Three-Dimensional Quantum Gravity,Chern-Simons Theory,and the A-Polynomial

We study three-dimensional Chern-Simons theory with complex gauge group SL(2,), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,) partition function and the colored Jones polynomial.     

Real polarization of the moduli space of flat connections on a Riemann surface

We prove that the moduli space of flatSU(2) connections on a Riemann surface has a real polarization, that is, a foliation by lagrangian subvarieties. This polarization may provide an alternative quantization of the Chern-Simons gauge theory in higher genus, in line with the results of [11] for genus one.Supported by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS 88-07291     

Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by “turns,” which are equivalence classes of directed great circle arcs on the unit sphere S ², in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group SU(1, 1)=Sp(2, R)=SL(2, R), the double cover of SO(2, 1). The present work develops a theory of turns for SL(2, C), the double and universal cover of SO(3, 1) and SO(3, C), rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation. PACS numbers: 02.20.-a     

A basis-free approach to time-reversal for symmetry groups

We develop a basis-free approach to time-reversal for the quantal angular momentum group,SU2, and apply these methods to the physical symmetrySU2_isospin,SU3_flavor,SU3_nuclear and the nuclear collective symmetry groupSL(3,R) of Gell-Mann and Tomonaga.     

<Emphasis Type="Italic">D</Emphasis>-Branes in the Euclidean <Emphasis Type="Italic">AdS</Emphasis><Subscript>3</Subscript> and <Emphasis Type="Italic">T</Emphasis>-Duality

We show that D-branes in the Euclidean AdS ₃ can be naturally associated to the maximally isotropic subgroups of the Lu–Weinstein double of SU(2). This picture makes very transparent the residual loop group symmetry of the D-brane configurations and gives also immediately the D-branes shapes and the σ-model boundary conditions in the de Sitter T-dual of the SL(2,C)/SU(2) WZW model.     

Three-Manifold Invariants and Their Relation with the Fundamental Group

We consider the 3-manifold invariant I(M) which is defined by means of the Chern–Simons quantum field theory and which coincides with the Reshetikhin–Turaev invariant. We present some arguments and numerical results supporting the conjecture that for nonvanishing I(M), the absolute value |I(M)| only depends on the fundamental group π₁ (M) of the manifold M. For lens spaces, the conjecture is proved when the gauge group is SU(2). In the case in which the gauge group is SU(3), we present numerical computations confirming the conjecture. Received: 15 November 1996 / Accepted: 17 June 1997     

A q-deformed Lorentz algebra

We derive a q-deformed version of the Lorentz algebra by deforming the algebraSL(2,C). The method is based on linear representations of the algebra on the complex quantum spinor space. We find that the generators usually identified withSL _q(2,C) generateSU _q (2) only. Four additional generators are added which generate Lorentz boosts. The full algebra of all seven generators and their coproduct is presented. We show that in the limitq→1 the generators are those of the classical Lorentz algebra plus an additionalU(1). Thus we have a deformation ofSL(2,C)×U(1).     

On the irreducible representations of the lorentz group

All continuous irreducible representations of the SL(2, C) group (as given by Naimark) are obtained by means of methods developed by Harish-Chandra and Kihlberg. The analysis is done in the SU(2) basis and a single closed expression for the matrix elements of the noncompact generators for an arbitrary irreducible representation of SL(2, C) is given. For the unitary irreducible representations the scalar product for each irreducible Hilbert space is found explicitly. The connection between the unitary irreducible representations of SL(2, C) and those of

is discussed by means of Inönü and Wigner contraction procedure and the Gell-Mann formula. Finally, due to physical interest, the addition of a four-vector operator to SL(2, C) unitary irreducible representations in a minimal way is considered; and all group extensions of the parity and time reversal operators by SL(2, C) are explicitly obtained and some aspects of their representations are treated.     

Group-Theoretical Approach to Extended Conformal Supersymmetry: Function Space Realizations and Invariant Differential Operators

We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup SU(2, 2 /N) induced from irreducible finite-dimensional Lorentz and SU(N) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Examples of higher order invariant differential operators are given.     

The Weyl Quantization and the Quantum Group Quantization of the Moduli Space of Flat <Emphasis Type="Italic">SU</Emphasis>(2)-Connections on the Torus are the Same

 We prove that, for the moduli space of flat SU(2)-connections on the 2-dimensional torus, the Weyl quantization and the quantization performed using the quantum group of SL(2,C) are the same. This is done by comparing the matrices of the operators associated through the two quantizations to cosine functions. We also discuss the *-product of the Weyl quantization and show that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus. Received: 27 January 2002 / Accepted: 9 September 2002 Published online: 19 December 2002 RID="*" ID="*" Research supported in part by the NSF, award No. DMS 0070690 Communicated by A. Connes     

On the quantum Lorentz group

The quantum analog of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indices we construct a quantum Minkowski metric. In this framework we show explicitly the correspondence between the SL(2,C) and Lorentz quantum groups. Five matrices of the quantum Lorentz group are constructed in terms of the R matrix of SL(2,C) group. These matrices satisfy Yang–Baxter equations and two of which have adequate properties tied to the quantum Minkowski space structure as the reality conditions of the coordinates and the symmetrization of the metric. It is also shown that the Minkowski metric leads to invariant and central lengths of four-vectors.     

