Defect Partition Function from TDLs in Commutant Pairs
aa r X i v : . [ h e p - t h ] J a n Prepared for submission to JHEP
Defect Partition Function from TDLs inCommutant Pairs
Subramanya Hegde, Dileep P. Jatkar
Harish-Chandra Research Institute, Homi Bhabha National InstituteChhatnag Road, Jhunsi, Allahabad, India
E-mail: [email protected] , [email protected] Abstract:
We study topological defect lines in two character rational conformalfield theories. Among them one set of two character theories are commutant pairsin E , conformal field theory. Using these defect lines we construct defect partitionfunction in the E theory. We find that the defects preserve only a part of the E current algebra symmetry. We also determine the defect partition function in c = 24CFT using these defects lines of 2 character theories and we find that these defectspreserve all current algebra symmetries of c = 24 CFT. ontents A , CFT/ E , CFT 63.2 A , CFT/ E , CFT 93.3 G , CFT/ F , CFT 113.4 Lee-Yang model CFT/ E . IVOA 143.5 SO (8) WZW CFT 15 c = 24 Meromorphic CFTs 165 Discussion 17
Two-dimensional conformal field theories (2DCFT) have played a pivotal role in un-derstanding a variety of problems in theoretical physics, ranging from string theory[1]to mesoscopic physics[2] to quantum information[3]. All these applications, in turn,have helped deepen our understanding of 2DCFT. However, classification and studyof 2DCFTs is an interesting problem in its own right[1]. A programme of classify-ing 2D rational conformal field theories (RCFTs) by the number of characters wasaddressed long ago [4, 5]. Mathur, Mukhi, and Sen(MMS) used the technique ofmodular differential equations to classify theories with 2 and 3 characters[5].With renewed interest in the bootstrap programme, Mukhi and collaboratorsrevisited the classification program of the RCFTs[6–9]. They found that the MMSclassification of 2-character theories is not complete. They explicitly constructednew 2-character theories with central charge c >
8. Later Chandra and Mukhi[10]showed that the new set combined with the MMS series completes the classificationof RCFTs with two characters.In this note, we will be focusing on 2-character RCFTs and in fact will consider asubset of them which belong to MMS series and its commutants in a single charactertheory with c = 24. We will study topological defect line(TDL)[11] in these theoriesand construct defect partition functions. Within the MMS series of 2-charactertheories, we study TDLs in each of these conformal field theories. In general, the– 1 –DLs can be invertible if they are associated with some global symmetry[11–13]. Onthe other hand, non-symmetry defects are typically non-invertible. Even within theMMS series we encounter TDLs of both type. While in case of the commutant pairslike ( A , E ), ( A , E ), or ( D , D ), we get invertible defects, in case of ( G , F ) and(Lee-Yang(L-Y), E . ) we have non-invertible defects.It is known that the E , character can be decomposed in terms of sums ofproducts of characters of the commutants. For example, χ E , = χ A , ˜ χ E , + χ A , / ˜ χ E , / . (1.1)Using the TDLs in the commutant pairs, we construct the defect character in E , CFT. All the TDLs in the commutant pairs, whether invertible or not, do not com-mute with the symmetry currents of E . As a result, the level 1 degeneracy of thedefect character is different from that of the E , .We also briefly discuss commutant pairs of the MMS series of RCFTs in c = 24CFT. The relevant c = 24 CFTs possess current algebra symmetry except for those c = 24 CFTs, which contain commutants of L-Y and E . . We find that whenever wehave a symmetry defect, it preserves the entire current algebra symmetry of c = 24CFT. This is in sharp contrast with what happens in the E , case where the defectpreserves only a part of the E symmetry. The origin of this difference lies in thefact that the dimensions of non-trivial primary operators in a given commutant pairof E add up to 1. Hence the defect in the commutant component affects the level1 degeneracy of the E theory. On the other hand, the commutant pairs in c = 24CFT has the dimensions of non-trivial primary adding up to 2. As a result the defectin the commutant component leaves the level 1 degeneracy invariant.This