Abstract
We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups \( {H}_{\overline{\partial}}^{\left(p,0\right)}(k) \) graded by their charge under the Reeb vector. We then introduce a new cohomology, η-cohomology, which is defined by a CR structure and a holomorphic function f with non-vanishing η ≡ df. It is the natural cohomology associated to a class of supersymmetric type IIB flux backgrounds that generalise the notion of a Sasaki-Einstein manifold. These geometries are dual to finite deformations of the 4d \( \mathcal{N} \) = 1 SCFTs described by conventional Sasaki-Einstein manifolds. As such, they are associated to Calabi-Yau algebras with a deformed superpotential. We show how to compute the η-cohomology in terms of the transverse Dolbeault cohomology of the undeformed Sasaki-Einstein space. The gauge-gravity correspondence implies a direct relation between the cyclic homologies of the Calabi-Yau algebra, or equivalently the counting of short multiplets in the deformed SCFT, and the η-cohomology groups. We verify that this relation is satisfied in the case of S5, and use it to predict the reduced cyclic homology groups in the case of deformations of regular Sasaki-Einstein spaces. The corresponding Calabi-Yau algebras describe non-commutative deformations of ℙ2, ℙ1 × ℙ1 and the del Pezzo surfaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
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. M. Figueroa-O’Farrill, C. M. 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].
R. Eager, J. Schmude and Y. Tachikawa, Superconformal Indices, Sasaki-Einstein Manifolds, and Cyclic Homologies, Adv. Theor. Math. Phys. 18 (2014) 129 [arXiv:1207.0573] [INSPIRE].
J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Annals Math. 81 (1965) 451.
V. Ginzburg, Calabi-Yau algebras, math/0612139 [INSPIRE].
D. Berenstein, V. Jejjala and R. G. Leigh, Marginal and relevant deformations of N = 4 field theories and noncommutative moduli spaces of vacua, Nucl. Phys. B 589 (2000) 196 [hep-th/0005087] [INSPIRE].
D. Berenstein and R. G. Leigh, Resolution of stringy singularities by noncommutative algebras, JHEP 06 (2001) 030 [hep-th/0105229] [INSPIRE].
D. Berenstein and M. R. Douglas, Seiberg duality for quiver gauge theories, hep-th/0207027 [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
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].
A. Gadde, L. Rastelli, S. S. Razamat and W. Yan, On the Superconformal Index of N = 1 IR Fixed Points: A Holographic Check, JHEP 03 (2011) 041 [arXiv:1011.5278] [INSPIRE].
R. G. Leigh and M. J. Strassler, Exactly marginal operators and duality in four-dimensional N = 1 supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 95 [hep-th/9503121] [INSPIRE].
M. Van den Bergh, Noncommutative homology of some three-dimensional quantum spaces, in Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), vol. 8, pp. 213–230 (1994) [DOI].
O. Lunin and J. M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].
J. P. Gauntlett, D. Martelli, J. Sparks and D. Waldram, Supersymmetric AdS5 solutions of type IIB supergravity, Class. Quant. Grav. 23 (2006) 4693 [hep-th/0510125] [INSPIRE].
A. Ashmore, M. Petrini, E. Tasker and D. Waldram, Holomorphicity, supergravity duals and marginal deformations, to appear.
A. Ashmore, M. Petrini and D. Waldram, The exceptional generalised geometry of supersymmetric AdS flux backgrounds, JHEP 12 (2016) 146 [arXiv:1602.02158] [INSPIRE].
A. Ashmore, M. Petrini, E. Tasker and D. Waldram, Exactly Marginal Deformations and their Supergravity Duals, arXiv:2112.08375 [INSPIRE].
S. Sasaki, On differentiable manifolds with certain structures which are closely related to almost contact structure, I, Tohoku Math. J. 12 (1960) 459.
J. Sparks, New results in Sasaki-Einstein geometry, in Conference on Riemannian Topology: Geometric Structures on Manifolds: A Celebration of Charles P. Boyer’s 65th Birthday, (2007) [math/0701518] [INSPIRE].
J. Sparks, Sasaki-Einstein Manifolds, Surveys Diff. Geom. 16 (2011) 265 [arXiv:1004.2461] [INSPIRE].
C. Boyer and K. Galicki, Sasakian geometry, Oxford University Press (2008) [DOI].
S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974) 219.
N. Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of n complex variables, J. Math. Soc. Jap. 14 (1962) 397.
N. Tanaka, On non-degenerate real hypersurfaces, graded lie algebras and cartan connections, Jap. J. Math. 2 (1976) 131.
S. Dragomir and G. Tomasssini, Differential Geometry and Analysis on CR Manifolds, vol. 246 of Progress in Mathematics, Birkhäuser, Boston, Basel, Berlin (2006) [DOI].
