We consider a class of two-dimensional Schrdinger operator with a singular interaction of the type and a fixed strength supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an AharonovBohm flux in the center. It is shown that if , there is a critical value such that the discrete spectrum has an accumulation point when , while for the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed and small enough.

In this sequel to [L-Y1],[LLSY], and [L-Y2](respectively arXiv: 0709.1515 [math. AG], arXiv: 0809.2121 [math. AG], and arXiv: 0901.0342 [math. AG]), we study a D-brane probe on a conifold from the viewpoint of the Azumaya structure on D-branes and toric geometry. The details of how deformations and resolutions of the standard toric conifold Y can be obtained via morphisms from Azumaya points are given. This should be compared with the quantum-field-theoretic/D-brany picture of deformations and resolutions of a conifold via a D-brane probe sitting at the conifold singularity in the work of Klebanov and Witten [KW](arXiv: hep-th/9807080) and Klebanov and Strasser [KS](arXiv: hep-th/0007191). A comparison with resolutions via noncommutative desingularizations is given in the end.

In this note we discuss a recent proof of the formula for the worldsheet instanton prepotential predicted by Mirror Symmetry for the quintics in <b>P</b>⁴. One of the key ingredients in the proof is the equivariant cohomology groups on the so-called linear sigma model moduli spaces. We introduce the notion of admissible data on the equivariant cohomology groups of the linear sigma model. An admissible data may be thought of as a sequence of equivariant classes satisfying certain algebraic conditions. It arises naturally from Kontsevich's stable map compactification of moduli spaces of maps from curves into projective manifolds. The structures of admissible data help reduce many counting problems to checking certain combinatorial structure of a compactification. The mirror transformation of Candelas et al turns out to be a transformation between two admissible data associated respectively to the linear and the non-linear

We consider for <i>j</i>=, a spherically symmetric, static system of (2<i>j</i>+1) Dirac particles, each having total angular momentum <i>j</i>. The Dirac particles interact via a classical gravitational and electromagnetic field.

Based on examples from superstring/D-brane theory since the work of Douglas and Moore on resolution of singularities of a superstring target-space Y via a D-brane probe, the richness and the complexity of the stack of punctual D0-branes on a variety, and as a guiding question, we lay down a conjecture that any resolution Y of a variety Y over Y can be factored through an embedding of Y into the stack Y of punctual D0-branes of rank Y on Y for Y in Y , where Y depends on the germ of singularities of Y . We prove that this conjecture holds for the resolution Y of a reduced singular curve Y over Y . In string-theoretical language, this says that the resolution Y of a singular curve Y always arises from an appropriate D0-brane aggregation on Y and that the rank of the Chan-Paton module of the D0-branes involved can be chosen to be arbitrarily large.