Abstract
We consider symmetries of K3 manifolds. Holomorphic symplectic automorphisms of K3 surfaces have been classified, and observed to be subgroups of the Mathieu group M23. More recently, automorphisms of K3 sigma models commuting with SU(2) × SU(2) R-symmetry have been classified by Gaberdiel, Hohenegger, and Volpato. These groups are all subgroups of the Conway group. We fill in a small gap in the literature and classify the possible hyperkähler isometry groups of K3 manifolds. There is an explicit list of 40 possible groups, all of which are realized in the moduli space. The groups are all subgroups of M23.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P.S. Aspinwall and D.R. Morrison, String theory on K 3 surfaces, AMS/IP Stud. Adv. Math. 1 (1996) 703 [hep-th/9404151] [INSPIRE].
P.S. Aspinwall, K 3 surfaces and string duality, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 96): Fields, Strings, and Duality, pp. 421–540 (1996) [hep-th/9611137] [INSPIRE].
N. Benjamin, S. Kachru, K. Ono and L. Rolen, Black holes and class groups, arXiv:1807.00797 [INSPIRE].
G. Bini and A. Garbagnati, Quotients of the Dwork pencil, J. Geom. Phys. 75 (2014) 173.
J.W.S. Cassels, Rational Quadratic Forms, Dover (1978).
J. Conway, N. Sloane, Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290, 3rd edition, Springer-Verlag, New York (1999).
S.K. Donaldson, Polynomial invariants for smooth manifolds, Topology 29 (1990) 257 [INSPIRE].
T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K 3 Surface and the Mathieu group M24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Symmetries of K 3 σ-models, Commun. Num. Theor. Phys. 6 (2012) 1 [arXiv:1106.4315] [INSPIRE].
M.R. Gaberdiel, A. Taormina, R. Volpato and K. Wendland, A K 3 σ-model with \( {\mathrm{\mathbb{Z}}}_2^8 \): \( {\mathbbm{M}}_{20} \) symmetry, JHEP 02 (2014) 022 [arXiv:1309.4127] [INSPIRE].
J.A. Harvey and G.W. Moore, An Uplifting Discussion of T-duality, JHEP 05 (2018) 145 [arXiv:1707.08888] [INSPIRE].
J.A. Harvey and G.W. Moore, Conway Subgroup Symmetric Compactifications of Heterotic String, J. Phys. A 51 (2018) 354001 [arXiv:1712.07986] [INSPIRE].
J.A. Harvey and G.W. Moore, Moonshine, superconformal symmetry, and quantum error correction, JHEP 05 (2020) 146 [arXiv:2003.13700] [INSPIRE].
Kenji Hashimoto, Finite Symplectic actions on the K3 lattice, Nagoya Math. J. 206 (2012) 99.
G. Hoehn and G. Mason, The 290 fixed-point sublattices of the Leech lattice, J. Algebra 448 (2016) 618 [arXiv:1505.06420] [INSPIRE].
G. Hoehn and G. Mason, Finite groups of symplectic automorphisms of hyperkähler manifolds of type K 3[2], arXiv:1409.6055 [INSPIRE].
D. Huybrechts, On derived categories of K 3 surfaces and Mathieu groups, in Development of moduli theory — Kyoto 2013, Adv. Stud. Pure Math. 69 (2016) 387 [arXiv:1309.6528] [INSPIRE].
D. Huybrechts, Lectures on K3 Surfaces, Cambridge University Press, Cambridge (2016).
S. Kondo, Niemeier lattices, Mathieu groups and finite groups of symplectic automorphisms of K3 surfaces, Duke Math. J. 92 (1998) 593.
M. Leurer, Y. Nir and N. Seiberg, Mass matrix models, Nucl. Phys. B 398 (1993) 319 [hep-ph/9212278] [INSPIRE].
G. Mason, Symplectic automorphisms of K3 surfaces, CWI Newslett. No. 13 (1986) 3.
R. Miranda and D.R. Morrison, Embeddings Of Integral Quadratic Forms, http://www.math.ucsb.edu/~drm/manuscripts/eiqf.pdf.
G.W. Moore, Attractors and arithmetic, hep-th/9807056 [INSPIRE].
G.W. Moore, Arithmetic and attractors, hep-th/9807087 [INSPIRE].
G.W. Moore, Strings and Arithmetic, in Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry, pp. 303–359 (2007) [DOI] [hep-th/0401049] [INSPIRE].
S. Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988) 183.
W. Nahm and K. Wendland, A Hiker’s guide to K 3: Aspects of N = (4, 4) superconformal field theory with central charge c = 6, Commun. Math. Phys. 216 (2001) 85 [hep-th/9912067] [INSPIRE].
V.V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 111.
K. Wendland, On Superconformal field theories associated to very attractive quartics, in Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry, pp. 223–244, 2007, DOI [hep-th/0307066] [INSPIRE].
G. Xiao, Galois covers between K3 surfaces, Annales Inst. Fourier 46 (1996) 73.
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: 2009.11769
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
Banerjee, A., Moore, G.W. Hyperkähler isometries of K3 surfaces. J. High Energ. Phys. 2020, 193 (2020). https://doi.org/10.1007/JHEP12(2020)193
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)193