Invertible defects and isomorphisms of rational CFTs

ZMP-HH/10-13 Ham burger Beitr¨ age zur Mathematik 374 In v ertible defects and isomo rphisms of rationa l CFTs Alexei Da vydov a , Liang Kong b,c , Ingo Runkel d , ∗ a Max Pl a n ck In stitut f ¨ ur Mathematik Vivatsgasse 7, 53111 Bonn, Germany b California In s titute of T e chnolo gy, Center for the Physics of Inform ation, Pasadena, CA 9112 5, USA c Institute for A dvanc e d S tudy (Scienc e Hal l) Tsinghua Univ e rs i ty, Beijing 100084, China d Dep artment Mathematik, Univ e rs i t¨ at Hambur g Bundesstr aße 5 5, 20146 Hambur g , Germany April 2010 Abstract Giv en tw o t wo-dimensional c onformal field theories, a domain wall – or defect line – b e- t w een th em is called inv ertible if there is another defect with wh ic h it fuses to th e iden tit y defect. A defect is called top ological if it is transparent to the stress tensor. A conformal isomorphism b et we en the t w o CFTs is a linear isomorphism b etw een their state spaces whic h preserve s the stress tensor and is compatible with the op erator pro duct expansion. W e sho w that for rational CFTs there is a one-to-one corresp ondence b et w een in v ertible top ological defects and conformal isomorph isms if b oth p reserv e the rational symmetry . This corresp on- dence is compatible with comp osition. ∗ Emails: da vydov @mpim -bonn.mpg.de, kong.f an.li ang@gmail.com, ingo. runke l@uni -hamburg.de Con ten ts 1 In tro duction 2 2 Conformal isomorphisms and defects 4 3 Pro of via algebras in mo dular categories 6 3.1 Mo dular categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 F rob enius algebras and mo dular inv ariance . . . . . . . . . . . . . . . . . . . . . . 7 3.3 The full centre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 Bimo dules and defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 Equiv alence of g roup oids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4 Examples 14 4.1 Simple curren t s mo dels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 Holomorphic orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 Conclusion 16 1 In t r o duction Dualities pla y an imp ortant role in understanding non-p erturba tiv e prop erties of mo dels in quan- tum field theory , statistical phy sics or string theory , b ecause they allo w to relate observ ables in a mo del at we ak coupling to those of the dual mo del at strong coupling. Some well kno wn exam- ples are K r a mers-W annier duality [KW], electric-magnetic duality [MO], T-duality [GPR], mirror symmetry [L VW, GP], and the AdS/CFT corresp ondence [Ma ]. By their ve ry nature, dualities a re hard to find and it is difficult to understand precisely how quan tities in the t w o dual descriptions are related. In man y examples, it has prov ed helpful to describe dualities b y a ‘dualit y domain w all’, a co-dimension one defect whic h separates the dual theories [F r2, FSW, FGRS, GW, K T]. It is then natura l to ask if in a ny sense al l dualities can b e described b y such defects. F or a particularly simple t yp e of dualit y defects – so-called inv ertible defects – in a par t icularly w ell understo o d class of quantum field theories, namely t w o-dimensional rational confo rmal field theories, we will answ er this question in the affirmative. Let us describ e the setting a nd the result of t his pap er in more detail. Generically , a dualit y transformation exc hanges lo cal fields and disorder fields. This is the case in the a r c het ypical example of suc h dualities, Kr a mers-W annier duality of the tw o-dimensional Ising mo del. In t he latt ice mo del, the duality exc hanges the lo cal spin-op erator with the non- lo cal disorder-op erator, whic h marks the endp oint of a frustration line o n the dual lattice. In the conformal field theory whic h describ es the critical p oin t of t he Ising mo del, the duality accordingly pro vides an automor phism on the space consisting of all lo cal fields a nd a ll disorder fields. In particular, the Kramers-W annier duality transforma t io n is not a n automorphism o n the space of lo cal fields alone. Ho w ev er, there is an esp ecially simple t yp e of dua lity whic h do es giv e rise to a n isomorphism b et w een the spaces of lo cal fields for the t w o mo dels related b y the duality . The conformal field theory description of T-dualit y and mirror symmetry on the string w orld sheet are examples of suc h dualities. Giv en tw o conformal field theories C A and C B , the dat a of suc h a dualit y consists 2 of an isomorphism b et w een their spaces of stat es H A and H B whic h resp ects the op erator pro duct expansion and whic h preserv es the v acuum and the stress tensor; w e will call this a c onformal isomorphism . The in finite symme try algebra of a conformal field theory is generated b y its conserv ed curren ts. It alwa ys includes the stress tensor, accounting for the Virasoro symmetry , but it ma y also con tain fields t ha t do not a rise via m ultiple op erator pro duct expansions o f the stress t ensor. A r ational CFT, roughly sp eaking, is a CFT whose symmetry algebra is large enough to decomp o se the space of states into a fini te direct sum of irreducible represen tations. Examples of ra t ional CFTs are the Virasoro minimal mo dels, ra tional toroidal compactifications of free b osons, W ess-Zumino-Witten mo dels and coset models obtained from affine Lie algebras a t p ositive in teger lev el, as well as appropriate orbifolds thereof. Supp ose that w e a re g iv en t w o CFTs C A and C B whic h ar e rational, hav e a unique v a cuum, ha v e isomorphic algebras o f holomorphic a nd anti-holomorphic conserv ed curren ts, and hav e a mo dular in v ariant partition function. W e will show that fo r each conformal isomorphism that preserv es the rat io nal symmetry , there exists (up to isomorphism) one and only one in v ertible defect, i.e. a duality domain w all, b etw een the CFTs C A and C B whic h implemen ts this duality . Con v ersely , eac h inv ertible defect g iv es rise to a conformal isomorphism. Alto g ether we sho w that for this class of mo dels: Ther e is