## The Århus Integral of Rational Homology 3-Spheres I: A Highly Non Trivial Flat Connection on S³ (2002)

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### BibTeX

@MISC{Bar-Natan02theårhus,

author = {Dror Bar-Natan and Stavros Garoufalidis and Lev Rozansky and Dylan P. Thurston},

title = {The Århus Integral of Rational Homology 3-Spheres I: A Highly Non Trivial Flat Connection on S³},

year = {2002}

}

### Years of Citing Articles

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### Abstract

Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly non-trivial mathematical theorems and theories. We argue that even though non-trivial at connections on S³ do not really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the non-existent Chern-Simons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finite-type invariant of rational homology spheres. We show that this invariant is equal (up to a normalization) to the LMO (Le-Murakami-Ohtsuki, [LMO]) invariant and that it recovers the Rozansky and Ohtsuki invariants. This is part I of a 4...

### Citations

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Citation Context ... to a non-existent measure, and show that the resulting averaged holonomies have all the right properties. Namely, we replace hA(L) by hg,k(L) := ∫ hB(L)dµk(B), where dµk(B) is the famed Chern-Simons =-=[Wi]-=- measure on the space of g-connections, depending on some integer parameter k: dµk(B) = exp ( ∫ ik 4π S3 tr B ∧ dB + 2 ) B ∧ B ∧ B DB 3 (DB denotes the path integral measure over the space of g-connec... |

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Citation Context ...mula for the map −→ m xy z x y on uni-trivalent diagrams. Let x and y be two elements in a free associative (but not-commutative) completed algebra. The Baker-Campbel-Hausdorf (BCH) formula (see e.g. =-=[Ja]-=-) measures the failure of the identity e x+y = e x e y to hold, in terms of Lie elements, or, what is the same, in terms of trees modulo the IHX and AS relation. The first few terms in the BCH formula... |

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Citation Context ...ne is integrable so is the other, and we have to prove the equality of the integrals. The case of knots: If n = 1 then the fact that A(↑) is isomorphic to A(�) (namely, the commutativity of A(↑), see =-=[B-N1]-=-) implies that −→ m xy z = −→ m yx z and there’s nothing to prove. The lucky case: If G1,2 are integrable with respect to E we can use proposition 2.13 and compute the integrals with respect to those ... |

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Citation Context ... = ˚A0(U+) −σ+˚A0(U−) −σ−˚A0(L), with all products and powers taken using the disjoint union product of A(∅). Theorem 1. ˚A is invariant under orientation flips and under both Kirby moves, and hence (=-=[Ki]-=-) it is an invariant of rational homology 3-spheres. Our second goal in this article is to prove that ˚A is a universal Ohtsuki invariant, and hence that all Q-valued finite-type invariants of integer... |

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Citation Context ...are true only when this (or a similar, see below) limit is taken. If one is ready to sacrifice some simplicity, all of these statements can be formulated without limits if the technology of qtangles (=-=[LM1]-=-) (or, what is nearly the same, non associative tangles ([B-N3])) is used instead of using specific Morse embeddings. Readers familiar with [LM1] and/or [B-N3] should have no difficulty translating ou... |

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Citation Context ...d in [˚A-I, section 2.2.2]. Definition 4.1. An invariant U of integer homology spheres with values in A(∅) is a “universal Ohtsuki invariant” if 1. The degree m part U (m) of U is of Ohtsuki type 3m (=-=[Oh]-=-). 2. If OGL denotes the Ohtsuki-Garoufalidis-Le map, defined in figure 6, from manifold diagrams to formal linear combinations of unit framed algebraically split links in S3 , and S denotes the surge... |

55 | An invariant of integral homology 3-spheres which is universal for all finite type invariants
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Citation Context ...rational homology 3-spheres. Very briefly, we have defined the pre-normalized ˚Arhus integral ˚A0 to be the composition ⎧ ⎫ ⎨regular⎬ ˚A0 : pure = RPT ⎩ tangles ⎭ In this formula, ˇZ −−−−−−−−−−−→ the =-=[LMMO]-=- version of the Kontsevich integral A(↑X) σ −−−−→ formal PBW B(X) ∫ F G −−−−−−−→ formal Gaussian integration A(∅). • RPT denotes the set of regular pure tangles whose components are marked by the elem... |

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Citation Context ...bras see, while the k → ∞ limit of the Witten invariants is practically limited to semi-simple super Lie algebras. On the other hand, there are Witten-like theories with finite gauge groups, see e.g. =-=[FQ]-=-, which have no parallel in the ˚A world. Question 3.4. What is the relation between the ˚Arhus integral and the Axelrod-Singer perturbative 3-manifold invariants [AS1, AS2] and/or the Kontsevich “con... |

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Citation Context ...he definitions of the two invariants are different, it is nice to have independent proofs of the main properties. From corollary 2.13 and the computation of the low degree parts of A(∅) in [B-N5] and =-=[Kn]-=- it follows that the low-degree dimensions of the associated graded of the space of Ohtsuki invariants are given by the table below. The last row of this table lists the dimensions of “primitives” — m... |

27 |
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Citation Context ...e ˚Arhus integral, still carries some affinity to Reshetikhin’s construction. Question 3.9. What is the relation between the ˚Arhus integral and the p-adic 3-manifold invariants considered by Ohtsuki =-=[Oh1]-=-? Answer: See corollary 2.14. Question 3.10. How powerful is ˚A? Answer: It is a “universal Ohtsuki invariant” (see section 2.2.2). In particular, as the Casson invariant is Ohtsuki-finite-type (see [... |