note is organised as follows: In section 2 we briefly recall the aspects oftopological defects. We focus on the TDLs associated with global symmetries as wellas those without any symmetry. In section 3, we take up the MMS series and studythe TDLs in each CFT in detail. We also show that these defects can be embeddedin the E theory, and the defect character in the E theory is computed using themodular S transformation of the character with the defect insertion. In section 4,we briefly discuss defects in MMS CFTs and their commutants in c = 24 CFT.These theories have been discussed in [6–8], and their c = 24 parent belongs the theSchellekens classification[14]. Although our analysis can be extended to the entireseries we focus only on two cases, namely A and its commutant in c = 24, and E and its commutant in c = 24. We conclude this note by summarising our results. Topological defects are generalisations of global symmetries. Consider a continuousglobal symmetry in a d dimensional field theory, the corresponding charge is given– 2 –y Q a = R d d − x n µ j µa . Action of the symmetry on the states is then implementedby, L θ = exp( iθ a Q a ), where θ a are the transformation parameters. In a CFT, thecharge and hence the operator L θ commute with the conformal generators. Theaction of the symmetry on the state is reflected on the corresponding operator viastate operator correspondence as an L θ line encircling the operator to lead to anew operator. As the line commutes with the conformal generators, the conformaldimension of the operator is preserved. In a two-dimensional CFT, the line operatorcommutes with the Virasoro generators and hence can be deformed as long as theyare no other insertions around them. Hence the operation is termed as acting onby a topological defect line (TDL). The operators L g are associated with symmetryelements g for continuous as well as discrete global symmetries. TDLs associatedwith global symmetries are known as invertible defect lines, as the operators onthe endpoint of defects have inverses under the fusion rules [11]. There are severalexamples of TDL that are not invertible and hence do not correspond to symmetriesof the theory. For example, duality defect in the Ising CFT.In diagonal modular invariant RCFTs, there is a TDL associated with eachprimary of the theory as [15, 16], L i = X k S ik S i P k , (2.1)where S ij are the elements of the symmetric and unitary modular S matrix for thetransformation of the characters and P k is the projector to the module generatedby k . It is important to note that the above operators commute with not only theVirasoro algebra but the full chiral algebra of the theory. TDLs given as aboveare known as Verlinde lines[15–18]. Their fusion follows that of the correspondingprimaries according to the Verlinde formula.Given a Verlinde line, one can define partition functions with Verlinde lines alongdifferent non-contractible cycles. Defect lines running along the space directions areencountered while taking the trace to compute the partition function as insertions ofoperators. Defect lines running along the time direction impose a boundary conditionon each spatial slice and hence modify the Hilbert space over which the trace isperformed. Given a Verlinde line of the form (2.1), we can obtain the partitionfunction with a defect insertion as, Z L i = T r ( L i e − q ( L − c ) ) = X k S ik S i ¯ χ k (¯ τ ) χ k ( τ ) . (2.2)We can obtain the partition function with the defect line inserted along the time– 3 –irection by a modular S transformation as [15], Z L i ( τ, ¯ τ ) = Z L i ( − τ , − τ )= X k,l,m S ∗ kl S ik S km S i ¯ χ l χ m = X l,m N lim ¯ χ l ¯ χ m , (2.3)where N lim = P k S ∗ kl S ik S km S i are the fusion coefficients given by the Verlinde lines andare hence integers. They give the degeneracy of the operators in the defect Hilbertspace with dimensions ( h l , ¯ h m ). Thus one can read off the operator content in thedefect Hilbert space by expressing Z L in terms of the characters. This utility extendsbeyond the Verlinde lines to more general TDLs.While partition functions for defects that commute with the full chiral algebraof the theory have been studied extensively in the literature, recently more generaldefects have been considered[13]. In [13], the authors considered defects inside theMonster CFT which do not commute with the Monster group. A key component inthe analysis was that the character for the Monster CFT can be expressed in termsof Ising and Baby Monster CFT characters as, χ M = χ Ising0 χ Baby0 + χ Ising χ Baby + χ Ising χ Baby , (2.4)where χ M = j ( τ ) −
