Abstract
We derive a large class of codimension-two defects of 4d \( \mathcal{N}=4 \) Super Yang-Mills (SYM) theory from the (2, 0) little string. The origin of the little string is type IIB theory compactified on an ADE singularity. The defects are D-branes wrapping the 2-cycles of the singularity. We use this construction to make contact with the description of SYM defects due to Gukov and Witten [1]. Furthermore, we provide a geometric perspective on the nilpotent orbit classification of codimension-two defects, and the connection to ADE-type Toda CFT. The only data needed to specify the defects is a set of weights of the algebra obeying certain constraints, which we give explicitly. We highlight the differences between the defect classification in the little string theory and its (2, 0) CFT limit.
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References
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
D. Gaiotto, \( \mathcal{N}=2 \) dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
E. Witten, Geometric Langlands From Six Dimensions, arXiv:0905.2720 [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].
A.K. Balasubramanian, Four dimensional \( \mathcal{N}=2 \) theories from six dimensions, Ph.D. Thesis, Texas University, U.S.A. (2014).
M. Aganagic and N. Haouzi, ADE Little String Theory on a Riemann Surface (and Triality), arXiv:1506.04183 [INSPIRE].
C. Vafa, Geometric origin of Montonen-Olive duality, Adv. Theor. Math. Phys. 1 (1998) 158 [hep-th/9707131] [INSPIRE].
T.A. Springer, The unipotent variety of a semi-simple group, in Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), pp. 373-391, Oxford University Press, London, (1969).
R. Steinberg, On the desingularization of the unipotent variety, Invent. Math. 36 (1976) 209.
B. Fu, Symplectic resolutions for nilpotent orbits, Invent. Math. 151 (2003) 167.
O. Bergman and G. Zafrir, Lifting 4d dualities to 5d, JHEP 04 (2015) 141 [arXiv:1410.2806] [INSPIRE].
H. Hayashi, Y. Tachikawa and K. Yonekura, Mass-deformed T N as a linear quiver, JHEP 02 (2015) 089 [arXiv:1410.6868] [INSPIRE].
Y. Tachikawa, Six-dimensional D(N) theory and four-dimensional SO-USp quivers, JHEP 07 (2009) 067 [arXiv:0905.4074] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
S. Kanno, Y. Matsuo, S. Shiba and Y. Tachikawa, N=2 gauge theories and degenerate fields of Toda theory, Phys. Rev. D 81 (2010) 046004 [arXiv:0911.4787] [INSPIRE].
N.J. Hitchin, The selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].
N. Seiberg, New theories in six-dimensions and matrix description of M-theory on T 5 and T 5/ℤ 2, Phys. Lett. B 408 (1997) 98 [hep-th/9705221] [INSPIRE].
E. Witten, Some comments on string dynamics, in the proceedings of Future perspectives in string theory, Strings’95, Los Angeles, U.S.A., March 13-18, 1995, hep-th/9507121 [INSPIRE].
A. Losev, G.W. Moore and S.L. Shatashvili, M & m’s, Nucl. Phys. B 522 (1998) 105 [hep-th/9707250] [INSPIRE].
O. Aharony, A brief review of ‘little string theories’, Class. Quant. Grav. 17 (2000) 929 [hep-th/9911147] [INSPIRE].
M. Reid, McKay correspondence, alg-geom/9702016 [INSPIRE].
M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
D. Gaiotto and E. Witten, S-duality of Boundary Conditions In N = 4 Super Yang-Mills Theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
A. Hanany and N. Mekareeya, Complete Intersection Moduli Spaces in N = 4 Gauge Theories in Three Dimensions, JHEP 01 (2012) 079 [arXiv:1110.6203] [INSPIRE].
S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, T σ ρ (G) theories and their Hilbert series, JHEP 01 (2015) 150 [arXiv:1410.1548] [INSPIRE].
A. Hanany and R. Kalveks, Quiver Theories for Moduli Spaces of Classical Group Nilpotent Orbits, JHEP 06 (2016) 130 [arXiv:1601.04020] [INSPIRE].
A. Malcev, On the representation of an algebra as a direct sum of the radical and a semi-simple subalgebra, C.R. (Doklady) Acad. Sci. URSS (N.S.) 36 (1942) 42.
F. Cachazo, S. Katz and C. Vafa, Geometric transitions and N = 1 quiver theories, hep-th/0108120 [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].
D. Nanopoulos and D. Xie, Hitchin Equation, Singularity and N = 2 Superconformal Field Theories, JHEP 03 (2010) 043 [arXiv:0911.1990] [INSPIRE].
D.H. Collingwood and W.M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, (1993).
N. Spaltenstein, Classes unipotentes et sous-groupes de Borel, vol. 946 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, (1982).
P. Bala and R.W. Carter, Classes of unipotent elements in simple algebraic groups. I, Math. Proc. Cambridge Philos. Soc. 79 (1976) 401.
P. Bala and R.W. Carter, Classes of unipotent elements in simple algebraic groups. II, Math. Proc. Cambridge Philos. Soc. 80 (1976) 1.
N. Haouzi and C. Schmid, Little String Defects and Bala-Carter Theory, arXiv:1612.02008 [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for the D N series, JHEP 02 (2013) 110 [arXiv:1106.5410] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
O. Chacaltana, J. Distler and A. Trimm, Tinkertoys for the E 6 theory, JHEP 09 (2015) 007 [arXiv:1403.4604] [INSPIRE].
E. Frenkel and N. Reshetikhin, Deformations of \( \mathcal{W} \) -algebras associated to simple Lie algebras, Comm. Math. Phys. 197 (1998) 1 [q-alg/9708006].
T. Kimura and V. Pestun, Quiver W-algebras, arXiv:1512.08533 [INSPIRE].
N. Drukker, D. Gaiotto and J. Gomis, The Virtue of Defects in 4D Gauge Theories and 2D CFTs, JHEP 06 (2011) 025 [arXiv:1003.1112] [INSPIRE].
P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept. 223 (1993) 183 [hep-th/9210010] [INSPIRE].
P. Bouwknegt and K. Pilch, On deformed W algebras and quantum affine algebras, Adv. Theor. Math. Phys. 2 (1998) 357 [math/9801112] [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].
K. Thielemans, A Mathematica package for computing operator product expansions, Int. J. Mod. Phys. C 2 (1991) 787 [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
S. Gukov and E. Witten, Rigid Surface Operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [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].
A. Hanany and M. Sperling, Coulomb branches for rank 2 gauge groups in 3d \( \mathcal{N}=4 \) gauge theories, JHEP 08 (2016) 016 [arXiv:1605.00010] [INSPIRE].
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Haouzi, N., Schmid, C. Little string origin of surface defects. J. High Energ. Phys. 2017, 82 (2017). https://doi.org/10.1007/JHEP05(2017)082
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DOI: https://doi.org/10.1007/JHEP05(2017)082