26/10/2005, 15:00 — 16:00 — Anfiteatro Pa2, Pavilhão de Matemática Sergei Gukov, Harvard University, Cambridge
Knot Homology and Topological Strings - I
We start with a brief introduction into knot
homology theories and categorification of polynomial knot
invariants. Of particular interest are homology theories of
Ozsvath-Szabo-Rasmussen and Khovanov-Rozansky which provide a
homological lift of the Alexander polynomial and the quantum sl(N)
invariant, respectively. Motivated by the ideas from physics, we
then present a framework for unifying the sl(N) Khovanov-Rozansky
homology (for all N) with the knot Floer homology. We argue that
this unification should be accomplished by a triply graded homology
theory which categorifies the HOMFLY polynomial. We also describe
the geometric meaning of the new knot invariants in terms of the
enumerative geometry of Riemann surfaces with boundaries in a
certain Calabi-Yau three-fold.
REFERENCES:
S. Gukov, A. Schwarz and C. Vafa, Khovanov-Rozansky
homology and topological strings, hep-th/0412243.
N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial
for Knot Homologies, math.gt/0505662.