Abstract
We study \( \mathcal{N}=2 \) compactifications of heterotic string theory on the CHL orbifold \( \left(K3\times {T}^2\right)/{\mathrm{\mathbb{Z}}}_N \) with N = 2, 3, 5, 7. \( {\mathrm{\mathbb{Z}}}_N \) acts as an automorphism on K3 together with a shift of 1/N along one of the circles of T 2. These compactifications generalize the example of the heterotic string on K3 × T 2 studied in the context of dualities in \( \mathcal{N}=2 \) string theories. We evaluate the new supersymmetric index for these theories and show that their expansion can be written in terms of the McKay-Thompson series associated with the \( {\mathrm{\mathbb{Z}}}_N \) automorphism embedded in the Mathieu group M 24. We then evaluate the difference in one-loop threshold corrections to the non-Abelian gauge couplings with Wilson lines and show that their moduli dependence is captured by Siegel modular forms related to dyon partition functions of \( \mathcal{N}=4 \) string theories.
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References
S. Kachru and C. Vafa, Exact results for N = 2 compactifications of heterotic strings, Nucl. Phys. B 450 (1995) 69 [hep-th/9505105] [INSPIRE].
M.B. Green, J.H. Schwarz and P.C. West, Anomaly free chiral theories in six-dimensions, Nucl. Phys. B 254 (1985) 327 [INSPIRE].
M.A. Walton, The heterotic string on the simplest Calabi-Yau manifold and its orbifold limits, Phys. Rev. D 37 (1988) 377 [INSPIRE].
L.J. Dixon, V. Kaplunovsky and J. Louis, Moduli dependence of string loop corrections to gauge coupling constants, Nucl. Phys. B 355 (1991) 649 [INSPIRE].
P. Mayr and S. Stieberger, Moduli dependence of one loop gauge couplings in (0, 2) compactifications, Phys. Lett. B 355 (1995) 107 [hep-th/9504129] [INSPIRE].
B. de Wit, V. Kaplunovsky, J. Louis and D. Lüst, Perturbative couplings of vector multiplets in N = 2 heterotic string vacua, Nucl. Phys. B 451 (1995) 53 [hep-th/9504006] [INSPIRE].
J.A. Harvey and G.W. Moore, Algebras, BPS states and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].
B. de Wit, G. Lopes Cardoso, D. Lüst, T. Mohaupt and S.-J. Rey, Higher order gravitational couplings and modular forms in N = 2, D = 4 heterotic string compactifications, Nucl. Phys. B 481 (1996) 353 [hep-th/9607184] [INSPIRE].
G. Lopes Cardoso, G. Curio and D. Lüst, Perturbative couplings and modular forms in N = 2 string models with a Wilson line, Nucl. Phys. B 491 (1997) 147 [hep-th/9608154] [INSPIRE].
S. Chaudhuri, G. Hockney and J.D. Lykken, Maximally supersymmetric string theories in D < 10, Phys. Rev. Lett. 75 (1995) 2264 [hep-th/9505054] [INSPIRE].
S. Chaudhuri and J. Polchinski, Moduli space of CHL strings, Phys. Rev. D 52 (1995) 7168 [hep-th/9506048] [INSPIRE].
P.S. Aspinwall, Some relationships between dualities in string theory, Nucl. Phys. Proc. Suppl. 46 (1996) 30 [hep-th/9508154] [INSPIRE].
S. Ferrara and C. Kounnas, Extended supersymmetry in four-dimensional type II strings, Nucl. Phys. B 328 (1989) 406 [INSPIRE].
J.H. Schwarz and A. Sen, Type IIA dual of the six-dimensional CHL compactification, Phys. Lett. B 357 (1995) 323 [hep-th/9507027] [INSPIRE].
I. Antoniadis, E. Gava and K.S. Narain, Moduli corrections to gravitational couplings from string loops, Phys. Lett. B 283 (1992) 209 [hep-th/9203071] [INSPIRE].
I. Antoniadis, E. Gava and K.S. Narain, Moduli corrections to gauge and gravitational couplings in four-dimensional superstrings, Nucl. Phys. B 383 (1992) 93 [hep-th/9204030] [INSPIRE].
S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A new supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [INSPIRE].
S. Cecotti and C. Vafa, Ising model and N = 2 supersymmetric theories, Commun. Math. Phys. 157 (1993) 139 [hep-th/9209085] [INSPIRE].
M.C.N. Cheng, X. Dong, J. Duncan, J. Harvey, S. Kachru and T. Wrase, Mathieu moonshine and N = 2 string compactifications, JHEP 09 (2013) 030 [arXiv:1306.4981] [INSPIRE].
J.R. David, D.P. Jatkar and A. Sen, Product representation of dyon partition function in CHL models, JHEP 06 (2006) 064 [hep-th/0602254] [INSPIRE].
C. Angelantonj, I. Florakis and B. Pioline, A new look at one-loop integrals in string theory, Commun. Num. Theor. Phys. 6 (2012) 159 [arXiv:1110.5318] [INSPIRE].
