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Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 154 (14 self)
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For a copy with the handdrawn figures please email
TwoDimensional Topological Quantum Field Theories And Frobenius Algebras
 J. Knot Theory Ramifications
, 1996
"... We characterize Frobenius algebras A as algebras having a comultiplication which is a map of Amodules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either ..."
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Cited by 64 (2 self)
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We characterize Frobenius algebras A as algebras having a comultiplication which is a map of Amodules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either "annihilator algebras"  algebras whose socle is a principal ideal  or field extensions. The relationship between twodimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable twodimensional topological quantum field theories. Keywords: topological quantum field theory, frobenius algebra, twodimensional cobordism, category theory 1. Introduction Topological Quantum Field Theories (TQFT's) were first described axiomatically by Atiyah in [1]. Since then, much work has been done ...
HigherDimensional Algebra I: Braided Monoidal 2Categories
 Adv. Math
, 1996
"... We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give lon ..."
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Cited by 54 (9 self)
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We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2category Z(C) as the `center' of a semistrict monoidal category C, in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `HigherDimensional Algebra and Topological Quantum Field Theory' [1], henceforth referred to as `HDA'. This program consists of generalizing algebraic concep...
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 50 (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.
Quantum geometry with intrinsic local causality
, 1997
"... The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact twodimensional surfaces. The space of states of the theory is the direct sum of the ..."
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Cited by 39 (17 self)
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The space of states and operators for a large class of background independent theories of quantum spacetime dynamics is defined. The SU(2) spin networks of quantum general relativity are replaced by labelled compact twodimensional surfaces. The space of states of the theory is the direct sum of the spaces of invariant tensors of a quantum group Gq over all compact (finite genus) oriented 2surfaces. The dynamics is background independent and locally causal. The dynamics constructs histories with discrete features of spacetime geometry such as causal structure and multifingered time. For SU(2) the theory satisfies the Bekenstein bound and the holographic hypothesis is recast in this formalism.
Causal evolution of spin networks
 Nucl. Phys
, 1997
"... A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum stat ..."
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Cited by 29 (6 self)
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A new approach to quantum gravity is described which joins the loop representation formulation of the canonical theory to the causal set formulation of the path integral. The theory assigns quantum amplitudes to special classes of causal sets, which consist of spin networks representing quantum states of the gravitational field joined together by labeled null edges. The theory exists in 3+1, 2+1 and 1+1 dimensional versions, and may also be interepreted as a theory of labeled timelike surfaces. The dynamics is specified by a choice of functions of the labelings of d+1 dimensional simplices,which represent elementary future light cones of events in these discrete spacetimes. The quantum dynamics thus respects the discrete causal structure of the causal sets. In the 1 + 1 dimensional case the theory is closely related to directed percolation models. In this case, at least, the theory may have critical behavior associated with percolation, leading to the existence of a classical limit.
Claspers and finite type invariants of links
, 2000
"... We introduce the concept of “claspers,” which are surfaces in 3–manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called “Ck–equivalence,” which is generated by surgery operatio ..."
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Cited by 26 (1 self)
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We introduce the concept of “claspers,” which are surfaces in 3–manifolds with some additional structure on which surgery operations can be performed. Using claspers we define for each positive integer k an equivalence relation on links called “Ck–equivalence,” which is generated by surgery operations of a certain kind called “Ck–moves”. We prove that two knots in the 3–sphere are Ck+1–equivalent if and only if they have equal values of Vassiliev–Goussarov invariants of type k with values in any abelian groups. This result gives a characterization in terms of surgery operations of the informations that can be carried by Vassiliev–Goussarov invariants. In the last section we also describe outlines of some applications of claspers to other fields in 3–dimensional topology.
Generalized Centers of Braided and Sylleptic Monoidal 2Categories
, 1997
"... Recent developments in higherdimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2categories, and a center construction for monoidal 2categories which gives a braided monoidal 2category. I give ge ..."
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Cited by 26 (3 self)
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Recent developments in higherdimensional algebra due to Kapranov and Voevodsky, Day and Street, and Baez and Neuchl include definitions of braided, sylleptic and symmetric monoidal 2categories, and a center construction for monoidal 2categories which gives a braided monoidal 2category. I give generalized center constructions for braided and sylleptic monoidal 2categories which give sylleptic and symmetric monoidal 2categories respectively, and I correct some errors in the original center construction for monoidal 2categories. 1 Introduction The initial motivation for the study of braided monoidal categories was twofold: from homotopy theory, where braided monoidal categories of a particular kind arise as algebraic 3types of arcconnected, simply connected spaces, and from higherdimensional category theory, where braided monoidal categories arise as one object monoidal bicategories [16]. These motivations have subsequently been brought together by the definition of tricategori...
A unified WittenReshetikhinTuraev invariant for integral homology spheres
, 2006
"... We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTurae ..."
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Cited by 21 (3 self)
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We construct an invariant JM of integral homology spheres M with values in a completion ̂ Z[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) WittenReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhinTuraev invariants of M. As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) as a function on ζ behaves like an “analytic function ” defined on the set of roots of unity. That is, the τζ(M) for all roots of unity are determined by a “Taylor expansion ” at any root of unity, and also by the values at infinitely many roots of unity of prime power orders. In particular, τζ(M) for all roots of unity are determined by the Ohtsuki series, which can be regarded as the Taylor expansion at q = 1.