## Category theory for conformal boundary conditions (2003)

Venue: | FIELDS INST. COMMUN. AMER. MATH. SOC., PROVIDENCE, RI |

Citations: | 54 - 14 self |

### BibTeX

@ARTICLE{Fuchs03categorytheory,

author = {Jürgen Fuchs and Christoph Schweigert},

title = {Category theory for conformal boundary conditions},

journal = { FIELDS INST. COMMUN. AMER. MATH. SOC., PROVIDENCE, RI},

year = {2003},

volume = {39},

pages = {25--70}

}

### OpenURL

### Abstract

... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a by-product we obtain results about the Frobenius-Schur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIM-rep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.

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Citation Context ...as. Yet another advantage is that similar categories also arise from 1 The properties which equip CRep A with the structure of a modular tensor category can be encoded in so-called Moore-Seiberg data =-=[62,29,3]-=-. It has actually not yet been proven that the representation category of every rational VOA indeed possesses all features of a modular tensor category. But this property has been established for seve... |

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Citation Context ...rge is then that over each point of the moduli space there lies a haploid Frobenius algebra. This picture is somewhat reminiscent of the role of Frobenius manifolds in the study of integrable systems =-=[23]-=-. Our category theoretic results are contained in sections 2 to 5. Readers less interested in the mathematical development are invited to start a first reading in section 6, where we summarize the sal... |

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Citation Context ... be present in the resulting boundary category only that particular structure needs to be assumed for C. For instance we do not have to require C to be modular, nor to be semisimple. As is well known =-=[20]-=-, semisimplicity of the representation category of a vertex algebra amounts to rationality of the theory in the sense that there are only finitely many inequivalent simple modules. Thus our framework ... |

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Citation Context ...provided by the unitary models with c <1, for which every irreducible Aext-module is finitely fully reducible as a Virasoro module; in this case all conformal boundary conditions are known explicitly =-=[2,42,7]-=-. More generally, in any rational CFT one may wish to restrict one’s attention to the subclass of ‘rational’ boundary conditions, for which by definition the preserved subalgebra A is a rational verte... |

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Citation Context ...algebra A leads directly to interesting structure on its representation category. For instance, a chiral algebra is expected to possess coproduct-like structures, see [29] for an early discussion and =-=[49,47]-=- for an approach in the context of vertex algebras. As the representation category of a chiral algebra, CRep A is therefore expected to carry the structure of a tensor category. For rational vertex op... |

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Citation Context ...ms of vertex operator algebras (section 10). Note added: As mentioned below, ideas similar to some of ours have been expressed a little earlier by A.A. Kirillov and V. Ostrik in [53,52] (compare also =-=[65]-=- for a continuation of their work). After completing this contribution, other related work has come to our attention. Pioneering results on algebras in braided categories were obtained by B. Pareigis ... |

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Citation Context ...ollowing. Vertex algebras constitute just one particular mathematical formalization of the physical idea of a chiral algebra. Other approaches exist which use different algebraic structures (see e.g. =-=[25, 64]-=-). They lead to categories that are, sometimes, equivalent to categories of representations of vertex algebras. A treatment on the level of category theory therefore allows to deal with aspects of con... |

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Citation Context ...er we investigate a proposal for a universal and model independent construction of all conformal boundary conditions. In fact, the construction encodes at the same time both a modular invariant for A =-=[11]-=-, i.e. a ‘CFT in the bulk’, and the conformal boundary conditions for that CFT. When combined with ideas from [27,28], it also opens the way to determine all correlation functions. And it sheds new li... |

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Citation Context .... provided by (id (X ∨ ) ∨⊗dX) ◦( ˜ bX ∨⊗idX), with inverse (idX⊗ ˜ dX ∨) ◦(bX⊗id (X ∨ ) ∨) – but still (X ∨ ) ∨ can be different from X. A notion similar to sovereignty is that of a pivotal category =-=[32,6]-=-, where the primary structure is the isomorphism between X and (X ∨ ) ∨ rather than the left duality. 3. The traces in a sovereign category are cyclic, so that the term ‘trace’ is appropriate. This pr... |

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Citation Context ...nerate invariant pairing; in the present setting an analogue of such a pairing is supplied by the morphism ǫ ◦m. 6. In the framework of *-categories, a special Frobenius algebra is known as a Qsystem =-=[57,59]-=-. In that case the product and coproduct, and the unit and counit, respectively, are *’s of each other, and the Frobenius axiom (2.10) can be derived from the other axioms. 7. In the vertex operator s... |

