Abstract
We apply the modular approach to computing the topological string partition function on non-compact elliptically fibered Calabi-Yau 3-folds with higher Kodaira singularities in the fiber. The approach consists in making an ansatz for the partition function at given base degree, exact in all fiber classes to arbitrary order and to all genus, in terms of a rational function of weak Jacobi forms. Our results yield, at given base degree, the elliptic genus of the corresponding non-critical 6d string, and thus the associated BPS invariants of the 6d theory. The required elliptic indices are determined from the chiral anomaly 4-form of the 2d worldsheet theories, or the 8-form of the corresponding 6d theories, and completely fix the holomorphic anomaly equation constraining the partition function. We introduce subrings of the known rings of Weyl invariant Jacobi forms which are adapted to the additional symmetries of the partition function, making its computation feasible to low base wrapping number. In contradistinction to the case of simpler singularities, generic vanishing conditions on BPS numbers are no longer sufficient to fix the modular ansatz at arbitrary base wrapping degree. We show that to low degree, imposing exact vanishing conditions does suffice, and conjecture this to be the case generally.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140] [INSPIRE].
S. Hosono, M.H. Saito and A. Takahashi, Holomorphic anomaly equation and BPS state counting of rational elliptic surface, Adv. Theor. Math. Phys. 3 (1999) 177 [hep-th/9901151] [INSPIRE].
M.-X. Huang and A. Klemm, Direct integration for general Ω backgrounds, Adv. Theor. Math. Phys. 16 (2012) 805 [arXiv:1009.1126] [INSPIRE].
A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].
M. Alim and E. Scheidegger, Topological strings on elliptic fibrations, Commun. Num. Theor. Phys. 08 (2014) 729 [arXiv:1205.1784] [INSPIRE].
M.-X. Huang, A. Klemm and M. Poretschkin, Refined stable pair invariants for E-, M- and [p, q]-strings, JHEP 11 (2013) 112 [arXiv:1308.0619] [INSPIRE].
M.-x. Huang, S. Katz and A. Klemm, Topological string on elliptic CY 3-folds and the ring of Jacobi forms, JHEP 10 (2015) 125 [arXiv:1501.04891] [INSPIRE].
S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [INSPIRE].
M. Reid, Young person’s guide to canonical singularities, in Algebraic geometry. Bowdoin 1985, J. Bloch, American Mathematical Society, Providence U.S.A. (1987).
S.S.-T. Yau and Y. Yu, Gorenstein quotient singularities in dimension three, Mem. Amer. Math. Soc. 105 (1993) 88.
G. Lockhart and C. Vafa, Superconformal partition functions and non-perturbative topological strings, arXiv:1210.5909 [INSPIRE].
B. Haghighat et al., M-strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].
B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].
S. Hohenegger and A. Iqbal, M-strings, elliptic genera and \( \mathcal{N} \) = 4 string amplitudes, Fortsch. Phys. 62 (2014) 155 [arXiv:1310.1325] [INSPIRE].
W. Nahm, Supersymmetries and their representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].
K. Kodaira, On compact analytic surfaces. II, Ann. Math. 77 (1963) 563.
K. Kodaira, On compact analytic surfaces. III, Ann. Math. 78 (1963) 1.
J.J. Heckman, D.R. Morrison and C. Vafa, On the classification of 6D SCFTs anD generalized ADE orbifolds, JHEP 05 (2014) 028 [Erratum ibid. 06 (2015) 017] [arXiv:1312.5746] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6D conformal matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Atomic classification of 6D SCFTs, Fortsch. Phys. 63 (2015) 468 [arXiv:1502.05405] [INSPIRE].
F. Apruzzi, M. Fazzi, D. Rosa and A. Tomasiello, All AdS 7 solutions of type-II supergravity, JHEP 04 (2014) 064 [arXiv:1309.2949] [INSPIRE].
D. Gaiotto and A. Tomasiello, Holography for (1, 0) theories in six dimensions, JHEP 12 (2014) 003 [arXiv:1404.0711] [INSPIRE].
L. Bhardwaj, Classification of 6d \( \mathcal{N} \) = (1, 0) gauge theories, JHEP 11 (2015) 002 [arXiv:1502.06594] [INSPIRE].
