Skip to main content
Log in

More Arnold’s \( \mathcal{N} = 2 \) superconformal gauge theories

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We study the \( \mathcal{N} = 2 \) gauge theories obtained by engineering the Type IIB superstring on the quasi-homogeneous elements of Arnold’s list of bimodal singularities. All these theories have finite BPS chambers and we describe, along the lines of arXiv:1107.5747, the algebraically obvious ones.

Our results leads to the prediction of 11 new periodic Y -systems, providing additional evidence for the correspondence in between thermodinamical Bethe ansatz periodic Y -systems and \( \mathcal{N} = 2 \) superconformal theories with a finite BPS chamber whose chiral primaries have dimensions of the form N/.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in \( \mathcal{N} = 2 \) supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in \( \mathcal{N} = 2 \) supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435

  4. T. Dimofte and S. Gukov, Refined, motivic and quantum, Lett. Math. Phys. 91 (2010) 1 [arXiv:0904.1420] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Dimofte, S. Gukov and Y. Soibelman, Quantum wall crossing in N = 2 gauge theories, Lett. Math. Phys. 95 (2011) 1 [arXiv:0912.1346] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. 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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].

  8. S. Cecotti and C. Vafa, BPS wall crossing and topological strings, arXiv:0910.2615 [INSPIRE].

  9. S. Cecotti, A. Neitzke and C. Vafa, R-twisting and 4 d/2d correspondences, arXiv:1006.3435 [INSPIRE].

  10. S. Cecotti and M. Del Zotto, On Arnold’s 14exceptional’ N = 2 superconformal gauge theories, JHEP 10 (2011) 099 [arXiv:1107.5747] [INSPIRE].

    Article  ADS  Google Scholar 

  11. S. Fomin and A. Zelevinsky, Cluster algebras I: foundations, math/0104151

  12. S. Fomin and A. Zelevinsky, Cluster algebras II: finite type classification, arXiv:math/0208229

  13. A. Berenstein, S. Fomin and A. Zelevinsky, Cluster algebras III: upper bounds and double Bruhat cells, math/0305434

  14. S. Fomin and A. Zelevinsky, Cluster algebras IV: coefficients, math/0602259

  15. A. Berenstein and A. Zelevinsky, Quantum cluster algebras, math/0404446

  16. V.V. Fock and A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm, arXiv:math/0311245

  17. V.V. Fock and A.B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, Inventiones Mathematicae 175 (2008) 223 [arXiv:math/0702397]

    Article  MathSciNet  ADS  Google Scholar 

  18. V.V. Fock and A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm II: the intertwiner, math/0702398

  19. S. Cecotti and C. Vafa, Classification of complete N = 2 supersymmetric theories in 4 dimensions, arXiv:1103.5832 [INSPIRE].

  20. B. Keller, On cluster theory and quantum dilogarithm identities, arXiv:1102.4148

  21. M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi, et al., BPS quivers and spectra of complete N = 2 quantum field theories, arXiv:1109.4941 [INSPIRE].

  22. B. Keller, The periodicity conjecture for pairs of Dynkin diagrams, arXiv:1001.1531

  23. V.I. Arnol’d, A.N. Varchenko, S.M. Gusein-Zade, Singularities of differentiable mappings, Vol. I, Nauka, Moscow Russia (1982) [Birkhauser, Basel Switzerland (1988)]

    Google Scholar 

  24. V.I. Arnol’d, Singularity theory I, Enc. Math. Sci. Dynamical Systems 6, Springer, Berlin Germany (1993)

    Google Scholar 

  25. W. Ebeling, Functions of several complex variables and their singularities, Graduate studies in mathematics 83, Am. Math. Soc. (2007)

  26. W. Ebeling, The monodromy groups of isolated singularities of complete intersections, LNM 1923, Springer-Verlag, Berlin Germany (1987).

    Google Scholar 

  27. A.M. Gabrielov, Dynkin diagrams for unimodal singularities, Funkt. Anal. Appl. 8 (1975) 192.

    Article  Google Scholar 

  28. A.M. Gabrielov, Bifurcations, Dynkin diagrams and modality of isolated singularities (in Russian), Funkt. Anal. Appl. 8 (1974).

