16 research outputs found
Localization for Involutions in Floer Cohomology
We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.National Science Foundation (U.S.) (grant DMS-0405516)National Science Foundation (U.S.) (grant DMS-065260)European Research Council (grant ERC-2007-StG-205349
Knot homology via derived categories of coherent sheaves II, sl(m) case
Using derived categories of equivariant coherent sheaves we construct a knot
homology theory which categorifies the quantum sl(m) knot polynomial. Our knot
homology naturally satisfies the categorified MOY relations and is
conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is
motivated by the geometric Satake correspondence and is related to Manolescu's
by homological mirror symmetry.Comment: 51 pages, 9 figure
Categorification of a linear algebra identity and factorization of Serre functors
We provide a categorical interpretation of a well-known identity from linear
algebra as an isomorphism of certain functors between triangulated categories
arising from finite dimensional algebras.
As a consequence, we deduce that the Serre functor of a finite dimensional
triangular algebra A has always a lift, up to shift, to a product of suitably
defined reflection functors in the category of perfect complexes over the
trivial extension algebra of A.Comment: 18 pages; Minor changes, references added, new Section 2.
Symplectic cohomology and q-intersection numbers
Given a symplectic cohomology class of degree 1, we define the notion of an
equivariant Lagrangian submanifold. The Floer cohomology of equivariant
Lagrangian submanifolds has a natural endomorphism, which induces a grading by
generalized eigenspaces. Taking Euler characteristics with respect to the
induced grading yields a deformation of the intersection number. Dehn twists
act naturally on equivariant Lagrangians. Cotangent bundles and Lefschetz
fibrations give fully computable examples. A key step in computations is to
impose the "dilation" condition stipulating that the BV operator applied to the
symplectic cohomology class gives the identity. Equivariant Lagrangians mirror
equivariant objects of the derived category of coherent sheaves.Comment: 32 pages, 9 figures, expanded introduction, added details of example
7.5, added discussion of sign
Surface Operators and Knot Homologies
Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot invariants. We illustrate the general construction with examples based on surface operators in N=2 and N=4 twisted gauge theories which lead to a categorification of the Alexander polynomial, the equivariant knot signature, and certain analogs of the Casson invariant
Defect Perturbations in Landau-Ginzburg Models
Perturbations of B-type defects in Landau-Ginzburg models are considered. In
particular, the effect of perturbations of defects on their fusion is analyzed
in the framework of matrix factorizations. As an application, it is discussed
how fusion with perturbed defects induces perturbations on boundary conditions.
It is shown that in some classes of models all boundary perturbations can be
obtained in this way. Moreover, a universal class of perturbed defects is
constructed, whose fusion under certain conditions obey braid relations. The
functors obtained by fusing these defects with boundary conditions are twist
functors as introduced in the work of Seidel and Thomas.Comment: 46 page
The bubble algebra: structure of a two-colour Temperley–Lieb Algebra
We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley–Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang–Baxter equations