L. Bhardwaj et al., F-theory and the classification of little strings, Phys. Rev. D 93 (2016) 086002 [arXiv:1511.05565] [INSPIRE].
F. Apruzzi and M. Fazzi, AdS 7 /CFT 6 with orientifolds, JHEP 01 (2018) 124 [arXiv:1712.03235] [INSPIRE].
B. Haghighat, G. Lockhart and C. Vafa, Fusing E-strings to heterotic strings: E + E → H, Phys. Rev. D 90 (2014) 126012 [arXiv:1406.0850] [INSPIRE].
J. Kim, S. Kim, K. Lee, J. Park and C. Vafa, Elliptic genus of E-strings, JHEP 09 (2017) 098 [arXiv:1411.2324] [INSPIRE].
B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of minimal 6d SCFTs, Fortsch. Phys. 63 (2015) 294 [arXiv:1412.3152] [INSPIRE].
A. Iqbal and K. Shabbir, Elliptic CY3-folds and non-perturbative modular transformation, Eur. Phys. J. C 76 (2016) 148 [arXiv:1510.03332] [INSPIRE].
S.-S. Kim, M. Taki and F. Yagi, Tao probing the end of the world, PTEP 2015 (2015) 083B02 [arXiv:1504.03672] [INSPIRE].
A. Gadde et al., 6d string chains, JHEP 02 (2018) 143 [arXiv:1504.04614] [INSPIRE].
S. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings from F-theory and flop transitions, JHEP 07 (2017) 112 [arXiv:1610.07916] [INSPIRE].
H.-C. Kim, S. Kim and J. Park, 6d strings from new chiral gauge theories, arXiv:1608.03919 [INSPIRE].
H. Hayashi and K. Ohmori, 5d/6d DE instantons from trivalent gluing of web diagrams, JHEP 06 (2017) 078 [arXiv:1702.07263] [INSPIRE].
B. Bastian, S. Hohenegger, A. Iqbal and S.-J. Rey, Dual little strings and their partition functions, arXiv:1710.02455 [INSPIRE].
B. Haghighat, W. Yan and S.-T. Yau, ADE string chains and mirror symmetry, JHEP 01 (2018) 043 [arXiv:1705.05199] [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].
J. Choi, S. Katz and A. Klemm, The refined BPS index from stable pair invariants, Commun. Math. Phys. 328 (2014) 903 [arXiv:1210.4403] [INSPIRE].
N.A. Nekrasov and A. Okounkov, Membranes and sheaves, Phys. Rev. A 91 (2015) 012106 [arXiv:1401.2323].
A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, hep-th/9607139 [INSPIRE].
J. Gu, M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, Refined BPS invariants of 6d SCFTs from anomalies and modularity, JHEP 05 (2017) 130 [arXiv:1701.00764] [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].
K. Wirthmüller, Root systems and Jacobi forms, Compositio Math. 82 (1992) 293.
M. Bertola, Jacobi groups, Jacobi forms and their applications, in Isomonodromic deformations and applications in physics, J.P Harnad ed., CRM Proc. Lecture Notes volume 31, American Mathematical Society, Providence U.S.A. (2002).
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, Anomaly polynomial of general 6d SCFTs, PTEP 2014 (2014) 103B07 [arXiv:1408.5572] [INSPIRE].
K. Intriligator, 6d, \( \mathcal{N} \) = (1, 0) Coulomb branch anomaly matching, JHEP 10 (2014) 162 [arXiv:1408.6745] [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Anomalies, renormalization group flows and the a-theorem in six-dimensional (1, 0) theories, JHEP 10 (2016) 080 [arXiv:1506.03807] [INSPIRE].
H. Shimizu and Y. Tachikawa, Anomaly of strings of 6d \( \mathcal{N} \) = (1, 0) theories, JHEP 11 (2016) 165 [arXiv:1608.05894] [INSPIRE].
N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir energy and the anomaly polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].
M. Del Zotto and G. Lockhart, On Exceptional Instanton Strings, JHEP 09 (2017) 081 [arXiv:1609.00310] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
A.S. Losev, A. Marshakov and N.A. Nekrasov, Small instantons, little strings and free fermions, hep-th/0302191 [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [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].
I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Worldsheet realization of the refined topological string, Nucl. Phys. B 875 (2013) 101 [arXiv:1302.6993] [INSPIRE].
