Abstract
We show how recent progress in computing elliptic genera of strings in six dimensions can be used to obtain expressions for elliptic genera of strings in five-dimensional field theories which have a six-dimensional parent. We further connect our results to recent mathematical results about sheaf counting on ruled surfaces.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
A. Boyarsky, J.A. Harvey and O. Ruchayskiy, A toy model of the M5-brane: Anomalies of monopole strings in five dimensions, Annals Phys. 301 (2002) 1 [hep-th/0203154] [INSPIRE].
N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, M5-Branes, D4-branes and Quantum 5D super-Yang-Mills, JHEP 01 (2011) 083 [arXiv:1012.2882] [INSPIRE].
B. Haghighat and S. Vandoren, Five-dimensional gauge theory and compactification on a torus, JHEP 09 (2011) 060 [arXiv:1107.2847] [INSPIRE].
B. Haghighat, J. Manschot and S. Vandoren, A 5d/2d/4d correspondence, JHEP 03 (2013) 157 [arXiv:1211.0513] [INSPIRE].
J.A. Minahan, D. Nemeschansky, C. Vafa and N.P. Warner, E strings and N = 4 topological Yang-Mills theories, Nucl. Phys. B 527 (1998) 581 [hep-th/9802168] [INSPIRE].
J. Manschot, The Betti numbers of the moduli space of stable sheaves of rank 3 on P 2, Lett. Math. Phys. 98 (2011) 65 [arXiv:1009.1775] [INSPIRE].
J. Manschot, BPS invariants of \( \mathcal{N}=4 \) gauge theory on Hirzebruch surfaces, Commun. Num. Theor. Phys. 6 (2012) 497 [arXiv:1103.0012] [INSPIRE].
J. Manschot, BPS invariants of semi-stable sheaves on rational surfaces, Lett. Math. Phys. 103 (2013) 895 [arXiv:1109.4861] [INSPIRE].
J. Manschot, Sheaves on P2 and generalized Appell functions, arXiv:1407.7785 [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, 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].
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].
B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of Minimal 6d SCFTs, Fortsch. Phys. 63 (2015) 294 [arXiv:1412.3152] [INSPIRE].
S. Mozgovoy, Invariants of moduli spaces of stable sheaves on ruled surfaces, arXiv:1302.4134 [INSPIRE].
H.-C. Kim, S. Kim, E. Koh, K. Lee and S. Lee, On instantons as Kaluza-Klein modes of M5-branes, JHEP 12 (2011) 031 [arXiv:1110.2175] [INSPIRE].
E. Witten, Nonperturbative superpotentials in string theory, Nucl. Phys. B 474 (1996) 343 [hep-th/9604030] [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].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
D. Joyce, Configurations in Abelian categories. IV. Changing stability conditions, math/0410268 [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].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
M. Del Zotto and A. Sen, About the Absence of Exotics and the Coulomb Branch Formula, arXiv:1409.5442 [INSPIRE].
S. Hosono, M.-H. Saito and A. Takahashi, Relative Lefschetz action and BPS state counting, math/0105148 [INSPIRE].
C. Vafa and E. Witten, A strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
A. Klemm, J. Manschot and T. Wotschke, Quantum geometry of elliptic Calabi-Yau manifolds, arXiv:1205.1795 [INSPIRE].
W.-y. Chuang, D.-E. Diaconescu, J. Manschot, G.W. Moore and Y. Soibelman, Geometric engineering of (framed) BPS states, Adv. Theor. Math. Phys. 18 (2014) 1063 [arXiv:1301.3065] [INSPIRE].
C. Callias, Index Theorems on Open Spaces, Commun. Math. Phys. 62 (1978) 213 [INSPIRE].
E. Diaconescu and G.W. Moore, Crossing the wall: Branes versus bundles, Adv. Theor. Math. Phys. 14 (2010) 1621 [arXiv:0706.3193] [INSPIRE].
B. Haghighat, A. Iqbal and A. Kahn, work in progress.
S. Hohenegger, A. Iqbal and S.-J. Rey, M-strings, monopole strings and modular forms, Phys. Rev. D 92 (2015) 066005 [arXiv:1503.06983] [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
ArXiv ePrint: 1502.06645
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Haghighat, B. From strings in 6d to strings in 5d. J. High Energ. Phys. 2016, 62 (2016). https://doi.org/10.1007/JHEP01(2016)062
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)062