Abstract
Motivated by the search for rational points in moduli spaces of two-dimensional conformal field theories, we investigate how points with enhanced symmetry algebras are distributed there. We first study the bosonic sigma-model with S1 target space in detail and uncover hitherto unknown features. We find for instance that the vanishing of the twist gap, though true for the S1 example, does not automatically follow from enhanced symmetry points being dense in the moduli space. We then explore the supersymmetric sigma-model on K3 by perturbing away from the torus orbifold locus. Though we do not reach a definite conclusion on the distribution of enhanced symmetry points in the K3 moduli space, we make several observations on how chiral currents can emerge and disappear under conformal perturbation theory.
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References
D. Friedan, Z.-a. Qiu and S. H. Shenker, Conformal Invariance, Unitarity and Two-Dimensional Critical Exponents, Phys. Rev. Lett. 52 (1984) 1575 [INSPIRE].
S. Gukov and C. Vafa, Rational conformal field theories and complex multiplication, Commun. Math. Phys. 246 (2004) 181 [hep-th/0203213] [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].
L. J. Dixon, Some world sheet properties of superstring compactifications, on orbifolds and otherwise, in Proceedings, Summer Workshop in High-energy Physics and Cosmology: Superstrings, Unified Theories and Cosmology, Trieste, Italy, June 29 – August 7, 1987.
W. Lerche, C. Vafa and N. P. Warner, Chiral Rings in N = 2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
J. L. Cardy, Continuously Varying Exponents and the Value of the Central Charge, J. Phys. A 20 (1987) L891.
M. R. Gaberdiel, A. Konechny and C. Schmidt-Colinet, Conformal perturbation theory beyond the leading order, J. Phys. A 42 (2009) 105402 [arXiv:0811.3149] [INSPIRE].
N. Behr and A. Konechny, Renormalization and redundancy in 2d quantum field theories, JHEP 02 (2014) 001 [arXiv:1310.4185] [INSPIRE].
Z. Komargodski, S. S. Razamat, O. Sela and A. Sharon, A Nilpotency Index of Conformal Manifolds, JHEP 10 (2020) 183 [arXiv:2003.04579] [INSPIRE].
G. W. Moore, Arithmetic and attractors, hep-th/9807087 [INSPIRE].
S. Hosono, B. H. Lian, K. Oguiso and S.-T. Yau, Classification of c = 2 rational conformal field theories via the Gauss product, Commun. Math. Phys. 241 (2003) 245 [hep-th/0211230] [INSPIRE].
K. Wendland, Moduli spaces of unitary conformal field theories, Ph.D. Thesis, Bonn University, Germany (2000).
N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, Twist gap and global symmetry in two dimensions, Phys. Rev. D 101 (2020) 106026 [arXiv:2003.02844] [INSPIRE].
H. Eberle, Twistfield perturbations of vertex operators in the Z(2) orbifold model, JHEP 06 (2002) 022 [hep-th/0103059] [INSPIRE].
P. H. Ginsparg, Applied Conformal Field Theory, hep-th/9108028 [INSPIRE].
J. Liouville, Sur des classes très-étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationelles algébriques, J. Math. Pures Appl. 16 (1851) 133.
K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955) 1 [Erratum ibid. 2 (1955) 16810].
J. C. Oxtoby, Measure and category, vol. 2 of Graduate Texts in Mathematics, second edition, Springer-Verlag, New York-Berlin (1980).
R. Dijkgraaf, E. P. Verlinde and H. L. Verlinde, On moduli spaces of conformal field theories with c ≥ 1, in Copenhagen 1987, proceedings, perspectives in string theory, (1987), pp. 117–137.
D. Kutasov, Geometry on the Space of Conformal Field Theories and Contact Terms, Phys. Lett. B 220 (1989) 153 [INSPIRE].
C. A. Keller and I. G. Zadeh, Conformal Perturbation Theory for Twisted Fields, J. Phys. A 53 (2020) 095401 [arXiv:1907.08207] [INSPIRE].
C. A. Keller and I. G. Zadeh, Lifting \( \frac{1}{4} \)-BPS States on K3 and Mathieu Moonshine, Commun. Math. Phys. 377 (2020) 225 [arXiv:1905.00035] [INSPIRE].
M. R. Gaberdiel, C. Peng and I. G. Zadeh, Higgsing the stringy higher spin symmetry, JHEP 10 (2015) 101 [arXiv:1506.02045] [INSPIRE].
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ArXiv ePrint: 2011.07062
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Benjamin, N., Keller, C.A., Ooguri, H. et al. On rational points in CFT moduli spaces. J. High Energ. Phys. 2021, 67 (2021). https://doi.org/10.1007/JHEP04(2021)067
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DOI: https://doi.org/10.1007/JHEP04(2021)067