744 is the character for the identity module of the MonsterCFT, and on the RHS the chiral characters are those of Ising modules for primarieswith dimensions 0 , , and Baby Monster primaries with dimensions 0 , , . Itwas shown that the above relation between the characters can be interpreted interms of fermionisation of the Monster theory. In particular, it was shown that onecan identify the fermionized Monster CFT as a tensor product of the fermionizedBaby Monster and the Majorana-Weyl CFT. This allowed the realisation of variousdefects of the Ising category inside Monster CFT. In particular, the duality defect ofthe bosonic Monster CFT was implemented as a Z defect of the fermionic Monsterwhere the Z corresponds to a sign flip of the Majorana-Weyl fermion. A usefulway indicated in [13] to obtain the partition functions for Monster CFT with defectsrunning along the different non-contractible cycle is the following. From (2.4), notethat as the LHS is modular invariant, the S matrix for the transformation of theIsing CFT and Baby Monster CFT are identical. Therefore, given the partitionfunction with an insertion of the Z Verlinde line corresponding to the primary withdimension 1 / Z Ising η = χ Ising0 χ Ising0 + χ Ising χ Ising − χ Ising χ Ising , (2.5)– 4 –e can write the corresponding partition function with insertion as, Z Mη = χ Baby0 χ Ising0 + χ Baby χ Ising − χ Baby χ Ising , (2.6)where each ¯ χ Ising is replaced by the corresponding χ Baby . As the modular S matricesare identical, the partition function for the Monster has the corresponding modulartransformation properties. In particular, the same replacement rule holds for thedefect running along the time direction. It is important to note that the defects η and N of the Ising category realised inside the Monster CFT in this way do notcommute with the full Monster group. They preserve only a part of the chiral algebraof the CFT. Bosonisation of the fermionic Baby Monster CFT extends the symmetryto its double cover 2 . B . The defects commute with this double cover, which is asubgroup of the Monster group. The elements in the defect partition function couldbe decomposed in terms of the dimensions of Baby Monster representations.In the following section, we will consider partition functions with defect insertionsrealised in the E , CFT and various c = 24 meromorphic CFTs by an analogousreplacement rule. We will see that the defects defined using such a rule commutewith a subgroup of the chiral algebra of the theory. In this section we will consider CFTs that appear in the Mathur-Mukhi-Sen seriesof two character rational CFTs [5]. These CFTs form a pair inside the E , WZWmodel by the relation χ E , = χ ˜ χ + N χ ˜ χ , (3.1)where N is an integer that denotes the degeneracy of the non-trivial primary. Thecharacters χ and ˜ χ correspond to pairs inside the series whose central charges addupto 8 and conformal dimensions of the non-trivial primary add upto 1. Because ofthe above relation, the modular S matrices of the pair are either identical, or complexconjugates of each other when the modular S matrix is complex. This results in thestructure of the partition functions with Verlinde lines to be identical. We will usethis to define defects inside the c = 8 theory by using a replacement rule. For thecases of A , /E , and A , /E , we will illustrate using the branching rules for E theaction of the defect and the subgroup of E that is left invariant by the defect.The MMS series CFTs have the following expressions for their characters [7, 9]which we will use in this section. χ ( τ ) = j c F ( 112 − h , − h , − h ; 1728 j ) χ h ( τ ) = |√ m |√ N j c − h F ( 112 + h ,
512 + h , h ; 1728 j ) , (3.2)– 5 –here √ m = (1728) h sin( π ( − h )) sin( π ( − h ))sin( π ( + h )) sin( π ( + h )) ! . Γ(1 − h )Γ( + h )Γ( + h )Γ(1 + h )Γ( − h )Γ( − h ) . (3.3)The factor |√ m |√ N is an integer for each of these theories. A , CFT/ E , CFT
