Turaev Viro Research Articles

BRAIDING AND ENTANGLEMENT IN SPIN NETWORKS: A COMBINATORIAL APPROACH TO TOPOLOGICAL PHASES

The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q = root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin network automata are able to perform efficiently approximate calculations of topological invarians of knots and 3-manifolds. The same algebraic background is shared by 2D lattice models supporting topological phases of matter that have recently gained much interest in condensed matter physics. These developments are motivated by the possibility to store quantum information fault-tolerantly in a physical system supporting fractional statistics since a part of the associated Hilbert space is insensitive to local perturbations. Most of currently addressed approaches are framed within a "double" quantum Chern–Simons field theory, whose quantum amplitudes represent evolution histories of local lattice degrees of freedom.We propose here a novel combinatorial approach based on "state sum" models of the Turaev–Viro type associated with SU(2)q-colored triangulations of the ambient 3-manifolds. We argue that boundary 2D lattice models (as well as observables in the form of colored graphs satisfying braiding relations) could be consistently addressed. This is supported by the proof that the Hamiltonian of the Levin–Wen condensed string net model in a surface Σ coincides with the corresponding Turaev–Viro amplitude on Σ × [0,1] presented in the last section.

Colored Turaev–Viro invariants of twist knots

In the present paper we give a formula for colored Turaev–Viro invariants of twist knots using special polyhedra derived from ( 1 , 1 ) -decomposition of the knots.

THE QUANTUM CONTENT OF THE NORMAL SURFACES IN A THREE-MANIFOLD

The formula for the Turaev–Viro invariant of a three-manifold depends on a complex parameter t. When t is not a root of unity, the formula becomes an infinite sum. This paper analyzes convergence of this sum when t does not lie on the unit circle, in the presence of an efficient triangulation of the three-manifold. The terms of the sum can be indexed by surfaces lying in the three-manifold. The contribution of a surface is largest when the surface is normal and when its genus is the lowest.

Ideal Turaev–Viro invariants

Turaev–Viro invariants are defined via state sum polynomials associated to a special spine or a triangulation of a compact 3-manifold. By evaluation of the state sum at any solution of the so-called Biedenharn–Elliott equations, one obtains a homeomorphism invariant of the manifold (“numerical Turaev–Viro invariant”). The Biedenharn–Elliott equations define a polynomial ideal. The key observation of this paper is that the coset of the state sum polynomial with respect to that ideal is a homeomorphism invariant of the manifold (“ideal Turaev–Viro invariant”), stronger than the numerical Turaev–Viro invariants. Using computer algebra, we obtain computational results on several examples of ideal Turaev–Viro invariants, for all closed orientable irreducible manifolds of complexity at most 9.

Formula omitted]-dimensional topological quantum field theory from subfactors and Dehn surgery formula for 3-manifold invariants

In this paper, we establish the general theory of ( 2 + 1 ) -dimensional topological quantum field theory (in short, TQFT) with a Verlinde basis. It is a consequence that we have a Dehn surgery formula for 3-manifold invariants for this kind of TQFT's. We will show that Turaev–Viro–Ocneanu unitary TQFT's obtained from subfactors satisfy the axioms of TQFT's with Verlinde bases. Hence, in a Turaev–Viro–Ocneanu TQFT, we have a Dehn surgery formula for 3-manifolds. It turns out that this Dehn surgery formula is nothing but the formula of the Reshetikhin–Turaev invariant constructed from a tube system, which is a modular category corresponding to the quantum double construction of a C * -tensor category. In Sato and Wakui (J. Knot Theory Ramif. 12 (2003) 543), we will exhibit computations of Turaev–Viro–Ocneanu invariants for several “basic 3-manifolds”. In the appendix, we discuss the relationship between the system of M ∞ - M ∞ bimodules arising from the asymptotic inclusion M ∨ M op ⊂ M ∞ constructed from N ⊂ M and the tube system obtained from a subfactor N ⊂ M .

Observables in three-dimensional quantum gravity and topological invariants

In this paper we report some results on the expectation values of a set of observables introduced for three-dimensional Riemannian quantum gravity with positive cosmological constant, that is, observables in the Turaev–Viro model. Instead of giving a formal description of the observables, we just formulate the paper by examples. This means that we just show how an idea works with particular cases and give a way to compute ‘expectation values’ in general by a topological procedure.

2D conformal field theories and holography

It is known that the chiral part of any 2D conformal field theory defines a 3D topological quantum field theory: quantum states of this TQFT are the CFT conformal blocks. The main aim of this paper is to show that a similar CFT/TQFT relation exists also for the full CFT. The 3D topological theory that arises is a certain “square” of the chiral TQFT. Such topological theories were studied by Turaev and Viro; they are related to 3D gravity. We establish an operator/state correspondence in which operators in the chiral TQFT correspond to states in the Turaev–Viro theory. We use this correspondence to interpret CFT correlation functions as particular quantum states of the Turaev–Viro theory. We compute the components of these states in the basis in the Turaev–Viro Hilbert space given by colored 3-valent graphs. The formula we obtain is a generalization of the Verlinde formula. The later is obtained from our expression for a zero colored graph. Our results give an interesting “holographic” perspective on conformal field theories in two dimensions.

