Skip to main content
SpringerLink
Log in

Hitchin Systems in \({\mathcal N}=2\) Field Theory

  • Chapter
  • First Online:
New Dualities of Supersymmetric Gauge Theories

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

This note is a short review of the way Hitchin systems appear in four-dimensional \({\mathcal N}=2\) supersymmetric field theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The fact that quantum correction are not suppressed around \({\mathcal B_{\mathrm {sing}}}\) is a good thing: exactly at this locus the naive metric becomes singular, and the quantum corrections smooth out these singularities, in such a way that the exact corrected metric extends to a complete space \(\mathcal M[R]\) which includes fibers over \({\mathcal B_{\mathrm {sing}}}\). This smoothing requires a correction which is of order 1, not suppressed in R.

  2. 2.

    These walls are known by various names: “BPS walls” in [29], “\(\mathcal K\)-walls” in [31], “walls of second kind” in [44], or parts of the “scattering diagram” in [32].

  3. 3.

    Actually, specifying \({\mathfrak g}\) does not quite determine the 5-dimensional theory; for that we should really specify a particular Lie group G with Lie algebra \({\mathfrak g}\). Which G we get depends on a subtle discrete choice which appears upon compactification, as described e.g. in [24], using some subtleties of the 6-dimensional \(S[{\mathfrak g}]\) explained in [58].

  4. 4.

    Slightly more generally, there are also simple line defects corresponding to mutually non-intersecting collections of closed curves on C, with nonnegative integer weights; the vev of such a defect is simply the product of the traces associated to the individual curves in the collection.

  5. 5.

    To be precise, consider the projection map \(\pi _*: H_1(\Sigma _u, \mathbb Z) \rightarrow H_1(C, \mathbb Z)\); the lattice \(\Gamma _u\) is the kernel of this projection.

  6. 6.

    The notation \(E_\mathbb C\) expresses the fact that the gauge group has been complexified; to avoid confusion we emphasize that the corresponding associated vector bundles do not get complexified.

  7. 7.

    We drop C here to emphasize that \(\mathcal M_{flat}\) can be defined without using the complex structure on C, e.g. as the space of representations \(\pi _1(M) \rightarrow G_\mathbb C\) up to equivalence, although of course it does still depend on the genus of C.

  8. 8.

    This is a consequence of Hodge theory for (0, 1)-forms on \(\Sigma \). One concrete way of thinking about it, in the case where \(\mathcal L\) has degree zero, is that every \(\mathcal L\) of degree zero admits a metric for which the Chern connection is flat, and this gives an isomorphism between the set of such \(\mathcal L\) and the set of unitary flat connections over \(\Sigma \), which is evidently a torus.

  9. 9.

    These singularities correspond to the punctures usually included in the definition of the theories of class S.

References

  1. 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, 113 (2010). arXiv:0909.0945 [hep-th]

  2. Alday, L.F., Gaiotto, D., Maldacena, J.: Thermodynamic bubble Ansatz (2009). arXiv:0911.4708 [hep-th]

  3. Alday, L.F., Maldacena, J.: Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space (2009). arXiv:0904.0663 [hep-th]

    Google Scholar 

  4. Alday, L.F., Maldacena, J., Sever, A., Vieira, P.: Y-system for scattering amplitudes (2010). arXiv:1002.2459 [hep-th]

    Google Scholar 

  5. Alvarez-Gaume, L., Freedman, D.Z.: Geometrical structure and ultraviolet finiteness in the supersymmetric sigma model. Commun. Math. Phys. 80, 443 (1981)

    Google Scholar 

  6. Bershadsky, M., Johansen, A., Sadov, V., Vafa, C.: Topological reduction of 4-d SYM to 2-d sigma models. Nucl. Phys. B448, 166–186 (1995). eprint: hep-th/9501096

