Abstract
We study renormalization group flows between six-dimensional superconformal field theories (SCFTs) using their geometric realizations as singular limits of F-theory compactified on elliptically fibered Calabi-Yau threefolds. There are two general types of flows: one corresponds to giving expectation values to scalars in the tensor multiplets (tensor branch flow) realized as resolving the base of the geometry. The other corresponds to giving expectation values to hypermultiplets (Higgs branch flow) realized as complex structure deformations of the geometry. To corroborate this physical picture we calculate the change in the anomaly polynomial for these theories, finding strong evidence for a flow from a UV fixed point to an IR fixed point. Moreover, we find evidence against non-trivial dualities for 6D SCFTs. In addition we find non-trivial RG flows for theories realizing small E 8 instantons on ALE spaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124] [INSPIRE].
E. Witten, Some comments on string dynamics, hep-th/9507121 [INSPIRE].
A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44 [hep-th/9512059] [INSPIRE].
E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 2, Nucl. Phys. B 476 (1996) 437 [hep-th/9603161] [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
N. Seiberg, Nontrivial fixed points of the renormalization group in six-dimensions, Phys. Lett. B 390 (1997) 169 [hep-th/9609161] [INSPIRE].
M. Bershadsky and A. Johansen, Colliding singularities in F-theory and phase transitions, Nucl. Phys. B 489 (1997) 122 [hep-th/9610111] [INSPIRE].
I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, JHEP 03 (1998) 003 [hep-th/9712143] [INSPIRE].
J.D. Blum and K.A. Intriligator, Consistency conditions for branes at orbifold singularities, Nucl. Phys. B 506 (1997) 223 [hep-th/9705030] [INSPIRE].
P.S. Aspinwall and D.R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].
K.A. Intriligator, New string theories in six-dimensions via branes at orbifold singularities, Adv. Theor. Math. Phys. 1 (1998) 271 [hep-th/9708117] [INSPIRE].
A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE].
J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs and generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
D. Gaiotto and A. Tomasiello, Holography for (1, 0) theories in six dimensions, JHEP 12 (2014) 003 [arXiv:1404.0711] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6D conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
J.J. Heckman, More on the matter of 6D SCFTs, Phys. Lett. B 747 (2015) 73 [arXiv:1408.0006] [INSPIRE].
M. Del Zotto, J.J. Heckman, D.R. Morrison and D.S. Park, 6D SCFTs and gravity, JHEP 06 (2015) 158 [arXiv:1412.6526] [INSPIRE].
J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].
L. Bhardwaj, Classification of 6D N = (1, 0) gauge theories, arXiv:1502.06594 [INSPIRE].
L.B. Anderson, J.J. Heckman and S. Katz, T-branes and geometry, JHEP 05 (2014) 080 [arXiv:1310.1931] [INSPIRE].
K. Ohmori, H. Shimizu and Y. Tachikawa, Anomaly polynomial of E-string theories, JHEP 08 (2014) 002 [arXiv:1404.3887] [INSPIRE].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6D SCFTs, Prog. Theor. Exp. Phys. 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
K. Intriligator, 6d, N = (1, 0) Coulomb branch anomaly matching, JHEP 10 (2014) 162 [arXiv:1408.6745] [INSPIRE].
Y. Hayakawa, Degeneration of Calabi-Yau manifold with Weil-Petersson metric, Ph.D. thesis, University of Maryland, College Park U.S.A. (1994) [alg-geom/9507016].
C.-L. Wang, On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds, Math. Res. Lett. 4 (1997) 157.
A. Grassi, On minimal models of elliptic threefolds, Math. Ann. 290 (1991) 287.
C. Beasley, J.J. Heckman and C. Vafa, GUTs and exceptional branes in F-theory — I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].
K.A. Intriligator and B. Wecht, The exact superconformal R-symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
M.J. Duff, J.T. Liu and R. Minasian, Eleven-dimensional origin of string-string duality: a one loop test, Nucl. Phys. B 452 (1995) 261 [hep-th/9506126] [INSPIRE].
D. Freed, J.A. Harvey, R. Minasian and G.W. Moore, Gravitational anomaly cancellation for M-theory five-branes, Adv. Theor. Math. Phys. 2 (1998) 601 [hep-th/9803205] [INSPIRE].
J.A. Harvey, R. Minasian and G.W. Moore, Non-Abelian tensor multiplet anomalies, JHEP 09 (1998) 004 [hep-th/9808060] [INSPIRE].
A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys. 6 (2012) 51 [arXiv:1109.0042] [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: 1505.00009
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
Heckman, J.J., Morrison, D.R., Rudelius, T. et al. Geometry of 6D RG flows. J. High Energ. Phys. 2015, 52 (2015). https://doi.org/10.1007/JHEP09(2015)052
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2015)052