Construction and Properties of the t-Invariant

Special spine theory is used for constructing a new invariant of compact 3-manifolds: the t-invariant. The behavior of the invariant under (boundary) connected sum is investigated. One of the Turaev―Viro invariants is expressed via the t-invariant. The t-invariant is interpreted from the point of view of TQFT. The values of the t-invariant are computed for lens spaces and for all closed oriented 3-manifolds of complexity at most six. It is proved that the set of values of the t-invariant on Seifert manifolds with fixed base (which is a closed surface) and fixed number of singular fibers is finite. Bibliography: 10 titles.

Properties of Viro--Turaev Representations of the Mapping Class Group of a Torus

Matrix realizations for the Turaev―Viro representations of the mapping class group of a torus are constructed. Each of the representations is generated by three involutive matrices. The finiteness of the representations is proved for the levels \(r = 3,4,5,6\). Bibliography: 13 titles.

Spin foam diagrammatics and topological invariance

We provide a simple proof of the topological invariance of the Turaev–Viro model (corresponding to simplicial 3D pure Euclidean gravity with cosmological constant) by means of a novel diagrammatic formulation of the state sum models for quantum BF theories. Moreover, we prove the invariance under more general conditions allowing the state sum to be defined on arbitrary cellular decompositions of the underlying manifold. Invariance is governed by a set of identities corresponding to local gluing and rearrangement of cells in the complex. Due to the fully algebraic nature of these identities our results extend to a vast class of quantum groups. The techniques introduced here could be relevant for investigating the scaling properties of non-topological state sums, proposed as models of quantum gravity in 4D, under refinement of the cellular decomposition.

Turaev–Viro invariant and 3nj symbols

We propose in this work a new method to construct the Turaev–Viro state sums, using a diagrammatic presentation describing surgery operation as Heegaard splittings. The resulting invariants can be connected with suitable 3nj symbols, and we evaluate them for the lens spaces.

SPIN TOPOLOGICAL QUANTUM FIELD THEORIES

Starting from the quantum group [Formula: see text], we construct operator invariants of 3-cobordisms with spin structure, satisfying the requirements of a topological quantum field theory and refining the Reshetikhin–Turaev and Turaev–Viro models. We establish the relationship between these two refined theories.

A simple proof of integrality of quantum invariants at prime roots of unity

Recently, Hitoshi Murakami has shown that the quantum SU(2)- and SO(3)-invariants of 3-manifolds at roots of unity of prime order are algebraic integers. Unfortunately, his proof is by a very complicated computation. Here, a quite different and very simple proof is presented, based on the second author's method to show that the Turaev–Viro invariant is the square of the modulus of the Reshetikhin–Turaev invariant.

ON A CLASS OF TOPOLOGICAL QUANTUM FIELD THEORIES IN THREE DIMENSIONS

We investigate the Chung–Fukuma–Shapere theory, or Kuperberg theory, of three-dimensional lattice topological field theory. We construct a functor which satisfies Atiyah’s axioms of topological quantum field theory by reformulating the theory as a Turaev–Viro type state sum theory on a triangulated manifold. This corresponds to giving the Hilbert space structure to the original theory. The theory can be extended to give a topological invariant of manifolds with boundary.

RATIONAL CONFORMAL FIELD THEORY AND MULTIWORMHOLE PARTITION FUNCTION IN THREE-DIMENSIONAL GRAVITY

We study the Turaev-Viro (TV) invariant as the Euclidean Chern-Simons-Witten gravity partition function with positive cosmological constant. After explaining why it can be identified as the partition function of three-dimensional gravity, we show that the initial data of the TV invariant can be constructed from the duality data of a certain class of rational conformal field theories, and that, in particular, the original TV initial data are associated with the Ak+1 modular invariant SU(2) WZW model. As a corollary we then show that the partition function Z(M) is bounded from above by [Formula: see text], where g is the smallest genus of handlebodies with which M can be presented by Hegaard splitting. Z(M) is generically very large near Λ~+0 if M is neither S3 nor a lens space, and many-wormhole configurations dominate near Λ~+0 in the sense that Z(M) generically tends to diverge faster as the “number of wormholes” g becomes larger.

Q-DEFORMED LATTICE GAUGE THEORY AND 3-MANIFOLD INVARIANTS

The notion of q-deformed lattice gauge theory is introduced. If the deformation parameter is a root of unity, the weak coupling limit of a 3-d partition function gives a topological invariant of a corresponding 3-manifold. It enables us to define the generalized Turaev–Viro invariant for cell complexes. It is shown that this invariant is determined by an action of a fundamental group on a universal covering of a complex. A connection with invariants of framed links in a manifold is also explored. A model giving a generating function of all simplicial complexes weighted with the invariant is investigated.