a bije ction b etwe en c on f o rmal isom orphisms and invertible d efe cts, b oth pr eserving the r ational symmetry. (1.1) The pro of relies o n the v ertex algebraic description of CFTs in [HK1, Ko1], on the relatio n b et w een t w o-dimensional CFT and three-dimensional to p ological field theory [F e1, FRS1, Fj2, F r3], and on results in categorial a lg ebra [KR1, KR2]. Given this bac kground, the pro of is actually quite short, and it is phrased as a result in catego r ical alg ebra. Let us briefly link the ph ysical concepts with their mat hematical coun terparts; more details and the pro o f will b e giv en in Section 3. The represen tations of the holo mo r phic c hiral algebra of a rational CFT (a v ertex op erator algebra) form a so-called mo dular category [MS , T u, Hu2], which w e denote by C . The bulk fields of a ratio na l CFT with unique v acuum and with isomorphic ho lomorphic and anti-holomorphic c hiral alg ebra giv e rise to a simple comm uta tiv e symme tric F rob enius alg ebra C in C ⊠ C [Ko1, Fj2]. Here, C ⊠ C is the pro duct of t w o copies of C , where the second cop y corr esp onds to represen ta t ions of the anti-c hiral alg ebra (so that the braiding and twis t there are replaced b y their inv erses). W e assume in addition that t he CFT is mo dular inv ariant. In this case the algebra C is maximal, a condition on the categorical dimension of C defined in Section 3. If the CFT is defined on the upp er half plane and the b oundary condition preserv es the rational chiral symmetry , the b oundary fields giv e rise to a simple sp ecial symmetric F rob enius algebra A in the mo dular catego ry C [FRS2, Ko2]. F rom A one can construct the ful l c entr e Z ( A ), a simple comm utative maximal sp ecial symmetric F rob enius algebra in C ⊠ C [Fj2]. It is prov ed in [Fj2, KR2] that C ∼ = Z ( A ) as algebras. Denote by C A | A the monoidal category of A - A - bimo dules in C . These bimo dules describ e to p ological defect lines of the CFT whic h preserv e the c hiral symm etry [FRS2, F r3]. In v ertible top ological defects corresp ond to in ve rtible A - A -bimo dules. Let Pic( A ) b e the Picard group of C A | A . The elemen ts of Pic( A ) are isomorphism classes of in v ertible ob jects in C A | A and the gro up o p eration is induced b y the tensor pro duct of C A | A . W e prov e that there is an isomorphism of g r oups Aut( Z ( A )) ∼ = Pic( A ) , (1.2) where Aut( Z ( A )) are the algebra automor phisms of Z ( A ) . In fact, w e will pro v e a group oid v ersion of this statemen t. The first group oid has a s ob jects simple sp ecial symmetric F rob enius 3 algebras in C and as morphisms isomorphism classes of inv ertible bimo dules. The second group oid has simple comm utativ e maximal sp ecial symmetric F rob enius algebras in C ⊠ C a s o b jects and its morphisms are algebra isomorphisms. W e pro v e the equiv alence of these t w o group o ids, whic h is the mathematical v ersion of the phys ical statemen t (1 .1 ). This pap er is o r ganised as f ollo ws. In Section 2 w e giv e a brief description of CFT and defect lines, a nd w e formulate the result o f the pap er this language. In Section 3, the result is restated in algebraic terms and pro v ed. Section 4 con tains tw o examples, and with Section 5 w e conclude. 2 Conformal iso morp hisms and defec t s Consider a CFT C A with space of states H A . By the state-field corresp o ndence, H A coincides with the space of fields of the CFT. The space of states con tains the states T A and ¯ T A , the holomorphic and an ti-holomo r phic comp onen ts of the stress tensor. Their mo des, L m and ¯ L m , giv e rise to tw o comm uting copies of the Virasoro algebra. Pic k a basis { φ i } of H A consisting of eigen vec tors 1 of L 0 and ¯ L 0 . Then we hav e the op erator pro duct expansion (OPE) φ i ( z ) φ j ( w ) = X k C A ij k ( z − w ) φ k ( w ) , (2.1) where z and w are t w o distinct p oints on the complex plane and eac h function C A ij k ( x ) is determined b y conformal co v ariance up to an o verall constan t; the OPE has to b e asso ciativ e and comm utativ e [BPZ], see [HK1] for the mathematical formulation w e will use in Section 3. Apart from an asso ciativ e comm utativ e OPE, we make the follo wing assumptions: Uniqueness of the vacuum: There is a unique elemen t 1 A ∈ H A , the v acuum v ector, whic h is annihilated b y L 0 , L ± 1 and ¯ L 0 , ¯ L ± 1 , and whic h has t he OPE 1 ( z ) 1 ( w ) = 1 ( w ). Non-de gener a cy: T ake the first basis vec tor to b e φ 1 = 1 A . Then h φ i , φ j i := C A ij 1 defines a non-degenerate pa ir ing on t he space of states H A . In o ther w ords, the tw o-p oin t correlator is non-degenerate. Mo dular invarianc e: The pa rtition function Z ( τ ) = tr H A q L 0 − c/ 24 ( q ∗ ) ¯ L 0 − ¯ c/ 24 is mo dular in v arian t, i.e. it ob eys Z ( τ ) = Z ( − 1 /τ ) = Z ( τ + 1). Here τ is a complex num b er with im( τ ) > 0, q = exp(2 π iτ ), and c and ¯ c are the left and the righ t cen tral ch arge. Supp ose no w w e are give n t w o CFTs C A and C B . By a c onformal isomorphis m f from C A to C B w e mean a linear isomorphism f : H A → H B whic h preserv es the v acuum, the stress tensor, and the OPE. This means that f ( 1 A ) = 1 B , f ( T A ) = T B , f ( ¯ T A ) = ¯ T B , and t ha t, if w e c ho ose a basis { φ i } of H A as ab ov e, and tak e φ ′ i = f ( φ i ) as basis fo r H B , then C A ij k ( x ) = C B ij k ( x ). Next w e giv e some bac kground on defects. Giv en tw o CFTs C A and C B , w e can consider domain w alls – or defects – b etw een C A and C B . T o b e sp ecific, ta k e the complex plane with a defect placed on t he real axis, and with CFT C A defined on the upp er half plane and CFT C B on the lo w er half plane. The defect is defined b y the b oundary conditions ob ey ed by the fields o f C A and C B on the real line. W e call a defect c o n formal iff the stress tensors satisfy 1 W e as sume here that L 0 and ¯ L 0 are diagonalisable, i.e . we exclude logarithmic CFTs from o ur treatment. W e also assume the common eigenspaces of L 0 and ¯ L 0 are finite-dimensio nal, a nd that their eig env alues form a countable set. The la tter condition excludes for example Liouville theory . 