Invariants of 3-manifolds via link polynomials and quantum groups

Reshetikhin, Nicolai; Turaev, Vladimir

doi:10.1007/bf01239527

Cited by 1,182 publications

(1,092 citation statements)

References 14 publications

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“…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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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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“…In [15], Witten conceived of combining the Jones polynomial with Chern-Simons quantum theory to produce a three-manifold invariant, which was subsequently realized by Reshetikhin and Turaev in [13]. Further study of these constructions led to what is now referred to as examples of SU(2) and SO(3) Topological Quantum Field Theories.…”

Section: Introductionmentioning

confidence: 99%

SO(3) quantum invariants are dense

Wong¹

2009

Math. Proc. Camb. Phil. Soc.

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“…Physical applications of quantum groups in two-dimensional CFT and three dimensional Chern Simons theory [33,34,35,36,37] show that, in addition to the quantum coproduct an interesting role is played by quantum traces. In the Mathematics literature quantum traces with different choices of Cartan elements have been shown to have interesting properties [38].…”

Section: Introductionmentioning

confidence: 99%

Strings on plane waves, super-Yang–Mills in four dimensions, quantum groups at roots of one

Corley

Ramgoolam

2004

Nuclear Physics B

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We show that the BMN operators in D = 4 N = 4 super Yang Mills theory proposed as duals of stringy oscillators in a plane wave background have a natural quantum group construction in terms of the quantum deformation of the SO(6) R symmetry. We describe in detail how a q-deformed U (2) subalgebra generates BMN operators, with q ∼ e 2iπ J . The standard quantum co-product as well as generalized traces which use q-cyclic operators acting on tensor products of Higgs fields are the ingredients in this construction. They generate the oscillators with the correct ( undeformed ) permutation symmetries of Fock space oscillators. The quantum group can be viewed as a spectrum generating algebra, and suggests that correlators of BMN operators should have a geometrical meaning in terms of spaces with quantum group symmetry.

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Invariants of 3-manifolds via link polynomials and quantum groups

Cited by 1,182 publications

References 14 publications

Positivity of the universal pairing in 3 dimensions

Positivity of the universal pairing in 3 dimensions

SO(3) quantum invariants are dense

Strings on plane waves, super-Yang–Mills in four dimensions, quantum groups at roots of one

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