Abstract
We provide a uniform solution to 4d \( \mathcal{N} = 2 \) gauge theory with a single gauge group G = A, D, E when the one-loop contribution to the beta function from any irreducible component R of the hypermultiplets is less than or equal to half of that of the adjoint representation. The solution is given by a non-compact Calabi-Yau geometry, whose defining equation is built from explicitly known polynomials W G and X R , associated respectively to the gauge group G and each irreducible component R. We provide many pieces of supporting evidence, for example by analyzing the system from the point of view of the 6d \( \mathcal{N} = \left( {2,0} \right) \) theory compactified on a sphere.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Seiberg and E. Witten, Monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087] [SPIRES].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [SPIRES].
A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [SPIRES].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [SPIRES].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [SPIRES].
E. D’Hoker and D.H. Phong, Seiberg-Witten theory and integrable systems, hep-th/9903068 [SPIRES].
H. Itoyama and A. Morozov, Integrability and Seiberg-Witten theory: curves and periods, Nucl. Phys. B 477 (1996) 855 [hep-th/9511126] [SPIRES].
R.Y. Donagi, Seiberg-Witten integrable systems, alg-geom/9705010 [SPIRES].
E. D’Hoker and D.H. Phong, Lectures on supersymmetric Yang-Mills theory and integrable systems, hep-th/9912271 [SPIRES].
S. Kachru and C. Vafa, Exact results for N = 2 compactifications of heterotic strings, Nucl. Phys. B 450 (1995) 69 [hep-th/9505105] [SPIRES].
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] [SPIRES].
M. Billó et al., Special geometry of Calabi-Yau compactifications near a rigid limit, Fortsch. Phys. 47 (1999) 133 [hep-th/9801140] [SPIRES].
A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Self-dual strings and N = 2 supersymmetric field theory, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034] [SPIRES].
S.H. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239] [SPIRES].
W. Lerche, Introduction to Seiberg-Witten theory and its stringy origin, Nucl. Phys. Proc. Suppl. 55B (1997) 83 [hep-th/9611190] [SPIRES].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [SPIRES].
A. Giveon and D. Kutasov, Brane dynamics and gauge theory, Rev. Mod. Phys. 71 (1999) 983 [hep-th/9802067] [SPIRES].
D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [SPIRES].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [SPIRES].
A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, Simple singularities and N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 344 (1995) 169 [hep-th/9411048] [SPIRES].
P.C. Argyres and A.E. Faraggi, The vacuum structure and spectrum of N = 2 supersymmetric SU(n) gauge theory, Phys. Rev. Lett. 74 (1995) 3931 [hep-th/9411057] [SPIRES].
A. Hanany and Y. Oz, On the quantum moduli space of vacua of N = 2 supersymmetric SU(N c ) gauge theories, Nucl. Phys. B 452 (1995) 283 [hep-th/9505075] [SPIRES].
P.C. Argyres, M.R. Plesser and A.D. Shapere, The Coulomb phase of N = 2 supersymmetric QCD, Phys. Rev. Lett. 75 (1995) 1699 [hep-th/9505100] [SPIRES].
A. Brandhuber and K. Landsteiner, On the monodromies of N = 2 supersymmetric Yang-Mills theory with gauge group SO(2n), Phys. Lett. B 358 (1995) 73 [hep-th/9507008] [SPIRES].
P.C. Argyres and A.D. Shapere, The vacuum structure of N = 2 SuperQCD with classical gauge groups, Nucl. Phys. B 461 (1996) 437 [hep-th/9509175] [SPIRES].
A. Hanany, On the quantum moduli space of N = 2 supersymmetric gauge theories, Nucl. Phys. B 466 (1996) 85 [hep-th/9509176] [SPIRES].
J.H. Brodie, Exact solutions of exceptional gauge theories from toric geometry, Nucl. Phys. B 506 (1997) 183 [hep-th/9705068] [SPIRES].
M. Aganagic and M. Gremm, Exact solutions for some N = 2 supersymmetric SO(N) gauge theories with vectors and spinors, Nucl. Phys. B 524 (1998) 207 [hep-th/9712011] [SPIRES].
S. Terashima and S.-K. Yang, Exceptional Seiberg-Witten geometry with massive fundamental matters, Phys. Lett. B 430 (1998) 102 [hep-th/9803014] [SPIRES].
S. Terashima and S.-K. Yang, Seiberg-Witten geometry with various matter contents, Nucl. Phys. B 537 (1999) 344 [hep-th/9808022] [SPIRES].
J. Hashiba and S. Terashima, Geometry and N = 2 exceptional gauge theories, JHEP 09 (1999) 020 [hep-th/9909032] [SPIRES].
P.C. Argyres, R. Maimon and S. Pelland, The M-theory lift of two O6-planes and four D6 branes, JHEP 05 (2002) 008 [hep-th/0204127] [SPIRES].
K. Landsteiner and E. Lopez, New curves from branes, Nucl. Phys. B 516 (1998) 273 [hep-th/9708118] [SPIRES].
E. D’Hoker and D.H. Phong, Spectral curves for super-Yang-Mills with adjoint hypermultiplet for general Lie algebras, Nucl. Phys. B 534 (1998) 697 [hep-th/9804126] [SPIRES].
S. Katz and D.R. Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Alg. Geom. 1 (1992) 449 [alg-geom/9202002].
T. Shioda, Construction of elliptic curves with high rank via the invariants of the W eyl groups, J. Math. Soc. Japan 43 (1991) 673.
M. Noguchi, S. Terashima and S.-K. Yang, N = 2 superconformal field theory with ADE global symmetry on a D3-brane probe, Nucl. Phys. B 556 (1999) 115 [hep-th/9903215] [SPIRES].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [SPIRES].
O. Chacaltana and J. Distler, Tinkertoys for the D N series, arXiv:1106.5410 [SPIRES].
W. Lerche and N.P. Warner, Exceptional SW geometry from ALE fibrations, Phys. Lett. B 423 (1998) 79 [hep-th/9608183] [SPIRES].
T.J. Hollowood, Strong coupling N = 2 gauge theory with arbitrary gauge group, Adv. Theor. Math. Phys. 2 (1998) 335 [hep-th/9710073] [SPIRES].
S.H. Katz and C. Vafa, Matter from geometry, Nucl. Phys. B 497 (1997) 146 [hep-th/9606086] [SPIRES].
D.R. Morrison and W. Taylor, Matter and singularities, arXiv:1106.3563 [SPIRES].
H. Rhedin, Seiberg-Witten theory for the asymptotic free rank three tensors of SU(N), hep-th/0010233 [SPIRES].
F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [SPIRES].
Y. Tachikawa, N = 2 S-duality via outer-automorphism twists, J. Phys. A 44 (2011) 182001 [arXiv:1009.0339] [SPIRES].
S. Gukov and E. Witten, Rigid surface operators, arXiv:0804.1561 [SPIRES].
A. Moreau, On the dimension of the sheets of a reductive Lie algebras, Journal of Lie Theory 18 (2008) 671 [arXiv:0711.2735].
D. Nanopoulos and D. Xie, N = 2 SU quiver with USP ends or SU ends with antisymmetric matter, JHEP 08 (2009) 108 [arXiv:0907.1651] [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Professor Sung-Kil Yang
ArXiv ePrint: 1108.2315
Rights and permissions
About this article
Cite this article
Tachikawa, Y., Terashima, S. Seiberg-Witten geometries revisited. J. High Energ. Phys. 2011, 10 (2011). https://doi.org/10.1007/JHEP09(2011)010
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2011)010