## CONFORMAL CORRELATION FUNCTIONS, FROBENIUS ALGEBRAS AND TRIANGULATIONS (2001)

Citations: | 35 - 18 self |

### BibTeX

@MISC{Fuchs01conformalcorrelation,

author = {Jürgen Fuchs and Ingo Runkel and Christoph Schweigert},

title = {CONFORMAL CORRELATION FUNCTIONS, FROBENIUS ALGEBRAS AND TRIANGULATIONS},

year = {2001}

}

### OpenURL

### Abstract

We formulate two-dimensional rational conformal field theory as a natural generalization of two-dimensional 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 Moore--Seiberg 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 T-duality. 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 NIM-reps of the fusion rules, respectively.

### Citations

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Citation Context ...mal blocks – to two-manifolds, and linear maps between these vector spaces to threemanifolds. When a path integral formulation is available, such as the Chern--Simons theory in the case of WZW models =-=[21]-=-, the vector spaces can be thought of as the spaces of possible initial conditions, while the linear maps describe transition amplitudes between given initial and final conditions. More precisely, to ... |

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Citation Context ...to the structure of a symmetric special Frobenius algebra Atop over the complex numbers. A is symmetric in the sense introduced above if and only if Atop is a symmetric Frobenius algebra in the usual =-=[14]-=- sense. For general modular tensor categories the situation is much more involved. But one haploid special Frobenius algebra is always present, namely the tensor unit 1. The associated torus partition... |

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Citation Context ...e detailed account of our results, including proofs, will appear elsewhere. 2 Frobenius algebras As already pointed out, our considerations are formulated in the language of modular tensor categories =-=[7]-=-, a formalization of Moore--Seiberg [8] data. A modular tensor category C may be thought of as the category of representations of some chiral algebra A, which in turn correspond to the primary fields ... |

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Citation Context ...uality of the two morphisms (7). This proves to be most reasonable, as symmetric special Frobenius algebras in the category of complex vector spaces are known to describe two-dimensional lattice TFTs =-=[6,12, 13]-=-. Since a topological field theory is in particular a (rather degenerate) conformal field theory, it is gratifying that our formalism covers this case. More generally, an algebra that is symmetric but... |

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Citation Context ...acuum on the sphere, which relates the s-channel and the t-channel, results in additional properties of the product on A. The formalization of these properties of A as an object of C reads as follows =-=[11]-=-. There is a multiplication morphism m ∈ Hom(A⊗A, A) that is associative and for which there exists a unit η ∈ Hom(1, A). There exists a coassociative coproduct ∆ ∈ Hom(A, A⊗A) as well, along with a c... |

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Citation Context ... → H(∂+M) . (12) These linear maps are multiplicative, compatible with the gluing of surfaces, and obey a few further functoriality and naturality axioms (see e.g. chapter 4 of [22]). As explained in =-=[23]-=-, correlation functions of a full conformal field theory on an arbitrary world sheet X are nothing but special elements in the space H( ˆ X) of conformal blocks on the complex cover ˆ X of X. This is ... |

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Citation Context ... WZW theory. For the E6-type modular invariant at level 10, one has A =(0) ⊕(6), for E7 at level 16, one finds A =(0) ⊕(8) ⊕ (16), and for the E8-type invariant at level 28, A = (0) ⊕(10) ⊕(18) ⊕(28) =-=[17]-=-. 3 Representations Just like for ordinary algebras, the next step to be taken in the analysis of the algebra A is the study of its representation theory. Precisely as in the case of vector spaces, an... |

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Citation Context ...ion functions of the full CFT. It would also be interesting to see whether this reconstruction is related to other algebraic structures, such as the double triangle algebras [29] which in the work of =-=[30, 31]-=- are regarded as a structure underlying every rational CFT. This would also allow for a comparison between the expressions for correlators obtained there and those which follow from our prescription. ... |

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Citation Context ...eld theories. In fact, rationality implies that the category C is semisimple. Generalizations of TFT for non semi-simple tensor categories have been studied in the literature, compare the recent book =-=[32]-=-. It can be expected that the concepts which underlie these generalizations will play a role in the analysis of non-rational theories. Finally, we point out that our results suggest an intimate relati... |

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Citation Context ...guide to the general theory, is the category of finite-dimensional vector spaces over the complex numbers. This category is also relevant to the analysis of two-dimensional lattice topological theory =-=[6]-=-, so that one may suspect a relation between those theories and our prescription. Indeed, one of our central observations is that much of the structure of (rational) conformal field theories can be un... |

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Citation Context ...ra over the abelian group Si. The twist can be computed explicitly in terms of the KSB (9) and of certain gauge invariant 6j-symbols. This way one recovers the list of boundary conditions proposed in =-=[19]-=-, which was used in [20] to compute the correct B-type boundary states in Gepner models. 74 Partition functions We now demonstrate how to extract partition functions, i.e. torus and annulus amplitude... |

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Citation Context ...n H) in each argument and that obey φ(J, J) =1 for all J ∈ H. Employing this result, the isomorphism class of the algebra structure m can be encoded in the Kreuzer--Schellekens bihomomorphism (KSB) Ξ =-=[16]-=-, which graphically is represented as J K Ξ(J, K) = c J⊗K ∨ ,K J⊗K ∨ θK J (9) (Here the twist morphism θK appears; were we drawing ribbons instead of lines, this would amount to a full 2π rotation of ... |

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Citation Context ...dition, then the vacuum 1 appears just once in the annulus, i.e. A M 1 M = 1. From the annuli, one can read off the boundary states, and show that their coefficients provide the ‘classifying algebra’ =-=[26]-=-. Performing a modular transformation, one obtains the annuli in the closed string channel; one can check that only fields i appear that are compatible with the torus partition function (15), and that... |

