Topological Auantum Field Theories derived from the Kauffman bracket

Blanchet, Christian; Habegger, Nathan; Masbaum, Gregor; Vogel, Pierre

doi:10.1016/0040-9383(94)00051-4

Cited by 316 publications

(701 citation statements)

References 27 publications

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“…(For n = 1 the specialized Kauffman polynomial is the Kauffman bracket, and we will recover the TQFT's obtained in [5].) If s is generic, then we can construct the idempotentỹ λ for λ in the set Γ(C n ) = {λ; λ ∨ 1 ≤ n + 1, λ ∨ 2 ≤ n} , and λ has non-vanishing quantum dimension (see formula (9)) if it belongs to Γ(C n ) = {λ; λ ∨ 1 ≤ n} .…”

Section: The Symplectic Casementioning

confidence: 99%

Modular categories of types B, C and D

Beliakova¹,

Blanchet²

2001

Comment. Math. Helv.

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Abstract.We construct four series of modular categories from the two-variable Kauffman polynomial, without use of the representation theory of quantum groups at roots of unity. The specializations of this polynomial corresponding to quantum groups of types B, C and D produce series of pre-modular categories. One of them turns out to be modular and three others satisfy Bruguières' modularization criterion. For these four series we compute the Verlinde formulas, and discuss spin and cohomological refinements. Mathematics Subject Classification (2000). 57M25, 57R56.

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Section: The Symplectic Casementioning

confidence: 99%

Modular categories of types B, C and D

Beliakova¹,

Blanchet²

2001

Comment. Math. Helv.

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“…half) vertices or edges to regular (resp. half) vertices or edges In words, two elements are indistinguishable with respect to criterion C 2 if they can be cut up into irreducible pieces in a combinatorially isomorphic way (respecting the intersection with S) and with the same S 3 and B 3 terms, but without paying attention (yet) to the homeomorphism types of the irreducible pieces which are not S 3 or B 3 . Figure 5 shows an example of a relative sum graph.…”

Section: Lemma 314 Let S Be Adequate For M the Number Of Fat Vertmentioning

confidence: 99%

“…As a specific and important example, given any compact Lie group G and level k > 0, the Reshetikhin-Turaev TQFT [38], denoted by V G,k , and as reconstructed by [3] fits into the following diagram: …”

Section: Z(a) Z(a)mentioning

confidence: 99%

Positivity of the universal pairing in 3 dimensions

Calegari

Freedman

Walker

2009

J. Amer. Math. Soc.

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Associated to a closed, oriented surface S S is the complex vector space with basis the set of all compact, oriented 3 3 -manifolds which it bounds. Gluing along S S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3 3 -manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary ( 2 + 1 ) (2+1) -dimensional TQFTs. The proof involves the construction of a suitable complexity function c c on all closed 3 3 -manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c ( A B ) ≤ max ( c ( A A ) , c ( B B ) ) c(AB) \le \max (c(AA),c(BB)) for all A , B A,B which bound S S , with equality if and only if A = B A=B . The complexity function c c involves input from many aspects of 3 3 -manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite-volume hyperbolic 3 3 -manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3 3 -manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3 3 -manifolds due to Agol-Storm-Thurston.

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“…However, in [BHMV2] was shown that there is always a canonical TQFT. In our case this is the one of Lickorish, Blanchet, Habegger, Masbaum and Vogel.…”

Section: Introductionmentioning

confidence: 99%