Abstract
We show that conformal manifolds in d ≥ 3 conformal field theories with at least 4 supercharges are Kähler-Hodge, thus extending to 3d \( \mathcal{N} \) = 2 and 4d \( \mathcal{N} \) = 1 similar results previously derived for 4d \( \mathcal{N} \) = 2 and \( \mathcal{N} \) = 4 and various types of 2d SCFTs. Conformal manifolds in SCFTs are equipped with a holomorphic line bundle ℒ, which encodes the operator mixing of supercharges under marginal deformations. Using conformal perturbation theory and superconformal Ward identities, we compute the curvature of ℒ at a generic point on the conformal manifold. We show that the Kähler form of the Zamolodchikov metric is proportional to the first Chern class of ℒ, with a constant of proportionality given by the two-point function coefficient of the stress tensor, CT. In cases where certain additional conditions about the nature of singular points on the conformal manifold hold, this implies a quantization condition for the total volume of the conformal manifold.
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References
K. Papadodimas, Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].
J. Gomis, P.-S. Hsin, Z. Komargodski, A. Schwimmer, N. Seiberg and S. Theisen, Anomalies, Conformal Manifolds, and Spheres, JHEP 03 (2016) 022 [arXiv:1509.08511] [INSPIRE].
A. Strominger, Special geometry, Commun. Math. Phys. 133 (1990) 163 [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, Aspects of Berry phase in QFT, JHEP 04 (2017) 062 [arXiv:1701.05587] [INSPIRE].
V. Asnin, On metric geometry of conformal moduli spaces of four-dimensional superconformal theories, JHEP 09 (2010) 012 [arXiv:0912.2529] [INSPIRE].
H. Osborn and A. C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
E. Barnes, E. Gorbatov, K. A. Intriligator, M. Sudano and J. Wright, The Exact superconformal R-symmetry minimizes tau(RR), Nucl. Phys. B 730 (2005) 210 [hep-th/0507137] [INSPIRE].
M. Buican and T. Nishinaka, Compact Conformal Manifolds, JHEP 01 (2015) 112 [arXiv:1410.3006] [INSPIRE].
M. Baggio, N. Bobev, S. M. Chester, E. Lauria and S. S. Pufu, Decoding a Three-Dimensional Conformal Manifold, JHEP 02 (2018) 062 [arXiv:1712.02698] [INSPIRE].
E. Perlmutter, L. Rastelli, C. Vafa and I. Valenzuela, A CFT distance conjecture, JHEP 10 (2021) 070 [arXiv:2011.10040] [INSPIRE].
E. Witten and J. Bagger, Quantization of Newton’s Constant in Certain Supergravity Theories, Phys. Lett. B 115 (1982) 202 [INSPIRE].
W. Gu and E. Sharpe, Bagger-Witten line bundles on moduli spaces of elliptic curves, Int. J. Mod. Phys. A 31 (2016) 1650188 [arXiv:1606.07078] [INSPIRE].
C. Cordova, T. T. Dumitrescu and K. Intriligator, Deformations of Superconformal Theories, JHEP 11 (2016) 135 [arXiv:1602.01217] [INSPIRE].
F. A. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [INSPIRE].
F. A. Dolan, On Superconformal Characters and Partition Functions in Three Dimensions, J. Math. Phys. 51 (2010) 022301 [arXiv:0811.2740] [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].
D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].
K. Ranganathan, H. Sonoda and B. Zwiebach, Connections on the state space over conformal field theories, Nucl. Phys. B 414 (1994) 405 [hep-th/9304053] [INSPIRE].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large N , Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].
E. D’Hoker, D. Z. Freedman and W. Skiba, Field theory tests for correlators in the AdS/CFT correspondence, Phys. Rev. D 59 (1999) 045008 [hep-th/9807098] [INSPIRE].
E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Extremal correlators in the AdS/CFT correspondence, hep-th/9908160 [INSPIRE].
K. A. Intriligator, Bonus symmetries of N = 4 superYang-Mills correlation functions via AdS duality, Nucl. Phys. B 551 (1999) 575 [hep-th/9811047] [INSPIRE].
K. A. Intriligator and W. Skiba, Bonus symmetry and the operator product expansion of N = 4 SuperYang-Mills, Nucl. Phys. B 559 (1999) 165 [hep-th/9905020] [INSPIRE].
B. Eden, P. S. Howe and P. C. West, Nilpotent invariants in N = 4 SYM, Phys. Lett. B 463 (1999) 19 [hep-th/9905085] [INSPIRE].
A. Petkou and K. Skenderis, A Nonrenormalization theorem for conformal anomalies, Nucl. Phys. B 561 (1999) 100 [hep-th/9906030] [INSPIRE].
P. S. Howe, C. Schubert, E. Sokatchev and P. C. West, Explicit construction of nilpotent covariants in N = 4 SYM, Nucl. Phys. B 571 (2000) 71 [hep-th/9910011] [INSPIRE].
P. J. Heslop and P. S. Howe, OPEs and three-point correlators of protected operators in N = 4 SYM, Nucl. Phys. B 626 (2002) 265 [hep-th/0107212] [INSPIRE].
M. Baggio, J. de Boer and K. Papadodimas, A non-renormalization theorem for chiral primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE].
V. Niarchos, Geometry of Higgs-branch superconformal primary bundles, Phys. Rev. D 98 (2018) 065012 [arXiv:1807.04296] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, tt∗ equations, localization and exact chiral rings in 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
J. Distler, Notes on N = 2 sigma models, hep-th/9212062 [INSPIRE].
J. de Boer, J. Manschot, K. Papadodimas and E. Verlinde, The Chiral ring of AdS3/CFT2 and the attractor mechanism, JHEP 03 (2009) 030 [arXiv:0809.0507] [INSPIRE].
M. Berger, Sur les groupes d’holonomie homogènes de variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955) 279.
C. Cordova, T. T. Dumitrescu and K. Intriligator, Multiplets of Superconformal Symmetry in Diverse Dimensions, JHEP 03 (2019) 163 [arXiv:1612.00809] [INSPIRE].
J. Gomis, Z. Komargodski, H. Ooguri, N. Seiberg and Y. Wang, Shortening Anomalies in Supersymmetric Theories, JHEP 01 (2017) 067 [arXiv:1611.03101] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
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Niarchos, V., Papadodimas, K. On the Kähler-Hodge structure of superconformal manifolds. J. High Energ. Phys. 2022, 104 (2022). https://doi.org/10.1007/JHEP09(2022)104
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DOI: https://doi.org/10.1007/JHEP09(2022)104