  7. Biquard, O., Boalch, P.: Wild nonabelian Hodge theory on curves (2002). eprint: math/0111098

  8. Cecotti, S., Neitzke, A., Vafa, C.: R-twisting and 4d/2d correspondences (2010). arXiv:1006.3435 [hep-th]

  9. Cherkis, S.A., Kapustin, A.: New hyperkaehler metrics from periodic monopoles. Phys. Rev. D 65, 084015 (2002). eprint: hep-th/0109141

  10. Cherkis, S.A., Kapustin, A.: Periodic monopoles with singularities and N = 2 super-QCD. Commun. Math. Phys. 234, 1–35 (2003). arXiv:hep-th/0011081

    Google Scholar 

  11. Cordova, C., Neitzke, A.: Line defects, tropicalization, and multi-centered quiver quantum mechanics (2013). arXiv:1308.6829 [hep-th]

  12. Corlette, K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28(3), 361–382 (1988)

    Google Scholar 

  13. Dimofte, T., Gukov, S.: Refined, motivic, and quantum. Lett. Math. Phys. 91, 1 (2010). arXiv:0904.1420 [hep-th]

    Google Scholar 

  14. Donagi, R.Y.: Seiberg-Witten integrable systems (1997). eprint: alg-geom/9705010

  15. Donagi, R.Y., Witten, E.: Supersymmetric Yang-Mills theory and integrable systems. Nucl. Phys. B460, 299–334 (1996). eprint: hep-th/9510101

    Google Scholar 

  16. Donagi, R., Wijnholt, M.: Higgs bundles and UV completion in F-theory (2009). eprint: 0904.1218

  17. Donaldson, S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. 3 55(1), 127–131 (1987)

    Google Scholar 

  18. Drukker, N., Gomis, J., Okuda, T., Teschner, J.: Gauge theory loop operators and Liouville theory (2009). arXiv:0909.1105 [hep-th]

  19. Drukker, N., Morrison, D.R., Okuda, T.: Loop operators and S-duality from curveson Riemann surfaces (2009). arXiv:0907.2593 [hep-th]

    Google Scholar 

  20. Feix, B.: Hypercomplex manifolds and hyperholomorphic bundles. Math. Proc. Camb. Philos. Soc. 133, 443–457 (2002)

    Google Scholar 

  21. Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Éc. Norm. Supér. (4) 42(6), 865–930 (2009). eprint: math/0311245

    Google Scholar 

  22. Fock, V., Goncharov, A.: Moduli spaces of local systems and higher Teichmüller theory. Publ. Math. Inst. Hautes Études Sci. (103), 1–211 (2006). eprint: math/0311149

    Google Scholar 

  23. Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002) (electronic)

    Google Scholar 

  24. Freed, D.S., Teleman, C.: Relative quantum field theory (2012). arXiv:1212.1692 [hep-th]

  25. Gaiotto, D.: N = 2 dualities (2009). arXiv:0904.2715 [hep-th]

  26. Gaiotto, D.: Surface operators in N = 2 4d gauge theories (2009). arXiv:0911.1316 [hep-th]

  27. Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation (2009). arXiv:0907.3987 [hep-th]

  28. Gaiotto, D., Moore, G.W., Neitzke, A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163–224 (2010). arXiv:0807.4723 [hep-th]

    Google Scholar 

  29. Gaiotto, D., Moore, G.W., Neitzke, A.: Framed BPS states (2010). arXiv:1006.0146 [hep-th]

  30. Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing in coupled 2d–4d systems. (2011). arXiv:1103.2598 [hep-th]

  31. Gaiotto, D., Moore, G.W., Neitzke, A.: Spectral networks (2012). arXiv:1204.4824 [hep-th]

  32. Gross, M., Siebert, B.: From real affine geometry to complex geometry. Ann. Math. 174, 1301–1428 (2011). eprint: math/0703822

    Google Scholar 

  33. Gross, M., Siebert, B.: Theta functions and mirror symmetry (2012). eprint: 1204.1991

  34. Gukov, S.: Surface operators and knot homologies (2007). eprint: 0706.2369

  35. Gukov, S., Witten, E.: Gauge theory, ramification, and the geometric Langlands program (2006). eprint: hep-th/0612073

