Abstract
We introduce a new perspective and a generalization of spectral networks for 4d \( \mathcal{N} \) = 2 theories of class S associated to Lie algebras \( \mathfrak{g} \) = A n , D n , E6, and E7. Spectral networks directly compute the BPS spectra of 2d theories on surface defects coupled to the 4d theories. A Lie algebraic interpretation of these spectra emerges naturally from our construction, leading to a new description of 2d-4d wall-crossing phenomena. Our construction also provides an efficient framework for the study of BPS spectra of the 4d theories. In addition, we consider novel types of surface defects associated with minuscule ccrepresentations of \( \mathfrak{g} \).
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References
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
Y. Tachikawa, Six-dimensional D(N) theory and four-dimensional SO-USp quivers, JHEP 07 (2009) 067 [arXiv:0905.4074] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
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] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
F. Ferrari and A. Bilal, The strong coupling spectrum of the Seiberg-Witten theory, Nucl. Phys. B 469 (1996) 387 [hep-th/9602082] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Spectral networks, Annales Henri Poincaré 14 (2013) 1643 [arXiv:1204.4824] [INSPIRE].
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-Crossing in Coupled 2d-4d Systems, JHEP 12 (2012) 082 [arXiv:1103.2598] [INSPIRE].
D. Gaiotto, S. Gukov and N. Seiberg, Surface Defects and Resolvents, JHEP 09 (2013) 070 [arXiv:1307.2578] [INSPIRE].
D. Galakhov, P. Longhi, T. Mainiero, G.W. Moore and A. Neitzke, Wild Wall Crossing and BPS Giants, JHEP 11 (2013) 046 [arXiv:1305.5454] [INSPIRE].
K. Maruyoshi, C.Y. Park and W. Yan, BPS spectrum of Argyres-Douglas theory via spectral network, JHEP 12 (2013) 092 [arXiv:1309.3050] [INSPIRE].
D. Galakhov, P. Longhi and G.W. Moore, Spectral Networks with Spin, Commun. Math. Phys. 340 (2015) 171 [arXiv:1408.0207] [INSPIRE].
N. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59.
G.W. Moore, 2012 Felix Klein Lecture Notes, http://www.physics.rutgers.edu/∼gmoore/.
A. Neitzke, Spectral networks and their uses, talk in Pisa, June 2014, http://www.ma.utexas.edu/users/neitzke/talks/spectral-networks-pisa.pdf .
A. Neitzke, Spectral networks, talk at kitp, August 2011, http://www.ma.utexas.edu/users/neitzke/talks/spectral-networks-kitp.pdf .
S. Cecotti and C. Vafa, On classification of N = 2 supersymmetric theories, Commun. Math. Phys. 158 (1993) 569 [hep-th/9211097] [INSPIRE].
K. Hori, C.Y. Park and Y. Tachikawa, 2d SCFTs from M2-branes, JHEP 11 (2013) 147 [arXiv:1309.3036] [INSPIRE].
R. Donagi, Decomposition of spectral covers, Journées de Geometrie Algébrique d’Orsay 218 (1993) 145.
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
M. Adler and P. van Moerbeke, Completely integrable systems, euclidean lie algebras, and curves, Adv. Math. 38 (1980) 267.
M. Adler and P. van Moerbeke, Linearization of hamiltonian systems, Jacobi varieties and representation theory, Adv. Math. 38 (1980) 318.
A. McDaniel, Representations of \( \frac{@}{@}\mathrm{s}\mathrm{l}\left(n,\mathbb{C}\right) \) and the toda lattice, Duke Math. J. 56 (1988) 47.
V. Kanev, Spectral curves, simple lie algebras, and Prym-Tjurin varieties, Part I, Proc. Symp. Pure Math. 49 (1989) 627.
V. Kanev, Spectral curves and Prym-Tjurin varieties, in Abelian Varieties: Proceedings of the International Conference, Held in Egloffstein, Germany, October 3-8, 1993 (1995) 151.
A. McDaniel and L. Smolinsky, A lie-theoretic galois theory for the spectral curves of an integrable system. I, Comm. Math. Phys. 149 (1992) 127.
A. McDaniel and L. Smolinsky, A lie-theoretic galois theory for the spectral curves of an integrable system. II, Trans. Amer. Math. Soc., 349 (1997) 713.
A. McDaniel and L. Smolinsky, Lax equations, weight lattices, and prym-tjurin varieties, Acta Mathematica 181 (1998) 283.
T.J. Hollowood, Strong coupling N = 2 gauge theory with arbitrary gauge group, Adv. Theor. Math. Phys. 2 (1998) 335 [hep-th/9710073] [INSPIRE].
M. Shifman and A. Yung, Quantum Deformation of the Effective Theory on Non-Abelian string and 2D-4D correspondence, Phys. Rev. D 89 (2014) 065035 [arXiv:1401.1455] [INSPIRE].
P. Longhi and C.Y. Park, ADE spectral networks and 2d coset models, to appear.
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
W. Lerche and N.P. Warner, Polytopes and solitons in integrable, N = 2 supersymmetric Landau-Ginzburg theories, Nucl. Phys. B 358 (1991) 571 [INSPIRE].
T. Eguchi and K. Sakai, Seiberg-Witten curve for the E string theory, JHEP 05 (2002) 058 [hep-th/0203025] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
U.H. Danielsson and B. Sundborg, The moduli space and monodromies of N = 2 supersymmetric SO(2r + 1) Yang-Mills theory, Phys. Lett. B 358 (1995) 273 [hep-th/9504102] [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].
M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, \( \mathcal{N} \) = 2 quantum field theories and their BPS quivers, Adv. Theor. Math. Phys. 18 (2014) 27 [arXiv:1112.3984] [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].
Y. Kazama and H. Suzuki, Characterization of N = 2 Superconformal Models Generated by Coset Space Method, Phys. Lett. B 216 (1989) 112 [INSPIRE].
W. Lerche, C. Vafa and N.P. Warner, Chiral Rings in N = 2 Superconformal Theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Spectral Networks and Snakes, Annales Henri Poincaré 15 (2014) 61 [arXiv:1209.0866] [INSPIRE].
J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, Cambridge University Press, (1992).
C. Procesi, Lie Groups: An Approach through Invariants and Representations, Springer (2007).
W. Lerche and N.P. Warner, Exceptional SW geometry from ALE fibrations, Phys. Lett. B 423 (1998) 79 [hep-th/9608183] [INSPIRE].
T. Eguchi, N.P. Warner and S.-K. Yang, ADE singularities and coset models, Nucl. Phys. B 607 (2001) 3 [hep-th/0105194] [INSPIRE].
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Longhi, P., Park, C.Y. ADE spectral networks. J. High Energ. Phys. 2016, 87 (2016). https://doi.org/10.1007/JHEP08(2016)087
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DOI: https://doi.org/10.1007/JHEP08(2016)087