Abstract
We study compactification of 6 dimensional (1,0) theories on T 2. We use geometric engineering of these theories via F-theory and employ mirror symmetry technology to solve for the effective 4d \( \mathcal{N}=2 \) geometry for a large number of the (1,0) theories including those associated with conformal matter. Using this we show that for a given 6d theory we can obtain many inequivalent 4d \( \mathcal{N}=2 \) SCFTs. Some of these respect the global symmetries of the 6d theory while others exhibit SL(2, ℤ) duality symmetry inherited from global diffeomorphisms of the T 2. This construction also explains the 6d origin of moduli space of 4d affine ADE quiver theories as flat ADE connections on T 2. Among the resulting 4d \( \mathcal{N}=2 \) CFTs we find theories whose vacuum geometry is captured by an LG theory (as opposed to a curve or a local CY geometry). We obtain arbitrary genus curves of class \( \mathcal{S} \) with punctures from toroidal compactification of (1, 0) SCFTs where the curve of the class \( \mathcal{S} \) theory emerges through mirror symmetry. We also show that toroidal compactification of the little string version of these theories can lead to class \( \mathcal{S} \) theories with no punctures on arbitrary genus Riemann surface.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Witten, Some comments on string dynamics, hep-th/9507121 [INSPIRE].
D. Gaiotto, N=2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
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. 1506 (2015) 017] [arXiv:1312.5746] [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].
L. Bhardwaj, Classification of 6d N=(1,0) gauge theories, arXiv:1502.06594 [INSPIRE].
A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional noncritical strings, hep-th/9607139 [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].
K. Ohmori, H. Shimizu, Y. Tachikawa and K. Yonekura, 6d \( \mathcal{N}=\kern0.6em \left(1,0\right) \) theories on T 2 and class S theories: Part I, JHEP 07 (2015) 014 [arXiv:1503.06217] [INSPIRE].
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
C. Vafa, Mirror symmetry and closed string tachyon condensation, hep-th/0111051 [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4 − D N = 2 gauge theories: 1., Adv. Theor. Math. Phys. 1 (1998) 53 [hep-th/9706110] [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
S. Cecotti and M. Del Zotto, Infinitely many N = 2 SCFT with ADE flavor symmetry, JHEP 01 (2013) 191 [arXiv:1210.2886] [INSPIRE].
S. Cecotti, M. Del Zotto and S. Giacomelli, More on the N = 2 superconformal systems of type D p (G), JHEP 04 (2013) 153 [arXiv:1303.3149] [INSPIRE].
C. Vafa, Topological mirrors and quantum rings, hep-th/9111017 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozaz, G. Lockhart and C. Vafa, M-Strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [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].
B. Haghighat, C. Kozcaz, G. Lockhart and C. Vafa, Orbifolds of M-strings, Phys. Rev. D 89 (2014) 046003 [arXiv:1310.1185] [INSPIRE].
T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].
D. Gaiotto and A. Tomasiello, Holography for (1,0) theories in six dimensions, JHEP 12 (2014) 003 [arXiv:1404.0711] [INSPIRE].
M. Del Zotto, J.J. Heckman, A. Tomasiello and C. Vafa, 6d Conformal Matter, JHEP 02 (2015) 054 [arXiv:1407.6359] [INSPIRE].
A. Hanany and A. Zaffaroni, Branes and six-dimensional supersymmetric theories, Nucl. Phys. B 529 (1998) 180 [hep-th/9712145] [INSPIRE].
I. Brunner and A. Karch, Branes at orbifolds versus Hanany Witten in six-dimensions, JHEP 03 (1998) 003 [hep-th/9712143] [INSPIRE].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
C. Kozcaz, S. Pasquetti and N. Wyllard, A and B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].
C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, Affine sl(N) conformal blocks from N =2 SU(N) gauge theories,JHEP 01 (2011) 045 [arXiv:1008.1412] [INSPIRE].
H. Hayashi, H.-C. Kim and T. Nishinaka, Topological strings and 5d T N partition functions, JHEP 06 (2014) 014 [arXiv:1310.3854] [INSPIRE].