25 |
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Citation Context ...ees the k → ∞ limit, and only “in the vicinity of the trivial connection”. But this means it sees a splitting (trivial vs. other flat connections) that the Witten invariants don’t see. Also, by Vogel =-=[Vo]-=-, we know that A(∅) “sees” more than all semi-simple super Lie algebras see, while the k → ∞ limit of the Witten invariants is practically limited to semi-simple super Lie algebras. On the other hand,... |

24 | the Kontsevich integral of the unknot - Bar-Natan, Garoufalidis, et al. |

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Citation Context ... 2.13. 1. ˚A is onto A(∅). 2. All Ohtsuki invariants factor through the map ˚A. 3. The dual of A(∅) is the associated graded of the space of Ohtsuki invariants (with degrees divided by 3; recall from =-=[GO]-=- that the associated graded of the space of Ohtsuki invariants vanishes in degrees not divisible by 3.). In view of Le [Le1], this theorem and corollary follow from the fact (discussed below) that the... |

22 | The trivial connection contribution to Witten’s invariant and finite type invariants of rational homology spheres, q-alg/9503011 preprint - Rozansky - 1995 |

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14 |
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Citation Context ...am in A(↑ {x,y,z}) and the STU relation. Both LMO0 and ˚A0 are defined as compositions of several maps. In both cases the first map is ˇ Z, the Kontsevich integral in its Le-Murakami-Murakami-Ohtsuki =-=[LMMO]-=- normalization (check [˚A-I, Definition 2.6] for the adaptation to pure tangles). If X is the set of components of a given regular pure tangle (or more elegantly, a set of “labels” or “colors” for the... |

13 |
homotopy string link invariants, Jour. of Knot Theory and its Ramifications 4
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Citation Context ... The existence of such a formula □THE ˚ARHUS INTEGRAL II: INVARIANCE AND UNIVERSALITY 21 is immediate from the definition of σ : A → B, and this existence was used in several places before (see e.g. =-=[B-N2]-=-), but we are not aware of a previous place where this formula was written explicitly. A similar formula is the “wheeling formula” of [BGRT]. Proof of proposition 5.4. Let A xy be the space of “plante... |

13 | The ˚Arhus integral of rational homology - Bar-Natan, Garoufalidis, et al. |

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10 |
diagrams and low-dimensional topology, First European
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Citation Context ...the ˚A world. Question 3.4. What is the relation between the ˚Arhus integral and the Axelrod-Singer perturbative 3-manifold invariants [AS1, AS2] and/or the Kontsevich “configuration space integrals” =-=[Ko2]-=-? Answer: We expect the ˚Arhus integral to be the same as Kontsevich’s configuration space integrals and as the formal (no-Lie-algebra) version of the Axelrod-Singer invariants, perhaps modulo some mi... |

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4 |
On a universal perturbative invariant of 3-manifolds, Topology
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Citation Context ...ontinuing the work started in [˚A-I] and [˚A-II], we prove the relationship between the ˚Arhus integral and the invariant Ω (henceforth called LMO) defined by T.Q.T. Le, J. Murakami and T. Ohtsuki in =-=[LMO]-=-. The basic reason for the relationship is that both constructions afford an interpretation as “integrated holonomies”. In the case of the ˚Arhus integral, this interpretation was the basis for everyt... |

4 |
invariants of rational homology spheres at prime values of K and trivial connection contribution
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(Show Context)
Citation Context ...ic of g, as in Section 1 and as in [B-N2], and �deg is the operator that multiplies each diagram D in A(∅) by � raised to the degree of D. Corollary 2.14. LMO recovers Rg for any g. In particular, by =-=[Ro2]-=-, LMO recovers the “p-adic” invariants of [Oh1, Oh2]. The last statement was proven in the case of g = sl(2) by Ohtsuki [Oh4]. 3. Frequently Asked Questions Let us answer some frequently asked questio... |

3 | invariants via formal geometry - Rozansky-Witten |

2 |
quantum method in 3-dimensional topology, Tokyo Inst. of
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Citation Context ... degree of D. Corollary 2.14. LMO recovers Rg for any g. In particular, by [Ro2], LMO recovers the “p-adic” invariants of [Oh1, Oh2]. The last statement was proven in the case of g = sl(2) by Ohtsuki =-=[Oh4]-=-. 3. Frequently Asked Questions Let us answer some frequently asked questions. Most questions will be answered anyway later in this series, but with the taxi driver already pushing you out the door, w... |

2 |
the Kontsevich integral of the unknot Hebrew
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Citation Context ... this existence was used in several places before (see e.g. [B-N2]), but we are not aware of a previous place where this formula was written explicitly. A similar formula is the “wheeling formula” of =-=[BGRT]-=-. Proof of proposition 5.4. Let A xy be the space of “planted forests” whose leaves are labeled ∂x and ∂y, modulo the usual STU (and hence AS and IHX) relations. A planted forest is simply a forest in... |

2 | Free Lie Algebras, Oxford Scientific - Reutenauer - 1993 |

1 | Chern-Simons theory with gauge group - Freed, Quinn - 1993 |

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1 |
of Vassiliev Invariants, electronic publication
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Citation Context ...t G be a non-degenerate perturbed Gaussian with covariance matrix (lxy), and let (lxy) be the inverse covariance matrix. Set ∫ 〈 ( FG G = exp ·∪ − 1 ∑ l 2 xy ) 〉 ∂x⌣∂y , PG . x,y 5 Strictly speaking, =-=[BGRT]-=- deals only with knots. But the generalization to links is obvious. 6 See [BGRT] for the conjectured value of this invariant, and [BLT] for the proof. 7 Notice that the normalization is different than... |