Abstract
We introduce new geometric objects called spectral networks. Spectral networks are networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional \({\mathcal{N} = 2}\) theories coupled to surface defects, particularly the theories of class S. In these theories, spectral networks provide a useful tool for the computation of BPS degeneracies; the network directly determines the degeneracies of solitons living on the surface defect, which in turn determines the degeneracies for particles living in the 4d bulk. Spectral networks also lead to a new map between flat \({{\rm GL}(K, \mathbb{C})}\) connections on a two-dimensional surface C and flat abelian connections on an appropriate branched cover \({\Sigma}\) of C. This construction produces natural coordinate systems on moduli spaces of flat \({{\rm GL}(K, \mathbb{C})}\) connections on C, which we conjecture are cluster coordinate systems.
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Witten, E.: Solutions of four-dimensional field theories via M-theory. Nucl. Phys. B500, 3–42 (1997). http://www.arXiv.org/abs/hep-th/9703166
Gaiotto, D.: \({\mathcal{N} = 2}\) dualities. http://www.arXiv.org/abs/0904.2715
Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin Systems, and the WKB Approximation. http://www.arXiv.org/abs/0907.3987
Gaiotto, D.: Surface Operators in \({\mathcal N=2}\) 4d Gauge Theories. http://www.arXiv.org/abs/0911.1316
Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-Crossing in Coupled 2d–4d Systems. http://www.arXiv.org/abs/1103.2598v1
Klemm, A., Lerche, W., Mayr, P., Vafa, C., Warner, N.P.: Self-dual strings and \({\mathcal{N} = 2}\) supersymmetric field theory. Nucl. Phys. B477, 746–766 (1996). http://www.arXiv.org/abs/hep-th/9604034
Brandhuber, A., Stieberger, S.: Self-dual strings and stability of BPS states in \({\mathcal{N} = 2 SU(2)}\) gauge theories. Nucl. Phys. B488, 199–222 (1997). http://www.arXiv.org/abs/hep-th/9610053
Aharony, O., Hanany, A., Kol, B.: Webs of (p, q) 5-branes, five dimensional field theories and grid diagrams. JHEP 01, 002 (1998). http://www.arXiv.org/abs/hep-th/9710116
Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations. http://www.arXiv.org/abs/0811.2435
Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: BPS Quivers and Spectra of Complete \({\mathcal{N} = 2}\) Quantum Field Theories. http://www.arXiv.org/abs/1109.4941
Alim, M., Cecotti, S., Cordova, C., Espahbodi, S., Rastogi, A., Vafa, C.: \({\mathcal{N} = 2}\) Quantum Field Theories and Their BPS Quivers. http://www.arXiv.org/abs/1112.3984
Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Éc. Norm. Supér. (4) 42(6), 865–930 (2009). http://www.arXiv.org/abs/math/0311245
Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. 103, 1–211 (2006). http://www.arXiv.org/abs/math/0311149
Fock, V.V., Goncharov, A.B.: The quantum dilogarithm and representations of quantum cluster varieties. http://www.arXiv.org/abs/math/0702397
Goncharov, A.: To appear
Hitchin N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55(1), 59–126 (1987)
Gaiotto, D., Moore, G.W., Neitzke, A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163–224 (2010). http://www.arXiv.org/abs/0807.4723
Delabaere É., Dillinger H., Pham F.: Résurgence de Voros et périodes des courbes hyperelliptiques. Ann. Inst. Fourier (Grenoble) 43(1), 163–199 (1993)
Voros A.: The return of the quartic oscillator: the complex WKB method. Ann. Inst. H. Poincaré Sect. A (N.S.) 39(3), 211–338 (1983)
Aoki, T., Kawai, T., Sasaki, S., Shudo, A., Takei, Y.: Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations. J. Phys. A 38(15), 3317–3336 (2005). http://www.arXiv.org/abs/math-ph/0409005
Gaiotto, D., Moore, G.W., Neitzke, A.: Framed BPS States. http://www.arXiv.org/abs/1006.0146v1
Zorich, A.: Flat surfaces. http://www.arXiv.org/abs/math/0609392
Kontsevich, M., Soibelman, Y.: Affine structures and non-Archimedean analytic spaces. In: The unity of mathematics. Progr. Math., vol. 244, pp. 321–385. Birkhäuser, Boston (2006). http://www.arXiv.org/abs/math.ag/0406564
Auroux D.: Mirror symmetry and T-duality in the complement of an anticanonical divisor. J. Gökova Geom. Topol. GGT 1, 51–91 (2007)
Gross, M., Siebert, B.: From real affine geometry to complex geometry. http://www.arXiv.org/abs/math/0703822
Gross, M., Pandharipande, R., Siebert, B.: The tropical vertex. http://www.arXiv.org/abs/0902.0779
Gross, M., Hacking, P., Keel, S.: Mirror symmetry for log Calabi–Yau surfaces I. http://www.arXiv.org/abs/1106.4977
Fraser, C., Hollowood, T.J.: On the weak coupling spectrum of N = 2 supersymmetric SU(n) gauge theory. Nucl. Phys. B490, 217–238 (1997). http://www.arXiv.org/abs/hep-th/9610142
Taylor, B.J.: On the strong coupling spectrum of pure SU(3) Seiberg–Witten theory. JHEP 0108, 031 (2001). http://www.arXiv.org/abs/hep-th/0107016
