From Singularities to Algebras to Pure Yang–Mills with Matter

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 36)


Since the advent of dualities in string theory, it has been well-known that codimension 4 orbifold singularities that appear in extra-dimensional spaces, such as Calabi–Yau or G 2 spaces, may be interpreted as ADE gauge theories. As to orbifold singularities of higher codimension, there has not been an analog of this interpretation. Here we show how the search for such an analog led us from the singularities to the creation of Lie Algebras of the Third Kind (“LATKes”). We introduce an example of a LATKe that arises from the singularity C 3Z 3, and prove it to be simple and unique. We explain that the uniqueness of the LATKe serves as a vacuum selection mechanism. We also show how the LATKe leads to a new kind of gauge theory in which the matter field arises naturally and which is tantalizingly close to the Standard Model of particle physics.


Gauge Theory Gauge Group Intersection Number Adjoint Representation Dynkin Diagram 



The author is grateful to the organizers, and especially to V. Dobrev, for putting together such a vibrant, stimulating, and enjoyable workshop.


  1. 1.
    Hull, C.M., Townsend, P.K.: Enhanced gauge symmetries in superstring theories. Nucl. Phys. B 451, 525 (1995) [arXiv:hep-th/9505073]MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Witten, E.: String theory dynamics in various dimensions. Nucl. Phys. B 443, 85 (1995) [arXiv:hep-th/9503124]MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Acharya, B.S.: M theory, Joyce orbifolds and super Yang-Mills. Adv. Theor. Math. Phys. 3, 227 (1999) [arXiv:hep-th/9812205]MathSciNetMATHGoogle Scholar
  4. 4.
    Acharya, B.S.: On realising N = 1 super Yang-Mills in M theory [arXiv:hep-th/0011089]Google Scholar
  5. 5.
    Atiyah, M., Witten, E.: M-theory dynamics on a manifold of G 2 holonomy. Adv. Theor. Math. Phys. 6, 1 (2003) [arXiv:hep-th/0107177]MathSciNetGoogle Scholar
  6. 6.
    Friedmann, T.: On the quantum moduli space of M theory compactifications. Nucl. Phys. B 635, 384 (2002) [arXiv:hep-th/0203256]MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Friedmann, T.: Physics through extra dimensions: on dualities, unification, and pair production. Ph.D. thesis, Princeton University, 2003, UMI-31-03026;
  8. 8.
    Friedmann, T., Witten, E.: Unification scale, proton decay, and manifolds of G 2 holonomy. Adv. Theor. Math. Phys. 7, 577 (2003) [arXiv:hep-th/0211269]MathSciNetMATHGoogle Scholar
  9. 9.
    Georgi, H., Glashow, S.L.: Unity of all elementary particle forces. Phys. Rev. Lett. 32, 438 (1974)CrossRefGoogle Scholar
  10. 10.
    T. Friedmann, Orbifold singularities, Lie algebras of the third kind (LATKes), and pure Yang-Mills with matter. J. Math. Phys. 52, 022304 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Klein, F.: Lectures on the Ikosahedron and the Solution of Equations of the Fifth Degree, Translated by G.G. Morrice. Trubner & Co., London (1888)Google Scholar
  12. 12.
    Du Val, P.: On isolated singularities of surfaces which do not affect the conditions of adjunctions. Proc. Camb. Philos. Soc. 30, 453–465, 483–491 (1933/1934)Google Scholar
  13. 13.
    Artin, M.: On isolated rational singularities of surfaces. Am. J. Math. 88, 129–136 (1966)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Steinberg, R.: Kleinian singularities and unipotent elements. Proc. Symp. Pure Math. 37, 265 (1980)CrossRefGoogle Scholar
  15. 15.
    Kac, V.: Lie superalgebras. Adv. Math. 26, 8 (1977)MATHCrossRefGoogle Scholar
  16. 16.
    Green, M.B., Schwarz, J.H.: Anomaly cancellation in supersymmetric D=10 gauge theory and superstring theory. Phys. Lett. B 149, 117 (1984)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gross, D.J., Harvey, J.A., Martinec, E.J., Rohm, R.: The heterotic string. Phys. Rev. Lett. 54, 502 (1985)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Candelas, P., Horowitz, G.T., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nucl. Phys. B 258, 46 (1985)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Douglas, M.R.: The statistics of string/M theory vacua. J. High Energy Phys. 0305, 046 (2003) [arXiv:hep-th/0303194]CrossRefGoogle Scholar

Copyright information

© Springer Japan 2013

Authors and Affiliations

  1. 1.University of RochesterRochesterUSA

Personalised recommendations