34 research outputs found
A New Matrix-Tree Theorem
The classical Matrix-Tree Theorem allows one to list the spanning trees of a
graph by monomials in the expansion of the determinant of a certain matrix. We
prove that in the case of three-graphs (that is, hypergraphs whose edges have
exactly three vertices) the spanning trees are generated by the Pfaffian of a
suitably defined matrix. This result can be interpreted topologically as an
expression for the lowest order term of the Alexander-Conway polynomial of an
algebraically split link. We also prove some algebraic properties of our
Pfaffian-tree polynomial.Comment: minor changes, 29 pages, version accepted for publication in Int.
Math. Res. Notice
Matrix factorizations and singularity categories for stacks
We study matrix factorizations of a section W of a line bundle on an
algebraic stack. We relate the corresponding derived category (the category of
D-branes of type B in the Landau-Ginzburg model with potential W) with the
singularity category of the zero locus of W generalizing a theorem of Orlov. We
use this result to construct push-forward functors for matrix factorizations
with relatively proper support.Comment: 29 page
Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations
We study the category of matrix factorizations for an isolated hypersurface
singularity. We compute the canonical bilinear form on the Hochschild homology
of this category. We find explicit expressions for the Chern character and the
boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula
for the Euler characteristic of the Hom-space between a pair of matrix
factorizations. We also establish G-equivariant versions of these results.Comment: v1: 45 pages, v2: added the generalized HRR theorem (Cardy condition)
and new examples with the boundary-bulk maps, v3: 54 pages, to appear in Duke
Math.
Chiral de Rham complex
The aim of this note is to define certain sheaves of vertex algebras on
smooth manifolds. For each smooth complex algebraic (or analytic) manifold ,
we construct a sheaf , called the {\bf chiral de Rham complex}
of . It is a sheaf of vertex algebras in the Zarisky (or classical)
topology, It comes equipped with a \BZ-grading by {\it fermionic charge}, and
the {\it chiral de Rham differential} , which is an endomorphism
of degree 1 such that . One has a canonical embedding of the
usual de Rham complex (\Omega_X, d_{DR})\hra (\Omega_X^{ch}, d_{DR}^{ch})
which is a quasiisomorphism.
If is Calabi-Yau then this sheaf admits an N=2 supersymmetry. For some
(for example, for curves or for the flag spaces ), one can construct
also a purely even analogue of this sheaf, a {\it chiral structure sheaf}
\CO^{ch}_X. For the projective line, the space of global sections of the last
sheaf is the irreducible vacuum \hsl(2)-module on the critical level.Comment: 37 pages, Tex. Completed and revised version, to appear in CMP (1999