189 research outputs found
Khovanov homology is an unknot-detector
We prove that a knot is the unknot if and only if its reduced Khovanov
cohomology has rank 1. The proof has two steps. We show first that there is a
spectral sequence beginning with the reduced Khovanov cohomology and abutting
to a knot homology defined using singular instantons. We then show that the
latter homology is isomorphic to the instanton Floer homology of the sutured
knot complement: an invariant that is already known to detect the unknot.Comment: 124 pages, 13 figure
Gauge theory and Rasmussen's invariant
A previous paper of the authors' contained an error in the proof of a key
claim, that Rasmussen's knot-invariant s(K) is equal to its gauge-theory
counterpart. The original paper is included here together with a corrigendum,
indicating which parts still stand and which do not. In particular, the
gauge-theory counterpart of s(K) is not additive for connected sums.Comment: This version bundles the original submission with a 1-page
corrigendum, indicating the error. The new version of the corrigendum points
out that the invariant is not additive for connected sums. 23 pages, 3
figure
Vortex Counting and Lagrangian 3-manifolds
To every 3-manifold M one can associate a two-dimensional N=(2,2)
supersymmetric field theory by compactifying five-dimensional N=2
super-Yang-Mills theory on M. This system naturally appears in the study of
half-BPS surface operators in four-dimensional N=2 gauge theories on one hand,
and in the geometric approach to knot homologies, on the other. We study the
relation between vortex counting in such two-dimensional N=(2,2) supersymmetric
field theories and the refined BPS invariants of the dual geometries. In
certain cases, this counting can be also mapped to the computation of
degenerate conformal blocks in two-dimensional CFT's. Degenerate limits of
vertex operators in CFT receive a simple interpretation via geometric
transitions in BPS counting.Comment: 70 pages, 29 figure
Cones, Tri-Sasakian Structures and Superconformal Invariance
In this note we show that rigid N=2 superconformal hypermultiplets must have
target manifolds which are cones over tri-Sasakian metrics. We comment on the
relation of this work to cone-branes and the AdS/CFT correspondence.Comment: 10 pages, Latex2
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