The pair of CFTs here are A . and E , WZW model with central charge c = 1and c = 7 respectively. As we will explain later in this section, they have the samemodular S matrix given by [1], S = 1 √ (cid:18) − (cid:19) . (3.4)Partition function for the A , theory is given as, Z A , = χ A , χ A , + χ A , χ A , , (3.5)where χ A , , χ A , (or χ A , , χ A , ) are the (anti-)chiral characters corresponding toKac-Moody primaries with scaling dimensions h L = h R = 0 and h L = h R = respectively. The non-trivial primary transforms under the representation of A ! , .The q expansion of the characters read, χ A , = q − (1 + 3 q + 4 q + 7 q + · · · ) χ A , = q (2 + 2 q + 6 q + 8 q + · · · ) . (3.6)Let us consider the Verlinde line corresponding to the non-trivial primary. The actionof the Verlinde line is given by,ˆ L | φ i = S S | φ i = | φ i , ˆ L | φ i = S S | φ i = −| φ i . (3.7)Therefore the partition function with L insertion is given according to (2.2) by, Z L = | χ | − | χ | . (3.8)Note that as the insertion keeps the vacuum invariant and flips the sign for thefundamental representation, the Z can be interpreted as the center of A . This hasbeen discussed recently in [19]. We will come back to this later in this section.– 6 –o obtain the defect Hilbert space partition function, where the defect runsalong the time direction, we perform a S modular transformation as explained in theprevious section. We obtain, Z L ( τ ) = Z L ( − /τ )= | χ ( − /τ ) | − | χ ( − /τ ) | = ¯ χ ( τ ) χ ( τ ) + ¯ χ ( τ ) χ ( τ ) . (3.9)Therefore the scaling dimensions of the operators in the defect Hilbert space is givenas ( h L , h R ) = (0 , ) and ( h L , h R ) = ( , T transforma-tions of the defect partition function, one can diagnose the ’t Hooft anomaly for thissymmetry defect. This was studied recently in [11, 12, 19, 20]. The partition func-tion Z L ( − /τ ) is invariant under Γ (4) congruent subgroup of SL (2 , Z ). Modularproperties of the defects in the MMS series can be inferred from the results in [21].For E WZW model, the primaries are of dimension h L = h R = 0 and h L = h R = . The non-trivial primary trasnforms under the representation of E , . We willdenote the corresponding holomorphic characters as ˜ χ E , and ˜ χ E , , and similarlyfor antiholomorphic characters whose q expansions are given as,˜ χ E , = q − (1 + 133 q + 1673 q + 11914 q + · · · )˜ χ E , = q (56 + 968 q + 7504 q + 42616 q + · · · ) . (3.10)As discussed in the beginning of this section, A , and E , WZW satisfy, χ E , = χ A , ˜ χ E , + χ A , ˜ χ E , , (3.11)where χ E , is the character for E , WZW model whose q expansion reads, χ E , = q − (1 + 248 q + 4124 q + 34752 q + · · · ) . (3.12)The above expression can be made sense of by using the branching rules for E representations into A and E representations. Let us note the following branchingrules [22]. ( ) = ( , ) + ( , ) + ( , ) , ( ) = ( , ) + ( , ) + ( , ) + ( , ) + ( , ) . (3.13)We can then write the coefficients as,248 = 3 × × × ×
1) + (2 ×
56) + (3 × × × . (3.14)– 7 –ote that as E , WZW model is a single character theory, its character is invariantunder the modular S transformation. Therefore, the modular S matrix for E theoryshould be the inverse transpose of the modular S matrix of SU (2) theory.However modular S -matrix of A , is a symmetric orthogonal matrix. Therefore,modular S matrix of the two theories are the same. Thus the partition functions withdefect L for E , have the same structure as the L partition functions consideredabove if we replace χ with ˜ χ and χ with ˜ χ . Therefore the scaling dimensions ofthe operators in the defect Hilbert space is given as ( h L , h R ) = (0 , ) and ( h L , h R ) =( , E , .Parition function of E , WZW model in terms of the characters defined aboveis, Z E , = | χ E , | = | χ A , ˜ χ E , + χ A , ˜ χ E , | . (3.15)From this and from (3.8), we maybe motivated to define the following partitionfunction for E , WZW model with an η insertion,( Z η , ) E , = | ( χ η , ) E | ≡ | χ A , ˜ χ E , − χ A , ˜ χ E , | , (3.16)which we obtained by considering the RHS of (3.8) and replacing one of the ¯ χ ineach term with the corresponding ˜ χ and then taking the modulus squared of theexpression. We have chosen to denote the defect as η , where the subscript standsfor the central charges of the A , and E , pair.The above twisted character ( χ η , ) E has the following q series expansion,( χ η , ) E = q − (1 + 24 q + 28 q + 192 q + · · · ) (3.17)Recall that we interpreted the defect as the center of A , . We already know that thefundamental representation changes the sign under the defect. We can use ClebschGordan addition to identify which representations change sign. For A , , as expectedall the even dimensional representations change sign. We then have the above coef-ficients as,24 = 3 × × − (2 × × − (2 ×