Siebenmann-type cobordisms with borders and topology changes by quantum tunneling

Siebenmann-type cobordisms are constructed to describe topology changes with the Seifert fibered homology spheres in in- and out-states. We study the problem of determining of topology-changing amplitudes for these quantum tunneling processes. The calculations are performed in the stationary phase approximation for Kodama wave functions. In this approximation the amplitudes are expressed in terms of Chern-Simons invariants of flatSU(2)-connections over the cobordism boundary components. The topology-change amplitudes found are factorized into the Kodama wave functions for the lens spaces. The results are compared with those for Fintushel-Stern-type cobordisms which have been previously investigated.     

Classification of qubit entanglement: SL(2,ℂ) versus SU(2) invariance

The role of SU(2) invariants for the classification of multiparty entanglement is discussed and exemplified for the Kempe invariant I ₅ of pure three-qubit states. It is found being an independent invariant only in presence of both W-type entanglement and three-tangle. In this case, constant I ₅ allows for a wide range of both three-tangle and concurrences. This means that I ₅ provides no information on the entanglement in the system in addition to that contained already in the tangles (concurrences and three-tangle) themselves. Furthermore, norm-preserving SL ^⊗3 orbits of states with equal tangles but continuously varying I ₅ are shown to exist. As a consequence, I ₅ can be increased (and decreased) by general local operations. Nevertheless, numerical analysis of random SLOCC’s has not shown any violation of the monotone property of I ₅. In case I ₅ finally turned out to being an entanglement monotone, this would imply that both SU(2) invariance and the stronger monotone property are too weak requirements for the characterization and quantification of entanglement for systems of three qubits, and that SL(2,ℂ) invariance is required. This conclusion can be extended to general multipartite systems (including higher local dimension) because the entanglement classes of three-qubit systems appear as subclasses.     

Spectral Triples and Associated Connes-de Rham Complex for the Quantum <Emphasis Type="Italic">SU</Emphasis>(2) and the Quantum Sphere

In this article, we construct spectral triples for the C^*-algebra of continuous functions on the quantum SU(2) group and the quantum sphere. There have been various approaches towards building a calculus on quantum spaces, but there seem to be very few instances of computations outlined in Chapter 6, [5]. We give detailed computations of the associated Connes-de Rham complex and the space of L₂-forms.The first author would like to acknowledge support from the National Board for Higher Mathematics, India.     

Left-covariant differential calculi on SLq(2) and SLq(3)

We study (N²−1)-dimensional left-covariant differential calculi on the quantum group SL_q(N) for which the generators of the quantum Lie algebras annihilate the quantum trace. In this way we obtain one distinguished calculus on SL_q(2) (which corresponds to Woronowicz' 3D-calculus on SU_q(2)) and two distinguished calculi on SL_q(3) such that the higher-order calculi give the ordinary differential calculus on SL(2) and SL(3), respectively, in the limit q → 1. Two new differential calculi on SL_q(3) are introduced and developed in detail.     

Integrable string models with constant <Emphasis Type="Italic">SU</Emphasis>(3) torsion

We used the local invariant chiral currents to obtain new integrable string equations for string WZW model type with SU(3) constant torsion. We solved Burgers equation of motion for first invariant current in. terms of Lambert function. We show that string model with SU(n), n > 3 constant torsion does not integrable, because procedure of decomposition of non-primitive invariant chiral currents to primitive currents is the procedure of introduction of infinite-dimensions matrix of second kind constraints in bi-Hamiltonian approach to integrable systems.     

Gauge Theories Labelled by Three-Manifolds

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional ${\mathcal{N} = 2}$ gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that ${S^{3}_{b}}$ partition functions of two mirror 3d ${\mathcal{N} = 2}$ gauge theories are equal. Three-dimensional ${\mathcal{N} = 2}$ field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional ${\mathcal{N} = 2}$ SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.     

Bicovariant differential geometry of the quantum group SL h (2)

There are only two quantum group structures on the space of two by two unimodular matrices, these are the SL _q (2) and the SL _h (2) quantum groups. The differential geometry of SL _q (2) is well known. In this Letter, we develop the differential geometry of SL _h (2), and show that the space of left invariant vector fields is three-dimensional.