A gerby deformation of complex tori and the homological mirror symmetry

Let $(X,\check{X})$ be a mirror pair of a complex torus $X$ and its mirror partner $\check{X}$. This mirror pair is described as the trivial special Lagrangian torus fibrations $X\rightarrow B$ and $\check{X}\rightarrow B$ on the same base space $B$ by SYZ construction. Then, we can associate a holomorphic line bundle $E(s,\mathcal{L})\rightarrow X$ to a pair $(s,\mathcal{L})$ of a Lagrangian section $s$ of $\check{X}\rightarrow B$ and a unitary local system $\mathcal{L}$ along it. In this paper, we first construct the deformation $X_{\mathcal{G}}$ of $X$ by a certain flat gerbe $\mathcal{G}$ and its mirror partner $\check{X}_{\mathcal{G}}$ from the mirror pair $(X,\check{X})$, and discuss deformations of objects $E(s,\mathcal{L})$ and $(s,\mathcal{L})$ over the deformed mirror pair $(X_{\mathcal{G}},\check{X}_{\mathcal{G}})$.Comment: 35 page

GEOMETRIC INTERPRETATION FOR EXACT TRIANGLES CONSISTING OF PROJECTIVELY FLAT BUNDLES ON HIGHER DIMENSIONAL COMPLEX TORI

Let (X^n, X^^ˇ^n) be a mirror pair of an n-dimensional complex torus X^n and its mirror partner X^^ˇ^n. Then, a simple projectively flat bundle E(L,) → X^n is constructed from each affine Lagrangian submanifold L in X^^ˇ^n with a unitary local system → L. In this paper, we first interpret these simple projectively flat bundles E(L,) in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles E(L,) and their shifts by focusing on the dimension of intersections of the corresponding affine Lagrangian submanifolds L. Finally, as an application of this geometric interpretation, we discuss whether such an exact triangle on X^n (n ≥ 2) is obtained as the pullback of an exact triangle on X^1 by a suitable holomorphic projection X^n → X^1

NMR Evidence for Antiferromagnetic Transition in the Single-Component Molecular System, [Cu(tmdt)2_{2}]

The magnetic state of the single-component molecular compound, [Cu(tmdt)$_{2}$], is investigated by means of $^{1}$H-NMR. An abrupt spectral broadening below 13 K and a sharp peak in nuclear spin-lattice relaxation rate, $T_{1}^{-1}$, at 13 K are observed as clear manifestations of a second-order antiferromagnetic transition, which is consistent with the previously reported magnetic susceptibility and EPR measurement. The ordered moment is estimated at $0.22-0.45$ ${\mu}_{\rm B}$/molecule. The temperature-dependence of $T_{1}^{-1}$ above the transition temperature indicates one-dimensional spin dynamics and supports that the spins are on the central part of the molecule differently from other isostructural compounds.Comment: 13pages, 5 figure