Taylor, B.J.: On the moduli space of SU(3) Seiberg–Witten theory with matter. JHEP 0212 040 (2002). http://www.arXiv.org/abs/hep-th/0211086
Chen, H.-Y., Dorey, N., Petunin, K.: Moduli space and wall-crossing formulae in higher-rank gauge theories. JHEP 1111, 020 (2011). http://www.arXiv.org/abs/1105.4584
Gaiotto, D., Moore, G.W., Neitzke, A.: Spectral networks and snakes (2013, to appear)
Alday, L.F., Gaiotto, D., Gukov, S., Tachikawa, Y., Verlinde, H.: Loop and surface operators in N = 2 gauge theory and Liouville modular geometry. JHEP 1001 (2010). http://www.arXiv.org/abs/0909.0945
Dimofte, T., Gukov, S., Soibelman, Y.: Quantum Wall Crossing in N = 2 Gauge Theories. http://www.arXiv.org/abs/0912.1346
Cecotti, S., Vafa, C.: BPS Wall Crossing and Topological Strings. http://www.arXiv.org/abs/0910.2615
Cecotti, S., Vafa, C.: 2d Wall-Crossing, R-twisting, and a Supersymmetric Index. http://www.arXiv.org/abs/1002.3638
Cecotti, S., Fendley, P., Intriligator, K.A., Vafa, C.: A new supersymmetric index. Nucl. Phys. B386, 405–452 (1992). http://www.arXiv.org/abs/hep-th/9204102
Cecotti, S., Vafa, C.: On classification of \({\mathcal{N} = 2}\) supersymmetric theories. Commun. Math. Phys. 158, 569–644 (1993). http://www.arXiv.org/abs/hep-th/9211097
Gaiotto, D., Witten, E.: Knot invariants from four-dimensional gauge theory. http://www.arXiv.org/abs/1106.4789
Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. JHEP 1111 (2011), http://www.arXiv.org/abs/hep-th/0702146
Andriyash, E., Denef, F., Jafferis, D.L., Moore, G.W.: Bound state transformation walls. http://www.arXiv.org/abs/1008.3555
Andriyash, E., Denef, F., Jafferis, D.L., Moore, G.W.: Wall-crossing from supersymmetric galaxies. JHEP 1201 (2012). http://www.arXiv.org/abs/1008.0030
Gaiotto, D., Moore, G.W., Neitzke, A.: Spectral network movies. http://www.ma.utexas.edu/users/neitzke/spectral-network-movies/
Strebel K.: Quadratic differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 5 [Results in Mathematics and Related Areas(3)]. Springer, Berlin (1984)
Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in \({\mathcal{N} = 2}\) supersymmetric Yang–Mills theory. Nucl. Phys. B426, 19–52 (1994). http://www.arXiv.org/abs/hep-th/9407087
Galakhov, D., Longhi, P., Mainiero, T., Moore, G. W., Neitzke, A.: To appear (2013)
Shapere, A.D., Vafa, C. BPS structure of Argyres–Douglas superconformal theories. http://www.arXiv.org/abs/hep-th/9910182
Cecotti, S., Neitzke, A., Vafa, C.: R-Twisting and 4d/2d Correspondences. http://www.arXiv.org/abs/1006.3435
Donagi, R.: Spectral covers. In: Current topics in complex algebraic geometry (Berkeley, CA, 1992/93). Math. Sci. Res. Inst. Publ., vol. 28, pp. 65–86. Cambridge Univ. Press, Cambridge (1995)
Donagi, R., Gaitsgory, D.: The gerbe of Higgs bundles. http://www.arXiv.org/abs/math/0005132
Atiyah M.F., Bott R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. Ser. A 308(1505), 523–615 (1983)
Boalch, P.P.: Symplectic Geometry and Isomonodromic Deformations. PhD thesis, Wadham College, Oxford (1999)
Alexandrov, S., Persson, D., Pioline, B.: Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence. JHEP 1112 (2011). http://www.arXiv.org/abs/1110.0466
Cecotti, S., Del Zotto, M.: On Arnold’s 14 ‘exceptional’ \({\mathcal{N} = 2}\) superconformal gauge theories. JHEP 1110, 099 (2011). http://www.arXiv.org/abs/1107.5747
Atiyah M.F.: The geometry and physics of knots. Lezioni Lincee. [Lincei Lectures]. Cambridge University Press, Cambridge (1990)
Corlette K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28(3), 361–382 (1988)
Donaldson S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. (3) 55(1), 127–131 (1987)
Simpson C.: Harmonic bundles on noncompact curves. J. Am. Math. Soc. 3, 713–770 (1990)
Biquard, O., Boalch, P.: Wild nonabelian Hodge theory on curves. http://www.arXiv.org/abs/math/0111098
Bender C.M., Wu T.T.: Anharmonic oscillator. Phys. Rev. 184, 1231–1260 (1969)
Zinn-Justin J.: Instantons in quantum mechanics: numerical evidence for a conjecture. J. Math. Phys. 25, 549 (1984)
Berk H.L., Nevins W.M., Roberts K.V.: New Stokes’ line in WKB theory. J. Math. Phys. 23(6), 988–1002 (1982)
Simpson, C.: Asymptotics for general connections at infinity. http://www.arXiv.org/abs/math/0311531
Simpson C.: Asymptotic behavior of monodromy. In: Lecture Notes in Mathematics, vol. 1502. Springer, Berlin (1991)
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Communicated by Marcos Marino.
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Gaiotto, D., Moore, G.W. & Neitzke, A. Spectral Networks. Ann. Henri Poincaré 14, 1643–1731 (2013). https://doi.org/10.1007/s00023-013-0239-7
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DOI: https://doi.org/10.1007/s00023-013-0239-7