Abstract
We present new anomalies in two-dimensional \( \mathcal{N}=\left(2,2\right) \) superconformal theories. They obstruct the shortening conditions of chiral and twisted chiral multiplets at coincident points. This implies that marginal couplings cannot be promoted to background superfields in short representations. Therefore, standard results that follow from \( \mathcal{N}=\left(2,2\right) \) spurion analysis are invalidated. These anomalies appear only if supersymmetry is enhanced beyond \( \mathcal{N}=\left(2,2\right) \). These anomalies explain why the conformal manifolds of the K3 and T 4 sigma models are not Kähler and do not factorize into chiral and twisted chiral moduli spaces and why there are no \( \mathcal{N}=\left(2,2\right) \) gauged linear sigma models that cover these conformal manifolds. We also present these results from the point of view of the Riemann curvature of conformal manifolds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [INSPIRE].
N. Seiberg, Observations on the moduli space of superconformal field theories, Nucl. Phys. B 303 (1988) 286 [INSPIRE].
D. Kutasov, Geometry on the space of conformal field theories and contact terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].
J.L. Cardy, Continuously varying exponents and the value of the central charge, J. Phys. A 20 (1987) L891 [INSPIRE].
M. Bershadsky, C. Vafa and V. Sadov, D strings on D manifolds, Nucl. Phys. B 463 (1996) 398 [hep-th/9510225] [INSPIRE].
S.H. Katz, D.R. Morrison and M.R. Plesser, Enhanced gauge symmetry in type-II string theory, Nucl. Phys. B 477 (1996) 105 [hep-th/9601108] [INSPIRE].
W. Lerche, C. Vafa and N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
L.J. Dixon, V. Kaplunovsky and J. Louis, On effective field theories describing (2,2) vacua of the heterotic string, Nucl. Phys. B 329 (1990) 27 [INSPIRE].
M. Dine and N. Seiberg, Microscopic knowledge from macroscopic physics in string theory, Nucl. Phys. B 301 (1988) 357 [INSPIRE].
A. Strominger, Special geometry, Commun. Math. Phys. 133 (1990) 163 [INSPIRE].
L.J. Dixon, Some World Sheet Properties Of Superstring Compactifications, On Orbifolds And Otherwise, proceedings of the 1987 ICTP Summer Workshop, Trieste, Italy, 29 June – 7 August 1987.
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [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].
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].
N. Seiberg, Naturalness versus supersymmetric nonrenormalization theorems, Phys. Lett. B 318 (1993) 469 [hep-ph/9309335] [INSPIRE].
V. Asnin, On metric geometry of conformal moduli spaces of four-dimensional superconformal theories, JHEP 09 (2010) 012 [arXiv:0912.2529] [INSPIRE].
K.S. Narain, New heterotic string theories in uncompactified dimensions < 10, Phys. Lett. B 169 (1986) 41 [INSPIRE].
S. Cecotti, \( \mathcal{N}=2 \) Landau-Ginzburg versus Calabi-Yau σ-models: Nonperturbative aspects, Int. J. Mod. Phys. A 6 (1991) 1749 [INSPIRE].
J. de Boer, J. Manschot, K. Papadodimas and E. Verlinde, The chiral ring of AdS 3 /CFT 2 and the attractor mechanism, JHEP 03 (2009) 030 [arXiv:0809.0507] [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].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press Princeton, U.S.A. (1992), 259 p.
D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press (2012).
L. Andrianopoli, R. D’Auria and S. Ferrara, Supersymmetry reduction of N extended supergravities in four-dimensions, JHEP 03 (2002) 025 [hep-th/0110277] [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
P.S. Aspinwall, K3 surfaces and string duality, hep-th/9611137 [INSPIRE].
N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact results in D = 2 supersymmetric gauge theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].
F. Benini and S. Cremonesi, Partition functions of \( \mathcal{N}=\left(2,2\right) \) gauge theories on S 2 and vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].
J. Gomis and S. Lee, Exact Kähler potential from gauge theory and mirror symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].
N. Doroud and J. Gomis, Gauge theory dynamics and Kähler potential for Calabi-Yau complex moduli, JHEP 12 (2013) 099 [arXiv:1309.2305] [INSPIRE].
H. Jockers, V. Kumar, J.M. Lapan, D.R. Morrison and M. Romo, Two-Sphere partition functions and Gromov-Witten invariants, Commun. Math. Phys. 325 (2014) 1139 [arXiv:1208.6244] [INSPIRE].
E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere partition functions and the Zamolodchikov metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
H. Osborn, \( \mathcal{N}=1 \) superconformal symmetry in four-dimensional quantum field theory, Annals Phys. 272 (1999) 243 [hep-th/9808041] [INSPIRE].
M.B. Green and N. Seiberg, Contact interactions in superstring theory, Nucl. Phys. B 299 (1988) 559 [INSPIRE].
J.J. Atick, L.J. Dixon and A. Sen, String calculation of Fayet-Iliopoulos d terms in arbitrary supersymmetric compactifications, Nucl. Phys. B 292 (1987) 109 [INSPIRE].
M. Dine, I. Ichinose and N. Seiberg, F terms and D terms in string theory, Nucl. Phys. B 293 (1987) 253 [INSPIRE].
E. Witten, Superstring perturbation theory revisited, arXiv:1209.5461 [INSPIRE].
M.J. Lighthill, An introduction to Fourier analysis and generalized functions, Cambridge Monographs on Mechanics, Cambridge University Press (1958).
Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, The axial anomaly and the bound state spectrum in confining theories, Nucl. Phys. B 177 (1981) 157 [INSPIRE].
K. Ranganathan, Nearby CFTs in the operator formalism: the role of a connection, Nucl. Phys. B 408 (1993) 180 [hep-th/9210090] [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].
D. Friedan and A. Konechny, Curvature formula for the space of 2-d conformal field theories, JHEP 09 (2012) 113 [arXiv:1206.1749] [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].
D.V. Alekseevsky and S. Marchiafava, A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold, Ann. Mat. Pura Appl. 184 (2005) 53.
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [INSPIRE].
K.A. Intriligator and W. Skiba, Bonus symmetry and the operator product expansion of N = 4 Super Yang-Mills, Nucl. Phys. B 559 (1999) 165 [hep-th/9905020] [INSPIRE].
Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Supersymmetry constraints and string theory on K3, JHEP 12 (2015) 142 [arXiv:1508.07305] [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: 1611.03101
Dedicated to John Schwarz on his 75th birthday
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
Gomis, J., Komargodski, Z., Ooguri, H. et al. Shortening anomalies in supersymmetric theories. J. High Energ. Phys. 2017, 67 (2017). https://doi.org/10.1007/JHEP01(2017)067
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2017)067