Abstract
We decompose the de Rham Laplacian on Sasaki-Einstein manifolds as a sum over mostly positive definite terms. An immediate consequence are lower bounds on its spectrum. These bounds constitute a supergravity equivalent of the unitarity bounds in dual superconformal field theories. The proof uses a generalisation of Kähler identities to the Sasaki-Einstein case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Huybrechts, Complex geometry: an introduction, Springer, Germany (2005).
C. Voisin, Hodge theory and complex algebraic geometry, volume 1, Cambridge University Press, Cambridge U.K. (2008).
H. Lewy, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. Math. 64 (1956) 514.
J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math. 81 (1965) 451.
G. Folland and J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Princeton U.S.A. (1972).
S. Yau, Kohn-Rossi cohomology and its application to the complex plateau problem. I, Ann. Math. 113 (1981) 67.
H. Luk and S. Yau, Kohn-Rossi cohomology and its application to the complex plateau problem. II, J. Diff. Geom. 77 (2007) 135.
R. Du and S. Yau, Kohn-Rossi cohomology and its application to the complex plateau problem. III, J. Diff. Geom. 90 (2012) 251.
A.M. Tievsky, Analogues of Kähler geometry on Sasakian manifolds, Ph.D. thesis, MIT, U.S.A. (2008).
J. Sparks, Sasaki-Einstein manifolds, Surveys Diff. Geom. 16 (2011) 265 [arXiv:1004.2461] [INSPIRE].
M. Atiyah, Elliptic operators and compact groups, Lecture Notes in Mathematics volume 401, Springer, Berlin, Germany (1974).
A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications, Comp. Math. 73 (1990) 57.
A. Kehagias, New type IIB vacua and their F-theory interpretation, Phys. Lett. B 435 (1998) 337 [hep-th/9805131] [INSPIRE].
I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].
B.S. Acharya, J. Figueroa-O’Farrill, C. Hull and B.J. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249 [hep-th/9808014] [INSPIRE].
D.R. Morrison and M.R. Plesser, Nonspherical horizons. 1, Adv. Theor. Math. Phys. 3 (1999) 1 [hep-th/9810201] [INSPIRE].
D. Martelli and J. Sparks, Toric geometry, Sasaki-Einstein manifolds and a new infinite class of AdS/CFT duals, Commun. Math. Phys. 262 (2006) 51 [hep-th/0411238] [INSPIRE].
S. Minwalla, Restrictions imposed by superconformal invariance on quantum field theories, Adv. Theor. Math. Phys. 2 (1998) 781 [hep-th/9712074] [INSPIRE].
R. Eager, J. Schmude and Y. Tachikawa, Superconformal indices, Sasaki-Einstein manifolds and cyclic homologies, arXiv:1207.0573 [INSPIRE].
R. Eager and J. Schmude, Superconformal indices and M 2-branes, arXiv:1305.3547 [INSPIRE].
A. Ceresole, G. Dall’Agata and R. D’Auria, KK spectroscopy of type IIB supergravity on AdS 5 × T 11, JHEP 11 (1999) 009 [hep-th/9907216] [INSPIRE].
C. Pope, Kähler manifolds and quantum gravity, J. Phys. A 15 (1982) 2455.
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, S. Minwalla and S. Raju, Indices for superconformal field theories in 3, 5 and 6 dimensions, JHEP 02 (2008) 064 [arXiv:0801.1435] [INSPIRE].
K. Pilch and I. Yoo, On perturbative instability of Pope-Warner solutions on Sasaki-Einstein manifolds, JHEP 09 (2013) 124 [arXiv:1305.0295] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1308.1027
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Schmude, J. Laplace operators on Sasaki-Einstein manifolds. J. High Energ. Phys. 2014, 8 (2014). https://doi.org/10.1007/JHEP04(2014)008
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2014)008