Torsion in Cohomology Groups of Configuration Spaces

• Authors: T. Maciążek , A. Sawicki
• Languages of publication: EN

Abstracts

EN

An important and surprising discovery in physics in the last fifty years is that if quantum particles are constrained to move in two rather than three dimensions, they can in principle exhibit new forms of quantum statistics, called anyons. Although anyons were initially only a theoretical concept, they quickly proved to be useful in explaining one of the most significant discoveries of condensed matter physics in the last century, i.e. fractional quantum Hall effect. Recently, it was shown that particles constrained to move on a graph can exhibit even more exotic forms of quantum statistics, depending on the topology of the graph. In this paper we discuss what possible new quantum signatures of topology may arise when one takes into account more complex topological information, called higher (co)homology groups, which may also be associated with graph configuration spaces. In particular we focus on the significance of a torsion component.

Keywords

EN

Quantum graphs; quantum statistics; vector bundles; homology groups

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2017
• Volume: 132
• Issue: 6
• Pages: 1695-1698

Dates

published

2017-12

Contributors

author

T. Maciążek

• Center for Theoretical Physics PAS, Al. Lotników 32/46, 02-668 Warsaw, Poland

author

A. Sawicki

• Center for Theoretical Physics PAS, Al. Lotników 32/46, 02-668 Warsaw, Poland

References

• [1] M. Fierz, Helvet. Phys. Acta 12, 3 (1939) http://inspirehep.net/record/45336?ln=en
• [2] W. Pauli, Phys. Rev. 58, 716 (1940), doi: 10.1103/PhysRev.58.716
• [3] J. Schwinger, Phys. Rev. 82, 914 (1951), doi: 10.1103/PhysRev.82.914
• [4] R. Feynman, Quantum Electrodynamics, Benjamin Cummings, Menlo Park (CA) 1961 http://inis.iaea.org/search/search.aspx?orig_q=RN:21029946
• [5] M.V. Berry, J.M. Robbins, Proc. R. Soc. Lond. A 453, 1771 (1997), doi: 10.1098/rspa.1997.0096
• [6] M.F. Atiyah, Philos. Trans. R. Soc. Lond. A 359, 1 (2001), doi: 10.1098/rsta.2001.0840
• [7] M.F. Atiyah, in: Surveys in Differential Geometry, Vol. 7, International Press, Somerville (MA) 2001, p. 1 http://intlpress.com/site/pub/pages/books/items/00000112/index.html
• [8] M. Atiyah, P. Sutcliffe, Proc. Math. Phys. Eng. Sci. 458, 1089 (2002), doi: 10.1098/rspa.2001.0913
• [9] J.M. Souriau, Structure des systèmes dynamiques, Dunod, Paris 1970, (in French) http://gabay-editeur.com/SOURIAU-Structure-des-systemes-dynamiques-1970
• [10] J.M. Leinaas, J. Myrheim, Nuovo Cim. 37B, 1 (1977), doi: 10.1007/BF02727953
• [11] J.S. Dowker, J. Phys. A Math. Gen. 18, 3521 (1985), doi: 10.1088/0305-4470/18/18/015
• [12] M.G.G. Laidlaw, C.M. DeWitt, Phys. Rev. D 3, 1375 (1971), doi: 10.1103/PhysRevD.3.1375
• [13] F. Wilczek, Fractional Statistics and Anyon Superconductivity, World Sci., Singapore 1990, doi: 10.1142/0961
• [14] J.M. Harrison, J.P. Keating, J.M. Robbins, A. Sawicki, Commun. Math. Phys. 330, 1293 (2014), doi: 10.1007/s00220-014-2091-0
• [15] A. Abrams, Ph.D. Thesis, UC Berkley, 2000 http://home.wlu.edu/~abramsa/publications/thesis.ps
• [16] K.H. Ko, H.W. Park, Discr. Comput. Geom. 48, 915 (2012), doi: 10.1007/s00454-012-9459-8
• [17] R. Forman, Adv. Math. 134, 90145 (1998), doi: 10.1006/aima.1997.1650
• [18] D. Farley, L. Sabalka, Algebr. Geom. Topol. 5, 1075 (2005), doi: 10.2140/agt.2005.5.1075
• [19] D. Farley, L. Sabalka, J. Pure Appl. Algebra 212, 53 (2008), doi: 10.1016/j.jpaa.2007.04.011
• [20] K. Barnett, M. Farber, arXiv: 0903.2180 (2009), http://arXiv.org/abs/0903.2180,
• [21] M. Farber, in: Algorithmic Foundations of Robotics VI, Eds. M. Erdmann, D. Hsu, M. Overmars, A. Frank van der Stappen, Springer, 2005, p. 123 http://springer.com/us/book/9783540257288#otherversion=9783642065132
• [22] A. Sawicki, J. Phys. A Math. Theor. 45, 505202 (2012), doi: 10.1088/1751-8113/45/50/505202
• [23] T. Maciążek, A. Sawicki, J. Math. Phys. 58, 062103 (2017), doi: 10.1063/1.4984309
• [24] J.K. Asbóth, L. Oroszlány, A. Pályi, in: Lecture Notes in Physics, 2016, p. 919, doi: 10.1007/978-3-319-25607-8
• [25] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002 http://lehmanns.de/shop/mathematik-informatik/2678068-9780521795401-algebraic-topology
• [26] A. Hatcher, Vector bundles and K theory, unfinished book http://math.cornell.edu/~hatcher/VBKT/VBpage.html
• [27] J. Cheeger, J. Simons, Differential characters and geometric invariants, Lecture Notes in Mathematics, Vol. 1167, Springer Verlag, 1985, p. 50, doi: 10.1007/BFb0075216

Identifiers

DOI

10.12693/APhysPolA.132.1695