C. Angelantonj, I. Florakis and B. Pioline, One-loop BPS amplitudes as BPS-state sums, JHEP 06 (2012) 070 [arXiv:1203.0566] [INSPIRE].
C. Angelantonj, I. Florakis and B. Pioline, Rankin-Selberg methods for closed strings on orbifolds, JHEP 07 (2013) 181 [arXiv:1304.4271] [INSPIRE].
C. Angelantonj, I. Florakis and B. Pioline, Threshold corrections, generalised prepotentials and Eichler integrals, Nucl. Phys. B 897 (2015) 781 [arXiv:1502.00007] [INSPIRE].
S. Stieberger, (0, 2) heterotic gauge couplings and their M-theory origin, Nucl. Phys. B 541 (1999) 109 [hep-th/9807124] [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Counting dyons in N = 4 string theory, Nucl. Phys. B 484 (1997) 543 [hep-th/9607026] [INSPIRE].
G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy, JHEP 12 (2004) 075 [hep-th/0412287] [INSPIRE].
D. Shih, A. Strominger and X. Yin, Recounting dyons in N = 4 string theory, JHEP 10 (2006) 087 [hep-th/0505094] [INSPIRE].
D. Gaiotto, Re-recounting dyons in N = 4 string theory, hep-th/0506249 [INSPIRE].
S. Hohenegger and S. Stieberger, BPS saturated string amplitudes: K3 elliptic genus and Igusa cusp form, Nucl. Phys. B 856 (2012) 413 [arXiv:1108.0323] [INSPIRE].
D.P. Jatkar and A. Sen, Dyon spectrum in CHL models, JHEP 04 (2006) 018 [hep-th/0510147] [INSPIRE].
J.R. David and A. Sen, CHL dyons and statistical entropy function from D1-D5 system, JHEP 11 (2006) 072 [hep-th/0605210] [INSPIRE].
V.V. Nikulin, Finite automorphism groups of Kähler K3 surfaces, Trans. Moscow. Math. Soc. 38 (1979) 71.
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46 [INSPIRE].
E. Witten, Symmetry breaking patterns in superstring models, Nucl. Phys. B 258 (1985) 75 [INSPIRE].
T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, gauge theories and differential geometry, Phys. Rept. 66 (1980) 213 [INSPIRE].
M.C.N. Cheng, K3 surfaces, N = 4 dyons and the Mathieu group M 24, Commun. Num. Theor. Phys. 4 (2010) 623 [arXiv:1005.5415] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Mathieu twining characters for K3, JHEP 09 (2010) 058 [arXiv:1006.0221] [INSPIRE].
M.R. Gaberdiel, S. Hohenegger and R. Volpato, Mathieu moonshine in the elliptic genus of K3, JHEP 10 (2010) 062 [arXiv:1008.3778] [INSPIRE].
T. Eguchi and K. Hikami, Note on twisted elliptic genus of K3 surface, Phys. Lett. B 694 (2011) 446 [arXiv:1008.4924] [INSPIRE].
T. Eguchi, H. Ooguri, A. Taormina and S.-K. Yang, Superconformal algebras and string compactification on manifolds with SU(N) holonomy, Nucl. Phys. B 315 (1989) 193 [INSPIRE].
T. Eguchi, H. Ooguri and Y. Tachikawa, Notes on the K3 surface and the Mathieu group M 24, Exper. Math. 20 (2011) 91 [arXiv:1004.0956] [INSPIRE].
S. Govindarajan and K. Gopala Krishna, BKM Lie superalgebras from dyon spectra in Z N CHL orbifolds for composite N, JHEP 05 (2010) 014 [arXiv:0907.1410] [INSPIRE].
T. Kawai, String duality and modular forms, Phys. Lett. B 397 (1997) 51 [hep-th/9607078] [INSPIRE].
G. Lopes Cardoso, Perturbative gravitational couplings and Siegel modular forms in D = 4, N = 2 string models, Nucl. Phys. Proc. Suppl. 56B (1997) 94 [hep-th/9612200] [INSPIRE].
T. Kawai, N = 2 heterotic string threshold correction, K3 surface and generalized Kac-Moody superalgebra, Phys. Lett. B 372 (1996) 59 [hep-th/9512046] [INSPIRE].
J.R. David, D.P. Jatkar and A. Sen, Dyon spectrum in N = 4 supersymmetric type II string theories, JHEP 11 (2006) 073 [hep-th/0607155] [INSPIRE].
C. Angelantonj, I. Florakis and M. Tsulaia, Universality of gauge thresholds in non-supersymmetric heterotic vacua, Phys. Lett. B 736 (2014) 365 [arXiv:1407.8023] [INSPIRE].
C. Angelantonj, I. Florakis and M. Tsulaia, Generalised universality of gauge thresholds in heterotic vacua with and without supersymmetry, Nucl. Phys. B 900 (2015) 170 [arXiv:1509.00027] [INSPIRE].
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Datta, S., David, J.R. & Lüst, D. Heterotic string on the CHL orbifold of K3. J. High Energ. Phys. 2016, 56 (2016). https://doi.org/10.1007/JHEP02(2016)056
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DOI: https://doi.org/10.1007/JHEP02(2016)056