56) + (3 × × − (2 × . (3.18)Thus we can interpret the operator η , inside E , as the center of A , . Note thatwe can also interpret the action of η , as the center of E , instead but not both.This is so because E has a maximal subgroup ( A ⊗ E ) / ( − , − A and E are identified due to the ( − , − η , inside E , is the equivalence class of– 8 – { , − } , {− , }} acting on appropriate representations inside A and E as per theabove decomposition. This element of course does not commute with the full E .When one expresses E as A ⊗ E / ( − , −
1) + 2 ⊗
56, then this decomposition hasa non-algebraic double cover. The defect commutes with this non algebraic doublecover inside E .Using the modular S transformation of the characters χ and ˜ χ , we find the defectpartition function, ( Z η ) E = | χ ˜ χ + χ ˜ χ | (3.19)This is consistent, if we take (3.9) and follow the same replacement rule as we haveused above to obtain the partition function with insertion. From the above formula,we can deduce the modular transformation property of the defect insertion partitionfunction (3.16). We see that (3.19) given above is invariant under T modular trans-formation. Therefore, (3.16) is invariant under ST S which belongs to the congruentsubgroup Γ (2). It can be checked that the invariance extends to the full Γ (2). A , CFT/ E , CFT
The A , and E , WZW model CFTs have central charge c = 2 and c = 6 respec-tively.For A , theory, the modular S matrix is given by [1], S = 1 √ − + i √ − − i √ − − i √ − + i √ . (3.20)Partition function is given as, Z = χ A , χ A , + 2 χ A , χ A , . (3.21)where χ A , , χ A , ( or χ A , , χ A , ) are (anti-)chiral characters corresponding to Kac-Moody primaries with scaling dimensions h L = h R = 0 and h L = h R = respectively.Primaries corresponding to and ¯3 representations of A , have the same character χ A , . Their q expansion is given as, χ A , = q − (1 + 8 q + 17 q + 46 q + · · · ) χ A , = 3 q (1 + 3 q + 9 q + 19 q + · · · ) (3.22)– 9 –et us consider the Verlinde line corresponding to the non-trivial primary that trans-forms as representation of A , . The action of the Verlinde line is given by,ˆ L | φ i = S S | φ i = | φ i , ˆ L | φ i = S S | φ i = − i √ | φ i , ˆ L | φ ¯3 i = S S | φ ¯3 i = − − i √ | φ ¯3 i (3.23)Therefore the partition function with L insertion is given by, Z L = | χ A , | − | χ A , | . (3.24)Note again that the above insertion can be interpreted as the center of A , .To obtain the defect Hilbert space partition function, we perform an S modulartransformation. Z L ( τ ) = Z L ( − /τ )= | χ A , ( − /τ ) | − | χ A , ( − /τ ) | = χ A , χ A , + χ A , χ A , + χ A , χ A , . (3.25)Therefore the scaling dimensions ( h L , h R ) of the operators in the defect Hilbert spaceis given as ( , ),(0 , ) and ( ,
0) which form representation of A , coming from ¯3 ⊗ ¯3 , ⊗ and ⊗ A , respresentations respectively. Note that there is also an L ¯3 Verlinde line with the same defect Hilbert space.For the E , WZW model CFT, the characters have the q expansion,˜ χ E , = q − (1 + 78 q + 729 q + 4382 q + · · · )˜ χ E , = q (27 + 378 q + 2484 q + 12312 q + · · · ) , (3.26)where ˜ χ E , , ˜ χ E , are the characters for the modules corresponding to identity andthe non-trivial primaries transforming as and ¯27 under E , . Modular S matrixfor this model is the complex conjugate of the A , model given above. However itcan easily be verified that the L defect partition functions have the same structureas the L defect partition functions and are reproduced on replacing the characters χ A , with the corresponding ˜ χ E , .The E , CFT character is given as [5] , χ E , = χ A , ˜ χ E , + 2 χ A , ˜ χ E , . (3.27) Note that we choose to write the degeneracy factor 2 outside the characters. Sometimes thefactor is