4 T A ( x ) − ¯ T A ( x ) = T B ( x ) − ¯ T B ( x ) for a ll x ∈ R . The defect is called top ol o gic al iff the stronger conditions T A ( x ) = T B ( x ) and ¯ T A ( x ) = ¯ T B ( x ) hold for all x ∈ R . T op ological defects are totally transmitting and tensionless. They can exist only if the central c harg es of the CFTs C A and C B are the same, and they can b e deformed on the complex plane without affecting the v alues of correlators, as long as they are not ta k en past field insertions or other defects. A trivial example of a t op ological defect is the id entity defe ct b et w een a giv en CFT and itself, which simply consists of no defect at all, i.e. all fields of the CFT a re contin uous across the real line. Conformal defects a re v ery difficult to classify , the only mo dels for whic h all conformal defects (with discrete sp ectrum) are know n are the Lee-Y ang mo del and the critical Ising mo del [OA, QR W]; eve n for the free b oson one kno ws only certain examples [BBDO, BB]. T op ological defects ha v e b een classified for Virasoro minimal mo dels [PZ, FRS2] and for the free b oson [FGRS]. F or t o p ological defects one can define the op eration of fusion [PZ, FRS2], whereb y o ne places a top ological defect R on the real line, and another top o lo gical defect S on the line R + iε , and considers the limit ε → 0. Since correlators are indep enden t of ε , this pro cedure is non-singular (whic h is not true fo r general confo rmal defects [BBDO, BB]), and it giv es a new top ological defect R ⋆ S on the real line. W e call a top ological defect b et w een CFTs C A and C B invertible , iff t here exists a defect b etw een C B and C A suc h that their fusion in b oth p ossible orders yields the iden tity defect of CFT C A and of CFT C B , resp ectiv ely . A top ological defect R b et w een CFT C A and CFT C B giv es rise to a linear op erato r D [ R ] : H A → H B . T his op erator is obtained by pla cing a field φ of CFT C A at the orig in 0 a nd the defect R o n the circle aro und 0 of radius ε . In the limit ε → 0 (a gain, all correlato rs are actually indep enden t o f ε ) one o bta ins a field ψ of CFT C B . This defines the action o f D [ R ] via ψ = D [ R ] φ . Since the defect is top ological, D [ R ] in t ert wines the Virasoro actions on H A and H B . The iden t it y defect induces the iden tity map, and the assignme n t is compatible with fusion of defects, D [ R ⋆ S ] = D [ R ] D [ S ]. In particular, in v ertible defects give rise to isomorphisms b etw een state spaces. Giv en tw o (non- trivial) CFTs C A and C B , it is not true tha t ev ery linear map from H A to H B can b e written as D [ R ] fo r an appropriate defect R . Indeed, a defect has to satisfy man y additional conditions. One w a y to for mulate this is to extend the axiomatic definition o f CFT in terms of sewing of surfaces [Se] to surfaces decorated by defect lines [RS]. F or example, in the setting of [RS ], one can sho w that a n in v ertible defect X b etw een C A and C B pro vides a conformal isomorphism Z ( X ) from C A to C B b y setting Z ( X ) = γ − 1 X D [ X ], where γ X ∈ C is defined via D [ X ] 1 A = γ X 1 B . Let us now restrict our atten tion t o ratio na l CFTs. More precisely , by a rational CFT C A w e mean that H A con tains a subspace V L consisting of holomorphic fields and ¯ V R of an ti-holomorphic fields, suc h that V L and V R are vertex op erator a lgebras (VO As) satisfying the conditions of [Hu2], and such t ha t V L ⊗ C ¯ V R is em b edded in H A (the bar in ¯ V R just reminds us that the fields in V R are an t i- holomorphic). This turns H A in to a V L ⊗ C ¯ V R -mo dule, and b y rat io nalit y of V L ⊗ C ¯ V R it is finitely reducible, see [HK1] for details. W e call C A a rational CFT o v er V L ⊗ C ¯ V R . Note that, while bulk fields in the image of V L ⊗ C ¯ V R can alwa ys b e written as a sum of (non- singular) OPEs o f a holomorphic and an a nti-holomorphic field in H A , the same is in general not true for an arbitrary field in H A . Giv en t w o rational CFTs C A and C B o v er V L ⊗ C ¯ V R , w e say a conformal isomorphism from H A to H B pr eserves the r ational symmetry iff it acts a s the identit y on V L ⊗ C ¯ V R . Similar ly w e sa y that a defect from C A to C B pr eserves the r a tion al symmetry iff all bulk fields in V L ⊗ C ¯ V R are 5 con tin uous a cro ss the defect line. Since T and ¯ T a re in V L ⊗ C ¯ V R , suc h a defect is in particular top olog ical. W e hav e no w ga t hered in more detail all the ingredien ts needed to stat e our ma in result: Giv en t w o rational CFTs C A and C B o v er V ⊗ C ¯ V (i.e. we demand that V L = V R = V ), for eac h conformal isomorphism f f rom C A to C B there exists a unique (up to isomorphism, see Section 3) in v ertible defect X suc h that f = Z ( X ). This a ssignmen t is compatible with comp o sition. As a sp ecial case of this result w e obtain that all automorphisms of a ratio nal CFT ov er V ⊗ C ¯ V whic h a ct as the iden tity o n V ⊗ C ¯ V are implemen ted by defects. The existing results in the literature [FRS5] imply that there is an injectiv e group homomorphism fro m (isomorphism classes o f ) in vertible defects of t he CFT t o itself to conformal automorphisms. Our result sho ws in addition that this map is surjectiv e. Let us stress that this is b y no means ob vious, as the defining conditions to b e satisfied b y confo rmal isomorphisms and defects a re very differen t: compatibilit y with the OPE v ersus sewing relatio ns for surfaces decorated by defect lines. 3 Pro of via algeb r as in mo dular cate gories The aim of this section is to prov e an equiv alence of group oids whic h is the a lgebraic coun terpart of the CFT result stat ed in (1.1) and detailed in t he previous section. W e will start by in tro ducing the necessary a lg ebraic ob jects – mo dular categories, certain F rob enius a lgebras, the full cen tre – and describe their relation to CFT in a series of remarks. 