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101
Quantum Geometry of Isolated Horizons and Black Hole Entropy
, 2000
"... Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a nonperturbative quantization of the sector of general relativity, coupled to matter, admitting nonrotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polym ..."
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Cited by 45 (3 self)
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Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a nonperturbative quantization of the sector of general relativity, coupled to matter, admitting nonrotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic geometry of the horizon is then described by the quantum ChernSimons theory of a U(1) connection on a punctured 2sphere, the horizon. Subtle mathematical features of the quantum ChernSimons theory turn out to be important for the existence of a coherent quantum theory of the horizon geometry. Heuristically, the intrinsic geometry is flat everywhere except at the punctures. The distributional curvature of the U(1) connection at the punctures gives rise to quantized deficit angles which account for the overall curvature. For macroscopic black holes, the logarithm of the number of these horizon microstates is proportional to the area, irrespective of the values of (nongravitational) charges. Thus, the black hole entropy can be accounted for entirely by the quantum states of the horizon geometry. Our analysis is applicable to all nonrotating black holes, including the astrophysically interesting ones which are very far from extremality. Furthermore, cosmological horizons (to which statistical mechanical considerations are known to apply) are naturally incorporated. An effort has been made to make the paper selfcontained by including short reviews of the background material.
Quantum geometry of algebra factorisations and coalgebra bundles
 Commun. Math. Phys
, 2000
"... We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 · CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame bundles and el ..."
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Cited by 34 (15 self)
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We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 · CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame bundles and elements of Riemannian geometry. As an example, we construct qmonopoles on all the Podle´s quantum spheres S 2 q,s. 1.
TFT CONSTRUCTION OF RCFT CORRELATORS V: PROOF OF MODULAR INVARIANCE AND FACTORISATION
, 2006
"... The correlators of twodimensional rational conformal field theories that are obtained in the TFT construction of [FRS I, FRS II, FRS IV] are shown to be invariant under ..."
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Cited by 33 (19 self)
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The correlators of twodimensional rational conformal field theories that are obtained in the TFT construction of [FRS I, FRS II, FRS IV] are shown to be invariant under
Twisting all the Way: from Classical Mechanics to Quantum Fields
, 2007
"... We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and sym ..."
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Cited by 20 (7 self)
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We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative spacetime, i.e. we establish a noncommutative correspondence principle from?Poisson brackets to?commutators. In particular commutation relations among creation and annihilation operators are deduced.
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 19 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 19 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
Duality Symmetries and Noncommutative Geometry of String Spacetime
 COMMUN. MATH. PHYS
, 1998
"... We examine the structure of spacetime symmetries of toroidally compactified string theory within the framework of noncommutative geometry. Following a proposal of Fröhlich and Gawedzki, we describe the noncommutative string spacetime using a detailed algebraic construction of the vertex operator alg ..."
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Cited by 15 (11 self)
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We examine the structure of spacetime symmetries of toroidally compactified string theory within the framework of noncommutative geometry. Following a proposal of Fröhlich and Gawedzki, we describe the noncommutative string spacetime using a detailed algebraic construction of the vertex operator algebra. We show that the spacetime duality and discrete worldsheet symmetries of the string theory are a consequence of the existence of two independent Dirac operators, arising from the chiral structure of the conformal field theory. We demonstrate that these Dirac operators are also responsible for the emergence of ordinary classical spacetime as a lowenergy limit of the string spacetime, and from this we establish a relationship between Tduality and changes of spin structure of the target space manifold. We study the automorphism group of the vertex operator algebra and show that spacetime duality is naturally a gauge symmetry in this formalism. We show that classical general covariance also becomes a gauge symmetry of the string spacetime. We explore some larger symmetries of the algebra in the context of a universal gauge group for string theory, and connect these symmetry groups with some of the algebraic structures which arise in the mathematical theory of vertex operator algebras, such as the Monster group. We also briefly describe how the classical topology of spacetime is modified by the string theory, and calculate the cohomology groups of the noncommutative spacetime. A selfcontained, pedagogical introduction to the techniques of noncommmutative geometry is also included.
HomLie admissible Homcoalgebras and HomHopf algebras, Published as Chapter 17, pp 189206
 Generalized Lie theory in Mathematics, Physics and Beyond
, 2008
"... Abstract. The aim of this paper is to generalize the concept of Lieadmissible coalgebra introduced in [2] to Homcoalgebras and to introduce HomHopf algebras with some properties. These structures are based on the Homalgebra structures introduced in [12]. ..."
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Cited by 15 (4 self)
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Abstract. The aim of this paper is to generalize the concept of Lieadmissible coalgebra introduced in [2] to Homcoalgebras and to introduce HomHopf algebras with some properties. These structures are based on the Homalgebra structures introduced in [12].