The Yang-Mills measure in the Kauffman bracket skein module

Bullock, Doug; Frohman, Charles; Kania-Bartoszynska, Joanna

doi:10.1007/s000140300000

Cited by 15 publications

(30 citation statements)

References 25 publications

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“…This object is a C-algebra equipped with a trace (known as Yang-Mills trace) and hence a complex bilinear form [5], see Section 6. We show here the following.…”

Section: 2mentioning

confidence: 99%

An analytic family of representations for the mapping class group of punctured surfaces

Costantino

Martelli

2014

Geom. Topol.

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Abstract. We use quantum invariants to define an analytic family of representations for the mapping class group Mod(Σ) of a punctured surface Σ. The representations depend on a complex number A with |A| 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A| < 1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from ±1 and ±i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations.The unitary representations in the interval [−1, 0] interpolate analytically between two natural geometric unitary representations, the SU(2)-character variety representation studied by Goldman and the multicurve representation induced by the action of Mod(Σ) on multicurves.The finite-dimensional representations converge analytically to the infinitedimensional ones. We recover Marché and Narimannejad's convergence theorem, and Andersen, Freedman, Walker and Wang's asymptotic faithfulness, that states that the image of a non-central mapping class is always non-trivial after some level r 0 . When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level r 0 in term of its dilatation.

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“…This object is a C-algebra equipped with a trace (known as Yang-Mills trace) and hence a complex bilinear form [5], see Section 6. We show here the following.…”

Section: 2mentioning

confidence: 99%

An analytic family of representations for the mapping class group of punctured surfaces

Costantino

Martelli

2014

Geom. Topol.

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“…Although it can be shown using the Symmetry Principle for the shadow world formula, we can see this more simply using the formulas from [3]. Note that in this section, subscripts of YM are used to indicate the value of the complex parameter t.…”

Section: The Shadow Worldmentioning

confidence: 99%

“…To compute the Yang-Mills measure of the skein s in K r (F ), first puncture the surface, to get a surface F ′ with boundary, and then take the sum over all the labels u on the blackboard framed knot ∂ parallel to the boundary [3], as follows:…”

Section: Remarkmentioning

confidence: 99%

“…In [3] we constructed a diffeomorphism invariant trace YM : K t (F ) → C, called the Yang-Mills measure, that is defined for all t with |t| = 1 and when t is a root of unity. When t = −1 this trace coincides with integration against the symplectic measure on the variety underlying the SU (2)-characters of π 1 (F ).…”

Section: Introductionmentioning

confidence: 99%

“…This paper is organized as follows. Section 2 contains definitions and recalls from [3] the construction of the Yang-Mills measure away from roots of unity. Section 3 derives the state sum formula for the Yang-Mills measure.…”

Section: Introductionmentioning

confidence: 99%

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Shadow world evaluation of the Yang–Mills measure

Frohman

Kania-Bartoszynska

2004

Algebr. Geom. Topol.

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A new state-sum formula for the evaluation of the Yang-Mills measure in the Kauffman bracket skein algebra of a closed surface is derived. The formula extends the Kauffman bracket to diagrams that lie in surfaces other than the plane. It also extends Turaev's shadow world invariant of links in a circle bundle over a surface away from roots of unity. The limiting behavior of the Yang-Mills measure when the complex parameter approaches −1 is studied. The formula is applied to compute integrals of simple closed curves over the character variety of the surface against Goldman's symplectic measure.

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Trace Functionals on Noncommutative Deformations of Moduli Spaces of Flat Connections

Roché¹,

Szenes

2002

Advances in Mathematics

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We describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a Riemann surface. The resulting algebra is a variant of the quantum moduli algebra introduced by Alekseev, Grosse, and Schomerus and Buffenoir and Roche. We construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of Fedosov and Nest and Tsygan via Verlinde's formula. © 2002 Elsevier Science (USA) Key Words: formal deformations; quantum groups; moduli spaces. Contents.0. Introduction. • the theory of star products or formal deformations of symplectic manifolds, more specifically, the index theorem of Fedosov and Nest and Tsygan (F-NT) [14,19,29].Our first result is a greatly simplified, invariant construction of the quantum moduli algebra. We fix a generic conjugacy class s. The quantum moduli algebra is a canonically defined noncommutative algebra A q , which is finitely generated over a subring D(q) of rational functions in q. The main focus of the present work is shown in the following commuting diagram:We are searching for a natural module K defined over D(q), a trace functional Tr q and a map ev which complete the diagram. In other words, we seek to lift the canonical trace from the infinitesimal world to the global q world.A natural solution to this lifting problem seems to be setting K to be D(q) and ev: D(q) Q CQ(R to be the Laurent expansion at q=1. Surprisingly, cyclic functionals on A Our results are subject to various restrictions and assumption which we have omitted in the discussion so far for the sake of clarity. These will be discussed below and in the main body of the paper.Contents of the paper. In Section 1 we recall the volume formula of Witten and Verlinde's formula. We compute the asymptotics of the q-volume series that we introduce by analogy. In Section 2 we recall the

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The Yang-Mills measure in the Kauffman bracket skein module

Cited by 15 publications

References 25 publications

An analytic family of representations for the mapping class group of punctured surfaces

An analytic family of representations for the mapping class group of punctured surfaces

Shadow world evaluation of the Yang–Mills measure

Trace Functionals on Noncommutative Deformations of Moduli Spaces of Flat Connections

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