3.1 Mo dular categories W e will emplo y the usual gr a phical notation for ribb on categories [JS],[T u , BK]. T o fix con ven tions, w e note that our diagrams are read from b ottom to top (the ‘optimistic’ w a y), and that the pictures for the bra iding and the dualit y morphisms are U V V U : U ⊗ V c U,V − − → V ⊗ U (3.1) and U ∨ U : U ∨ ⊗ U d U − → 1 , U U ∨ : U ⊗ U ∨ ˜ d U − → 1 , U U ∨ : 1 ˜ d U − → U ⊗ U ∨ , U ∨ U : 1 ˜ b U − → U ∨ ⊗ U, The t wist is denoted b y θ U : U → U . F or f : U → U , the t r ace is defined as tr( f ) = ˜ d U ◦ ( f ⊗ id U ∨ ) ◦ b U ∈ End( 1 ). Definition 3.1 ([T u, BK ]) . A mo dular c a te gory is a ribb on category , whic h is C -linear, ab elian, semi-simple, whic h has a simple tensor unit, a nd a finite n umber of isomorphism classes of simple ob jects. If { U i | i ∈ I } denotes a c hoice of represen tativ es for these classes, in addition the complex |I |×| I | -matr ix s i,j defined b y s i,j id 1 = t r ( c U i ,U j ◦ c U j ,U i ) is in vertible. 6 Remark 3.2. F o r a V O A V whic h satisfies the reductiv eness and finiteness conditions stated in [Hu2], it is pro v ed in [Hu2, Thm. 4.6] that the category Rep V of V -mo dules is mo dular. W e will refer to a VO A satisfying these conditions as r ational . Let C b e a mo dular category . The dimens ion of U ∈ C is defined as dim( U )id 1 = t r (id U ), and the glob al d i m ension o f C is defined to b e Dim C = X i ∈I (dim U i ) 2 . (3.2) The dimensions dim ( U i ) of the simple ob jects are non-zero and r eal [ENO1, Thm. 2.3 & Prop. 2.9], so that in part icular D im C ≥ 1. If C is a mo dular category , then ¯ C denotes the mo dular category obtained fr om C b y replacing braiding and tw ist by their in vers es. Giv en tw o mo dular categories C and D , denote b y C ⊠ D their Deligne-pro duct [De, BK ], whic h in this case amoun ts to ta king pairs of ob jects U ⊠ V and tensor pro ducts of Hom spaces, and completing with resp ect to direct sums. Eve ry monoidal (and in particular eve ry mo dular) category is equiv alen t to a strict one (whic h has trivial asso ciator and unit isomorphisms). W e will w ork with strict mo dular catego ries without f urther men tio n. 3.2 F rob enius algebras and mo dular in v ariance The definitions giv en b elow only require some o f the structure of a mo dular category , but rather than giving a minimal set of assumptions in each case, let us t ak e C to b e a mo dular catego r y in this section. An algeb r a in C is an ob ject A ∈ C equipped with t w o morphisms m A : A ⊗ A → A and η A : 1 → A satisfying the usual a sso ciativit y and unit prop erties (more details for this and the follo wing can b e found e.g. in [FS]). An A - left mo d ule is an ob ject M ∈ C equipp ed with a morphism ρ M : A ⊗ M → M compatible with unit and multiplication of A . Accordingly one defines righ t mo dules a nd bimo dules, as w ell as in tert winers of mo dules. A c o algebr a is an o b ject A ∈ C equipped with t w o morphisms ∆ A : A → A ⊗ A and ǫ : A → 1 satisfying the usual coasso ciativit y and counit prop erties. A F r ob enius algebr a A = ( A, m, η , ∆ , ǫ ) is an algebra and a coalg ebra suc h that (id A ⊗ m ) ◦ (∆ ⊗ id A ) = ∆ ⊗ m = ( m ⊗ id A ) ◦ (id A ⊗ ∆) , (3.3) i.e. the copro duct is an intert winer of A - A -bimo dules. W e will use the following graphical repre- sen tat io n for the morphisms of a F rob enius algebra: m = A A A , η = A , ∆ = A A A , ǫ = A . A F rob enius algebra A in C is called • haploid iff dim Hom( 1 , A ) = 1 , • simple iff it is simple as a bimo dule ov er itself, 7 • sp e cial iff m ◦ ∆ = ζ id A and ǫ ◦ η = ξ id 1 for nonzero constants ζ , ξ ∈ C , • symmetric iff A A ∨ = A A ∨ . • c ommutative iff m ◦ c A,A = m , • maximal iff dim A = (dim C ) 1 2 , pro vided A is also haploid and commutativ e, • mo dular invariant iff θ A = id A and for all W ∈ C we ha v e A A W W = X i ∈I dim( U i ) (dim C ) 1 2 A A W W U i . (3.4) All t he sp ecial symmetric F rob enius a lgebras that will a pp ear here are in fa ct ‘normalised’ sp ecial in the sense that ζ = 1, which then implies ξ = dim( A ). W e will not men tion the qualifier ‘normalised’ explicitly b elow. As an aside, we no te that the name ‘maximal’ is motiv ated a s follows . A A -left mo dule M is called lo c al iff ρ M ◦ c M ,A ◦ c A,M = ρ M (see [KiO] or [F r1, Sect. 3.4]) . W e call a commutativ e algebra maximal iff its category of lo cal mo dules is monoidally equiv alent to the category of vec tor spaces. If a comm utativ e maximal algebra A is contained in another commutativ e algebra B as a subalgebra, then B is isomorphic to a direct sum of copies of A as an A -mo dule. Th us, if A is haploid, it cannot b e a subalgebra of a larger comm utativ e haploid algebra. In this sense, A is ‘maximal’. If A is a haploid commutativ e F rob enius algebra of non-zero dimension, then A is maximal iff dim( A ) = (Dim C ) 1 2 [KiO, Thm. 4.5], hence the simplified definition ab o v e. The mo dular inv ariance condition ab o v e is the least standa r d (and the most complicated) notion. It was introduced in [Ko2] (see [KR2, L em. 3 .2] for the relation to the definition ab ov e), and we included it f or the sak e of R emark 3.4 b elo w. F ortunately , for the case of in terest to us it can b e replaced by a muc h simpler condition: Theorem 3.3 ([KR2, Thm. 3 .4 ]) . L et A b e a haploid c om mutative symmetric F r ob enius alge b r a in C . Th e n A is mo dular invariant iff it is maxim al. In ei ther c ase, A is in addition sp e cial. Remark 3.4. There are man y approac hes to axiomatise prop erties of conformal field theories, see e.g. [BPZ, F rS, Bo, V a, F L M, MS, Se, Hu1, G G, KaO]. W e will use those dev elop ed in [HK1, HK2, Ko2] and [F R S1, Fj1, Fj2]. Let V L and V R b e tw o rat io nal V O As suc h that c L − c R ≡ 0 mo d 24. A CFT ov er V L ⊗ C V R in the sense of Section 2 , is – in the nomenclature of [HK2, Ko2] – a conformal full field alg ebra ov er V L ⊗ C V R with non-degenerate inv ariant bilinear form, whic h 8 is mo dular in v arian t a nd has a unique v acuum. L et C L = Rep V L and C R = Rep V R . It is sho wn in [Ko2, Thm. 6.7] that CFTs ov er V L ⊗ C V R are in one-to- one corresp ondence with ha ploid comm uta tiv e symmetric