, 137–141 (2001), arXiv-hep-th/0105181; Stability of massive cosmological gravitons, Phys. Lett. B 508 (2001), 347–353, arXiv-hep-th/0103255; Arbitrary spin representations in de Sitter from dS / CFT with applications to dS supergravity, Nucl. Phys. B 662 (2003), 379–392, arXiv-hep-th/0301068; Conformal invariance of partially massless higher spins, Phys. Lett. B 603 (2004), 30–39, arXiv-hep-th/0408155; Partially Massless Spin 2 Electrodynamics, Phys. Rev. D 74 (2006), 084036, arXiv-hep-th/0609113.
• [16] Michael Eastwood, Notes on conformal differential geometry, The Proceedings of the 15th Winter School “Geometry and Physics” (Srní, 1995), 1996, pp. 57–76. MR 1463509
• [17] M.G. Eastwood and J.W. Rice, Conformally invariant differential operators on Minkowski space and their curved analogues, Comm. Math. Phys. 109, 207 (1987); loc. cit. 144, 213, 1992.
• [18] C. Fefferman, and C.R. Graham, Conformal invariants in: The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque 1985, Numero Hors Serie, 95–116. page(s): 3, 78
• [19] Charles Fefferman and C. Robin Graham, The ambient metric, Annals of Mathematics Studies, vol. 178, Princeton University Press, Princeton, NJ, 2012. MR 2858236
• [20] C. Fefferman and K. Hirachi, Ambient metric construction of Q-curvature in conformal and CR geometries, Math. Res. Lett. 10, (2003) 819–831, arXiv:math.DG/0303184.
• [21] A. Rod Gover, Aspects of parabolic invariant theory, Rend. Circ. Mat. Palermo (2) Suppl. 59 (1999), 25–47. The 18th Winter School “Geometry and Physics” (Srní, 1998). MR 1692257
• [22] A. Rod Gover, Invariant theory and calculus for conformal geometries, Adv. Math. 163 (2001), no. 2, 206–257. MR 1864834, 10.1006/aima.2001.1999
• [23] A. R. Gover, Laplacian operators and $Q$-curvature on conformally Einstein manifolds, Math. Ann. 336 (2006), no. 2, 311–334. MR 2244375, 10.1007/s00208-006-0004-z
• [24] A.R. Gover, Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 100, arXiv:0710.2585.
• [25] A. Rod Gover, Almost conformally Einstein manifolds and obstructions, Differential geometry and its applications, Matfyzpress, Prague, 2005, pp. 247–260. MR 2268937
• [26] A.R. Gover, Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature, J. Geometry and Physics, 60 (2010), 182–204, arXiv:0803.3510.
• [27] A.R. Gover and P. Nurowski, Obstructions to conformally Einstein metrics in n dimensions, J. Geom. Phys. 56 (2006), 45–484, arXiv:math/0405304.
• [28] A. Rod Gover and Lawrence J. Peterson, Conformally invariant powers of the Laplacian, $Q$-curvature, and tractor calculus, Comm. Math. Phys. 235 (2003), no. 2, 339–378. MR 1969732, 10.1007/s00220-002-0790-4
• [29] A. Rod Gover and Lawrence J. Peterson, The ambient obstruction tensor and the conformal deformation complex, Pacific J. Math. 226 (2006), no. 2, 309–351. MR 2247867, 10.2140/pjm.2006.226.309
• [30] A. R. Gover, A. Shaukat, and A. Waldron, Tractors, mass, and Weyl invariance, Nuclear Phys. B 812 (2009), no. 3, 424–455. MR 2502272, 10.1016/j.nuclphysb.2008.11.026
• [31] A.R. Gover and A. Waldron, Boundary calculus for conformally compact manifolds, arXiv:1104.2991.
• [32] C. Robin Graham, Volume and area renormalizations for conformally compact Einstein metrics, The Proceedings of the 19th Winter School “Geometry and Physics” (Srní, 1999), 2000, pp. 31–42. MR 1758076
• [33] C. Robin Graham, Ralph Jenne, Lionel J. Mason, and George A. J. Sparling, Conformally invariant powers of the Laplacian. I. Existence, J. London Math. Soc. (2) 46 (1992), no. 3, 557–565. MR 1190438, 10.1112/jlms/s2-46.3.557
• [34] C. Robin Graham and John M. Lee, Einstein metrics with prescribed conformal infinity on the ball, Adv. Math. 87 (1991), no. 2, 186–225. MR 1112625, 10.1016/0001-8708(91)90071-E
• [35] C. Robin Graham and Maciej Zworski, Scattering matrix in conformal geometry, Invent. Math. 152 (2003), no. 1, 89–118. MR 1965361, 10.1007/s00222-002-0268-1
• [36] D. Grant, A conformally invariant third order Neumann-type operator for hyper surfaces, M.Sc. Thesis, University of Auckland, 2003; http://www.math.auckland.ac.nz/mathwiki/images /5/51/GrantMSc.pdf.
