Skip to main content
SpringerLink
Log in

K-theoretic classification of fermionic operator mixings in holographic conformal field theories

  • Article
  • Physics & Astronomy
  • Published:
Science Bulletin

Abstract

In this paper, we apply the K-theory scheme of classifying the topological insulators/superconductors to classify the topological classes of the massive multi-flavor fermions in anti-de Sitter (AdS) space. In the context of AdS/conformal field theory (CFT) correspondence, the multi-flavor fermionic mass matrix is dual to the pattern of operator mixing in the boundary CFT. Thus, our results classify the possible patterns of operator mixings among fermionic operators in the holographic CFT.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. The key difference in adopting the above scheme for the high energy and condensed matters is that in the former \(\alpha ^i\)’s do not mix with the gauge and flavor structure. On the other hand, in the condensed matter (5) is the low energy effective model so that \(\alpha ^i\) may be mixed with other non-space-time structures.

  2. We assume our theory is CPT invariant so that P symmetry will be determined by C and T.

  3. Our d is different from the one in Ref. [27] where \(d=2k+2\).

  4. There are also the other boundary actions such as the ones considered in Ref. [29] for holographic flat band. Different boundary actions yield different boundary CFTs which are related by the RG flows caused by the double trace operators [29, 30].

References

  1. Hasan MZ, Kane CL (2010) Topological insulators. Rev Mod Phys 82:3045

    Article  Google Scholar 

  2. Qi XL, Zhang SC (2011) Topological insulators and superconductors. Rev Mod Phys 83:1057

    Article  Google Scholar 

  3. Altland A, Zirnbauer MR (1997) Novel symmetry classes in mesoscopic normal-superconducting hybrid structures. Phys Rev B 55:1142

    Article  Google Scholar 

  4. Schnyder AP, Ryu S, Furusaki A et al (2009) Classification of topological insulators and superconductors. AIP Conf Proc 1134:10

    Article  Google Scholar 

  5. Chen X, Gu ZC, Liu ZX et al (2012) Symmetry protected topological orders and the group cohomology of their symmetry group. Science 338:1604

    Article  Google Scholar 

  6. Kitaev A (2009) Periodic table for topological insulators and superconductors. arXiv:0901.2686

  7. Wen XG (2012) Symmetry protected topological phases in non-interacting fermion systems. Phys Rev B 85:085103

    Article  Google Scholar 

  8. Ho SH, Lin FL, Wen XG (2012) Majorana zero-modes and topological phases of multi-flavored Jackiw–Rebbi model. J High Energy Phys 1212:074

    Article  Google Scholar 

  9. Jackiw R, Rebbi C (1976) Solitons with Fermion number 1/2. Phys Rev D 13:3398

    Article  Google Scholar 

  10. McGreevy J, Swingle B (2011) Non-Abelian statistics versus the Witten anomaly. Phys Rev D 84:065019

    Article  Google Scholar 

  11. Witten E (1982) An \(SU(2)\) anomaly. Phys Lett B 117:324

    Article  Google Scholar 

  12. Ryu S, Takayanagi T (2010) Topological insulators and superconductors from D-branes. Phys Lett B 693:175

    Article  Google Scholar 

  13. Ryu S, Takayanagi T (2010) Topological insulators and superconductors from string theory. Phys Rev D 82:086014

    Article  Google Scholar 

  14. Shankar R, Vishwanath A (2011) Equality of bulk wave functions and edge correlations in topological superconductors: a spacetime derivation. Phys Rev Lett 107:106803

    Article  Google Scholar 

  15. Maldacena J (1999) The large \(N\) limit of superconformal field theories and supergravity. Int J Theor Phys 38:1113–1133

    Article  Google Scholar 

  16. Maldacena J (1998) The large \(N\) limit of superconformal field theories and supergravity. Adv Theor Math Phys 2:231–252

    Article  Google Scholar 

  17. Gubser SS, Klebanov IR, Polyakov AM (1998) Gauge theory correlators from noncritical string theory. Phys Lett B 428:105

    Article  Google Scholar 

  18. Witten E (1998) Anti-de sitter space and holography. Adv Theor Math Phys 2:253–291

    Article  Google Scholar 

  19. Hartnoll SA (2007) Anyonic strings and membranes in AdS space and dual Aharonov–Bohm effects. Phys Rev Lett 98:111601

    Article  Google Scholar 

  20. Kawamoto S, Lin FL (2010) Holographic anyons in the ABJM theory. J High Energy Phys 1002:059

    Article  Google Scholar 

  21. Korchemsky GP (2016) On level crossing in conformal field theories. J High Energy Phys 1603:212

    Article  Google Scholar 

  22. Constable NR, Freedman DZ, Headrick M et al (2002) Operator mixing and the BMN correspondence. J High Energy Phys 0210:068

    Article  Google Scholar 

  23. Iqbal N, Liu H (2009) Real-time response in AdS/CFT with application to spinors. Fortsch Phys 57:367–384

    Article  Google Scholar 

  24. Kaplan DB, Sun S (2012) Spacetime as a topological insulator: mechanism for the origin of the fermion generations. Phys Rev Lett 108:181807

    Article  Google Scholar 

  25. Peskin ME, Schroeder DV (1995) An introduction to quantum field theory. Westview Press, Boulder

    Google Scholar 

  26. Wilczek F (2009) Majorana returns. Nat Phys 5:614–618

    Article  Google Scholar 

  27. Polchinski J (1998) Appendix B in “String theory (Cambridge monographs on mathematical physics), Vol 2”, 1st edn. Cambridge University Press, Cambridge

  28. Klebanov IR, Witten E (1999) AdS/CFT correspondence and symmetry breaking. Nucl Phys B 556:89–114

    Article  Google Scholar 

  29. Laia JN, Tong D (2011) A holographic flat band. J High Energy Phys 1111:125

    Article  Google Scholar 

  30. Laia JN, Tong D (2011) Flowing between fermionic fixed points. J High Energy Phys 1111:131

    Article  Google Scholar 

Download references

Acknowledgments

We thank Xiao-Gang Wen for the discussion on the issue of K-theory analysis of models in high energy physics. FLL is supported by Taiwan’s Ministry of Science and Technology (MoST) Grants (100-2811-M-003-011 and 100-2918-I-003-008). We thank the support of National Center for Theoretical Sciences (NCTS).

Author information

Authors and Affiliations

  1. Department of Physics, Taiwan Normal University, Taipei, 116, Taiwan, China

    Shih-Hao Ho & Feng-Li Lin

Authors
  1. Shih-Hao Ho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feng-Li Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng-Li Lin.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ho, SH., Lin, FL. K-theoretic classification of fermionic operator mixings in holographic conformal field theories. Sci. Bull. 61, 1115–1125 (2016). https://doi.org/10.1007/s11434-016-1110-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11434-016-1110-2

Keywords

Search

Navigation