Torsion in Cohomology Groups of Configuration Spaces

T. Maciążek and A. Sawicki Center for Theoretical Physics PAS, Al. Lotników 32/46, 02-668 Warsaw, Poland

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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.

DOI: 10.12693/APhysPolA.132.1695 topics: Quantum graphs, quantum statistics, vector bundles, homology groups