F rob enius alg ebras in C L ⊠ ¯ C R whic h are mo dular in v arian t. 3.3 The full cen tre Fix a mo dular category C . The braiding on C allow s to endo w the functor T : C ⊠ C → C , given b y the t ensor pro duct on C , with the structure of a tensor functor. This can b e done in t w o wa ys, and we choose the con v ention of [KR2, Sect. 2.4 ]. The functor T has an adjo in t R : C → C ⊠ C , that is, there is a bi-natural fa mily of isomorphisms ˆ χ Y , V : Hom C ( T ( Y ) , V ) − → Hom C ⊠ C ( Y , R ( V )) . (3.5) In fa ct, R is b oth left and righ t adjoint to T , but w e will not need this. Denote the t wo natural transformations asso ciated to the adjunction b y id C ⊠ C ˆ δ − → RT and T R ˆ ρ − → id C . (3.6) They ar e ˆ δ Y = ˆ χ (id T ( Y ) ) and ˆ ρ V = ˆ χ − 1 (id R ( V ) ) for V ∈ C , Y ∈ C ⊠ C . Explicit express ions fo r ˆ χ , ˆ δ and ˆ ρ are giv en in [K R2, Sect. 2.4]. The functor R ob eys R ( 1 ) = M i ∈I U ∨ i ⊠ U i , R ( V ) ∼ = ( V ⊠ 1 ) ⊗ R ( 1 ) . (3.7) Prop osition 3.5 ([K R 2, Pro p. 2.16, 2.24, 2.25]) . L et A ∈ C and B ∈ C ⊠ C b e algebr as. (i) If A and B ar e sp e cial symmetric F r ob enius, so ar e R ( A ) ∈ C ⊠ C a n d T ( B ) ∈ C . (ii) A morphism f : T ( B ) → A is an algebr a homomorphis m iff ˆ χ ( f ) : B → R ( A ) is an algebr a homomorphism . The structure morphisms for R ( A ) and T ( B ) in part (i) a re give n in [KR2, Sect. 2.2]. P art (ii) sho ws in particular that ˆ ρ A : T R ( A ) → A is a n algebra map. F or an algebra in a braided category one can define a left and a right cen tre [VZ , Os1]. W e will only need the left cen tre. Given a n algebra A in C , its left c e ntr e C l ( A ) ֒ → A is the largest sub ob ject o f A suc h that the comp osition C l ( A ) ⊗ A → A ⊗ A c A,A − − → A ⊗ A m A − − → A (3.8) coincides with the comp osition C l ( A ) ⊗ A → A ⊗ A m A − − → A . (3.9) If A is sp ecial symmetric F rob enius (and C ab elian), the left centre exists and can b e written as the image of an idemp otent defined in terms of m A , ∆ A , c A,A and the dualit y morphisms, see [F r1, Sect. 2.4] for details. Definition 3.6 ( [F j2, Def. 4.9 ]) . The ful l c e n tr e of a sp ecial symmetric F rob enius algebra A in a mo dular category C is Z ( A ) = C l ( R ( A )) ∈ C ⊠ C . 9 The full cen tre ha s a natural generalisation to algebras in general monoidal categories, in whic h case it pro vides a comm utativ e algebra in the monoidal cen tre of the category and is c har a cterised b y a univ ersal prop ert y [Da2]. Denote the sub ob ject embedding and restriction morphisms b y e Z : Z ( A ) ֒ → R ( A ) and r Z : R ( A ) ։ Z ( A ) . (3.10) They ob ey r Z ◦ e Z = id Z ( A ) , i.e. Z ( A ) is a direct summand of R ( A ). By construction of the algebra structure on Z ( A ), the map e Z is an algebra homomo r phism. Theorem 3.7 ([KR1, Prop. 2.7] a nd [KR2, Thm. 3.22]) . L et C b e a mo dular c ate gory. (i) The ful l c e n tr e of a s i m ple sp e cial symmetric F r ob en ius algebr a in C is a haploid c ommutative maximal sp e cial symmetric F r ob enius alge b r a in C ⊠ C . (ii) Every haploid c o m mutative maximal sp e cial symmetric F r ob e n ius a lgebr a in C ⊠ C is isomor- phic as an algebr a to the ful l c entr e of some simp le sp e ci a l symmetric F r ob enius algebr a in C . 3.4 Bimo d u les and defects Fix a mo dular category C . Let A, B , C b e algebras in C . An A - B - bim o dule X is an A -left mo dule and a B -right mo dule suc h that the tw o actions comm ute. Giv en a B - C - bimo dule Y , w e define the A - C -bimo dule X ⊗ B Y a s a cok ernel in the usual w a y . If B is sp ecial symmetric F rob enius, X ⊗ B Y can b e written as the image of an idemp oten t on X ⊗ Y , and so in this case X ⊗ B Y ֒ → X ⊗ Y is a direct summand ( a s a bimo dule). W e denote the em b edding and restriction maps as e B : X ⊗ B Y ֒ → X ⊗ Y , r B : X ⊗ Y ։ X ⊗ B Y , (3.11) suc h that r B ◦ e B = id X ⊗ B Y . T o k eep the notation at ba y , we will not include lab els for X a nd Y . Remark 3.8. In the approa c h to CFT correlators via three-dimensional top ological field theory giv en in [F RS1, Fj 1, Fj2], a CFT is sp ecified b y a sp ecial symmetric F rob enius alg ebra A in Rep V . In this a pproac h, one automatically obta ins an op en/closed CFT whic h satisfies gen us 0 and genus 1 consistency conditions (and, sub j ect to mo dular functor prop erties o f higher gen us conformal blo c ks, is in fact w ell-defined on surfaces of arbitr a ry gen us). The bulk CFT one finds in this w a y is the CFT ov er V ⊗ C V described b y Z ( A ) via Remark 3.4, see [Fj2, Sect. 4.3]. In the TFT approa ch, o ne can also describ e CFTs in the presence of top olo gical defect lines whic h resp ect t he V ⊗ C V symmetry [F r3]. Differen t patches of the CFT w orld sheet are lab elled b y sp ecial symmetric F rob enius algebras and the defects (o r domain walls) b et w een them by bimo dules. The fusion of defect lines translates into the tensor pro duct of bimo dules ov er their in termediate algebra. In this wa y , CFTs o v er V ⊗ C V b ecome a bicategory [SFR], where ob jects are CFTs, 1- morphisms are top ological defects preserving V ⊗ C V , and 2- morphisms are ‘defect fields’ in the v acuum represen tation (described b y intert winers o f bimo dules). 3.5 Equiv alence of group oids Definition 3.9. Let C b e a mo dular category . (i) P ( C ) is the group oid whose ob jects are simple sp ecial symmetric F ro b enius algebras A, B , . . . in C and whose morphisms A → B are isomorphism classes o f inv ertible B - A -bimo dules. 