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215
Relativistic Spin Networks and Quantum Gravity
 J. Math Phys
, 1998
"... Abstract. Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) × SU(2). Relativistic quantum spins are related to the geometry of the 2dimensional faces of a 4simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geome ..."
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Cited by 131 (14 self)
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Abstract. Relativistic spin networks are defined by considering the spin covering of the group SO(4), SU(2) × SU(2). Relativistic quantum spins are related to the geometry of the 2dimensional faces of a 4simplex. This extends the idea of Ponzano and Regge that SU(2) spins are related to the geometry of the edges of a 3simplex. This leads us to suggest that there may be a 4dimensional state sum model for quantum gravity based on relativistic spin networks which parallels the construction of 3dimensional quantum gravity from ordinary spin networks.
Twisted representations of vertex operator algebras
"... Abstract. Let V be a vertex operator algebra and g an automorphism of finite order. We construct an associative algebra Ag(V) and a pair of functors between the category of Ag(V)modules and a certain category of admissible gtwisted Vmodules. In particular, these functors exhibit a bijection betwe ..."
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Cited by 116 (43 self)
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Abstract. Let V be a vertex operator algebra and g an automorphism of finite order. We construct an associative algebra Ag(V) and a pair of functors between the category of Ag(V)modules and a certain category of admissible gtwisted Vmodules. In particular, these functors exhibit a bijection between the simple modules in each category. We give various applications, including the fact that the complete reducibility of admissible gtwisted modules implies both the finitedimensionality of homogeneous spaces and the finiteness of the number of simple gtwisted modules. 1.
Simulation of topological field theories by quantum computers
 Comm.Math.Phys.227
"... Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has ..."
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Cited by 80 (12 self)
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Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models ” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is twofold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm. 1.
On fusion categories
 Annals of Mathematics
"... Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Gr ..."
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Cited by 76 (17 self)
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Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger. Dedicated to Leonid Vainerman on the occasion of his 60th birthday 1. introduction The theory of fusion categories arises in many areas of mathematics such as representation theory, quantum groups, operator algebras and topology. The representation categories of semisimple (quasi) Hopf algebras are important examples of fusion categories. Fusion categories have been studied extensively in the literature,
Boundary Liouville Field Theory: Boundary three point function”, Nucl. Phys
 B622 (2002) 309, hepth/0110244. 33 K. Hosomichi, ”BulkBoundary Propagator in Liouville Theory on a Disc”, JHEP 0111 044
, 2001
"... Liouville theory seems to be a universal building block that appears in various contexts such as noncritical string theory, twodimensional gravity or Dbrane physics. It is also closely related to the SL(2) or SL(2)/U(1) WZNW models which are interesting as solvable models for string theory on nonc ..."
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Cited by 71 (8 self)
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Liouville theory seems to be a universal building block that appears in various contexts such as noncritical string theory, twodimensional gravity or Dbrane physics. It is also closely related to the SL(2) or SL(2)/U(1) WZNW models which are interesting as solvable models for string theory on noncompact curved backgrounds. From a more general point of view one may
Spiders for rank 2 Lie algebras
 Commun. Math. Phys
, 1996
"... Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or grouplike object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point o ..."
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Cited by 62 (1 self)
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Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or grouplike object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namely A2, B2, and G2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig’s canonical bases, and they are useful for computing quantities such as generalized 6jsymbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger. 1.
Liouville bootstrap via harmonic analysis on a noncompact quantum group
"... Abstract. The purpose of this short note is to announce results that amount to a verification of the bootstrap for Liouville theory in the generic case under certain assumptions concerning existence and properties of fusion transformations. Under these assumptions one may characterize the fusion and ..."
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Cited by 58 (8 self)
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Abstract. The purpose of this short note is to announce results that amount to a verification of the bootstrap for Liouville theory in the generic case under certain assumptions concerning existence and properties of fusion transformations. Under these assumptions one may characterize the fusion and braiding coefficients as solutions of a system of functional equations that follows from the combination of consistency requirements and known results. This system of equations has a unique solution for irrational central charge c> 25. The solution is constructed by solving the ClebschGordan problem for a certain continuous series of quantum group representations and constructing the associated Racahcoefficients. This gives an explicit expression for the fusion coefficients. Moreover, the expressions can be continued into the strong coupling region 1 < c < 25, providing a solution of the bootstrap also for this region. 1.
Operator Algebras and Conformal Field Theory
 COMMUNICATIONS MATHEMATICAL PHYSICS
, 1993
"... We define and study twodimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite typ ..."
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Cited by 55 (2 self)
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We define and study twodimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III 1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the TomitaTakesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a "backgroundindependent" formulation of conformal field theories.
and Category: is quantum gravity algebraic
 Journal of Mathematical Physics
, 1995
"... ABSTRACT: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing Topological Quantum Field theories. The algebraic structures are related to ideas about the reinterpretati ..."
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Cited by 51 (3 self)
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ABSTRACT: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing Topological Quantum Field theories. The algebraic structures are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context. I.
Category theory for conformal boundary conditions
 FIELDS INST. COMMUN. AMER. MATH. SOC., PROVIDENCE, RI
, 2003
"... ... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the d ..."
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Cited by 50 (14 self)
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... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIMrep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.