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Register a new userA Groenewold-Van Hove Theorem for S^2
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We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold S^2 which is irreducible on the subalgebra generated by the components {S_1,S_2,S_3} of the spin vector. We also show that there does not exist such a quantization of the Poisson subalgebra P consisting of polynomials in {S_1,S_2,S_3}. Furthermore, we show that the maximal Poisson subalgebra of P containing {1,S_1,S_2,S_3} that can be so quantized is just that generated by {1,S_1,S_2,S_3}.
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An assessment of the magnitude of the rearrangement contribution to the Fermi energy and to the binding energy per particle is carried out in symmetric nuclear matter by extending the G-matrix framework. The restoration of the thermodynamic consistency or, equivalently, the fulfillment of the Hugenholtz-Van Hove theorem, is discussed.
We study the Seiberg-Witten invariant $lambda_{rm{SW}} (X)$ of smooth spin $4$-manifolds $X$ with integral homology of $S^1times S^3$ defined by Mrowka, Ruberman, and Saveliev as a signed count of irreducible monopoles amended by an index-theoretic correction term. We prove a splitting formula for this invariant in terms of the Fr{o}yshov invariant $h(X)$ and a certain Lefschetz number in the reduced monopole Floer homology of Kronheimer and Mrowka. We apply this formula to obstruct existence of metrics of positive scalar curvature on certain 4-manifolds, and to exhibit new classes of integral homology $3$-spheres of Rohlin invariant one which have infinite order in the homology cobordism group.
Let $Sigma$ be a compact Riemann surface and $h_{d,k}(Sigma)$ denote the space of degree $dgeq 1$ holomorphic maps $Sigmara CP^k$. In theoretical physics this arises as the moduli space of charge $d$ lumps (or instantons) in the $CP^k$ model on $Sigma$. There is a natural Riemannian metric on this moduli space, called the $L^2$ metric, whose geometry is conjectured to control the low energy dynamics of $CP^k$ lumps. In this paper an explicit formula for the $L^2$ metric on of $h_{d,k}(Sigma)$ in the special case $d=1$ and $Sigma=S^2$ is computed. Essential use is made of the kahler property of the $L^2$ metric, and its invariance under a natural action of $G=U(k+1)times U(2)$. It is shown that {em all} $G$-invariant kahler metrics on $h_{1,k}(S^2)$ have finite volume for $kgeq 2$. The volume of $h_{1,k}(S^2)$ with respect to the $L^2$ metric is computed explicitly and is shown to agree with a general formula for $h_{d,k}(Sigma)$ recently conjectured by Baptista. The area of a family of twice punctured spheres in $h_{d,k}(Sigma)$ is computed exactly, and a formal argument is presented in support of Baptistas formula for $h_{d,k}(S^2)$ for all $d$, $k$, and $h_{2,1}(T^2)$.
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