10 (ii) A ( C ) is the g r o up oid whose ob jects are simple comm utat ive maximal sp ecial symmetric F rob e- nius alg ebras C , D , . . . in C and whose morphisms C → D are algebra isomorphisms from C to D . In the r emainder of this section w e will prov e the statemen t announced in the in tr o duction, namely that the tw o g roup oids P ( C ) and A ( C ⊠ C ) a re equiv alent (Theorem 3.14 b elo w). The pro of will b e split in to sev eral lemmas. W e start b y constructing a functor Z : P ( C ) → A ( C ⊠ C ) . On ob jects it is giv en b y taking the full cen tre (hence the not a tion ‘ Z ’), Z ( A ) = C l ( R ( A )) for A ∈ P ( C ) . (3.12) In o rder to define t he functor Z on morphisms, w e need some more notat ion. Fix t w o ob jects A, B ∈ P ( C ), i.e. tw o simple sp ecial symmetric F rob enius algebras. Giv en a B - A -bimo dule X , we define a morphism φ X : Z ( A ) → Z ( B ) a s in [FRS5] a nd [KR1 , Lem. 3 .2 ], φ X = dim( X ) dim( A ) X ⊠ 1 B ⊠ 1 A ⊠ 1 R ( 1 ) Z ( B ) Z ( A ) e Z r Z , (3.13) where e Z and r Z ha v e b een introduced in (3 .10). W e define the functor Z on morphisms of P ( C ) as Z ( X ) = φ X for X : A → B . (3.14) The follo wing lemma implies that Z is w ell-defined and functorial. Lemma 3.10 ([KR1, Lem. 3.1, 3.2, 3 .3]) . L e t A, B , C b e sim p l e sp e cial symmetric F r ob en i us alge- br as in C and let X , X ′ b e C - B -bimo d ules and Y a B - A -bimo dule. Then (i) If X ∼ = X ′ as bimo dules, then φ X = φ X ′ . (ii) φ A = id Z ( A ) . (iii) φ X ◦ φ Y = φ X ⊗ B Y . (iv) I f X ∨ ⊗ B X ∼ = A or X ⊗ A X ∨ ∼ = B as bimo dules, then φ X is an algebr a isomorphism . In the following w e will giv e a series of lemmas whic h will sho w that the functor Z is full, faithful and essen tially surjectiv e. Let J A b e a lab el set for the isomorphism classes of simple left A -mo dules and let { M κ | κ ∈ J A } b e a c hoice of represen tativ es. Define T A = M λ ∈J A M ∨ λ ⊗ A M λ , (3.15) 11 Eac h o f the ob jects M ∨ λ ⊗ A M λ is naturally a haploid algebra in C (see e.g. [KR1, Lem. 4 .2]), and th us also T A is an algebra (no n- haploid in g eneral). Define the morphisms ι κ : M ∨ κ ⊗ A M κ ֒ → T A , e κ := T A π κ ։ M ∨ κ ⊗ A M κ e A ֒ → M ∨ κ ⊗ M κ , π κ : T A ։ M ∨ κ ⊗ A M κ , r κ := M ∨ κ ⊗ M κ r A ։ M ∨ κ ⊗ A M κ ι κ ֒ → T A , (3.16) where e A and r A where g iv en in (3.11). Note that by definition of the alg ebra structure on T A , π κ is an algebra map, while ι κ resp ects the m ult iplicatio n but not necessarily the unit. F rom Prop osition 3 .5 w e know that T ( Z ( A )) is a sp ecial symmetric F rob enius algebra (b ecause A is), and from (3.7) we hav e T ( R ( A )) = L i ∈I A ⊗ U ∨ i ⊗ U i . Using the ma ps ( 3.10) w e can define e i = T Z ( A ) T ( e Z ) ֒ → T R ( A ) ։ A ⊗ U ∨ i ⊗ U i r i = A ⊗ U ∨ i ⊗ U i ֒ → T R ( A ) T ( r Z ) ։ T Z ( A ) . (3.17) Using these ing redien ts w e define t w o morphisms ϕ : T Z ( A ) → T A and ¯ ϕ : T A → T Z ( A ) by ϕ = X i ∈I X κ ∈J A M κ U i A r κ e i T A T Z ( A ) , ¯ ϕ = X i ∈I X κ ∈J A dim( U i ) dim( M κ ) Dim C M κ U i A r i e κ T Z ( A ) T A . (3.18) It has b een shown in [KR1, Prop. 4.3 & Lem. 4.6 , 4.7] that ϕ and ¯ ϕ are inv erse to each other, and that t hey are algebra isomorphisms. Fix another simple sp ecial symmetric F rob enius algebra B and let ρ : M λ ⊗ B → M λ b e a righ t B - action on a simple left A -mo dule M λ whic h comm utes with the left A -action. Denote the resulting A - B - bimo dule b y M λ ( ρ ) and define the morphism g λ ( ρ ) = dim( M λ ) dim( A ) M ∨ λ ⊗ A M λ B B M λ ( ρ ) e A . (3.19) One quic kly c heck s that g λ ( ρ ) is a n intert winer of B - B - bimo dules. Lemma 3.11. The fol l o wing e quality of morphi s m s Z ( A ) → Z ( B ) holds: φ M λ ( ρ ) ∨ = r Z ◦ ˆ χ ( g λ ( ρ ) ◦ π λ ◦ ϕ ) . (3.20) Pr o of. The iden tit y can b e established by comp osing the gr a phical expressions of ϕ , g λ ( ρ ) and ˆ χ (see [KR2, Eqn. (2.43)]) and comparing the result to the gra phical expression (3.13) for φ X . 12 Lemma 3.12. L et A, B ∈ P ( C ) b e haploid. Given an alge b r a is omorphism f : Z ( A ) → Z ( B ) , ther e exist λ f ∈ J A and a right B -action ρ f on M λ f such that (i) M λ f ( ρ f ) is an invertible A - B -bimo d ule, (ii) Z ( M λ f ( ρ f ) ∨ ) = f . Pr o of. Given the algebra isomorphism f and an index λ ∈ J A , w e can define the map h ( f , λ ) : M ∨ λ ⊗ A M λ ι λ ֒ → T A ¯ ϕ − → T Z ( A ) T ( f ) − − → T Z ( B ) T e Z − − → T R ( B ) ˆ ρ − → B . (3.21) It is sho wn in part e) of the pro of of [KR 1, Thm. 1.1] that there exists a unique λ f ∈ J A suc h that h ( f , λ f ) 6 = 0. W e hav e already seen that all the individual maps ab o v e resp ect the algebra m ultiplication. The map ι κ do es in general not preserv e the unit, but b ecause M ∨ λ ⊗ A M λ and B are haploid, the comp osite map h ( f , λ ) do es. This a mo unts to the argumen t in part b) and e) of the pro of of [KR1, Thm. 1.1], whic h sho ws that h ( f , λ f ) is an algebra isomorphism. W e can use the isomorphism h ( f , λ f ) to define a righ t B - a ction on M ≡ M λ f b y setting ρ f : M ⊗ B id ⊗ h ( f ,λ f ) − 1 − − − − − − − → M ⊗ M ∨ ⊗ A M id ⊗ e A − − − → M ⊗ M ∨ ⊗ M ˜ d M ⊗ id − − − − → M (3.22) By construction, w e no w hav e h ( f , λ f ) : M ∨ ⊗ A M ∼ − → B as B - B -bimo dules, whic h implies tha t M is an inv ertible A - B -bimo dule (see e.g. [F r3, Lem. 3.4]). This prov es part (i). Let us now turn to part (ii). W e first claim t ha t g λ f ( ρ f ) = h ( f , λ f ) . (3.23) T o see this iden tity first note that b oth sides are in tert winers of B - B - bimo dules. F urthermore, M ∨ ⊗ A M and B are b oth simple as B - B -bimo dules (b ecause B is simple). Th us g λ f ( ρ f ) = ξ h ( f , λ f ) fo r some ξ ∈ C . T o determine ξ we let b oth sides act o n the unit r A ◦ ˜ b M of M ∨ ⊗ A M . As h ( f , λ f ) is an algebra map, it g iv es η B . F o r the left hand side one uses the explicit form (3.1 9 ) together with [KR1, L em. 3.3 & Eqns. (3.4), (3.7)] to find that it is also equal to η B . Th us ξ = 1. Next consider the equalities h ( f , λ f ) ◦ π λ f ◦ ϕ (1) = X λ ∈J A h ( f , λ ) ◦ π λ ◦ ϕ (2) = X λ ∈J A ˆ ρ ◦ T ( e Z ◦ f ) ◦ ¯ ϕ ◦ ι λ ◦ π λ ◦ ϕ (3) = ˆ ρ ◦ T ( e Z ◦ f ) (4) = ˆ χ − 1 ( e Z ◦ f ) . (3.24) Step (1) uses that h ( f , λ ) is only no n-zero for λ = λ f , in step(2) w e inserted t he definition (3.21) of h ( f , λ ), and step (3) amounts to the identit y P λ ι λ ◦ π λ = id T A and the fact that ¯ ϕ is the in verse of ϕ . Finally , step (4) fo llo ws from the definition of ˆ ρ and naturalit y of ˆ χ , see [K R2, Eqn. (2.53)]. By Lemma 3.1 1, (3.23) and (3.24) w e hav e Z ( M λ f ( ρ f ) ∨ ) = r Z ◦ ˆ χ ( h ( f , λ f ) ◦ π λ f ◦ ϕ ) = r Z ◦ ˆ χ ( ˆ χ − 1 ( e Z ◦ f )) = f . (3.25) This sho ws part (ii) . Lemma 3.13. L et A, B ∈ P ( C ) b e haploid. L et X b e an invertible B - A -bim o dule and let f = Z ( X ) : Z ( A ) → Z ( B ) b e the c orr esp ondi n g algebr a isomorphism (L em ma 3.10 ( iv)). Cho ose λ f and ρ f as in L emma 3.12. Then X ∼ = M λ f ( ρ f ) ∨ as B - A -bimo dules. 