Communications in Mathematical Physics

, Volume 330, Issue 3, pp 1293–1326 | Cite as

n-Particle Quantum Statistics on Graphs

  • J. M. Harrison
  • J. P. Keating
  • J. M. Robbins
  • A. Sawicki
Open Access
Article

Abstract

We develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs.

References

  1. 1.
    Leinaas J.M., Myrheim J.: On the theory of identical particles. Nuovo Cim. 37B, 1–23 (1977)ADSCrossRefGoogle Scholar
  2. 2.
    Souriau J.M.: Structure des systmes dynamiques. Dunod, Paris (1970)Google Scholar
  3. 3.
    Wilczek, F. (ed.): Fractional Statistics and Anyon Superconductivity. World Scientific, Singapore (1990)Google Scholar
  4. 4.
    Dowker J.S.: Remarks on non-standard statistics. J. Phys. A Math. Gen. 18, 3521 (1985)ADSCrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Sawicki A.: Discrete Morse functions for graph configuration spaces. J. Phys. A Math. Theor. 45, 505202 (2012)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Harrison J.M., Keating J.P., Robbins J.M.: Quantum statistics on graphs. Proc. R. Soc. A 467(2125), 212–23 (2011)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Balachandran A.P., Ercolessi E.: Statistics on networks. Int. J. Mod. Phys. A 07, 4633 (1992)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Mathematical Surveys and Monographs, vol. 186. AMS, Providence (2013)Google Scholar
  9. 9.
    Bolte J., Kerner J.: Quantum graphs with singular two-particle interactions. J. Phys. A Math. Theor. 46, 045206 (2013)ADSCrossRefMathSciNetGoogle Scholar
  10. 10.
    Forman R.: Morse theory for cell complexes. Adv. Math. 134, 90145 (1998)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Farley D., Sabalka L.: Discrete Morse theory and graph braid groups. Algebr. Geom. Topol. 5, 1075–1109 (2005)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Ko K.H., Park H.W.: Characteristics of graph braid groups. Discrete Comput. Geom. 48, 915–963 (2012)MATHMathSciNetGoogle Scholar
  13. 13.
    Tutte, W.T.: Graph Theory. Cambridge University Press, New York (2001)Google Scholar
  14. 14.
    Holberg W.: The decomposition of graphs into k-connected components. Discrete Math. 109(1–3), 133–145 (1992)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Abrams, A.: Configuration spaces and braid groups of graphs. Ph.D. thesis, UC Berkeley (2000)Google Scholar
  16. 16.
    Nakahara, M.: Geometry, Topology, and Physics. Hilger, London (1990)Google Scholar
  17. 17.
    Kim J.H., Ko K.H., Park H.W.: Graph braid groups and right-angled Artin groups. Trans. Am. Math. Soc. 364, 309–360 (2012)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Kuratowski K.: Sur le problème des courbes gauches en topologie. Fundam. Math. 15, 271–283 (1930)MATHGoogle Scholar
  19. 19.
    Farley D., Sabalka L.: On the cohomology rings of tree braid groups. J. Pure Appl. Algebra 212, 53–71 (2008)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Farley D., Sabalka L.: Presentations of graph braid groups. Forum Math. 24, 827–859 (2012)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  • J. M. Harrison
    • 1
  • J. P. Keating
    • 2
  • J. M. Robbins
    • 2
  • A. Sawicki
    • 2
    • 3
  1. 1.Department of MathematicsBaylor UniversityWacoUSA
  2. 2.School of MathematicsUniversity of BristolBristolUK
  3. 3.Center for Theoretical PhysicsPolish Academy of SciencesWarsawPoland

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