A rational surgery formula for the LMO invariant

Bar-Natan, Dror; Lawrence, Ruth

doi:10.1007/bf02786626

Cited by 39 publications

(72 citation statements)

References 19 publications

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“…As we have already seen, for r 3 the only possible exceptional fiber data in this case are (2, 2, p), (2,3,4) and (2,3,5). The corresponding Seifert fibered are precisely the quotients of S 3 by the free action of a group of isometries.…”

Section: Orbifold Euler Characteristicmentioning

confidence: 78%

“…For instance, it could happen (but it is not automatic) that the right multiplication by an element h ∈ G leaves the orbit stable. (4) If the solution itself has a priori some symmetries (like (2.19)), they should also appear in P. (5) We know what is the ramification data of x on : the solutions of P(y, x) = ∂ y P(y, x) = 0 must all be of the form (y b , g(x b )) for some g ∈ G and x b an edge of , and they must also satisfy…”

Section: More Information On the Riemann Surfacementioning

confidence: 99%

“…As a matter of fact, (2.1) first appeared in [5] for the Lê-Murakami-Ohtsuki invariant [39] on Seifert spaces. If M is a closed, framed 3-manifold obtained by surgery on a link in S 3 , the LMO invariant Z LMO (M) is a graph-valued formal power series inh [39].…”

Section: Avatars Of Chern-simons Theorymentioning

confidence: 99%

“…If M is a closed, framed 3-manifold obtained by surgery on a link in S 3 , the LMO invariant Z LMO (M) is a graph-valued formal power series inh [39]. Its relation to the Kontsevich integral-which is a universal formal series of finite-type invariants-and Aarhus integral for rationally framed links was exposed in [2][3][4], see also [5]. For any choice of compact Lie algebra g, it can be evaluated to a g-dependent, formal power series inh.…”

Section: Avatars Of Chern-simons Theorymentioning

confidence: 99%

“…• Bar-Natan [5] has computed the LMO invariant of Seifert rational homology spheres, via the Kontsevich integral.…”

Section: Exact Evaluationsmentioning

confidence: 99%

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Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

Borot

Eynard

Weiße

2016

Sel. Math. New Ser.

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We study in detail the large N expansion of SU(N ) and SO(N )/Sp(2N ) Chern-Simons partition function Z N (M) of 3-manifolds M that are either rational homology spheres or more generally Seifert fibered spaces. This partition function admits a matrix model-like representation, whose spectral curve can be characterized in terms of a certain scalar, linear, non-local Riemann-Hilbert problem (RHP). We develop tools necessary to address a class of such RHPs involving finite subgroups of PSL 2 (C). We associate with such problems a (maybe infinite) root system and describe the relevance of the orbits of the Weyl group in the construction of its solutions. These techniques are applied to the RHP relevant for Chern-Simons theory on Seifert spaces. When π 1 (M) is finite-i.e., for manifolds M that are quotients of S 3 by a finite isometry group of type ADE-we find that the Weyl group associated with the RHP is finite and the spectral curve is algebraic and can be in principle computed. We then show that the large N expansion of Z N (M) is computed by the topological recursion. This has consequences for the analyticity properties of SU/SO/Sp perturbative invariants of knots along fibers in M.Mathematics Subject Classification 14Hxx · 45Exx · 51Pxx · 57M27 · 60B20 · 81T45

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Section: Orbifold Euler Characteristicmentioning

confidence: 78%

Section: More Information On the Riemann Surfacementioning

confidence: 99%

Section: Avatars Of Chern-simons Theorymentioning

confidence: 99%

Section: Avatars Of Chern-simons Theorymentioning

confidence: 99%

“…• Bar-Natan [5] has computed the LMO invariant of Seifert rational homology spheres, via the Kontsevich integral.…”

Section: Exact Evaluationsmentioning

confidence: 99%

See 3 more Smart Citations

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

Borot

Eynard

Weiße

2016

Sel. Math. New Ser.

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Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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This paper develops a method to carry out the large-N asymptotic analysis of a class of Ndimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size N, but in the present problem, two scales 1/N α and 1/N naturally occur. In our case, the equilibrium measure is N α -dependent and characterised by means of the solution to a 2×2 Riemann-Hilbert problem, whose large-N behaviour is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-N behaviour of the free energy explicitly up to o(1). The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales 1/N α and 1/N, thus waiving the analyticity assumptions often used in random matrix theory.1 gborot@mpim-bonn.mpg.de 2 guionnet@math.mit.edu 3 karol.kozlowski@ens-lyon.fr 2 An opening discussionThe present work develops techniques enabling one to carry out the large-N asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh model: [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations. Furthermore, a degeneration of z N [W] arises as a multiple integral representation for the partition function of the six-vertex model subject to domain wall boundary conditions [93]. In the context of quantum integrable systems, the number N of integrals defining z N is related to the number of sites in a model (as, e.g. in the case of the compact or non-compact XXZ chains or the lattice regularisations of the Sinh or Sine-Gordon models) or to the number of particles (as, e.g. in the case of the quantum Toda chain). From the point of view of applications, one is mainly interested in the thermodynamic limit of the model, which is attained by sending N to +∞. For instance, in the case of an integrable lattice discretisation of some quantum field theory, one obtains in this way an exact and non-perturbative description of a quantum field theory in 1 + 1 dimensions and in finite volume. This limit, at the level of z N [W], translates itself in the need to extract the large N-asymptotic expansion of ln z N [W] up to o(1). It is, in fact, the constant term in the expansion of ln zwith W ′ some deformation of W that gives rise to the correlation functions of the underlying quantum field theory in finite volume. These applications to physics constitute the first motivation for our analysis. From the purely mathematical side, the motivation of our works stems ...

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Chern-Simons Theory, Matrix Integrals, and Perturbative Three-Manifold Invariants

Mariño

2004

Commun. Math. Phys.

259

489

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Abstract:The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some previous results of Lawrence and Rozansky, the Chern-Simons partition function with arbitrary simply-laced group for these spaces is written in terms of matrix integrals. The analysis of the perturbative expansion amounts to the evaluation of averages in a Gaussian ensemble of random matrices. As a result, explicit expressions for the universal perturbative invariants of Seifert homology spheres up to order five are presented.

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A rational surgery formula for the LMO invariant

Cited by 39 publications

References 19 publications

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

Root systems, spectral curves, and analysis of a Chern-Simons matrix model for Seifert fibered spaces

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Chern-Simons Theory, Matrix Integrals, and Perturbative Three-Manifold Invariants

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