L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi, Non-Lagrangian Theories from Brane Junctions, JHEP 01 (2014) 175 [arXiv:1310.3841] [INSPIRE].
M. Aganagic, N. Haouzi, C. Kozcaz and S. Shakirov, Gauge/Liouville Triality, arXiv:1309.1687 [INSPIRE].
M. Aganagic, N. Haouzi and S. Shakirov, A n -Triality, arXiv:1403.3657 [INSPIRE].
M. Aganagic and S. Shakirov, Gauge/Vortex duality and AGT, arXiv:1412.7132 [INSPIRE].
S.-S. Kim and F. Yagi, 5d E n Seiberg-Witten curve via toric-like diagram, JHEP 06 (2015) 082 [arXiv:1411.7903] [INSPIRE].
C. Vafa, Supersymmetric Partition Functions and a String Theory in 4 Dimensions, arXiv:1209.2425 [INSPIRE].
C. Vafa, The String landscape and the swampland, hep-th/0509212 [INSPIRE].
O. Aharony, M. Berkooz, D. Kutasov and N. Seiberg, Linear dilatons, NS five-branes and holography, JHEP 10 (1998) 004 [hep-th/9808149] [INSPIRE].
O. Aharony, A Brief review of ’little string theories’, Class. Quant. Grav. 17 (2000) 929 [hep-th/9911147] [INSPIRE].
J. Kim, S. Kim and K. Lee, Little strings and T-duality, arXiv:1503.07277 [INSPIRE].
A. Sen, F theory and orientifolds, Nucl. Phys. B 475 (1996) 562 [hep-th/9605150] [INSPIRE].
K. Dasgupta and S. Mukhi, F theory at constant coupling, Phys. Lett. B 385 (1996) 125 [hep-th/9606044] [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].
J. Erler, Anomaly cancellation in six-dimensions, J. Math. Phys. 35 (1994) 1819 [hep-th/9304104] [INSPIRE].
V. Sadov, Generalized Green-Schwarz mechanism in F-theory, Phys. Lett. B 388 (1996) 45 [hep-th/9606008] [INSPIRE].
M. Bershadsky and C. Vafa, Global anomalies and geometric engineering of critical theories in six-dimensions, hep-th/9703167 [INSPIRE].
J.D. Blum and K.A. Intriligator, Consistency conditions for branes at orbifold singularities, Nucl. Phys. B 506 (1997) 223 [hep-th/9705030] [INSPIRE].
F. Riccioni and A. Sagnotti, Consistent and covariant anomalies in six-dimensional supergravity, Phys. Lett. B 436 (1998) 298 [hep-th/9806129] [INSPIRE].
A. Grassi and D.R. Morrison, Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds, Commun. Num. Theor. Phys. 6 (2012) 51 [arXiv:1109.0042] [INSPIRE].
E. Witten, Phase transitions in M-theory and F-theory, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150] [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].
J.J. Heckman, More on the Matter of 6D SCFTs, Phys. Lett. B 747 (2015) 73 [arXiv:1408.0006] [INSPIRE].
M. Bershadsky and A. Johansen, Colliding singularities in F-theory and phase transitions, Nucl. Phys. B 489 (1997) 122 [hep-th/9610111] [INSPIRE].
P.S. Aspinwall and D.R. Morrison, Point - like instantons on K3 orbifolds, Nucl. Phys. B 503 (1997) 533 [hep-th/9705104] [INSPIRE].
E. Witten, Small instantons in string theory, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030] [INSPIRE].
O.J. Ganor and A. Hanany, Small E 8 instantons and tensionless noncritical strings, Nucl. Phys. B 474 (1996) 122 [hep-th/9602120] [INSPIRE].
N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
D.R. Morrison and N. Seiberg, Extremal transitions and five-dimensional supersymmetric field theories, Nucl. Phys. B 483 (1997) 229 [hep-th/9609070] [INSPIRE].
