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42
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 140 (14 self)
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For a copy with the handdrawn figures please email
The Århus Integral of Rational Homology 3Spheres I: A Highly Non Trivial Flat Connection on S³
, 2002
"... Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly nontrivial mathematical theorems and theories. We argue that even though nontri ..."
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Cited by 20 (4 self)
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Path integrals do not really exist, but it is very useful to dream that they do and figure out the consequences. Apart from describing much of the physical world as we now know it, these dreams also lead to some highly nontrivial mathematical theorems and theories. We argue that even though nontrivial at connections on S³ do not really exist, it is beneficial to dream that one exists (and, in fact, that it comes from the nonexistent ChernSimons path integral). Dreaming the right way, we are led to a rigorous construction of a universal finitetype invariant of rational homology spheres. We show that this invariant is equal (up to a normalization) to the LMO (LeMurakamiOhtsuki, [LMO]) invariant and that it recovers the Rozansky and Ohtsuki invariants. This is part I of a 4...
Finite groups, spherical 2categories, and 4manifold invariants. arXiv:math.QA/9903003
"... In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], althou ..."
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Cited by 16 (5 self)
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In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the statesum invariants of Birmingham and Rakowski [11, 12, 13], who studied DijkgraafWitten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are nontrivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3types, such as [15], for example. 1 1
A Combinatorial Approach to Topological Quantum Field Theories and Invariants of Graphs
, 1993
"... : The combinatorial state sum of Turaev and Viro for a compact 3manifold in terms of quantum 6jsymbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techn ..."
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Cited by 15 (3 self)
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: The combinatorial state sum of Turaev and Viro for a compact 3manifold in terms of quantum 6jsymbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques gives the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any Riemann surface with arbitrary coloured punctures. * Supported by DFG, SFB 288 "Differentialgeometrie und Quantenphysik" 1 email: karowski@vax1.physik.fuberlin.dbp.de 2 email: schrader@vax1.physik.fuberlin.dbp.de 1 1. Introduction Since the early days of topological quantum field theories there was the question whether such field theories have a lattice formulation analogous to lattice gauge theory. The reason is that one would like to work in a context with mathematically well defined quantities instead of more or less formal functional integrals. This qu...
On exotic modular tensor categories
 Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Cited by 13 (7 self)
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 nontrivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 nontrivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.
Mixed Hodge polynomials of character varieties
"... We calculate the Epolynomials of certain twisted GL(n,C)character varietiesMn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lietype GL(n,Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geomet ..."
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Cited by 13 (6 self)
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We calculate the Epolynomials of certain twisted GL(n,C)character varietiesMn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lietype GL(n,Fq) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)character variety. The calculation also leads to several conjectures about the cohomology of Mn: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain
The local GromovWitten theory of curves
, 2008
"... We study the equivariant GromovWitten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the GromovWitten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven ..."
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Cited by 12 (3 self)
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We study the equivariant GromovWitten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the GromovWitten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven jointly with C. Faber and A. Okounkov in the appendix.
The twisted Drinfeld double of a finite group via gerbes and finite groupoids
"... Abstract. The twisted Drinfeld double (or quasiquantum double) of a finite group with a 3cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3 ..."
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Cited by 12 (0 self)
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Abstract. The twisted Drinfeld double (or quasiquantum double) of a finite group with a 3cocycle is identified with a certain twisted groupoid algebra. The groupoid is the loop (or inertia) groupoid of the original group and the twisting is shown geometrically to be the loop transgression of the 3cocycle. The twisted representation theory of finite groupoids is developed and used to derive properties of the Drinfeld double, such as representations being classified by their characters. This is all motivated by gerbes and 3dimensional quantum field theory. In particular the representation category of the twisted Drinfeld double is viewed as the ‘space of sections ’ associated to a transgressed gerbe over the loop groupoid.
Orbifolding Frobenius algebras
, 2000
"... Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and ax ..."
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Cited by 9 (1 self)
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Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super–graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi–homogeneous singularities and their symmetries.