Topology of multiple log transforms of 4-manifolds

Given a 4-manifold X and an imbedding of T^{2} x B^2 into X, we describe an algorithm X --> X_{p,q} for drawing the handlebody of the 4-manifold obtained from X by (p,q)-logarithmic transforms along the parallel tori. By using this algorithm, we obtain a simple handle picture of the Dolgachev surface E(1)_{p,q}, from that we deduce that the exotic copy E(1)_{p,q} # 5(-CP^2) of E(1) # 5(-CP^2) differs from the original one by a codimension zero simply connected Stein submanifold M_{p,q}, which are therefore examples of infinitely many Stein manifolds that are exotic copies of each other (rel boundaries). Furthermore, by a similar method we produce infinitely many simply connected Stein submanifolds Z_{p} of E(1)_{p,2} # 2(-CP^2)$ with the same boundary and the second Betti number 2, which are (absolutely) exotic copies of each other; this provides an alternative proof of a recent theorem of the author and Yasui [AY4]. Also, by using the description of S^2 x S^2 as a union of two cusps glued along their boundaries, and by using this algorithm, we show that multiple log transforms along the tori in these cusps do not change smooth structure of S^2 x S^2.Comment: Updated, with 17 pages 21 figure

Exact dimer ground state of the two dimensional Heisenberg spin system SrCu_2(BO_3)_2

The two dimensional Heisenberg model for SrCu_2(BO_3)_2 has the exact dimer ground state which was proven by Shastry and Sutherland almost twenty years ago. The critical value of the quantum phase transition from the dimer state to the N\'{e}el ordered state is determined. Analysis of the experimental data shows that SrCu_2(BO_3)_2 has the dimer ground state but is close to the transition point, which leads to the unusual temperature dependence of the susceptibility. Almost localized nature of the triplet excitations explains the plateaus observed in the magnetization curve.Comment: 4 pages, 5 figures, to appear in PR