13 Pr o of. Since X is inv ertible, it is neces sarily simple as a B - A - bimo dule (see e.g. [F r3, Lem. 3.4]). In fa ct, it is ev en simple as a right A -mo dule, b ecause, if X ∼ = M ⊕ N as righ t A -mo dules, then X ⊗ A X ∨ = M ⊗ A M ∨ ⊕ N ⊗ A N ∨ ⊕ · · · would not b e haploid. But X ⊗ A X ∨ ∼ = B , as X is in v ertible, and so X ⊗ A X ∨ is haploid. Th us there is a λ 0 ∈ J A suc h that M λ 0 ∼ = X ∨ as left A -mo dules. W e will no w sho w that λ 0 = λ f . Denote b y ρ the righ t B -a ction on M λ 0 induced b y the isomorphism M λ 0 ∼ = X ∨ . By Lemma 3.11 we hav e Z ( X ) = r Z ◦ ˆ χ ( g λ 0 ( ρ ) ◦ π λ 0 ◦ ϕ ). The n, h ( Z ( X ) , κ ) (1) = ˆ ρ ◦ T ( e Z ◦ Z ( X )) ◦ ¯ ϕ ◦ ι κ (2) = ˆ χ − 1 ( e Z ◦ r Z ◦ ˆ χ ( g λ 0 ( ρ ) ◦ π λ 0 ◦ ϕ )) ◦ ¯ ϕ ◦ ι κ (3) = ˆ χ − 1 ( ˆ χ ( g λ 0 ( ρ ) ◦ π λ 0 ◦ ϕ )) ◦ ¯ ϕ ◦ ι κ (4) = g λ 0 ( ρ ) ◦ π λ 0 ◦ ι κ (5) = δ λ 0 ,κ g λ 0 ( ρ ) , (3.26) where step (1) is the definition o f the map h fro m (3.21), in step (2) we inserted the expression fo r Z ( X ) just obtained, step (3) amounts to [KR1, Lem. 3.1 (iv)], step (4) uses t ha t ¯ ϕ is t he inv erse of ϕ , and step ( 5 ) is just the definition of the maps π λ 0 and ι κ in (3.16). In the pro of of Lemma 3.12, λ f is defined t o b e the unique elemen t of J A for whic h h ( f , λ ) is non-zero. Th us the ab ov e calculatio n sho ws λ f = λ 0 . On the other hand, it follow s fro m (3.23) and the ab ov e calculation that g λ f ( ρ f ) = g λ 0 ( ρ ) . (3.27) This equalit y in turn implies that ρ = ρ f , and th us the right B -actions on M λ f ( ρ f ) a nd M λ 0 ∼ = X ∨ agree, i.e. M λ f ( ρ f ) ∼ = X ∨ as A - B -bimo dules. W e hav e now g a thered the necessary ingredien ts t o prov e our main result. Theorem 3.14. L et C b e a mo dular c a te gory. The gr oup oids P ( C ) a n d A ( C ⊠ C ) give n in Defini- tion 3.9 ar e e quivalen t. Pr o of. By Theorem 3.7 (ii) the functor Z is essen tially surjectiv e. Fix tw o ob jects A, B ∈ P ( C ). W e need to show tha t Z pro vides a n isomorphism b et w een the morphism spaces A → B and Z ( A ) → Z ( B ). By [K R1, Prop. 4.10] there exist haploid algebras A ′ , B ′ ∈ P ( C ) and in v ertible bimo dules X : A → A ′ and Y : B → B ′ . It is th us enough to sho w that Z ( − ) : Hom P ( C ) ( A ′ , B ′ ) − → Hom A ( C ⊠ C ) ( Z ( A ′ ) , Z ( B ′ )) (3.28) is an isomorphism. By Lemma 3.12 , Z ( − ) is full, and b y Lemma 3.13, it is faithful. 4 Examples 4.1 Simple current s mo dels Let V b e a rational V OA with the pro p ert y that C = Rep V is p ointe d , i.e. ev ery simple ob ject of C is in v ertible. In other w o rds, C is g enerated b y simple curren ts. A large class of examples of suc h V OAs are provide d by lat t ice VO As (see for example [FLM]). 14 A p ointe d bra ided monoidal category C is c haracterised b y a finite ab elian group A (the group of simple curren ts) together with a quadratic function q : A → C ∗ enco ding their braid statistics [JS, Sect. 3]. C is mo dular if the quadratic function is non-de ge n er ate , i.e. if the asso ciated bi- m ultiplicativ e function σ : A × A → C ∗ defined b y σ ( a, b ) = q ( ab ) q ( a ) − 1 q ( b ) − 1 (4.1) is non-degenerate in the sense that f or eac h a 6 = 1 the homomorphism σ ( a, − ) : A → C ∗ is non-trivial. The structure of a mo dular category is enco ded in the group of (isomorphism classes of ) simple ob jects A , a 3- co cycle α ∈ Z 3 ( A, C ∗ ), whic h con trols the asso ciativit y constraint and a certain function c : A × A → C ∗ , controlling the braiding (see [JS, Sect. 3] for the conditions on c ) . The pair ( α, c ) is known as an a b elian 3-c o cycle of A with co efficien ts in C ∗ . It w a s shown in [EM] that the group of classes of a b elian 3-co cycles mo dulo cob oundaries coincides with the group of quadratic functions. In other words up to a braided equiv alence a p ointed catego ry depends only o n the quadratic function q , defined b y q ( a ) = c ( a, a ) (see [JS, Sect. 3]). W e will denote a represen tative of this class b y C ( A, q ) . Isomorphism classes of haploid special symmetric F r ob enius algebras (also called Schel lekens algebr as in this con text [FRS3, Def. 3.7]) a r e lab elled b y pairs ( B , β ), where B ⊂ A is a subgroup and β : B × B → C ∗ is a symmetric bi-multiplicativ e function suc h that β ( b, b ) = q ( b ) for b ∈ B [FRS3, Def. 3.17, Prop. 3.2 2]. A Sc hellek ens algebra corresp onding to ( B , β ) is comm utative iff β = 1. This means that comm utative Sc hellek ens algebras corresp o nd to isotr opic subgroups (subgroups on whic h q restricts trivially). The details of the following discussion will app ear elsewhere. A comm utativ e Sche llek ens algebra is maximal iff the corresp onding subgroup is maximal isotropic, i.e. L agr angia n . In particular, commutativ e maximal Schelle k ens algebras in C ( A, q ) ⊠ C ( A, q ) = C ( A, q ) ⊠ C ( A, q − 1 ) = C ( A × A, q × q − 1 ) corresp ond to subgroups in A × A , Lagrang ia n with resp ect to q × q − 1 . The full cen tre of a Sc hellek ens algebra R = R ( B , β ) in C ( A, q ) for a pair ( B , β ) corresp onds to the Lagrangia n subgroup Γ( B , β ) = { ( a, a − 1 b ) | a ∈ A, b ∈ B , suc h that σ ( c, a ) = β ( c, b ) ∀ c ∈ B } (4.2) in A × A . The construction of Γ giv es an isomorphism b et w een the set of pairs ( B , β ) and the set of L agrangian subgro ups in A × A . This also prov ides the isomorphism [KR2, Cor. 3.25] b etw een the set of Mor it a classes of simple sp ecial symmetric F rob enius algebras in C ( A, q ) and the set of isomorphism classes of simple comm utat ive maximal special symmetric F rob enius algebras in C ( A, q ) ⊠ C ( A, q )). The automorphism group o f a Sc hellek ens algebra R corresp onding t o ( B , β ) is the dual group ˆ B = Ho m ( B , C ∗ ) (the group of c haracters). In part icular the automorphism g r o up of the full cen tre of R is the group \ Γ( B , β ). This is in agreemen t with [F r3, Prop. 5.14], where it w as established that the group Pic( R ) of isomorphism classes of inv ertible R - R -bimo dules fits in to a short exact sequence B → ˆ B × A → Pic( R ) , (4.3) where the first map sends b ∈ B in to ( β ( − , b ) − 1 , b ). It is easy to see that the group Γ( B , β ) fits in to a short exact sequence Γ( B , β ) → A × B → ˆ B , (4.4) where the first map is ( u, v ) 7→ ( u, uv ), and the second map sends ( a, b ) into σ ( − , a ) β ( − , b ) − 1 . The sequence (4.3) is isomorphic t o t he sequence dual to (4.4). 