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

Cohomological Hall algebra of a symmetric quiver

In the paper \cite{KS}, Kontsevich and Soibelman in particular associate to each finite quiver $Q$ with a set of vertices $I$ the so-called Cohomological Hall algebra \cH, which is $\Z_{\geq 0}^I$-graded. Its graded component \cH_{\gamma} is defined as cohomology of Artin moduli stack of representations with dimension vector $\gamma.$ The product comes from natural correspondences which parameterize extensions of representations. In the case of symmetric quiver, one can refine the grading to $\Z_{\geq 0}^I\times\Z,$ and modify the product by a sign to get a super-commutative algebra (\cH,\star) (with parity induced by $\Z$-grading). It is conjectured in \cite{KS} that in this case the algebra (\cH\otimes\Q,\star) is free super-commutative generated by a $\Z_{\geq 0}^I\times\Z$-graded vector space of the form V=V^{prim}\otimes\Q[x], where $x$ is a variable of bidegree $(0,2)\in\Z_{\geq 0}^I\times\Z,$ and all the spaces $\bigoplus\limits_{k\in\Z}V^{prim}_{\gamma,k},$ $\gamma\in\Z_{\geq 0}^I.$ are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs $(\gamma,k)$ for which $V^{prim}_{\gamma,k}\ne 0$ (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson-Thomas invariants, which was used by S. Mozgovoy to prove Kac's conjecture for quivers with sufficiently many loops \cite{M}. Finally, we mention a connection with the paper of Reineke \cite{R}.Comment: 16 pages, no figures; a reference adde