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Nonassociativity, Dirac monopoles and Aharonov-Bohm effect
The Aharonov-Bohm (AB) effect for the singular string associated with the
Dirac monopole carrying an arbitrary magnetic charge is studied. It is shown
that the emerging difficulties in explanation of the AB effect may be removed
by introducing nonassociative path-dependent wave functions. This provides the
absence of the AB effect for the Dirac string of magnetic monopole with an
arbitrary magnetic charge.Comment: Revised version. Typos corrected. References adde
Resonance varieties and Dwyer-Fried invariants
The Dwyer-Fried invariants of a finite cell complex X are the subsets
\Omega^i_r(X) of the Grassmannian of r-planes in H^1(X,\Q) which parametrize
the regular \Z^r-covers of X having finite Betti numbers up to degree i. In
previous work, we showed that each \Omega-invariant is contained in the
complement of a union of Schubert varieties associated to a certain subspace
arrangement in H^1(X,\Q). Here, we identify a class of spaces for which this
inclusion holds as equality. For such "straight" spaces X, all the data
required to compute the \Omega-invariants can be extracted from the resonance
varieties associated to the cohomology ring H^*(X,\Q). In general, though,
translated components in the characteristic varieties affect the answer.Comment: 39 pages; to appear in "Arrangements of Hyperplanes - Sapporo 2009,"
Advanced Studies in Pure Mathematic
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