N. Tanaka, A differential geometric study on strongly pseudo-convex manifolds, vol. 9 of Lectures in mathematics, Kinokuniya Book-store Co. (1975).
D. V. Alekseevsky, V. Cortes, K. Hasegawa and Y. Kamishima, Homogeneous locally conformally Kähler and Sasaki manifolds, Int. J. Math. 26 (2015) 1541001 [arXiv:1403.3268].
A. Tievsky, Analogues of Kähler geometry on Sasakian manifolds, Ph.D. Thesis, Massachusetts Institute of Technology (2008).
C. Stromenger, Sasakian Manifolds: Differential Forms, Curvature and Conformal Killing Forms, Ph.D. Thesis, University of Cologne (2010).
J. Schmude, Laplace operators on Sasaki-Einstein manifolds, JHEP 04 (2014) 008 [arXiv:1308.1027] [INSPIRE].
P. Tondeur, Geometry of foliations, vol. 90, Birkhäuser, Basel (1997) [DOI].
S. S.-T. Yau, Kohn-Rossi cohomology and its application to the complex plateau problem, I, Annals Math. 113 (1981) 67.
S. Katmadas and A. Tomasiello, Gauged supergravities from M-theory reductions, JHEP 04 (2018) 048 [arXiv:1712.06608] [INSPIRE].
R. Eager and J. Schmude, Superconformal Indices and M2-Branes, JHEP 12 (2015) 062 [arXiv:1305.3547] [INSPIRE].
Z. Wang and X. Zhou, CR eigenvalue estimate and Kohn-Rossi cohomology, arXiv:1905.03474.
C. Bar, Real Killing Spinors and Holonomy, Commun. Math. Phys. 154 (1993) 509 [INSPIRE].
D. Martelli, J. Sparks and S.-T. Yau, Sasaki-Einstein manifolds and volume minimisation, Commun. Math. Phys. 280 (2008) 611 [hep-th/0603021] [INSPIRE].
S. Bochner, Vector fields and Ricci curvature, Bull. Am. Math. Soc. 52 (1946) 776.
D. Perrone, On the minimal eigenvalue of the Laplacian operator for p-forms in conformally flat Riemannian manifolds, Proc. Am. Math. Soc. 86 (1982) 103.
S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. 54 (1975) 259.
S. Tachibana, On Killing tensors in Riemannian manifolds of positive curvature operator, Tohoku Math. J. 28 (1976) 177.
A. Ashmore, M. Gabella, M. Graña, M. Petrini and D. Waldram, Exactly marginal deformations from exceptional generalised geometry, JHEP 01 (2017) 124 [arXiv:1605.05730] [INSPIRE].
A. Ashmore and D. Waldram, Exceptional Calabi-Yau spaces: the geometry of \( \mathcal{N} \) = 2 backgrounds with flux, Fortsch. Phys. 65 (2017) 1600109 [arXiv:1510.00022] [INSPIRE].
B. Kol, On conformal deformations, JHEP 09 (2002) 046 [hep-th/0205141] [INSPIRE].
D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, Exactly Marginal Deformations and Global Symmetries, JHEP 06 (2010) 106 [arXiv:1005.3546] [INSPIRE].
G. Tian, On Kähler-Einstein metrics on certain Kähler manifolds with c1 (M) > 0, Invent. Math. 89 (1987) 225.
G. Tian and S.-T. Yau, Kähler-Einstein Metrics on Complex Surfaces With c1 > 0, Commun. Math. Phys. 112 (1987) 175 [INSPIRE].
R. Eager, Superconformal field theories and cyclic homology, Proc. Symp. Pure Math. 93 (2015) 141 [arXiv:1510.04078] [INSPIRE].
D. Z. Freedman and U. Gürsoy, Comments on the beta-deformed N = 4 SYM theory, JHEP 11 (2005) 042 [hep-th/0506128] [INSPIRE].
K. Madhu and S. Govindarajan, Chiral primaries in the Leigh-Strassler deformed N = 4 SYM: A Perturbative study, JHEP 05 (2007) 038 [hep-th/0703020] [INSPIRE].
S. Benvenuti and A. Hanany, Conformal manifolds for the conifold and other toric field theories, JHEP 08 (2005) 024 [hep-th/0502043] [INSPIRE].
D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
S. Franco, A. Hanany and P. Kazakopoulos, Hidden exceptional global symmetries in 4-D CFTs, JHEP 07 (2004) 060 [hep-th/0404065] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2112.09167
Rights and permissions
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.
About this article
Cite this article
Tasker, E.L. From β to η: a new cohomology for deformed Sasaki-Einstein manifolds. J. High Energ. Phys. 2022, 75 (2022). https://doi.org/10.1007/JHEP04(2022)075
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2022)075