absorbed into the characters as √ – 10 –o interpret the above, we need the branching rules of E , into A , and E , . Letus note the following branching rules [22],( ) = ( , ) + ( , ) + ( , ) + ( ¯3 , ¯27 ) , ( ) = ( , ) + ( , ) + +( ¯6 , ) + ( , ¯27 ) + ( , ) + ( , )+ ( , ) + ( ¯3 , ¯351 ) . (3.28)This explains the coefficients in the q expansion of the E , CFT in terms of A , and E , representations in an analogous manner as the previous subsection.Using the replacement rule on (3.24), the twisted chracter for E , CFT corre-sponding to the above defect is given by,( χ η , ) E = χ A , ˜ χ E , − χ A , ˜ χ E , = q − (1 + 5 q − q + 3 q + · · · ) (3.29)To interpret this as the center of A , inside E , , note that the center of A , actswith ω on and ω on ¯3 , where ω = − i √ is a cube root of 1. Using ClebschGordan addition, we can find that has trivial scaling while and ¯6 scale as ω and ω respectively. Thus the coefficients above can be written as,5 = (8 ×
1) + (1 ×
78) + ω (3 ×
37) + ω (3 × , = (8 ×
1) + (1 × − (3 × , (3.30)and, − ×
1) + (8 ×
1) + ( ω + ω )(6 ×
27) + (8 × × ω + ω )(3 × ×
1) + (8 × − (6 ×
27) + (8 × × − (3 × E , instead. The character withthe defect along the time direction is given as,( χ η , ) E = ˜ χ E , χ A , + ˜ χ E , χ A , + ˜ χ E , χ A , . (3.32)The corresponding ( Z η , ) E = | ( χ η , ) E | can be seen to invariant under Γ (3). G , CFT/ F , CFT
The G , and F , WZW CFTs have central charge c = and c = respectively.They share the same modular S matrix given by [1], S = r (cid:18) sin π sin π sin π − sin π (cid:19) . (3.33)– 11 –artition function is given as, Z = χ G , χ G , + χ G , χ G , . (3.34)where χ G , , χ G , (or χ G , , χ G , ) are (anti-)chiral characters corresponding to Kac-Moody primaries with scaling dimensions h L = h R = 0 and h L = h R = respectively.Their q expansion reads, χ G , = q − (1 + 14 q + 42 q + 140 q + · · · ) χ G , = q (7 + 34 q + 119 q + 322 q + · · · ) (3.35)Let us consider the Verlinde line corresponding to the second primary. Theaction of the Verlinde line is given by,ˆ L | φ i = S S | φ i = α | φ i , ˆ L | φ i = S S | φ i = − α | φ i , (3.36)where α = (1 + √
5) is the golden ratio. Therefore the partition function with ˆ L insertion is given by, Z L = α | χ G , | − α | χ G , | . (3.37)To obtain the defect Hilbert space partition function, we perform an S modulartransformation. Z L ( τ ) = Z L ( − /τ )= α | χ G , ( − /τ ) | − α | χ G , ( − /τ ) | = χ G , χ G , + χ G , χ G , + χ G , χ G , . (3.38)Therefore the scaling dimensions ( h L , h R ) of the operators in the defect Hilbert spaceis given as ( , ),(0 , ) and ( , G , WZW model dictates the following fusion ruleon L , L × L = + L . (3.39)Therefore the defect does not have an inverse under the fusion rule as there is noother primary operator to act as the inverse.– 12 –ust as in the previous sections, the discussion for the defect L in G , carriesover to the defect L in F , on the replacement of χ G , by the corresponding ˜ χ F , whose q expansions read,˜ χ F , = q − (1 + 52 q + 377 q + 1976 q + · · · )˜ χ F , = q (26 + 299 q + 1702 q + 7475 q + · · · ) (3.40)The character for E , is given as, χ E , = χ G , ˜ χ F , + χ G , ˜ χ F , , (3.41)We need the following branching rules [23], = ( , ) + ( , ) + ( , ) = ( , ) + ( , ) + ( , ) + ( , ) + ( , )+ ( , ) + ( , ) . (3.42)The defect η , insertion is given as follows,( χ η , ) E = αχ G , ˜ χ F , − α χ G , ˜ χ F , = q − (1 + (124 − √ q + (2062 − √ q + · · · ) . (3.43)Using the decomposition in terms of the characters above and using the branchingrules, we deduce that the representations which receive a contribution of − α fromthe insertion are, ( , ) + ( , ) + ( , ) + ( , ) , (3.44)while the rest of the representations carry a factor of α . However, the Clebsch Gordanaddition for the representation in G reads, ⊗ = ⊕ ⊕ ⊕ . (3.45)Thus one can not explain the insertion of the factors in terms of Clebsch Gordanaddition as we did earlier, already at the level of the G , CFT. In fact note that itis this curious feature of G that leads to the defect fusion rule (3.39). Unlike thecases considered so far where on Clebsch-Gordan addition of the fundamental repre-sentation only the identity survived on the RHS at level one, here the fundamentalrepresentation itself appears at the RHS hence making the defect non-invertible.Thus we have a non-invertible defect inside the E , WZW CFT. We have indicatedthe decomposition inside E which would give the character with defect insertionwhile further investigation is needed to obtain a clear understanding of this defectinside E , CFT. The defect partition function,( Z η , ) E = | ( χ η , ) E | = | χ G , ˜ χ F , + ˜ χ F , χ G , + ˜ χ F , χ G , | , (3.46)is invariant under Γ (5). – 13 – .4 Lee-Yang model CFT/ E . IVOA
