Abstract
We focus on 4D \( \mathcal{N} \) = 2 string vacua described both by perturbative Heterotic theory and by Type IIA theory; a Calabi-Yau three-fold XIIA in the Type IIA language is further assumed to have a regular K3-fibration. It is well-known that one can assign a modular form Φ to such a vacuum by counting perturbative BPS states in Heterotic theory or collecting Noether-Lefschetz numbers associated with the K3-fibration of XIIA. In this article, we expand the observations and ideas (using gauge threshold correction) in the literature and formulate a modular form Ψ with full generality for the class of vacua above, which can be used along with Φ for the purpose of classification of those vacua. Topological invariants of XIIA can be extracted from Φ and Ψ, and even a pair of diffeomorphic Calabi-Yau’s with different Kähler cones may be distinguished by introducing the notion of “the set of Ψ’s for Higgs cascades/for curve classes”. We illustrated these ideas by simple examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
J.A. Harvey and A. Strominger, The heterotic string is a soliton, Nucl. Phys. B 449 (1995) 535 [Erratum ibid. B 458 (1996) 456] [hep-th/9504047] [INSPIRE].
C. Vafa, Gas of D-branes and Hagedorn density of BPS states, Nucl. Phys. B 463 (1996) 415 [hep-th/9511088] [INSPIRE].
M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222] [INSPIRE].
C. Vafa, Instantons on D-branes, Nucl. Phys. B 463 (1996) 435 [hep-th/9512078] [INSPIRE].
S. Kachru and C. Vafa, Exact results for N = 2 compactifications of heterotic strings, Nucl. Phys. B 450 (1995) 69 [hep-th/9505105] [INSPIRE].
S. Ferrara, J.A. Harvey, A. Strominger and C. Vafa, Second quantized mirror symmetry, Phys. Lett. B 361 (1995) 59 [hep-th/9505162] [INSPIRE].
A. Klemm, W. Lerche and P. Mayr, K3 Fibrations and heterotic type-II string duality, Phys. Lett. B 357 (1995) 313 [hep-th/9506112] [INSPIRE].
C. Vafa and E. Witten, Dual string pairs with N = 1 and N = 2 supersymmetry in four-dimensions, Nucl. Phys. Proc. Suppl. 46 (1996) 225 [hep-th/9507050] [INSPIRE].
P.S. Aspinwall and J. Louis, On the ubiquity of K3 fibrations in string duality, Phys. Lett. B 369 (1996) 233 [hep-th/9510234] [INSPIRE].
J.A. Harvey and G.W. Moore, Algebras, BPS states and strings, Nucl. Phys. B 463 (1996) 315 [hep-th/9510182] [INSPIRE].
D. Maulik and R. Pandharipande, Gromov-Witten theory and Noether-Lefschetz theory, arXiv:0705.1653.
R.E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998) 491 [alg-geom/9609022] [INSPIRE].
T. Banks and L.J. Dixon, Constraints on String Vacua with Space-Time Supersymmetry, Nucl. Phys. B 307 (1988) 93 [INSPIRE].
T. Banks, L.J. Dixon, D. Friedan and E.J. Martinec, Phenomenology and Conformal Field Theory Or Can String Theory Predict the Weak Mixing Angle?, Nucl. Phys. B 299 (1988) 613 [INSPIRE].
J. Lauer, D. Lüst and S. Theisen, Supersymmetric String Theories, Superconformal Algebras and Exceptional Groups, Nucl. Phys. B 309 (1988) 771 [INSPIRE].
A.P. Braun and T. Watari, Heterotic-Type IIA Duality and Degenerations of K3 Surfaces, JHEP 08 (2016) 034 [arXiv:1604.06437] [INSPIRE].
I. Antoniadis, S. Ferrara, E. Gava, K.S. Narain and T.R. Taylor, Perturbative prepotential and monodromies in N = 2 heterotic superstring, Nucl. Phys. B 447 (1995) 35 [hep-th/9504034] [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].
I. Antoniadis, K.S. Narain and T.R. Taylor, Higher genus string corrections to gauge couplings, Phys. Lett. B 267 (1991) 37 [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, K.S. Narain and T.R. Taylor, Superstring threshold corrections to Yukawa couplings, Nucl. Phys. B 407 (1993) 706 [hep-th/9212045] [INSPIRE].
T. Eguchi and A. Taormina, Character formulas for the N = 4 superconformal algebra, Phys. Lett. B 200 (1988) 315.
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].
C.M. Hull, A Geometry for non-geometric string backgrounds, JHEP 10 (2005) 065 [hep-th/0406102] [INSPIRE].
C. Hull, D. Israel and A. Sarti, Non-geometric Calabi-Yau Backgrounds and K3 automorphisms, JHEP 11 (2017) 084 [arXiv:1710.00853] [INSPIRE].
Y. Gautier, C.M. Hull and D. Israël, Heterotic/type-II Duality and Non-Geometric Compactifications, JHEP 10 (2019) 214 [arXiv:1906.02165] [INSPIRE].
V.S. Kulikov, Degenerations of K3 surfaces and Enriques surfaces, Math. USSR Izv. 11 (1977) 957 [Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977) 1008].
R. Borcherds, The Gross-Kohnen-Zagier theorem in higher dimensions, Duke Math. J. 97 (1999) 219 [alg-geom/9710002].
A. Gholampour and A. Sheshmani, Donaldson-Thomas invariants of 2-dimensional sheaves inside threefolds and modular forms, Adv. Math. 326 (2018) 79 [arXiv:1309.0050] [INSPIRE].
V. Bouchard, T. Creutzig, D.-E. Diaconescu, C. Doran, C. Quigley and A. Sheshmani, Vertical D4-D2-D0 Bound States on K3 Fibrations and Modularity, Commun. Math. Phys. 350 (2017) 1069 [arXiv:1601.04030] [INSPIRE].
