Abstract
We extend the modular orbits method of constructing a two-dimensional orbifold conformal field theory to higher genus Riemann surfaces. We find that partition functions on surfaces of arbitrary genus can be constructed by a straightforward generalization of the rules that one would apply to the torus. We demonstrate how one can use these higher genus objects to compute correlation functions and OPE coefficients in the underlying theory. In the case of orbifolds of free bosonic theories by subgroups of continuous symmetries, we can give the explicit results of our procedure for symmetric and asymmetric orbifolds by cyclic groups.
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References
D. Friedan and S.H. Shenker, The Analytic Geometry of Two-Dimensional Conformal Field Theory, Nucl. Phys. B 281 (1987) 509 [INSPIRE].
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds. 2., Nucl. Phys. B 274 (1986) 285 [INSPIRE].
D. Robbins and T. Vandermeulen, Orbifolds from Modular Orbits, arXiv:1911.05172 [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, c = 1 Conformal Field Theories on Riemann Surfaces, Commun. Math. Phys. 115 (1988) 649 [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
J.L. Cardy, Operator content and modular properties of higher dimensional conformal field theories, Nucl. Phys. B 366 (1991) 403 [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP 08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP 10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP 09 (2018) 061 [arXiv:1608.06241] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman and A. Tajdini, Fast Conformal Bootstrap and Constraints on 3d Gravity, JHEP 05 (2019) 087 [arXiv:1903.06272] [INSPIRE].
J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP 10 (2017) 136 [arXiv:1705.05855] [INSPIRE].
C.A. Keller, G. Mathys and I.G. Zadeh, Bootstrapping Chiral CFTs at Genus Two, Adv. Theor. Math. Phys. 22 (2018) 1447 [arXiv:1705.05862] [INSPIRE].
M. Cho, S. Collier and X. Yin, Genus Two Modular Bootstrap, JHEP 04 (2019) 022 [arXiv:1705.05865] [INSPIRE].
J. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Springer (1973).
E. Verlinde, Conformal field theory and its application to strings, https://lccn.loc.gov/89193268 (1988).
E.P. Verlinde and H.L. Verlinde, Chiral Bosonization, Determinants and the String Partition Function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].
N. Behera, R.P. Malik and R.K. Kaul, Genus Two Correlators for Critical Ising Model, Phys. Rev. D 40 (1989) 1993 [INSPIRE].
G. Mason and M.P. Tuite, On genus two Riemann surfaces formed from sewn tori, Commun. Math. Phys. 270 (2007) 587 [math/0603088] [INSPIRE].
P. Di Francesco, H. Saleur and J.B. Zuber, Critical Ising Correlation Functions in the Plane and on the Torus, Nucl. Phys. B 290 (1987) 527 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Graduate texts in contemporary physics, New York, NY, Springer (1997) [INSPIRE].
G. Massuyeau, Lectures on mapping class groups, braid groups and formality, http://massuyea.perso.math.cnrs.fr/notes/formality.pdf (2015).
C. Vafa, Modular Invariance and Discrete Torsion on Orbifolds, Nucl. Phys. B 273 (1986) 592 [INSPIRE].
C. Bachas, J. de Boer, R. Dijkgraaf and H. Ooguri, Permeable conformal walls and holography, JHEP 06 (2002) 027 [hep-th/0111210] [INSPIRE].
C. Bachas and I. Brunner, Fusion of conformal interfaces, JHEP 02 (2008) 085 [arXiv:0712.0076] [INSPIRE].
D. Gaiotto, Domain Walls for Two-Dimensional Renormalization Group Flows, JHEP 12 (2012) 103 [arXiv:1201.0767] [INSPIRE].
T. Quella and V. Schomerus, Symmetry breaking boundary states and defect lines, JHEP 06 (2002) 028 [hep-th/0203161] [INSPIRE].
I. Brunner, M. Herbst, W. Lerche and B. Scheuner, Landau-Ginzburg realization of open string TFT, JHEP 11 (2006) 043 [hep-th/0305133] [INSPIRE].
I. Brunner and D. Roggenkamp, B-type defects in Landau-Ginzburg models, JHEP 08 (2007) 093 [arXiv:0707.0922] [INSPIRE].
A. Konechny, Fusion of conformal interfaces and bulk induced boundary RG flows, JHEP 12 (2015) 114 [arXiv:1509.07787] [INSPIRE].
K. Graham and G.M.T. Watts, Defect lines and boundary flows, JHEP 04 (2004) 019 [hep-th/0306167] [INSPIRE].
J. Fuchs and C. Schweigert, Surface defects and symmetries, J. Phys. Conf. Ser. 597 (2015) 012002 [INSPIRE].
C. Bachas, I. Brunner and D. Roggenkamp, Fusion of Critical Defect Lines in the 2D Ising Model, J. Stat. Mech. 1308 (2013) P08008 [arXiv:1303.3616] [INSPIRE].
M. Becker, Y. Cabrera and D. Robbins, Conformal interfaces between free boson orbifold theories, JHEP 09 (2017) 148 [arXiv:1706.03802] [INSPIRE].
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ArXiv ePrint: 1911.06306
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Robbins, D., Vandermeulen, T. Modular orbits at higher genus. J. High Energ. Phys. 2020, 113 (2020). https://doi.org/10.1007/JHEP02(2020)113
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DOI: https://doi.org/10.1007/JHEP02(2020)113