15 4.2 Holomorphic orbifolds Let V b e a holomor phic V OA, i.e. a V O A whose only simple mo dule is V it self. Supp ose a finite group G is acting on V b y V OA automorphisms. Then the fixed p oin t set V G is again a V O A, the orbifold VO A. It w as argued in [Ki] that the category of mo dules o f V G is equiv alen t to a (twis ted) group- theoretic mo dular category Z ( G, α ) where α is a 3-co cycle on G . W e assume for simplicit y that α is trivial. Th us our mo dular categor y is Z ( G ). This category can b e describ ed as the category of represen tations of the D rinfeld do uble D ( G ), see [Ka, Sec. IX.4.3, XI I I.5] or [D a1, Sec. 3.1] for an explicit description of Z ( G ). Morita equiv alence classes o f simple sp ecial symme tric F rob enius algebras in Z ( G ) w ere clas- sified in [Os2]. T hey ar e in one-to- o ne corresp ondence with conjugacy classes of pairs ( H , γ ), where H ⊂ G × G is a subgroup and γ ∈ H 2 ( H , C ∗ ) is a 2- co cycle. Simple comm utativ e maxi- mal sp ecial symmetric F rob enius algebras in Z ( G ) ⊠ Z ( G ) = Z ( G × G ) w ere describ ed in [Da1, Thm. 3.5.1 & 3.5 .3]. They a r e lab elled b y the same data (aga in making explicit the isomorphism [KR2, Cor. 3.25]). The details of the follow ing will ag a in app ear elsewhere. The a ut o morphism g r o up Γ( H , γ ) of the simple comm uta tiv e maximal algebra in Z ( G × G ) corresp onding t o the pair ( H , γ ) is an extension ˆ H → Γ( H , γ ) → S t N G × G ( H ) /H ( γ ) , (4.5) where N G × G ( H ) is the normaliser of H in G × G . The quotien t N G × G ( H ) /H has a w ell-defined action on the cohomology H 2 ( H , C ∗ ) b y conjugation in eac h argument; S t N G × G ( H ) /H ( γ ) is the stabiliser o f the class γ with resp ect to this action. In particular – and in contrast with the previous example – the automo r phism group Γ( H , γ ) is often non- a b elian. 5 Conclus ion W e hav e sho wn that in a pa rticularly w ell-understo o d class of quan tum field theories, namely tw o- dimensional rational conformal field theories, a ll inv ertible duality tra nsfor ma t io ns – whic h are nothing but conf o rmal isomorphisms – can b e implemen ted b y o ne-dimensional domain w a lls (i.e. defect lines) pro vided b oth are compatible with the rat io nal symmetry . In fa ct, giv en a ratio nal V O A V with category of represen tations C = Rep V , in Theorem 3.14 w e prov ed a n equiv a lence of g roup oids b etw een - CFTs ov er V ⊗ ¯ V and conformal isomor phisms a cting as the iden tit y on V ⊗ ¯ V (the group o id A ( C ⊠ C ) in the algebraic form ulation), and - CFTs ov er V ⊗ ¯ V and (isomorphism classes of ) inv ertible defect lines which preserv e V ⊗ ¯ V (the group oid P ( C ) in the alg ebraic fo rm ulation). W e w ould also lik e to note that this equiv alence of g r oup oids has an application ev en for t he b est studied class of rational confo r ma l field theories, the Virasoro minimal mo dels [BPZ]. There, it is in principle p ossible to compute all bulk structure constants for all minimal mo dels in the A-D-E classification of [CIZ] using the metho ds o f [FRS1, FRS4]. But these are cum b ersome to work with, and their confo rmal auto morphisms hav e not b een computed directly . Our result allo ws to instead compute fusion rules f o r bimo dules, whic h is m uc h easier to do (nonetheless they 16 ha v e not app eared in print explicitly for all minimal models). Our result also allo ws to mak e con tact with [R V], where mo dular prop erties where used to inv estigate automo r phisms of unitary minimal mo dels. It tur ns out that our main result is not an isolat ed phenomenon. A result analo g ous to ours, but one categorical lev el higher, has recen tly been pro v ed in [ENO2, KK]. In [ENO2], a fully faithful em b edding of 2 -group oids w as obtained, where the role of P ( C ) is tak en b y the 2-group oid of fusion categories, bimo dule categories, and isomorphism classes o f equiv alences of bimodule categories, and the role of A ( C ⊠ C ) is tak en b y braided fusion categor ies, braided equiv alences, and isomorphisms of bra ided equiv alences. The functor is pro vided b y the mono idal cen t r e. This hin ts at a corresp onding statemen t for T uraev-Viro theories. Although an axiomatic treatmen t of T uraev-Viro theories with do main w alls is not y et av ailable, a Hamiltonian v ersion of T uraev-Viro theories – the so-called Levin-W en mo dels [L W] – is carefully studied in [KK]. It is sho wn there that a bimo dule category o v er t w o unita ry tensor categories determines a doma in wall b et w een t wo bulk phases in a lattice mo del, a nd the monoidal centre describ es any on excitations in eac h bulk phase. Again, o ne has a one-to - one correspo ndence b etw een in v ertible defects and equiv alences (as braided tensor catego ries) b etw een excitations in the bulk. Ev en when sta ying within tw o-dimensional mo dels, an imp ortant unansw ered question is how m uc h, if an ything, o f our analysis carries o v er from the maximally we ll-b eha v ed class o f mo dels studied here to more complicated theories. F or example, it w ould b e v ery in teresting (at least to us) to in ves tigate lo garithmic conforma l field theories (see e.g. [Ga]) o r top o logical conformal field theories [Co]. Ac kno wledgemen t : The authors w ould lik e to thank J ¨ urgen F uc hs for helpful commen ts on a draft of this pap er. AD thanks Max Planc k Institut f ¨ ur Mathematik (Bonn) for ho spitalit y and excellen t w orking conditions. 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