A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Exact and asymptotic degeneracies of small black holes, JHEP 08 (2005) 021 [hep-th/0502157] [INSPIRE].
M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222] [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
M. Dedushenko and E. Witten, Some Details On The Gopakumar-Vafa and Ooguri-Vafa Formulas, Adv. Theor. Math. Phys. 20 (2016) 1 [arXiv:1411.7108] [INSPIRE].
S.H. Katz, A. Klemm and C. Vafa, M theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999) 1445 [hep-th/9910181] [INSPIRE].
A. Klemm, M. Kreuzer, E. Riegler and E. Scheidegger, Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections, JHEP 05 (2005) 023 [hep-th/0410018] [INSPIRE].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, JHEP 11 (2011) 129 [hep-th/0702146] [INSPIRE].
A. Klemm, D. Maulik, R. Pandharipande and E. Scheidegger, Noether-Lefschetz theory and the Yau-Zaslow conjecture, J. Am. Math. Soc. 23 (2010) 1013 [arXiv:0807.2477].
W. McGraw, The rationality of vector valued modular forms associated with the Weil representation, Math. Ann. 326 (2003) 105.
S. Kachru, A. Klemm, W. Lerche, P. Mayr and C. Vafa, Nonperturbative results on the point particle limit of N = 2 heterotic string compactifications, Nucl. Phys. B 459 (1996) 537 [hep-th/9508155] [INSPIRE].
P.H. Ginsparg, Gauge and Gravitational Couplings in Four-Dimensional String Theories, Phys. Lett. B 197 (1987) 139 [INSPIRE].
B. Haghighat and A. Klemm, Solving the Topological String on K3 Fibrations, JHEP 01 (2010) 009 [arXiv:0908.0336] [INSPIRE].
J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].
I. Antoniadis and H. Partouche, Exact monodromy group of N = 2 heterotic superstring, Nucl. Phys. B 460 (1996) 470 [hep-th/9509009] [INSPIRE].
C.F. Doran, A. Harder, A.Y. Novosltsev and A. Thompson, Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces, Math. Z. 294 (2019) 783 [arXiv:1701.03279].
J.H. Bruinier, Borcherds products on O(2, ℓ) and Chern classes of Heegner divisors, Springer, New York U.S.A. (2000).
S. Hellerman and C. Schmidt-Colinet, Bounds for State Degeneracies in 2D Conformal Field Theory, JHEP 08 (2011) 127 [arXiv:1007.0756] [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. Stieberger, (0, 2) heterotic gauge couplings and their M-theory origin, Nucl. Phys. B 541 (1999) 109 [hep-th/9807124] [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].
V. Kaplunovsky and J. Louis, On Gauge couplings in string theory, Nucl. Phys. B 444 (1995) 191 [hep-th/9502077] [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].
V. Kaplunovsky, J. Louis and S. Theisen, Aspects of duality in N = 2 string vacua, Phys. Lett. B 357 (1995) 71 [hep-th/9506110] [INSPIRE].
C.T. Wall, Classification problems in differential topology V. On certain 6-manifolds, Invent. Math. 1 (1966) 355.
P.S. Aspinwall, B.R. Greene and D.R. Morrison, Multiple mirror manifolds and topology change in string theory, Phys. Lett. B 303 (1993) 249 [hep-th/9301043] [INSPIRE].
P.S. Aspinwall, B.R. Greene and D.R. Morrison, Calabi-Yau moduli space, mirror manifolds and space-time topology change in string theory, Nucl. Phys. B 416 (1994) 414 [hep-th/9309097] [INSPIRE].
D.R. Morrison and C. Vafa, Compactifications of F-theory on Calabi-Yau threefolds. 1, Nucl. Phys. B 473 (1996) 74 [hep-th/9602114] [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].
M. Bershadsky, K.A. Intriligator, S. Kachru, D.R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200].
T. Kawai, String duality and modular forms, Phys. Lett. B 397 (1997) 51 [hep-th/9607078] [INSPIRE].
A. Kanazawa, Study of Calabi-Yau geometry, Ph.D Thesis, the University of British Columbia, Vancouver Canada (2014).
M. Eichler and D. Zagier, The theory of Jacobi forms, Birkhäuser, Boston U.S.A. (1985).
A. Dabholkar, S. Murthy and D. Zagier, Quantum Black Holes, Wall Crossing and Mock Modular Forms, arXiv:1208.4074 [INSPIRE].
R. Borcherds, Automorphic forms on Os+2,2(R) and infinite products, Invent. Math. 120 (1995) 161.
M. Mariño and G.W. Moore, Counting higher genus curves in a Calabi-Yau manifold, Nucl. Phys. B 543 (1999) 592 [hep-th/9808131] [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].
W. Lerche, A.N. Schellekens and N.P. Warner, Lattices and Strings, Phys. Rept. 177 (1989) 1 [INSPIRE].
A. Chattopadhyaya and J.R. David, \( \mathcal{N} \) = 2 heterotic string compactifications on orbifolds of K3 × T2 , JHEP 01 (2017) 037 [arXiv:1611.01893] [INSPIRE].
A. Chattopadhyaya and J.R. David, Gravitational couplings in \( \mathcal{N} \) = 2 string compactifications and Mathieu Moonshine, JHEP 05 (2018) 211 [arXiv:1712.08791] [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
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1911.09934
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Enoki, Y., Watari, T. Modular forms as classification invariants of 4D \( \mathcal{N} \) = 2 Heterotic-IIA dual vacua. J. High Energ. Phys. 2020, 21 (2020). https://doi.org/10.1007/JHEP06(2020)021
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2020)021