  36. Hanany, A., Pioline, B.: (Anti-)instantons and the Atiyah-Hitchin manifold. JHEP 07, 001 (2000). eprint: hep-th/0005160

    Google Scholar 

  37. Harvey, J. A., Moore, G.W., Strominger, A.: Reducing S duality to T duality. Phys. Rev. D 52, 7161–7167 (1995). eprint: hep-th/9501022

    Google Scholar 

  38. Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality, and the Hitchin system (2002). eprint: math.AG/0205236

  39. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. (3) 55(1), 59–126 (1987)

    Google Scholar 

  40. Hitchin, N.J., Karlhede, A., Lindstrom, U., Roček, M.: Hyperkähler metrics and supersymmetry. Commun. Math. Phys. 108, 535 (1987)

    Google Scholar 

  41. Hollands, L., Neitzke, A.: Spectral networks and Fenchel-Nielsen coordinates (2013). arXiv:1312.2979 [math.GT]

  42. Ito, Y., Okuda, T., Taki, M.: Line operators on \(S^{1}\times R^{3}\) and quantization of the Hitchin moduli space (2011). eprint: 1111.4221

  43. Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program (2006). eprint: hep-th/0604151

  44. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008). eprint: 0811.2435

  45. Le, I.: Higher laminations and affine. Build. eprint: 1209.0812 (2012)

  46. Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc. 3(2), 447–498 (1990)

    Google Scholar 

  47. Martinec, E.J., Warner, N.P.: Integrable systems and supersymmetric gauge theory. Nucl. Phys. B459, 97–112 (1996). eprint: hep-th/9509161

    Google Scholar 

  48. Nekrasov, N., Witten, E.: The omega deformation, branes, integrability, and Liouville theory (2010). arXiv:1002.0888 [hep-th]

  49. Okuda, T.: Line operators in supersymmetric gauge theories and the 2d–4d relation (2014). arXiv:1412.7126 [hep-th]

  50. Seiberg, N., Witten, E.: Electric-magnetic duality, monopole condensation, and confinement in \(\cal N\) = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B426, 19–52 (1994). eprint: hep-th/9407087

  51. Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in \(\cal N\) = 2 supersymmetric QCD. Nucl. Phys. B431, 484–550 (1994). eprint: hep-th/9408099

  52. Seiberg, N., Witten, E.: Gauge dynamics and compactification to three dimensions (1996). eprint: hep-th/9607163

  53. Sikora, A.S.: Generating sets for coordinate rings of character varieties (2011). eprint: 1106.4837

  54. Simpson, C.T.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1(4), 867–918 (1988)

    Google Scholar 

  55. Simpson, C.T.: Harmonic bundles on noncompact curves. J. Am. Math. Soc. 3(3), 713–770 (1990)

    Google Scholar 

  56. Teschner, J.: Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I. Adv. Theor. Math. Phys. 15, 471–564 (2011). arXiv:1005.2846 [hep-th]

    Google Scholar 

  57. Verbitsky, M.: Hyperholomorphic bundles over a hyper-Kähler manifold. J. Algebraic Geom. 5(4), 633–669 (1996). eprint: alg-geom/9307008

  58. Witten, E.: Geometric Langlands from six dimensions (2009). arXiv:0905.2720 [hep-th]

  59. Xie, D.: Network, cluster coordinates and \(N\) = 2 theory I. (2012). eprint: 1203.4573

  60. Xie, D.: Higher laminations, webs and \(N\) = 2 line operators (2013). arXiv:1304.2390 [hep-th]

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, US

    Andrew Neitzke

Authors
  1. Andrew Neitzke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrew Neitzke .

Editor information

Editors and Affiliations

  1. String Theory Group, DESY String Theory Group, Hamburg, Hamburg, Germany

    Jörg Teschner

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Neitzke, A. (2016). Hitchin Systems in \({\mathcal N}=2\) Field Theory. In: Teschner, J. (eds) New Dualities of Supersymmetric Gauge Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-18769-3_3

Download citation

Publish with us

Policies and ethics

Search

Navigation