Higher residues and canonical pairing on the twisted de Rham cohomology

We describe an explicit formula of the canonical pairing on the twisted de Rham cohomology associated with the category of local matrix factorizations and by characterizing its relation to Saito's higher residue pairings, we reprove the conjecture of Shklyarov.Comment: This is the final revised version: some sections reorganized, typos corrected, references adde

Branes at \C^4/\Ga Singularity from Toric Geometry

We study toric singularities of the form of \C^4/\Ga for finite abelian groups \Ga \subset SU(4). In particular, we consider the simplest case \Ga=\Z_2 \times \Z_2 \times \Z_2 and find explicitly charge matrices for partial resolutions of this orbifold by extending the method by Morrison and Plesser. We obtain three kinds of algebraic equations, $z_1 z_2 z_3 z_4=z_5^2, z_1 z_2 z_3=z_4^2 z_5$ and $z_1 z_2 z_5 = z_3 z_4$ where $z_i$'s parametrize \C^5. When we put $N$ D1 branes at this singularity, it is known that the field theory on the worldvolume of $N$ D1 branes is T-dual to $2 \times 2 \times 2$ brane cub model. We analyze geometric interpretation for field theory parameters and moduli space.Comment: 1 figure, 4 tables, latex file and 26 pages:v1 added mathematical results on projective crepant resolutions by Dais et al and refs added:v2 typos corrected and the beginning paragrphs in section 3 clarifie

Symmetry of Quantum Torus with Crossed Product Algebra

In this paper, we study the symmetry of quantum torus with the concept of crossed product algebra. As a classical counterpart, we consider the orbifold of classical torus with complex structure and investigate the transformation property of classical theta function. An invariant function under the group action is constructed as a variant of the classical theta function. Then our main issue, the crossed product algebra representation of quantum torus with complex structure under the symplectic group is analyzed as a quantum version of orbifolding. We perform this analysis with Manin's so-called model II quantum theta function approach. The symplectic group Sp(2n,Z) satisfies the consistency condition of crossed product algebra representation. However, only a subgroup of Sp(2n,Z) satisfies the consistency condition for orbifolding of quantum torus.Comment: LaTeX 17pages, changes in section 3 on crossed product algebr

Morita Equivalence of Noncommutative Supertori

In this paper we study the extension of Morita equivalence of noncommutative tori to the supersymmetric case. The structure of the symmetry group yielding Morita equivalence appears to be intact but its parameter field becomes supersymmetrized having both body and soul parts. Our result is mainly in the two dimensional case in which noncommutative supertori have been constructed recently: The group $SO(2,2,V_{\Z}^0)$, where $V_{\Z}^0$ denotes Grassmann even number whose body part belongs to ${\Z}$, yields Morita equivalent noncommutative supertori in two dimensions.Comment: LaTeX 18 pages, the version appeared in JM