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Citation Context ...osal that is based on the observations in [44,74], which in turn were motivated by a comparison of concrete information about boundary conditions in specific models with results from subfactor theory =-=[58,59,83,11,12]-=- and about the modularisation of premodular categories [15,63]. The basic idea is to realize the elementary boundary conditions preserving A as the (absolutely) simple objects of a suitable category, ... |

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Citation Context ...ts. Let us start with the relation between algebra objects and modular invariants. Most of the known modular invariants of conformal field theories can be described in terms of a group G of so-called =-=[73]-=- simple currents J – that is, simple objects of a modular tensor category that have dimension one – together with an element of H 2 (G, C × ) [54], called ‘discrete torsion’. The obvious candidate for... |

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Citation Context ...resentation theoretic level, such maps correspond to twisted intertwiners. They play an important role in the treatment of orbifold theories (see e.g. [9]); their traces, so-called twining characters =-=[40, 36]-=- frequently show an interesting behavior under modular transformations. To endow the collection of A-modules in C with more structure, it can be useful to require A to possess an additional property. ... |

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Citation Context ...osal that is based on the observations in [44,74], which in turn were motivated by a comparison of concrete information about boundary conditions in specific models with results from subfactor theory =-=[58,59,83,11,12]-=- and about the modularisation of premodular categories [15,63]. The basic idea is to realize the elementary boundary conditions preserving A as the (absolutely) simple objects of a suitable category, ... |

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Citation Context ... non-negative integral entries satisfying R(X ∨ ) = (R(X)) t and R(I) = 11 . (9.1)64 Jürgen Fuchs and Christoph Schweigert This concept of a NIM-rep has appeared in the search for modular invariants =-=[18]-=-, in the analysis of annulus amplitudes in conformal field theory (see e.g. [71,43, 7]), and in the study of the general structure of fusion rings [45]. It is worth noting, however, that the classific... |

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Citation Context ... serve as a source of inspiration for general constructions. (For instance, one can hope that non-compact quantum groups will improve our understanding of certain non-compact conformal field theories =-=[68]-=-.) Once one adopts this point of view, one is lead to the following proposal that is based on the observations in [44,74], which in turn were motivated by a comparison of concrete information about bo... |

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Citation Context ...76, 77] have developped an algebraic approach to two-dimensional topological field theories on surfaces with boundary that is closely related to the programme we have meanwhile pursued with I. Runkel =-=[37,38]-=- for conformal field theories; roughly speaking, their results correspond to specializing the constructions in [37, 38] to the modular tensor category of finite-dimensional complex vector spaces. 2 Al... |

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Citation Context ...osal that is based on the observations in [44,74], which in turn were motivated by a comparison of concrete information about boundary conditions in specific models with results from subfactor theory =-=[58,59,83,11,12]-=- and about the modularisation of premodular categories [15,63]. The basic idea is to realize the elementary boundary conditions preserving A as the (absolutely) simple objects of a suitable category, ... |

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Citation Context .... Because of (7.6) ˜c= cA A must be of the form ˜c = ∑ J,K∈G ∗ with scalars ζJ,K, and because of (7.7) these are restricted by ζJ,K cXJ,XK (7.9) ζJ,K ζK,J = 1 . (7.10) When combined with lemma 7.6 of =-=[53]-=-, this implies immediately that the object I⊕Xℓ of the modular tensor category of the sl(2) WZW model at level ℓ does not possess any ˜c-commutative algebra structure when ℓ is odd (and hence that the... |

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Citation Context ...h morphisms have been studied in [52]. At the representation theoretic level, such maps correspond to twisted intertwiners. They play an important role in the treatment of orbifold theories (see e.g. =-=[9]-=-); their traces, so-called twining characters [40, 36] frequently show an interesting behavior under modular transformations. To endow the collection of A-modules in C with more structure, it can be u... |

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Citation Context ...ns. In fact, the construction encodes at the same time both a modular invariant for A [11], i.e. a ‘CFT in the bulk’, and the conformal boundary conditions for that CFT. When combined with ideas from =-=[27,28]-=-, it also opens the way to determine all correlation functions. And it sheds new light on the classification problem (both of modular invariants and of boundary conditions) as well; given a chiral alg... |

