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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 140 (14 self)
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For a copy with the handdrawn figures please email
CONFORMAL CORRELATION FUNCTIONS, FROBENIUS ALGEBRAS AND TRIANGULATIONS
, 2001
"... We formulate twodimensional rational conformal field theory as a natural generalization of twodimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A ..."
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Cited by 36 (18 self)
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We formulate twodimensional rational conformal field theory as a natural generalization of twodimensional lattice topological field theory. To this end we lift various structures from complex vector spaces to modular tensor categories. The central ingredient is a special Frobenius algebra object A in the modular category that encodes the MooreSeiberg data of the underlying chiral CFT. Just like for lattice TFTs, this algebra is itself not an observable quantity. Rather, Morita equivalent algebras give rise to equivalent theories. Morita equivalence also allows for a simple understanding of Tduality. We present a construction of correlators, based on a triangulation of the world sheet, that generalizes the one in lattice TFTs. These correlators are modular invariant and satisfy factorization rules. The construction works for arbitrary orientable world sheets, in particular for surfaces with boundary. Boundary conditions correspond to representations of the algebra A. The partition functions on the torus and on the annulus provide modular invariants and NIMreps of the fusion rules, respectively.
Direct sum decompositions and indecomposable TQFTs
 J. Math. Phys
, 1995
"... Abstract. The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT’s in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is onedimensional, and i ..."
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Cited by 9 (0 self)
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Abstract. The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT’s in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is onedimensional, and indecomposable twodimensional theories are classified. 1.
Discrete Phase Transitions Associated to Topological Lattice Field Theories in Dimension D ≥ 2
, 1994
"... We investigate the neighborhood of Topological Lattice Field Theories (TLFTs) in the parameter space of general lattice field theories in dimension D ≥ 2, and discuss the phase structures associated to them. We first define a volumedependent TLFT, and discuss its decomposition to a direct sum of ir ..."
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We investigate the neighborhood of Topological Lattice Field Theories (TLFTs) in the parameter space of general lattice field theories in dimension D ≥ 2, and discuss the phase structures associated to them. We first define a volumedependent TLFT, and discuss its decomposition to a direct sum of irreducible TLFTs, which cannot be decomposed anymore. Using this decomposed form, we discuss phase structures and renormalization group flows of volumedependent TLFTs. We find that TLFTs are on multiple first order phase transition points as well as on fixed points of the flow. The phase structures are controlled by the physical states on (D − 1)sphere of TLFTs. The flow agrees with the NienhuisNauenberg criterion. We also discuss the neighborhood of a TLFT in general directions by a perturbative method, socalled cluster expansion. We investigate especially the Zp analogue of the TuraevViro model, and find that the TLFT is in general on a higher order discrete phase transition point. The phase structures depend on the topology of the base manifold and are controlled by the physical states on topologically nontrivial surfaces. We also discuss the correlation lengths of local fluctuations, and find longrange modes propagating along topological defects. Thus various discrete phase transitions are associated to TLFTs.