For Lee-Yang model CFT and E . IVOA, the central charges are c eff = ( c = c eff − h = − ) and c = . The modular S matrix is given by, S = r (cid:18) − sin π sin π sin π sin π (cid:19) . (3.47)Partition function is given as, Z = χ LY0 χ LY0 + χ LY − χ LY − . (3.48)where χ , χ − are characters corresponding to primaries with scaling dimensions h L = h R = 0 and h L = h R = − respectively. Let us consider the Verlinde linecorresponding to the second primary. The action of the Verlinde line is given by,ˆ L − | φ i = S S | φ i = − α | φ i , ˆ L − | φ − i = S S | φ − i = α | φ − i , (3.49)where α = (1 + √
5) is the golden ratio. Therefore the partition function with η insertion is given by, Z L − = − α | χ LY0 | + α | χ LY − | . (3.50)To obtain the defect Hilbert space partition function, we perform an S modulartransformation. Z L − ( τ ) = Z L − ( − /τ )= − α | χ LY0 ( − /τ ) | + α | χ LY − ( − /τ ) | = χ LY − χ LY − + χ LY0 χ LY − + χ LY − χ LY0 . (3.51)Therefore the scaling dimensions ( h L , h R ) of the operators in the defect Hilbert spaceis given as ( − , − ),(0 , − ) and ( − ,
0) [11]. We can similary construct the defectfor the E . IVOA. The defect insertion character inside E , is given as,( χ η , ) E , = − α χ LY0 ˜ χ E . + αχ LY − ˜ χ E . , (3.52)which is a non-invertible defect. The defect Hilbert space partition function,( Z η , ) E , = | ( χ η , ) E , | = | χ LY − ˜ χ E . + χ LY − ˜ χ E . + χ LY0 ˜ χ E . | (3.53)is invariant under Γ (5) . Note that for the moular transformation purpose, one should treat the primary in the Lee-Yangmodel with dimension − as a primary in the A . CFT with effective dimension . – 14 – .5 SO (8) WZW CFT
For SO (8) theory the central charge is c = 4. The theory is self-dual inside E , CFT. The modular S matrix is given by, S = 12 − − − − − − . (3.54)Partition function is given as, Z = χ SO (8) χ SO (8) + 3 χ SO (8) χ SO (8) . (3.55)where χ , χ are characters corresponding to Kac-Moody primaries with scaling di-mensions h L = h R = 0 and h L = h R = respectively. Let us consider the Verlindeline corresponding to the second primary. The action of the Verlinde line is given by,ˆ L v | φ i = S S | φ i = | φ i , ˆ L v | φ v i = S S | φ v i = | φ v i , ˆ L v | φ s i = S S | φ s i = −| φ s i , ˆ L v | φ c i = S S | φ c i = −| φ c i . (3.56)Therefore the partition function with L v insertion is given by, Z L v = | χ SO (8) | − | χ SO (8) | . (3.57)To obtain the defect Hilbert space partition function, we perform an S modulartransformation. Z L v ( τ ) = Z L v ( − /τ )= | χ SO (8) ( − /τ ) | − | χ SO (8) ( − /τ ) | = 2 χ SO (8) χ SO (8) + χ SO (8) χ SO (8) + χ SO (8) χ SO (8) . (3.58)Therefore the scaling dimensions ( h L , h R ) of the operators in the defect Hilbert spaceis given as ( , ), ( , ), (0 , ) and ( , Z × Z defectseen from the action on the primaries above. The defect insertion character in E , is, ( χ η , ) E , = (cid:16) χ SO (8) (cid:17) − (cid:16) χ SO (8) (cid:17) . (3.59)The defect Hilbert space partition function, | ( χ η , ) E , | = 2 χ SO (8) ( χ SO (8) + χ SO (8) ) , (3.60)is invariant under Γ (2). – 15 – Topological defects for c = 24 Meromorphic CFTs
In this section, we will discuss briefly on how to write down the twisted charactersfor c = 24 meromorphic CFTs which contain coset pairs involving the MMS seriesCFTs [7]. In Table-4, the pairs are listed where c, h and ˜ c, ˜ h are the central chargeand the conformal dimension of the non-trivial primary in the MMS series and itscoset dual respectively. The entry ˜ m denotes the number of currents in the cosetdual theory and M denotes the number of currents in the c = 24 meromorphic CFT.MMS CFT c h ˜ c ˜ h ˜ m MA ,
69 72 A ,
88 96 G ,
106 120 SO (8)
140 168 F ,
188 240 E ,
234 312 E ,