43 |
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Citation Context ...ensed matter physics to the theory of open strings. Such boundary conditions are partially characterized by the maximal vertex operator subalgebra A of the bulk chiral algebra Abulk that they respect =-=[43, 75]-=-. That A is respected by a boundary condition means that the correlation functions in the presence of the boundary condition satisfy the Ward identities for A and accordingly can be expressed through ... |

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Citation Context ...ight duality, it follows that in an autonomous category tensoring with an object is two-sided exact. 6. Spherical categories allow the construction of invariants of links in the threesphere (see e.g. =-=[5]-=-), which explains their name. The condition on the equality of traces implies that the invariant of a link in three dimensions does not depend on the two-dimensional projection of the link. Now the ca... |

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Citation Context ...morphism classes λ ∈ {0, 1, 2, ... , 10} of simple objects, and A to be in the class 0⊕6. This corresponds to the E6-type modular invariant of A (1) 1 and has been studied from various points of view =-=[69,83,11,7,44]-=-. The object A=I ⊕X6 is a commutative algebra, as follows [53] by the existence of a conformal embedding A ˆ= (A (1) 1 ) 10 ⊂ (B(1) 2 ) 1 ˆ= Aext. Since A is the direct sum of only two simple subobjec... |

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Citation Context ...otivated by a comparison of concrete information about boundary conditions in specific models with results from subfactor theory [58,59,83,11,12] and about the modularisation of premodular categories =-=[15,63]-=-. The basic idea is to realize the elementary boundary conditions preserving A as the (absolutely) simple objects of a suitable category, which we like to call the boundary category. Non-simple object... |

37 | Conformal correlation functions, Frobenius algebras and triangulations
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Citation Context ...76, 77] have developped an algebraic approach to two-dimensional topological field theories on surfaces with boundary that is closely related to the programme we have meanwhile pursued with I. Runkel =-=[37,38]-=- for conformal field theories; roughly speaking, their results correspond to specializing the constructions in [37, 38] to the modular tensor category of finite-dimensional complex vector spaces. 2 Al... |

36 | Modular data: the algebraic combinatorics of conformal field theory, Mar 2001, e-Print Archive: math.qa/0103044
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Citation Context ...as appeared in the search for modular invariants [18], in the analysis of annulus amplitudes in conformal field theory (see e.g. [71,43, 7]), and in the study of the general structure of fusion rings =-=[45]-=-. It is worth noting, however, that the classification of NIM-reps is not the same as the classification of good algebra objects. Rather, a NIM-rep can be unphysical, i.e. not correspond to any consis... |

36 |
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Citation Context ...A is a direct sum of simple subobjects each of which is a ‘simple current’ (see section 7), this reality condition should reduce to demanding that the simple currents belong to the ‘effective center’ =-=[46,54]-=-. 2. The dimensions provide us with distinguished numbers DimL(M) and DimR(M) for every module M, and with just a single Dim(M) when CA is spherical. In the56 Jürgen Fuchs and Christoph Schweigert pa... |

33 |
On relevant boundary perturbations of unitary minimal models
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Citation Context ...y, where the decomposition of a semisimple object into simple subobjects specifies the so-called Chan--Paton multiplicities, which in turn are a source of gauge symmetries in space-time. According to =-=[70]-=- they also appear in the study of renormalization group flows on boundary conditions, even when the starting point corresponds to a simple boundary condition. (A category theoretic approach to certain... |

33 |
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Citation Context ...that boundary categories are modular. The absence of this property should be compensated by additional structures on the three-manifold, which are similar to those of spin- [10,8,72] and so-called π- =-=[81]-=- manifolds. Lemma 3.7 Every object A of a sovereign category C that can be written in the form A = AX,X := X ⊗ X ∨ (3.17) is a special Frobenius algebra in C, with parameters βA = dimL(X), βI = dimR(X... |

31 |
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Citation Context ...situation where the twists do no longer form a group. 10 For simple current extensions of the modular tensor category that is associated to an affine Lie algebra at positive integral level A is known =-=[19]-=- to have the structure of a so-called abelian intertwining algebra. But except when this happens to be even a vertex operator algebra, it is not the structure we are looking for.68 Jürgen Fuchs and C... |

31 |
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Citation Context ...resentation theoretic level, such maps correspond to twisted intertwiners. They play an important role in the treatment of orbifold theories (see e.g. [9]); their traces, so-called twining characters =-=[40, 36]-=- frequently show an interesting behavior under modular transformations. To endow the collection of A-modules in C with more structure, it can be useful to require A to possess an additional property. ... |