I. Antoniadis, I. Florakis, S. Hohenegger, K.S. Narain and A. Zein Assi, Non-perturbative Nekrasov partition function from string theory, Nucl. Phys. B 880 (2014) 87 [arXiv:1309.6688] [INSPIRE].
M. Bershadsky et al., Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200] [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d \( \mathcal{N} \) = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
N. Benjamin, E. Dyer, A.L. Fitzpatrick and S. Kachru, Universal bounds on charged states in 2d CFT and 3d gravity, JHEP 08 (2016) 041 [arXiv:1603.09745] [INSPIRE].
D.R. Morrison and W. Taylor, Classifying bases for 6D F-theory models, Central Eur. J. Phys. 10 (2012) 1072 [arXiv:1201.1943] [INSPIRE].
M. Del Zotto, J.J. Heckman and D.R. Morrison, 6D SCFTs and phases of 5D theories, JHEP 09 (2017) 147 [arXiv:1703.02981] [INSPIRE].
M. Esole, R. Jagadeesan and M.J. Kang, The geometry of G 2 , Spin(7) and Spin(8)-models, arXiv:1709.04913 [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and \( 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} \), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403 [hep-th/9602022] [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].
V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301 [hep-th/9308122] [INSPIRE].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys. B 433 (1995) 501 [hep-th/9406055] [INSPIRE].
S. Hosono, A. Klemm and S. Theisen, Lectures on mirror symmetry, Lect. Notes Phys. 436 (1994) 235 [hep-th/9403096] [INSPIRE].
M. Bertola, Jacobi groups, Jacobi forms and their applications, in Isomonodromic deformations and applications in physics, J.P. Harnad ed., CRM Proc. Lecture Notes volume 31, American Mathematical Society, Providence U.S.A (2002).
V. Gritsenko, Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms, math/9906190 [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, Instanton counting and Chern-Simons theory, Adv. Theor. Math. Phys. 7 (2003) 457 [hep-th/0212279] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, SU(N ) geometries and topological string amplitudes, Adv. Theor. Math. Phys. 10 (2006) 1 [hep-th/0306032] [INSPIRE].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition function, math/0505553 [INSPIRE].
C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of instantons and W-algebras, JHEP 03 (2012) 045 [arXiv:1111.5624] [INSPIRE].
N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].
D. Gaiotto and S.S. Razamat, Exceptional indices, JHEP 05 (2012) 145 [arXiv:1203.5517] [INSPIRE].
A. Hanany, N. Mekareeya and S.S. Razamat, Hilbert series for moduli spaces of two instantons, JHEP 01 (2013) 070 [arXiv:1205.4741] [INSPIRE].
R. Dijkgraaf, G.W. Moore, E.P. Verlinde and H.L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197 [hep-th/9608096] [INSPIRE].
C.F. Cota, A. Klemm and T. Schimannek, Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds, JHEP 01 (2018) 086 [arXiv:1709.02820] [INSPIRE].
G. Oberdieck and A. Pixton, Gromov-Witten theory of elliptic fibrations: Jacobi forms and holomorphic anomaly equations, arXiv:1709.01481 [INSPIRE].
O.J. Ganor, D.R. Morrison and N. Seiberg, Branes, Calabi-Yau spaces and toroidal compactification of the N = 1 six-dimensional E 8 theory, Nucl. Phys. B 487 (1997) 93 [hep-th/9610251] [INSPIRE].
M. Eichler and D. Zagier, The theory of Jacobi forms, Progress in Mathematics volume 55, 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].
K. Sakai, E n Jacobi forms and Seiberg-Witten curves, arXiv:1706.04619 [INSPIRE].
K. Sakai, Topological string amplitudes for the local \( \frac{1}{2} \) K3 surface, PTEP 2017 (2017) 033B09 [arXiv:1111.3967] [INSPIRE].
N. Bourbaki, Groupes et algèbres de Lie. Chapitres 4, 5 et 6. Éléments de mathématique, Masson, Paris France (1981).
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
ArXiv ePrint: 1712.07017
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Del Zotto, M., Gu, J., Huang, Mx. et al. Topological strings on singular elliptic Calabi-Yau 3-folds and minimal 6d SCFTs. J. High Energ. Phys. 2018, 156 (2018). https://doi.org/10.1007/JHEP03(2018)156
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2018)156