Advances in Applied Clifford Algebras

, Volume 27, Issue 3, pp 2457–2471 | Cite as

Four-Dimensional Conformal Field Theory using Quaternions

Article
  • 43 Downloads

Abstract

We have built a constrained four-dimensional quaternion-parametrized conformal field theory using quaternion holomorphic functions as the generators of quaternionic conformal transformations. With the two-dimensional complex-parametrized conformal field theory as our model, we study the stress tensor, the conserved charge, the symmetry generators, the quantization conditions and several operator product expansions. Future applications are also addressed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson, J.: Hyperbolic Geometry. Springer, Berlin (2008)Google Scholar
  2. 2.
    Bischoff, M., Meise, D., Rehren, K.-H., Wagner, I.: Conformal quantum field theory in various dimensions. Bulg. J. Phys. 36, 170–185 (2009). arXiv:0908.3391 [math-ph]
  3. 3.
    Blumenhagen, R., Plauschinn, E.: Introduction to conformal field theory. Lect. Notes Phys. 779, 1–256 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chatterjee, A., Lowe, D.A.: Holographic operator mapping in dS/CFT and cluster decomposition. Phys. Rev. D92, 084038 (2015). arXiv:1503.07482 [hep-th]
  5. 5.
    Chen, C.-M., Kim, S.P., Lin, I.-C., Sun, J.-R., Wu, M.-F.: Spontaneous pair production in Reissner–Nordstrom black holes. Phys. Rev. D 85, 124041 (2012). arXiv:1202.3224 [hep-th]ADSCrossRefGoogle Scholar
  6. 6.
    Deavours, C.A.: Quaternion calculus. Am. Math. Mon. 80, 995–1008 (1973)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    De Leo, S., Giardino, S.: Dirac solutions for quaternionic potentials. J. Math. Phys. 55, 022301 (2014). arXiv:1311.6673 [math-ph]
  8. 8.
    De Leo, S., Ducati, G., Giardino, S.: Quaternioninc Dirac scattering. J. Phys. Math. 6, 1000130 (2015). arXiv:1505.01807 [math-ph]MATHGoogle Scholar
  9. 9.
    De Leo, S., Rotelli, P.: Quaternionic analyticity. Appl. Math. Lett. 16, 1077–1081 (2003)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Elkhidir, E., Karateev, D., Serone, M.: General three-point functions in 4D CFT. JHEP 01, 133 (2015). arXiv:1412.1796 [hep-th]ADSCrossRefGoogle Scholar
  11. 11.
    Evans, M., Gursey, F., Ogievetsky, V.: From 2-D conformal to 4-D selfdual theories: quaternionic analyticity. Phys. Rev. D 47, 3496–3508 (1993). arXiv:hep-th/9207089 ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gentili, G., Stoppato, C., Struppa, D.C.: Regular Functions of a Quaternionic Variable. Springer Monographs in Mathematics (2013). doi: 10.1007/978-3-642-33871-7
  13. 13.
    Giardino, S.: Möbius transformation for left-derivative quaternion holomorphic functions. Adv. Appl. Clifford Algebras (2016). doi: 10.1007/s00006-016-0673-y. arXiv:1508.01933 [math-ph]
  14. 14.
    Giardino, S.: Quaternionic particle in a relativistic box. Found. Phys. 46(4), 473–483 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Giardino, S., Teotonio-Sobrinho, P.: A non-associative quaternion scalar field theory. Mod. Phys. Lett. A 28(35), 1350163 (2013). arXiv:1211.5049 [math-ph]
  16. 16.
    Gilmore, R.: Lie Groups, Lie Algebras and Some of Their Applications. Dover, New York (2005)MATHGoogle Scholar
  17. 17.
    Ginsparg, P.H.: Applied conformal field theory. Les Houches Summer School (1988). arXiv:hep-th/9108028
  18. 18.
    Gursey, F., Tze, H.C.: Complex and quaternionic analyticity in chiral and gauge theories. Part 1. Ann. Phys. 128, 29 (1980)ADSCrossRefMATHGoogle Scholar
  19. 19.
    Hall, B.: Lie Groups, Lie Algebras and Representations: An Elementary Introduction. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  20. 20.
    Moon, S., Lee, S.-J., Lee, J., Oh, J.-H.: Electric-magnetic duality implies (global) conformal invariance. J. Korean Phys. Soc. 67(3), 427–432 (2015). arXiv:1405.4934 [hep-th]ADSCrossRefGoogle Scholar
  21. 21.
    Pomoni, E., Rastelli, L.: Large N field theory and AdS tachyons. JHEP 04, 020 (2009). arXiv:0805.2261 [hep-th]ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Popov, A.D.: Holomorphic Chern–Simons–Witten theory: from 2-D to 4-D conformal field theories. Nucl. Phys. B 550, 585–621 (1999). arXiv:hep-th/9806239 ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rattazzi, R., Rychkov, V.S., Tonni, E., Vichi, A.: Bounding scalar operator dimensions in 4D CFT. JHEP 12, 031 (2008). arXiv:0807.0004 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Sudbery, A.: Quaternionic analysis. Math. Proc. Camb. Philos. Soc. 85, 199–225 (1979)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Vos, G.: Generalized additivity in unitary conformal field theories. Nucl. Phys. B 899, 91–111 (2015). arXiv:1405.7941 [hep-th]ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Departamento de Física, Centro de Matemática e AplicaçõesUniversidade da Beira InteriorCovilhãPortugal

Personalised recommendations