5 research outputs found
A TQFT associated to the LMO invariant of three-dimensional manifolds
We construct a Topological Quantum Field Theory (in the sense of Atiyah)
associated to the universal finite-type invariant of 3-dimensional manifolds,
as a functor from the category of 3-dimensional manifolds with parametrized
boundary, satisfying some additional conditions, to an algebraic-combinatorial
category. It is built together with its truncations with respect to a natural
grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The
TQFT(s) induce(s) a (series of) representation(s) of a subgroup of
the Mapping Class Group that contains the Torelli group. The N=1 truncation
produces a TQFT for the Casson-Walker-Lescop invariant.Comment: 28 pages, 13 postscript figures. Version 2 (Section 1 has been
considerably shorten, and section 3 has been slightly shorten, since they
will constitute a separate paper. Section 4, which contained only announce of
results, has been suprimated; it will appear in detail elsewhere.
Consequently some statements have been re-numbered. No mathematical changes
have been made.
A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
We construct an invariant J_M of integral homology spheres M with values in a
completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at
each root of unity \zeta gives the the SU(2) Witten-Reshetikhin-Turaev
invariant \tau_\zeta(M) of M at \zeta. Thus J_M unifies all the SU(2)
Witten-Reshetikhin-Turaev invariants of M. As a consequence, \tau_\zeta(M) is
an algebraic integer. Moreover, it follows that \tau_\zeta(M) as a function on
\zeta behaves like an ``analytic function'' defined on the set of roots of
unity. That is, the \tau_\zeta(M) for all roots of unity are determined by a
"Taylor expansion" at any root of unity, and also by the values at infinitely
many roots of unity of prime power orders. In particular, \tau_\zeta(M) for all
roots of unity are determined by the Ohtsuki series, which can be regarded as
the Taylor expansion at q=1.Comment: 66 pages, 8 figure
A unified quantum SO(3) invariant for rational homology 3-spheres
Given a rational homology 3-sphere M with |H 1 (M,ℤ)|=b and a link L inside M, colored by odd numbers, we construct a unified invariant I M,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten-Reshetikhin-Turaev invariants of the pair (M,L). If b=1 and L=∅,I M coincides with Habiro's invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined by I M . Important applications are the new Ohtsuki series (perturbative expansions of I M ) dominating quantum SO(3) invariants at roots of unity whose order is not a power of a prime. These series are not known to be determined by the LMO invariant