31 | Galois theory for braided tensor categories and the modular closure
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(Show Context)
Citation Context ...otivated by a comparison of concrete information about boundary conditions in specific models with results from subfactor theory [58,59,83,11,12] and about the modularisation of premodular categories =-=[15,63]-=-. The basic idea is to realize the elementary boundary conditions preserving A as the (absolutely) simple objects of a suitable category, which we like to call the boundary category. Non-simple object... |

29 |
Vertex Algebras and Algebraic
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Citation Context ...: From the algebraic and algebro-geometric data of a chiral conformal field theory – a vertex operator algebra, its category of representations and the associated system of conformal blocks (see e.g. =-=[31, 75]-=-) – we have abstracted a tensor category C, and investigated the categories CA of modules of suitable algebra objects in C. An obvious question at this point is how to interpret the objects of CA at t... |

28 | Frobenius-Schur indicator in conformal field theory, Phys - Bantay - 1997 |

26 |
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Citation Context ...gert particular case of boundary conditions preserving the full chiral algebra, this number is known, in conformal field theory, as the ‘boundary entropy’ of the boundary condition corresponding to M =-=[1]-=-. The dimension provides a natural generalization of this concept to symmetry breaking boundary conditions; it shares the property of the boundary entropy to behave multiplicatively under the tensor p... |

21 |
Correlation functions and boundary conditions
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Citation Context ...ely related, and they should be treated simultaneously. Partial answers to both questions are known (see e.g. [45] for the classification problem, [43] for various aspects of boundary conditions, and =-=[28]-=- for the construction of correlation functions), but a general solution is still to be found. In general the number of conformally invariant boundary conditions is infinite. That one deals with only f... |

21 | On the structure of unitary conformal field theory - Felder, Frohlich, et al. - 1989 |

21 |
Intertwining operator algebras and vertex tensor categories for affine Lie algebras, Duke
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Citation Context ...ven that the representation category of every rational VOA indeed possesses all features of a modular tensor category. But this property has been established for several classes of VOAs, compare e.g. =-=[48,50]-=-, and it is commonly expected that possible exceptions should better be accounted for by an appropriate refinement of the qualification ‘rational’.Category theory for conformal boundary conditions 27... |

20 |
Topological structures in string theory
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Citation Context ...ries were obtained by B. Pareigis in [66,67]. A. Wassermann [82] has considered, in the context of quantum subgroups, (braided-)commutative algebras in *-tensor categories. G. Moore [61] and G. Segal =-=[76, 77]-=- have developped an algebraic approach to two-dimensional topological field theories on surfaces with boundary that is closely related to the programme we have meanwhile pursued with I. Runkel [37,38]... |

19 |
A matrix S for all simple current extensions, Nucl.Phys
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Citation Context ...remodular, and when applied to that subcategory, our construction is nothing but the modularisation procedure of [15, 63], which in turn in physics terminology amounts to a ‘simple current extension’ =-=[73, 41]-=- (since the associated modular invariant is then of pure extension type). In this situation, in particular the category C0 A induced from the 1-graded subcategory of C is again modular, and in fact sh... |

17 |
Open descendants in conformal field theory, Fortschr. Phys
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Citation Context ...1)64 Jürgen Fuchs and Christoph Schweigert This concept of a NIM-rep has appeared in the search for modular invariants [18], in the analysis of annulus amplitudes in conformal field theory (see e.g. =-=[71,43, 7]-=-), and in the study of the general structure of fusion rings [45]. It is worth noting, however, that the classification of NIM-reps is not the same as the classification of good algebra objects. Rathe... |

16 |
Intertwining operator algebras, genus-zero modular functors and genus-zero conformal field theories, in: Operads
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Citation Context ...ven that the representation category of every rational VOA indeed possesses all features of a modular tensor category. But this property has been established for several classes of VOAs, compare e.g. =-=[48,50]-=-, and it is commonly expected that possible exceptions should better be accounted for by an appropriate refinement of the qualification ‘rational’.Category theory for conformal boundary conditions 27... |

16 | K(N)-local duality for finite groups and groupoids
- Strickland
(Show Context)
Citation Context ...y for a so-called holomorphic orbifold conformal field theory with orbifold group G. 5. For symmetric tensor categories a definition of Frobenius object similar to the one used here has been given in =-=[79]-=-. 4 The motivation in [79] is that the spectrum of any finite groupoid provides an example of such an object. In the category of vector spaces, one usually defines a Frobenius algebra as an algebra wi... |