M.R. Douglas, S.H. Katz and C. Vafa, Small instantons, Del Pezzo surfaces and type-I-prime theory, Nucl. Phys. B 497 (1997) 155 [hep-th/9609071] [INSPIRE].
J. Polchinski and E. Witten, Evidence for heterotic - type-I string duality, Nucl. Phys. B 460 (1996) 525 [hep-th/9510169] [INSPIRE].
S. Cecotti and C. Vafa, Classification of complete N = 2 supersymmetric theories in 4 dimensions, Surveys in differential geometry 18 (2013) [arXiv:1103.5832] [INSPIRE].
S. Cecotti, Categorical Tinkertoys for N = 2 Gauge Theories, Int. J. Mod. Phys. A 28 (2013) 1330006 [arXiv:1203.6734] [INSPIRE].
M. Buican, S. Giacomelli, T. Nishinaka and C. Papageorgakis, Argyres-Douglas Theories and S-duality, JHEP 02 (2015) 185 [arXiv:1411.6026] [INSPIRE].
A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Selfdual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [INSPIRE].
B.R. Greene, C. Vafa and N.P. Warner, Calabi-Yau Manifolds and Renormalization Group Flows, Nucl. Phys. B 324 (1989) 371 [INSPIRE].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [Erratum ibid. B 608 (2001) 477-478] [hep-th/9906070] [INSPIRE].
A.D. Shapere and C. Vafa, BPS structure of Argyres-Douglas superconformal theories, hep-th/9910182 [INSPIRE].
H. Ooguri and C. Vafa, Two-dimensional black hole and singularities of CY manifolds, Nucl. Phys. B 463 (1996) 55 [hep-th/9511164] [INSPIRE].
E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [INSPIRE].
B. Haghighat, A. Klemm, G. Lockhart and C. Vafa, Strings of Minimal 6d SCFTs, Fortsch. Phys. 63 (2015) 294 [arXiv:1412.3152] [INSPIRE].
T. Eguchi, K. Hori, K. Ito and S.-K. Yang, Study of N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 471 (1996) 430 [hep-th/9603002] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the E 6 theory, JHEP 09 (2015) 007 [arXiv:1403.4604] [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d N=(2,0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [arXiv:1203.2930] [INSPIRE].
M. Del Zotto, J.J. Heckman, D.S. Park and T. Rudelius, On the Defect Group of a 6D SCFT, arXiv:1503.04806 [INSPIRE].
D. Gaiotto and S.S. Razamat, \( \mathcal{N}=1 \) theories of class \( {\mathcal{S}}_k \) , JHEP 07 (2015) 073 [arXiv:1503.05159] [INSPIRE].
S. Franco, H. Hayashi and A. Uranga, Charting Class \( {\mathcal{S}}_k \) Territory, Phys. Rev. D 92 (2015) 045004 [arXiv:1504.05988] [INSPIRE].
F. Apruzzi, M. Fazzi, A. Passias, A. Rota and A. Tomasiello, Six-Dimensional Superconformal Theories and their Compactifications from Type IIA Supergravity, Phys. Rev. Lett. 115 (2015) 061601 [arXiv:1502.06616] [INSPIRE].
F. Apruzzi, M. Fazzi, A. Passias and A. Tomasiello, Supersymmetric AdS 5 solutions of massive IIA supergravity, JHEP 06 (2015) 195 [arXiv:1502.06620] [INSPIRE].
P. Karndumri, RG flows from (1,0) 6D SCFTs to N = 1 SCFTs in four and three dimensions, JHEP 06 (2015) 027 [arXiv:1503.04997] [INSPIRE].
P.C. Argyres, K. Maruyoshi and Y. Tachikawa, Quantum Higgs branches of isolated N = 2 superconformal field theories, JHEP 10 (2012) 054 [arXiv:1206.4700] [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: 1504.08348
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
Del Zotto, M., Vafa, C. & Xie, D. Geometric engineering, mirror symmetry and \( 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} \) . J. High Energ. Phys. 2015, 123 (2015). https://doi.org/10.1007/JHEP11(2015)123
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2015)123