• [37] Maxim Grigoriev and Andrew Waldron, Massive higher spins from BRST and tractors, Nuclear Phys. B 853 (2011), no. 2, 291–326. MR 2835508, 10.1016/j.nuclphysb.2011.08.004
• [38] Colin Guillarmou and Frédéric Naud, Wave decay on convex co-compact hyperbolic manifolds, Comm. Math. Phys. 287 (2009), no. 2, 489–511. MR 2481747, 10.1007/s00220-008-0706-z
• [39] M. Henningson and K. Skenderis, The holographic Weyl anomaly, J. High Energy Phys. 7 (1998), Paper 23, 12 pp. (electronic). MR 1644988, 10.1088/1126-6708/1998/07/023
• [40] Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
• [41] Karl Hallowell and Andrew Waldron, The symmetric tensor Lichnerowicz algebra and a novel associative Fourier-Jacobi algebra, SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 089, 12. MR 2366933, 10.3842/SIGMA.2007.089
• [42] K. Hallowell and A. Waldron, Constant curvature algebras and higher spin action generating functions, Nuclear Phys. B 724 (2005), no. 3, 453–486. MR 2163016, 10.1016/j.nuclphysb.2005.06.021
• [43] Peter D. Hislop, Peter A. Perry, and Siu-Hung Tang, CR-invariants and the scattering operator for complex manifolds with boundary, Anal. PDE 1 (2008), no. 2, 197–227. MR 2472889, 10.2140/apde.2008.1.197
• [44] M. Joshi, A. Sá Barreto, Inverse scattering on asymptotically hyperbolic manifolds, Acta Math. 184, 41 (2000), arXiv:math/9811118. Cited on page(s): 4
• [45] Andreas Juhl, Families of conformally covariant differential operators, $Q$-curvature and holography, Progress in Mathematics, vol. 275, Birkhäuser Verlag, Basel, 2009. MR 2521913
• [46] C. R. LeBrun, ${\cal H}$-space with a cosmological constant, Proc. Roy. Soc. London Ser. A 380 (1982), no. 1778, 171–185. MR 652038, 10.1098/rspa.1982.0035
• [47] Per-Olov Löwdin, Angular momentum wavefunctions constructed by projector operators, Rev. Modern Phys. 36 (1964), 966–976. MR 0191466
• [48] Juan Maldacena, The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), no. 2, 231–252. MR 1633016, 10.4310/ATMP.1998.v2.n2.a1
• [49] R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Differential Geom., 28 (1988), 309–339.
• [50] Rafe Mazzeo, Unique continuation at infinity and embedded eigenvalues for asymptotically hyperbolic manifolds, Amer. J. Math. 113 (1991), no. 1, 25–45. MR 1087800, 10.2307/2374820
• [51] Rafe R. Mazzeo and Richard B. Melrose, Meromorphic extension of the resolvent on complete spaces with asymptotically constant negative curvature, J. Funct. Anal. 75 (1987), no. 2, 260–310. MR 916753, 10.1016/0022-1236(87)90097-8
• [52] Richard Melrose, Antônio Sá Barreto, and András Vasy, Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space, Comm. Partial Differential Equations 39 (2014), no. 3, 512–529. MR 3169793, 10.1080/03605302.2013.866958
• [53] R. R. Metsaev, Anomalous conformal currents, shadow fields and massive AdS fields, arXiv:1110.3749. Cited on page(s): 4
• [54] Roger Penrose and Wolfgang Rindler, Spinors and space-time. Vol. 1, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1984. Two-spinor calculus and relativistic fields. MR 776784
• [55] A. Proca, Sur la theórie ondulatoire des électrons positifs et négatifs, J. Phys. Radium 7, (1936) 347; Sur la theórie du positon, C. R. Acad. Sci. Paris 202 (1936), 1366.
• [56] S. Sasaki, On the spaces with normal conformal connexions whose groups of holonomy fix a point or a hypersphere II, Jap. J. Math. 18, (1943) 623–633.
• [57] R. Stafford, Tractor Calculus and Invariants for Conformal Sub-Manifolds, M.Sc. Thesis, University of Auckland, 2005; www.math.auckland.ac.nz/mathwiki/images/c/cf/StaffordMSc.pdf.
• [58] The Dirichlet problem at infinity for a negatively curved manifold, J. Diff. Geom. 18, 723 (1983).
• [59] Leonard Susskind, The world as a hologram, J. Math. Phys. 36 (1995), no. 11, 6377–6396. MR 1355913, 10.1063/1.531249
• [60] T.Y. Thomas, On conformal geometry, Proc. Natl. Acad. Sci. USA 12, 352–359 (1926).
• [61] G. 't Hooft, Dimensional reduction in quantum gravity. arXiv:gr-qc/9310026.
• [62] Valeriy N. Tolstoy, Fortieth anniversary of extremal projector method for Lie symmetries, Noncommutative geometry and representation theory in mathematical physics, Contemp. Math., vol. 391, Amer. Math. Soc., Providence, RI, 2005, pp. 371–384. MR 2184036, 10.1090/conm/391/07342
• [63] András Vasy, The wave equation on asymptotically de Sitter-like spaces, Adv. Math. 223 (2010), no. 1, 49–97. MR 2563211, 10.1016/j.aim.2009.07.005