323 456As observed in [7], the conformal dimensions of the pairs add upto 2. The charactersabove once again satisfy relations, χ Meromorphic = j −
744 + M = χ ˜ χ + N χ ˜ χ , (4.1)where N denotes the degeneracy in the characters, when present.The coset pairs of the MMS series CFTs have the following expressions for theircharacters [7]. χ ( τ ) = j ˜ c F ( − − ˜ h , − ˜ h , − ˜ h ; 1728 j ) χ h ( τ ) = |√ ˜ m |√ N j ˜ c − ˜ h F ( −
112 + ˜ h ,
712 + ˜ h , h ; 1728 j ) , (4.2)where √ m = (1728) h sin( π ( + ˜ h )) sin( π ( − ˜ h ))sin( π ( − ˜ h )) sin( π ( + ˜ h )) ! . Γ(1 − ˜ h )Γ( + ˜ h )Γ( + ˜ h )Γ(1 + ˜ h )Γ( − ˜ h )Γ( − ˜ h ) . (4.3)We will discuss the defects in the c = 24 CFTs with M = 72 and M = 456and make some comments on their interpretation. For the M = 72 CFT, the pairconsists of A , CFT whose characters we considered in (3.6). The corresponding c = 23 characters have the q expansion,˜ χ c =230 = q − (1 + 69 q + 131905 q + · · · ) , ˜ χ c =23 = q (32384 + 23493120 q + · · · ) . (4.4)– 16 –sing the replacement rule, the defect insertion chracter for the c = 24 CFT reads,( χ η , ) c =24 = χ A , ˜ χ c =230 − χ A , ˜ χ c =23 = 1 q + 72 + 67348 q + · · · (4.5)Thus we see that the defect has no effect on the dimension one contribution to thecharacter which corresponds to the currents. This is so because the second term inthe first equality above has contributes two powers of q more compared to the firstterm due to the conformal dimensions of the non-trivial primaries adding to 2 in thecoset pair. The meromorphic CFTs in the coset pair were found to be those givenin the list in [14]. For M = 72, one of the possibilities for the current algebra is( A , ) which is the No. 15 entry in the list. Here we can see that the above defectcan be interpreted as the center of either A , from the MMS series or the center ofthe ( A , ) which is the centraliser of the former inside the c = 24 CFT.Let us consider the example of M = 456 CFT. We considered the characters ofthe E , CFT in (3.10). The coset dual has characters,˜ χ c =170 = q − (1 + 323 q + 60860 q + · · · ) , ˜ χ c =17 = q (1632 + 162656 q + 4681120 q + · · · ) . (4.6)Therefore, ( χ η , ) c =24 = χ E , ˜ χ c =170 − χ E , ˜ χ c =17 = 1 q + 456 + 14100 q + · · · , (4.7)therefore as expected the defect has no effect on the dimension one contribution whichis 456. In both the cases above, the defect changes the dimension two contributions. We have studied the topological defect lines in 2 character rational conformal fieldtheories. We mainly focused on the MMS series of CFTs, which form commutantpairs in the E , conformal field theory. Using the TDLs, we constructed defectcharacters of the latter theory. The defects do not preserve E symmetry, whichshowed up as the reduction in the degeneracies at various levels in the characters.In particular, the reduction in the degeneracy at level one indicated the symmetriespreserved by the defect.We also analysed the commutant pairs of the MMS CFTs in c = 24 meromorphicCFT. We found that the defects preserve dimension 1 symmetries of the c = 24meromorphic CFT. We attributed this contrasting behaviour of the TDLs to their– 17 –mbedding in dimension 1 operators in the case of E theory, and in dimension 2operators in case of c = 24 CFT. It can be confirmed by seeing that the defect doesreduce the degeneracy at level 2 in the c = 24 CFT. Although we have not done anexhaustive study of defects in the commutant pairs in c = 24 theory, our results canbe generalised to other pairs in a straightforward manner.It would be interesting to generalise this method of deriving defect characters andpartition functions to three character theories. It would be curious to see how thisgeneralises to triple or multiple commutant cases. Recently novel coset relations havebeen proposed between the four-point functions of the currents in the E , theoryand conformal blocks in the commutant pairs inside E , [9]. It will be interesting tosee the consequence of the defect partition functions we have defined on these four-point function of currents. We hope to address some of these questions in the future. Acknowledgments
We thank Pramath A V for many useful discussions. We thank Ratul Mahanta forcollaboration in the early stages of the project and discussions.
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