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Citation Context ...erefore as a disappointment that there exist modular invariants that obey all the usual constraints – positivity, integrality, and uniqueness of the vacuum – but are nevertheless unphysical (see e.g. =-=[1,2]-=-). Thus, albeit a mathematically wellposed problem, classifying modular invariants is not exactly what is desired from a physical point of view. The study of the open string field content of conformal... |

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Citation Context ...ntations of the fusion rules by matrices with non-negative integral entries. Again, this classification yields (plenty of) spurious solutions that cannot appear in a consistent conformal field theory =-=[5]-=-. Motivated by these observations we pose the following questions: First, what is the correct structure that allows to classify full rational conformal field theories with given Moore--Seiberg data? A... |

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Citation Context ...ss of generality, we do this in the following manner. First, only trivalent vertices may appear. (But we allow for arbitrary polygonal faces rather than just triangles, which is completely equivalent =-=[25]-=- to the case with only triangular faces; for brevity we still use the term ‘triangulation’). Moreover, each segment of the boundary must contain the end point of an edge of the triangulation, and each... |

18 |
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Citation Context ...uding proofs, will appear elsewhere. 2 Frobenius algebras As already pointed out, our considerations are formulated in the language of modular tensor categories [7], a formalization of Moore--Seiberg =-=[8]-=- data. A modular tensor category C may be thought of as the category of representations of some chiral algebra A, which in turn correspond to the primary fields of a chiral conformal field theory. Acc... |

16 |
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Citation Context ...odular invariants is not exactly what is desired from a physical point of view. The study of the open string field content of conformal field theories resulted in the formulation of a similar problem =-=[3, 4]-=-: Classify NIM-reps, that is, representations of the fusion rules by matrices with non-negative integral entries. Again, this classification yields (plenty of) spurious solutions that cannot appear in... |

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Citation Context ...phisms cV,WcW,V with simple V, W. These data are subject to a number of axioms that can be understood as formalizations of various properties of primary fields in rational CFT (see e.g. appendix A of =-=[9]-=-). Essentially, the axioms guarantee that these morphisms can be visualized via ribbons and that the graphs obtained by their composition share the properties of the corresponding ribbon graphs, and t... |

15 |
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Citation Context ...p Si. The twist can be computed explicitly in terms of the KSB (9) and of certain gauge invariant 6j-symbols. This way one recovers the list of boundary conditions proposed in [19], which was used in =-=[20]-=- to compute the correct B-type boundary states in Gepner models. 74 Partition functions We now demonstrate how to extract partition functions, i.e. torus and annulus amplitudes, from a given symmetri... |

11 |
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Citation Context ...erefore as a disappointment that there exist modular invariants that obey all the usual constraints – positivity, integrality, and uniqueness of the vacuum – but are nevertheless unphysical (see e.g. =-=[1,2]-=-). Thus, albeit a mathematically wellposed problem, classifying modular invariants is not exactly what is desired from a physical point of view. The study of the open string field content of conformal... |

10 |
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Citation Context ...odular invariants is not exactly what is desired from a physical point of view. The study of the open string field content of conformal field theories resulted in the formulation of a similar problem =-=[3, 4]-=-: Classify NIM-reps, that is, representations of the fusion rules by matrices with non-negative integral entries. Again, this classification yields (plenty of) spurious solutions that cannot appear in... |

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Citation Context ...pecial. Together with the naturality properties of 3-d topological field theory, the independence from the triangulation implies that the correlators are invariant under the relative modular group of =-=[27]-=-. Furthermore, the correct factorization rule for boundary fields follows directly from dominance properties of the category C. Bulk factorisation requires in addition a surgery move on the connecting... |

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6 |
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Citation Context ...uality of the two morphisms (7). This proves to be most reasonable, as symmetric special Frobenius algebras in the category of complex vector spaces are known to describe two-dimensional lattice TFTs =-=[6,12, 13]-=-. Since a topological field theory is in particular a (rather degenerate) conformal field theory, it is gratifying that our formalism covers this case. More generally, an algebra that is symmetric but... |

5 |
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Citation Context ...ion functions of the full CFT. It would also be interesting to see whether this reconstruction is related to other algebraic structures, such as the double triangle algebras [29] which in the work of =-=[30, 31]-=- are regarded as a structure underlying every rational CFT. This would also allow for a comparison between the expressions for correlators obtained there and those which follow from our prescription. ... |

3 |
Modular categories and orbifold models, preprint math.QA/0104242
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Citation Context ...ions describe symmetry breaking boundary conditions. The standard representation theoretic tools, like induced modules and reciprocity theorems, generalize to the category theoretic setting (see e.g. =-=[17, 18, 11]-=-) and allow to work out the representation theory in concrete examples. The case of the E6 modular invariant of the sl(2) WZW theory has been presented in [17, 11]; here we briefly comment on the repr... |

2 |
notes by S. Goto), Paths on Coxeter diagrams: From Platonic solids and singularities to minimal models and subfactors
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Citation Context ...ome collection of correlation functions of the full CFT. It would also be interesting to see whether this reconstruction is related to other algebraic structures, such as the double triangle algebras =-=[29]-=- which in the work of [30, 31] are regarded as a structure underlying every rational CFT. This would also allow for a comparison between the expressions for correlators obtained there and those which ... |

1 |
An introduction to noncommutative geometry, preprint physics/9709045
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Citation Context ...MB) = B . (23) Using these relations, one can replace a triangulation of the world sheet X that is labelled by A-lines with the dual triangulation labelled by B-lines. Moreover, by standard arguments =-=[28]-=- it follows that Morita equivalent algebras have the same representation theory, so that A-modules and B-modules, and hence boundary conditions, are in one-to-one correspondence. Since the correlation... |