  29. A.M. Gabrielov and A.G. Kushnirenko, Description of deformations with constant Milnor number for homogeneous functions (in Russian), Funkt. Anal. Appl. 9 (1975) 67 [Engl. transl. ibid. 329]

    Article  Google Scholar 

  30. S. Cecotti and C. Vafa, On classification of N = 2 supersymmetric theories, Commun. Math. Phys. 158 (1993) 569 [hep-th/9211097] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. S. Cecotti and C. Vafa, 2 d wall-crossing, R-twisting and a supersymmetric index, arXiv:1002.3638 [INSPIRE].

  32. S. Cecotti, Trieste lectures on wall-crossing invariants, lectures given at the ICTP school and workshop on D-branes instantons, wall-crossing and microstate counting, Trieste, November 15–20, 2010 [http://people.sissa.it/~cecotti/].

  33. K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].

  34. B. Keller, Cluster algebras, quiver representations and triangulated categories, arXiv:0807.1960

  35. S. Fomin, and A. Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003)977.

  36. A.M. Gabrielov, Polar curves and intersection matrices of singularities, Inv. Math. 54 (1979) 15.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. A.D. Shapere and C. Vafa, BPS structure of Argyres-Douglas superconformal theories, hep-th/9910182 [INSPIRE].

  38. S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four folds, Nucl. Phys. B 584 (2000) 69 [hep-th/9906070] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. W. Lerche, C. Vafa and N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B 324 (1989) 427 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. C. Vafa, and N. Warner, Catastrophes and the classification of conformal theories, Phys. Lett. B 324 (1989) 427 .

    MathSciNet  Google Scholar 

  41. S. Cecotti, Geometry of N = 2 Landau-Ginzburg families, Nucl. Phys. B 355 (1991) 755 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  42. A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2d field theory, JETP letters 43(1986) 730.

    MathSciNet  ADS  Google Scholar 

  43. J. Milnor, Singular points of complex hypersurfaces, Ann. Math. Stud. 61 (1968).

  44. N. Bourbaki, Elements of mathematics, algebra I, chapters 1–3, Springer, Berlin Germany

  45. P. Seidel, and R. Thomas, Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001) 37.

    Article  MathSciNet  MATH  Google Scholar 

  46. H. Lenzing and J.A. de la Peña, Extended canonical algebras and Fuchsian singularities, math/0611532

  47. B. Keller, Quiver mutation in Java, http://www.institut.math.jussieu.fr/~keller/quivermutation.

  48. A. Kuniba, T. Nakanishi and J. Suzuki, Functional relations in solvable lattice models. 1: functional relations and representation theory, Int. J. Mod. Phys. A 9 (1994) 5215 [hep-th/9309137] [INSPIRE]

    MathSciNet  ADS  Google Scholar 

  49. A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  50. F. Gliozzi and R. Tateo, ADE functional dilogarithm identities and integrable models, Phys. Lett. B 348 (1995) 84 [hep-th/9411203] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  51. F. Gliozzi and R. Tateo, Thermodynamic Bethe ansatz and threefold triangulations, Int. J. Mod. Phys. A 11 (1996) 4051 [hep-th/9505102] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  52. R. Caracciolo, F. Gliozzi and R. Tateo, A topological invariant of RG flows in 2-D integrable quantum field theories, Int. J. Mod. Phys. B 13 (1999) 2927 [hep-th/9902094] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  53. I. Assem, D. Simson and A. Skowronski, Elements of the representation theory of associative algebras, vol. 1: techniques of representation theory, Cambridge University Press, Cambridge U.K. (2006).

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michele Del Zotto.

Additional information

ArXiv ePrint: 1110.3826

Rights and permissions

Reprints and permissions

About this article

Cite this article

Del Zotto, M. More Arnold’s \( \mathcal{N} = 2 \) superconformal gauge theories. J. High Energ. Phys. 2011, 115 (2011). https://doi.org/10.1007/JHEP11(2011)115

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP11(2011)115

Keywords

Navigation