Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Unified quantum invariants for integral homology spheres associated with
  simple Lie algebras   For each finite dimensional, simple, complex Lie algebra $\mathfrak g$ and
each root of unity $\xi$ (with some mild restriction on the order) one can
define the Witten-Reshetikhin-Turaev (WRT) quantum invariant $\tau_M^{\mathfrak
g}(\xi)\in \mathbb C$ of oriented 3-manifolds $M$. In the present paper we
construct an invariant $J_M$ of integral homology spheres $M$ with values in
the cyclotomic completion $\widehat {\mathbb Z [q]}$ of the polynomial ring
$\mathbb Z [q]$, such that the evaluation of $J_M$ at each root of unity gives
the WRT quantum invariant of $M$ at that root of unity. This result generalizes
the case ${\mathfrak g}=sl_2$ proved by the first author. It follows that $J_M$
unifies all the quantum invariants of $M$ associated with $\mathfrak g$, and
represents the quantum invariants as a kind of "analytic function" defined on
the set of roots of unity. For example, $\tau_M(\xi)$ 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. It follows that
WRT quantum invariants $\tau_M(\xi)$ for all roots of unity are determined by
the Ohtsuki series, which can be regarded as the Taylor expansion at $q=1$, and
hence by the Le-Murakami-Ohtsuki invariant. Another consequence is that the WRT
quantum invariants $\tau_M^{ \mathfrak g}(\xi)$ are algebraic integers. The
construction of the invariant $J_M$ is done on the level of quantum group, and
does not involve any finite dimensional representation, unlike the definition
of the WRT quantum invariant. Thus, our construction gives a unified,
"representation-free" definition of the quantum invariants of integral homology
spheres.

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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: stat
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: ao-sci