Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134)

Kauffman, Louis H.; Lins, Sóstenes

doi:10.1515/9781400882533

Cited by 389 publications

(794 citation statements)

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“…Recall that the value of d q together with the deformed binor identity are the two ingredients for the combinatorial definition of the Turaev-Viro invariant according to the formulation of [27,28]. This is encouraging as it indicates that perhaps a strict correspondence between LQG and the Tuarev-Viro invariant can be established if one appropriately implements the next step: quantizing and imposing the curvature constraint (1.4).…”

Section: Discussionmentioning

confidence: 92%

“…Now, if the previous statement makes sense, the Turaev-Viro amplitudes would have to be related to the kinematical states of the canonical theory, namely classical SU(2) spin networks. In contrast the Turaev-Viro amplitudes are constructed from the combinatorics of q-deformed spin networks [27,28]. This would imply that the understanding of the relationship between the Turaev-Viro invariants and quantum gravity requires the understanding of the dynamical interplay between classical spin-network states and q-deformed amplitudes.…”

Section: Jhep10(2011)036mentioning

confidence: 99%

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Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Noui

Pérez

Pranzetti

2011

J. High Energ. Phys.

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In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant Λ > 0 in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection A and the canonically conjugate triad field e. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of A ± = A± √ Λe. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection A λ = A + λe (for λ ∈ R) on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity (with q = exp(i λ/2)). The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman's identity). We discuss the possible implications of our result.

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Section: Discussionmentioning

confidence: 92%

Section: Jhep10(2011)036mentioning

confidence: 99%

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Noui

Pérez

Pranzetti

2011

J. High Energ. Phys.

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“…This invariant is closely related to that of Kauffman and Lins [18], but has two main differences. The first one is the choice in the definition of the "Wilson net" gauge invariant object.…”

mentioning

confidence: 78%

“…This was first attempted by Witten [15] and further developed by Martin [17]. These invariants are related to families of invariants constructed using braid group techniques by Kauffman and Lins [18]. In a recent paper we have considered related invariants as possible candidates for states of the quantum gravitational field [16].…”

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confidence: 99%

A spin network generalization of the Jones polynomial and Vassiliev invariants

Gambini¹,

Griego²,

Pullin

1998

Physics Letters B

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We apply the ideas of Alvarez and Labastida to the invariant of spin networks defined by Witten and Martin based on Chern-Simons theory. We show that it is possible to define ambient invariants of spin networks that (for the case of SU (2)) can be considered as extensions to spin networks of the Jones polynomial. Expansions of the coefficients of the polynomial yield primitive Vassiliev invariants. The resulting invariants are candidates for solutions of the Wheeler-DeWitt equations in the spin network representation of quantum gravity.CGPG-97/10-4 q-alg/9711014Witten [1] showed that the expectation value of a Wilson loop in an SU (2) Chern-Simons theory,is a knot invariant closely related with the Jones polynomial. More precisely, it is given by the Kauffman bracket [2] knot polynomial. The latter differs from the Jones polynomial by an overall "phase factor" given by the exponential of the writhe of the knot. This factor accounts for the fact that the Kauffman bracket is a regular isotopy invariant (it is not invariant when one removes twists, i.e., it is an invariant of framed knots) whereas the Jones polynomial is an ambient isotopic invariant. The explicit relation between the Kauffman bracket and the expectation value we introduced above is given by K(γ, q) =< W (γ) > with q = exp(2πi/k), where q is the variable of the polynomial and k is the coupling constant of the Chern-Simons theory 1 . The expression of the invariants one obtains from the path integral appear as power series expansions in terms of the parameter 2πi/k. It is tantamount to evaluating the polynomials for q = exp(2πi/k) and expanding the exponential in power series. If one does this for the Jones polynomial, the resulting coefficients are Vassiliev invariants [5][6][7].In a separate context, it was also noticed that the resulting Vassiliev invariants are candidates for wavefunctions in the loop representation of quantum gravity [8][9][10][11]. In this context, however, one is not interested in arbitrary knot invariants, but actually in those that arise as counterparts of states in the "connection representation",This poses a series of restrictions on the possible knot invariants ψ(γ) that one could consider as wavefunctions of a quantum theory in the loop representation. These restrictions are known as "Mandelstam identities". If one concentrates of wavefunctions depending on a single loop, they are linear, loop dependent, relations i c i ψ(γ i ) = 0. They arise from the fact that the Wilson loop W (γ, A) is a trace of a group element [12]. Of all the Vassiliev invariants that one constructs using the techniques described above, we only know for a fact that two of them are compatible with these relations. This makes it particularly difficult to construct candidates for quantum states of the gravitational field in terms of these invariants. Implicit in this discussion is the fact that in quantum gravity one is interested in loops with generic intersections, otherwise the spectra of various quantum operators is not appropriately realized [1...

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“…These transfer matrices are not actually matrices but rather diagrammatic objects (tangles [38,39]) defined in the corresponding planar TL algebra [2,3]. Matrix representations are obtained by acting with these tangles on suitable vector spaces of link states [1,35].…”

Section: Lm(pmentioning

confidence: 99%

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

2014

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A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width N ∈ N, the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TL N (β) with loop fugacity β = 2 cos λ, λ ∈ R. Similarly on a cylinder, the single-row transfer tangle T (u) is an element of the so-called enlarged periodic TL algebra. The logarithmic minimal models LM(p, p ′ ) comprise a subfamily of the TL loop models for which the crossing parameter λ = (p ′ − p)π/p ′ is a rational multiple of π parameterised by coprime integers 1 ≤ p < p ′ . For these special values, additional symmetries allow for particular degeneracies in the spectra that account for the logarithmic nature of these theories. For critical dense polymers LM(1, 2), with β = 0, D(u) and T (u) satisfy inversion identities that have led to the exact determination of the eigenvalues in any representation and for arbitrary finite system size N . The generalisation for p ′ > 2 takes the form of functional relations for D(u) and T (u) of polynomial degree p ′ . These derive from fusion hierarchies of commuting transfer tangles D m,n (u) and T m,n (u) where D(u) = D 1,1 (u) and T (u) = T 1,1 (u). The fused transfer tangles are constructed from (m, n)-fused face operators involving Wenzl-Jones projectors P k on k = m or k = n nodes. Some projectors P k are singular for k ≥ p ′ , but we argue that D m,n (u) and T m,n (u) are nonsingular for every m, n ∈ N in certain cabled link state representations. For generic λ, we derive the fusion hierarchies and the associated T -and Y -systems. For the logarithmic theories, the closure of the fusion hierarchies at n = p ′ translates into functional relations of polynomial degree p ′ for D m,1 (u) and T m,1 (u). We also derive the closure of the Y -systems for the logarithmic theories. The T -and Y -systems are the key to exact integrability and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.

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Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds (AM-134)

Cited by 389 publications

References 0 publications

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

A spin network generalization of the Jones polynomial and Vassiliev invariants

Fusion hierarchies,T-systems, andY-systems of logarithmic minimal models

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