Secular determinants of random unitary matrices.

We consider the characteristic polynomials of random unitary matrices U drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The variances of these `secular coefficients' are given explicitly for arbitrary dimension and continued analytically to arbitrary values of the level repulsion exponent beta. The latter secular coefficients are related to the traces of powers of U by Newton's well known formulae. While the traces tend to have Gaussian distributions and to be statistically independent among one another in the limit as the matrix dimension grows large, the secular coefficients exhibit strong mutual correlations due to Newton's mixing of traces to coefficients. These results might become relevant for current efforts at combining semiclassics and random-matrix theory in quantum treatments of classically chaotic dynamics

Kondo Behavior of U in CaB6_6

Replacing U for Ca in semiconducting CaB$_6$ at the few at.% level induces metallic behaviour and Kondo-type phenomena at low temperatures, a rather unusual feature for U impurities in metallic hosts. For Ca$_{0.992}$U$_{0.008}$B$_6$, the resistance minimum occurs at $T$ = 17 K. The subsequent characteristic logarithmic increase of the resistivity with decreasing temperature merges into the expected $T^2$ dependence below 0.8 K. Data of the low-temperature specific heat and the magnetization are analyzed by employing a simple resonance-level model. Analogous measurements on LaB$_6$ with a small amount of U revealed no traces of Kondo behavior, above 0.4 K.Comment: 4 pages, 4 figures, submitted for publication to Europhysics Letter

Feigin-Frenkel center in types B, C and D

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of Feigin and Frenkel about two decades ago. However, only recently simple formulas for the generators of the center were found for the Lie algebras of type A following Talalaev's discovery of explicit higher Gaudin Hamiltonians. We give explicit formulas for generators of the centers of the affine vertex algebras V_{crit}(g) associated with the simple Lie algebras g of types B, C and D. The construction relies on the Schur-Weyl duality involving the Brauer algebra, and the generators are expressed as weighted traces over tensor spaces and, equivalently, as traces over the spaces of singular vectors for the action of the Lie algebra sl_2 in the context of Howe duality. This leads to explicit constructions of commutative subalgebras of the universal enveloping algebras U(g[t]) and U(g), and to higher order Hamiltonians in the Gaudin model associated with each Lie algebra g. We also introduce analogues of the Bethe subalgebras of the Yangians Y(g) and show that their graded images coincide with the respective commutative subalgebras of U(g[t]).Comment: 29 pages, constructions of Pfaffian-type Sugawara operators and commutative subalgebras in universal enveloping algebras are adde